INTERACTIONS BETWEEN OXYGEN MOLECULES, FOR ATMOSPHERIC APPLICATIONS By Sasha Caroen Brookhouse North         A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements  for the degree of Chemistry-Doctor of Philosophy 2016   ABSTRACT INTERACTIONS BETWEEN OXYGEN MOLECULES, FOR ATMOSPHERIC APPLICATIONS  By Sasha Caroen Brookhouse North  Understanding the energy transfer processes between oxygen molecules is important for the elucidation of the reactions occurring in the Earth's atmosphere112. Even though two interacting oxygen molecules is a relatively small system, each diatomic has two unpaired electrons in antibonding, degenerate molecular orbitals, leading to an open shell system that forms a dense manifold of excited electronic states. This type of situation requires a multireference method. For this reason, progress on theoretical calculations for interacting oxygen molecules has been slow, since the discovery of collision-induced absorption in oxygen gas by Crawford, Welsh and Locke3 in 1949. In this work, interaction energy curves have been generated using the CASPT277,78 method, and the aug-cc-pVQZ basis set of Dunning76. In addition, electronic properties have been obtained for the oxygen molecule, and are used to obtain the dipole moments using the long-range approximation.  Changes in the electric charge distribution during a collision between the oxygen molecules can induce a net dipole and render the molecules infrared active, thus contributing to the absorption and emission in the IR and far-IR spectrum69. These dipole moments have also been calculated using ab initio methods, for many orientations of the oxygen molecules, and can be used to model the collision-induced spectra. iii  TABLE OF CONTENTS  LIST OF TABLES..........................................................................................................................iv  LIST OF FIGURES.....................................................................................................................xxv  CHAPTER 1: Introduction..............................................................................................................1  CHAPTER 2: The Interaction Energy of O2-O2 Collisional Complexes......................................15  2.1 Introduction.............................................................................................................................15  2.2 Basis Set Superposition Error (BSSE)....................................................................................17 2.3 Comparison of Interaction Energy to Literature.....................................................................30  CHAPTER 3: Multipole Moments and (hyper)polarizabilites of the O2 Molecule.......................39 3.1 Introduction..............................................................................................................................39 3.2 Methodology............................................................................................................................41 3.3. Properties at the Equilibrium Bond Length, Obtained using the Singly Augmented Basis Sets.................................................................................................................................................44 3.4  Properties at the Equilibrium Bond Length, Obtained using the Doubly Augmented Basis Sets.................................................................................................................................................54 3.5  Properties Obtained at non-Equilibrium Bond Lengths.........................................................66 3.6 Estimate of Complete Basis Set Limit...................................................................................130 3.7 Rovibrational Averages.........................................................................................................136 3.8 Conclusion.............................................................................................................................139  CHAPTER 4: Long-Range Approximation for the Interaction Dipole Moments of the O2-O2 Supermolecule, for Collision-Induced Absorption Applications.................................................142 4.1 Introduction: The Use of Electric Properties in the Long-Range Approximation.............................................................................................................................142 4.2 Contributions to the Collision-Induced Dipole Moment from Hexadecapolar Induction,  Quadrupolar induction, E-tensor Induction, Back Induction and Dispersion for Four Main Geometries...................................................................................................................................163 4.3 Contributions to the Dipole Moment at an Intermolecular Distance of 10 bohr..............................................................................................................................................172 4.4 Discussion and Summary.......................................................................................................178  CHAPTER 5: Ab Initio Versus Long-Range Approximation for the Dipole Moments..............181 5.1 Introduction: Methodology Used in Obtaining ab initio Dipole Moments......................................................................................................................................181 5.2 Comparison of Ab Initio and Long-Range Dipole  Moments...............................................182 5.3 Conclusions and Future Directions........................................................................................237  APPENDIX..................................................................................................................................239 REFERENCES............................................................................................................................323 iv  LIST OF TABLES  Table 1. Electronic bands of gaseous oxygen studied by McKellar, Rich, and Welsh23.................6 Table 2. BSSE and interaction energy  for the triplet spin state of the T-shape geometry............18 Table 3. BSSE and interaction energy for the singlet spin state of the T-shape geometry............18 Table 4. BSSE and interaction energy for the quintet spin state of the T-shape geometry...........19 Table 5. BSSE and interaction energy for the triplet spin state of the linear geometry.................19 Table 6. BSSE and interaction energy for the quintet spin state of the linear geometry...............19 Table 7. BSSE and interaction energy for the singlet spin state of the linear geometry...............20 Table 8. BSSE and interaction energy for the singlet spin state of the X-shape geometry...........20 Table 9. BSSE and interaction energy for the quintet spin state of the X-shape geometry...........20 Table 10. BSSE and interaction energy for the triplet spin state of the X-shape geometry..........21 Table 11. BSSE and interaction energy for the triplet spin state of the H-shape geometry..........21 Table 12. BSSE and interaction energy for the singlet spin state of the H-shape geometry.........21 Table 13. BSSE and interaction energy for the quintet spin state of the H-shape geometry.........22 Table 14. Position of the PEC minimum for the singlet state and T-shape geometry...................24 Table 15. Position of the PEC minimum for the triplet state and T-shape geometry....................24 Table 16. Position of the PEC minimum for the quintet state and T-shape geometry..................25 Table 17. Position of the PEC minimum for the singlet state and linear geometry.......................25 Table 18. Position of the PEC minimum for the triplet state and linear geometry........................26 Table 19. Position of the PEC minimum for the quintet state and linear geometry......................26 Table 20. Position of the PEC minimum for the singlet state and H-shape geometry..................27 Table 21. Position of the PEC minimum for the triplet state and H-shape geometry....................27 v  Table 22. Position of the PEC minimum for the quintet state and H-shape geometry..................28 Table 23. Position of the PEC minimum for the singlet state and X-shape geometry..................28 Table 24. Position of the PEC minimum for the triplet state and X-shape geometry....................29 Table 25. Position of the PEC minimum for the quintet state and X-shape geometry..................29 Table 26. Positions of BSSE corrected minima for the four main geometries studied.................29 Table 27. The position and depth of the interaction potential energy curve for the linear geometry of O2-O2.........................................................................................................................................36  Table 28. The position and depth of the interaction potential energy curve for the T-shape geometry of O2-O2.........................................................................................................................36  Table 29. The position and depth of the interaction potential energy curve for the H-shape geometry of O2-O2.........................................................................................................................37   Table 30. The position and depth of the interaction potential energy curve for the X-shape geometry of O2-O2.........................................................................................................................38  Table 31. Electronic properties (in a.u.) of O2 at the equilibrium bond length, r(O-O)=2.28187  bohr, using the MRCI and CASPT2/aug-cc-pVXZ levels of theory.............................................47  Table 32. Electronic properties (in a.u.) of O2 at the equilibrium bond length, r(O-O)=2.28187 bohr, using the MRCI and CASPT2/d-aug-cc-pVXZ levels of theory..........................................56  Table 33. CBS Limit Estimated Values (B-tensors and C-tensors) Obtained with the MRCI Method.........................................................................................................................................133  Table 34. CBS Limit Estimated Values (polarizabilities, quadrupole moments, hexadecapole moments and E-tensors) obtained with the MRCI method..........................................................133  Table 35. CBS Limit Estimated Values (B-tensors and C-tensors)  obtained with the CASPT2 method..........................................................................................................................................134  Table 36. CBS Limit Estimated Values (polarizabilities, quadrupole moments, hexadecapole moments and E-tensors)  obtained with the CASPT2 method....................................................135  Table 37. Vibrational averages obtained for the MRCI properties, at the d-aug-cc-pV5Z basis set level, unless otherwise noted.......................................................................................................137    Table 38. Vibrational averages obtained for the MRCI properties, at the d-aug-cc-pV5Z basis set level..............................................................................................................................................137  vi  Table 39. Vibrational averages obtained for the CASPT2 properties, at the d-aug-cc-pV5Z basis set level, unless otherwise noted.................................................................................................138  Table 40. Vibrational averages obtained for the CASPT2 properties, at the d-aug-cc-pV5Z basis set level........................................................................................................................................138  Table 41. Direct multipolar induction, back-induction, and dispersion terms in the collision-induced dipole coefficients  for rB=2 bohr, at the CASPT2 level, with the d-aug-cc-pVQZ basis set.............................................................................................................................147  Table 42. Direct multipolar induction, back-induction, and dispersion terms in the collision-induced dipole coefficients  for rB=2.1 bohr, at the CASPT2 level, with the d-aug-cc-pVQZ basis set.............................................................................................................................148  Table 43. Direct multipolar induction, back-induction, and dispersion terms in the collision-induced dipole coefficients  for rB=2.2 bohr, at the CASPT2 level, with the d-aug-cc-pVQZ basis set.............................................................................................................................149  Table 44. Direct multipolar induction, back-induction, and dispersion terms in the collision-induced dipole coefficients  for rB=2.28187 bohr, at the CASPT2 level, with the d-aug-cc-pVQZ  basis set.......................................................................................................................150  Table 45. Direct multipolar induction, back-induction, and dispersion terms in the collision-induced dipole coefficients  for rB=2.296 bohr, at the CASPT2 level, with the d-aug-cc-pVQZ basis set.............................................................................................................................151  Table 46. Direct multipolar induction, back-induction, and dispersion terms in the collision-induced dipole coefficients  for rB=2.36 bohr, at the CASPT2 level, with the d-aug-cc-pVQZ basis set.............................................................................................................................152  Table 47. Direct multipolar induction, back-induction, and dispersion terms in the collision-induced dipole coefficients  for rB=2.457 bohr, at the CASPT2 level, with the d-aug-cc-pVQZ basis set.............................................................................................................................153  Table 48. Direct multipolar induction, back-induction, and dispersion terms in the collision-induced dipole coefficients  for rB=2.646 bohr, at the CASPT2 level, with the d-aug-cc-pVQZ basis set.............................................................................................................................154  Table 49. Direct multipolar induction, back-induction, and dispersion terms in the collision-induced dipole coefficients  for rB=2.929 bohr, at the CASPT2 level, with the d-aug-cc-pVQZ basis set.............................................................................................................................155  vii  Table 50. Direct multipolar induction, back-induction, and dispersion terms in the collision-induced dipole coefficients  for rB=3.123 bohr, at the CASPT2 level, with the d-aug-cc-pVQZ basis set.............................................................................................................................156  Table 51. Direct multipolar induction, back-induction, and dispersion terms in the collision-induced dipole coefficients  for rB=3.3 bohr, at the CASPT2 level, with the d-aug-cc-pVQZ basis set.............................................................................................................................157  Table 52. Contributions to the long-range dipole moment, for four main O2-O2 orientations, with rA=req and rB =2 bohr....................................................................................................................163  Table 53. Contributions to the long-range dipole moment, for four main O2-O2 orientations, with rA=req and rB =2.1 bohr.................................................................................................................165  Table 54. Contributions to the long -range dipole moment, for four main O2-O2 orientations, with rA=req and rB =2.2 bohr.................................................................................................................166  Table 55. Contributions to the long-range dipole moment, for four main O2-O2 orientations, with rA=req and rB =req..........................................................................................................................166  Table 56. Contributions to the long -range dipole moment, for four main O2-O2 orientations, with rA=req and rB =2.296 bohr............................................................................................................167  Table 57. Contributions to the long-range dipole moment, for four main O2-O2 orientations, with rA=req and rB =2.36 bohr..............................................................................................................168  Table 58. Contributions to the long-range dipole moment, for four main O2-O2 orientations, with rA=req and rB =2.457 bohr............................................................................................................169  Table 59. Contributions to the long-range dipole moment, for four main O2-O2 orientations, with rA=req and rB =2.646....................................................................................................................170  Table 60. Contributions to the long-range dipole moment, for four main O2-O2 orientations, with rA=req and rB =2.929.....................................................................................................................170  Table 61. Contributions to the long-range dipole moment, for four main O2-O2 orientations, with rA=req and rB =3.213.....................................................................................................................171  Table 62. Contributions to the long-range dipole moment, for four main O2-O2 orientations, with rA=req and rB =3.3.........................................................................................................................171  Table 63. Orientations with largest contribution from quadrupolar induction, hexadecapolar induction, E-tensor induction, back-induction and dispersion for rB=2-3.3 bohr........................172  viii  Table 64.  Geometries used in the Calculation of Ab Initio Dipole Moments............................182  Table 65. The z-component of the collision-induced dipole moment at an internuclear distance of  2 bohr for geometry 0-0-0-0 (linear)...........................................................................................240  Table 66. The z-component of the collision-induced dipole moment at an internuclear distance of 2.1 bohr for geometry 0-0-0-0 (linear).........................................................................................240  Table 67. The z-component of the collision-induced dipole moment at an internuclear distance of 2.2 bohr for geometry 0-0-0-0 (linear).........................................................................................240  Table 68. The z-component of the collision-induced dipole moment at an internuclear distance of 2.296 bohr for geometry 0-0-0-0 (linear)....................................................................................241  Table 69. The z-component of the collision-induced dipole moment at an internuclear distance of 2.36 bohr for geometry 0-0-0-0 (linear).......................................................................................241  Table 70. The z-component of the collision-induced dipole moment at an internuclear distance of 2.457 bohr for geometry 0-0-0-0 (linear).....................................................................................241  Table 71. The z-component of the collision-induced dipole moment at an internuclear distance of 2.646 bohr for geometry 0-0-0-0 (linear).....................................................................................242  Table 72. The z-component of the collision-induced dipole moment at an internuclear distance of 2.929 bohr for geometry 0-0-0-0 (linear).....................................................................................242  Table 73. The z-component of the collision-induced dipole moment at an internuclear distance of 3.213 bohr for geometry 0-0-0-0 (linear).....................................................................................242  Table 74. The z-component of the collision-induced dipole moment at an internuclear distance of 3.3 bohr for geometry 0-0-0-0 (linear).........................................................................................243  Table 75. The z-component of the collision-induced dipole moment at an internuclear distance of 2 bohr for geometry 90-0-0-0 (T-shaped)....................................................................................243  Table 76. The z-component of the collision-induced dipole moment at an internuclear distance of 2.1 bohr for geometry 90-0-0-0 (T-shaped).................................................................................243  Table 77. The z-component of the collision-induced dipole moment at an internuclear distance of 2.2 bohr for geometry 90-0-0-0 (T-shaped).................................................................................244  Table 78. The z-component of the collision-induced dipole moment at an internuclear distance of 2.296 bohr for geometry 90-0-0-0 (T-shaped).............................................................................244  Table 79. The z-component of the collision-induced dipole moment at an internuclear distance of 2.36 bohr for geometry 90-0-0-0 (T-shaped)...............................................................................244 ix  Table 80. The z-component of the collision-induced dipole moment at an internuclear distance of 2.457 bohr for geometry 90-0-0-0 (T-shaped).............................................................................245  Table 81. The z-component of the collision-induced dipole moment at an internuclear distance of 2.646 bohr for geometry 90-0-0-0 (T-shaped).............................................................................245  Table 82. The z-component of the collision-induced dipole moment at an internuclear distance of 2.929 bohr for geometry 90-0-0-0 (T-shaped).............................................................................245  Table 83. The z-component of the collision-induced dipole moment at an internuclear distance of 3.213 bohr for geometry 90-0-0-0 (T-shaped)............................................................................246  Table 84. The z-component of the collision-induced dipole moment at an internuclear distance of 3.3 bohr for geometry 90-0-0-0 (T-shaped).................................................................................246  Table 85. The z-component of the collision-induced dipole moment at an internuclear distance of 2 bohr for geometry 90-0-90-0 (H-shaped).................................................................................246  Table 86. The z-component of the collision-induced dipole moment at an internuclear distance of 2.1 bohr for geometry 90-0-90-0 (H-shaped)..............................................................................247  Table 87. The z-component of the collision-induced dipole moment at an internuclear distance of 2.2 bohr for geometry 90-0-90-0 (H-shaped)..............................................................................247  Table 88. The z-component of the collision-induced dipole moment at an internuclear distance of 2.296 bohr for geometry 90-0-90-0 (H-shaped)..........................................................................247  Table 89. The z-component of the collision-induced dipole moment at an internuclear distance of 2.36 bohr for geometry 90-0-90-0 (H-shaped)............................................................................248  Table 90. The z-component of the collision-induced dipole moment at an internuclear distance of 2.457 bohr for geometry 90-0-90-0 (H-shaped)..........................................................................248  Table 91. The z-component of the collision-induced dipole moment at an internuclear distance of 2.646 bohr for geometry 90-0-90-0 (H-shaped)..........................................................................248  Table 92. The z-component of the collision-induced dipole moment at an internuclear distance of 2.929 bohr for geometry 90-0-90-0 (H-shaped)..........................................................................249  Table 93. The z-component of the collision-induced dipole moment at an internuclear distance of 3.213 bohr for geometry 90-0-90-0 (H-shaped)..........................................................................249  Table 94. The z-component of the collision-induced dipole moment at an internuclear distance of 3.3 bohr for geometry 90-0-90-0 (H-shaped)..............................................................................249  x  Table 95. The z-component of the collision-induced dipole moment at an internuclear distance of 2 bohr for geometry 90-0-90-90 (X-shaped)...............................................................................250  Table 96. The z-component of the collision-induced dipole moment at an internuclear distance of 2.1 bohr for geometry 90-0-90-90 (X-shaped)............................................................................250  Table 97. The z-component of the collision-induced dipole moment at an internuclear distance of 2.2 bohr for geometry 90-0-90-90 (X-shaped)............................................................................250  Table 98. The z-component of the collision-induced dipole moment at an internuclear distance of 2.296 bohr for geometry 90-0-90-90 (X-shaped)........................................................................251  Table 99. The z-component of the collision-induced dipole moment at an internuclear distance of 2.36 bohr for geometry 90-0-90-90 (X-shaped)..........................................................................251  Table 100. The z-component of the collision-induced dipole moment at an internuclear distance of 2.457 bohr for geometry 90-0-90-90 (X-shaped)....................................................................251  Table 101. The z-component of the collision-induced dipole moment at an internuclear distance of 2.646 bohr for geometry 90-0-90-90 (X-shaped)....................................................................252  Table 102. The z-component of the collision-induced dipole moment at an internuclear distance of 2.929 bohr for geometry 90-0-90-90 (X-shaped)....................................................................252  Table 103. The z-component of the collision-induced dipole moment at an internuclear distance of 3.213 bohr for geometry 90-0-90-90 (X-shaped)....................................................................252  Table 104. The z-component of the collision-induced dipole moment at an internuclear distance of 3.3 bohr for geometry 90-0-90-90 (X-shaped)........................................................................253  Table 105. The z-component of the collision-induced dipole moment at an internuclear distance of 2 bohr for geometry 30-0-0-0 .................................................................................................253  Table 106. The z-component of the collision-induced dipole moment at an internuclear distance of 2.1 bohr for geometry 30-0-0-0...............................................................................................253  Table 107. The z-component of the collision-induced dipole moment at an internuclear distance of 2.2 bohr for geometry 30-0-0-0...............................................................................................254  Table 108. The z-component of the collision-induced dipole moment at an internuclear distance of 2.28187 bohr for geometry 30-0-0-0.......................................................................................254   Table 109. The z-component of the collision-induced dipole moment at an internuclear distance of 2.296 bohr for geometry 30-0-0-0...........................................................................................254   xi  Table 110. The z-component of the collision-induced dipole moment at an internuclear distance of 2.36 bohr for geometry 30-0-0-0.............................................................................................255   Table 111. The z-component of the collision-induced dipole moment at an internuclear distance of 2.457 bohr for geometry 30-0-0-0...........................................................................................255   Table 112. The z-component of the collision-induced dipole moment at an internuclear distance of 2.646 bohr for geometry 30-0-0-0...........................................................................................255   Table 113. The z-component of the collision-induced dipole moment at an internuclear distance of 2.929 bohr for geometry 30-0-0-0...........................................................................................256   Table 114. The z-component of the collision-induced dipole moment at an internuclear distance of 3.213 bohr for geometry 30-0-0-0...........................................................................................256   Table 115. The z-component of the collision-induced dipole moment at an internuclear distance of 3.3 bohr for geometry 30-0-0-0...............................................................................................256   Table 116. The z-component of the collision-induced dipole moment at an internuclear distance of 2 bohr for geometry 60-0-0-0..................................................................................................257   Table 117. The z-component of the collision-induced dipole moment at an internuclear distance of 2.1 bohr for geometry 60-0-0-0...............................................................................................257   Table 118. The z-component of the collision-induced dipole moment at an internuclear distance of 2.2 bohr for geometry 60-0-0-0..............................................................................................257   Table 119. The z-component of the collision-induced dipole moment at an internuclear distance of 2.28187 bohr for geometry 60-0-0-0.......................................................................................258   Table 120. The z-component of the collision-induced dipole moment at an internuclear distance of 2.296 bohr for geometry 60-0-0-0...........................................................................................258   Table 121. The z-component of the collision-induced dipole moment at an internuclear distance of 2.36 bohr for geometry 60-0-0-0.............................................................................................258   Table 122. The z-component of the collision-induced dipole moment at an internuclear distance of 2.457 bohr for geometry 60-0-0-0...........................................................................................259   Table 123. The z-component of the collision-induced dipole moment at an internuclear distance of 2.646 bohr for geometry 60-0-0-0...........................................................................................259   Table 124. The z-component of the collision-induced dipole moment at an internuclear distance of 2.929 bohr for geometry 60-0-0-0...........................................................................................259   xii  Table 125. The z-component of the collision-induced dipole moment at an internuclear distance of 3.213 bohr for geometry 60-0-0-0......................................................................................... .260  Table 126. The z-component of the collision-induced dipole moment at an internuclear distance of 3.3 bohr for geometry 60-0-0-0...............................................................................................260   Table 127. The z-component of the collision-induced dipole moment at an internuclear distance of 2 bohr for geometry 105-0-60-60............................................................................................260   Table 128. The z-component of the collision-induced dipole moment at an internuclear distance of 2.1 bohr for geometry 105-0-60-60.........................................................................................261   Table 129. The z-component of the collision-induced dipole moment at an internuclear distance of 2.2 bohr for geometry 105-0-60-60.........................................................................................261  Table 130. The z-component of the collision-induced dipole moment at an internuclear distance of 2.28187 bohr for geometry 105-0-60-60.................................................................................261   Table 131. The z-component of the collision-induced dipole moment at an internuclear distance of 2.296 bohr for geometry 105-0-60-60.....................................................................................262   Table 132. The z-component of the collision-induced dipole moment at an internuclear distance of 2.36 bohr for geometry 105-0-60-60.......................................................................................262   Table 133. The z-component of the collision-induced dipole moment at an internuclear distance of 2.457 bohr for geometry 105-0-60-60.....................................................................................262   Table 134. The z-component of the collision-induced dipole moment at an internuclear distance of 2.646 bohr for geometry 105-0-60-60.....................................................................................263  Table 135. The z-component of the collision-induced dipole moment at an internuclear distance of 2.929 bohr for geometry 105-0-60-60.....................................................................................263   Table 136. The z-component of the collision-induced dipole moment at an internuclear distance of 3.213 bohr for geometry 105-0-60-60.....................................................................................263   Table 137. The z-component of the collision-induced dipole moment at an internuclear distance of 3.3 bohr for geometry 105-0-60-60.........................................................................................264   Table 138. The z-component of the collision-induced dipole moment at an internuclear distance of 2 bohr for geometry 40-0-115-45............................................................................................264  Table 139. The z-component of the collision-induced dipole moment at an internuclear distance of 2.1 bohr for geometry 40-0-115-45.........................................................................................264  Table 140. The z-component of the collision-induced dipole moment at an internuclear distance of 2.2 bohr for geometry 40-0-115-45.........................................................................................265 xiii  Table 141. The z-component of the collision-induced dipole moment at an internuclear distance of 2.28187 bohr for geometry 40-0-115-45.................................................................................265  Table 142. The z-component of the collision-induced dipole moment at an internuclear distance of 2.296 bohr for geometry 40-0-115-45.....................................................................................265  Table 143. The z-component of the collision-induced dipole moment at an internuclear distance of 2.36 bohr for geometry 40-0-115-45.......................................................................................266  Table 144. The z-component of the collision-induced dipole moment at an internuclear distance of 2.457 bohr for geometry 40-0-115-45.....................................................................................266  Table 145. The z-component of the collision-induced dipole moment at an internuclear distance of 2.646 bohr for geometry 40-0-115-45.....................................................................................266  Table 146. The z-component of the collision-induced dipole moment at an internuclear distance of 2.929 bohr for geometry 40-0-115-45.....................................................................................267  Table 147. The z-component of the collision-induced dipole moment at an internuclear distance of 3.213 bohr for geometry 40-0-115-45.....................................................................................267  Table 148. The z-component of the collision-induced dipole moment at an internuclear distance of 3.3 bohr for geometry 40-0-115-45.........................................................................................267  Table 149. The z-component of the collision-induced dipole moment at an internuclear distance of 2 bohr for geometry 20-0-115-45............................................................................................268  Table 150. The z-component of the collision-induced dipole moment at an internuclear distance of 2.1 bohr for geometry 20-0-115-45........................................................................................268  Table 151. The z-component of the collision-induced dipole moment at an internuclear distance of 2.2 bohr for geometry 20-0-115-45.........................................................................................268  Table 152. The z-component of the collision-induced dipole moment at an internuclear distance of 2.28187 bohr for geometry 20-0-115-45.................................................................................269  Table 153. The z-component of the collision-induced dipole moment at an internuclear distance of 2.296 bohr for geometry 20-0-115-45.....................................................................................269  Table 154. The z-component of the collision-induced dipole moment at an internuclear distance of 2.36 bohr for geometry 20-0-115-45.......................................................................................269  Table 155. The z-component of the collision-induced dipole moment at an internuclear distance of 2.457 bohr for geometry 20-0-115-45.....................................................................................270  xiv  Table 156. The z-component of the collision-induced dipole moment at an internuclear distance of 2.646 bohr for geometry 20-0-115-45.....................................................................................270  Table 157. The z-component of the collision-induced dipole moment at an internuclear distance of 2.929 bohr for geometry 20-0-115-45.....................................................................................270  Table 158. The z-component of the collision-induced dipole moment at an internuclear distance of 3.213 bohr for geometry 20-0-115-45.....................................................................................271  Table 159. The z-component of the collision-induced dipole moment at an internuclear distance of 3.3 bohr for geometry 20-0-115-45.........................................................................................271  Table 160. The z-component of the collision-induced dipole moment at an internuclear distance of 2 bohr for geometry 15-0-75-30..............................................................................................271  Table 161. The z-component of the collision-induced dipole moment at an internuclear distance of 2.1 bohr for geometry 15-0-75-30...........................................................................................272  Table 162. The z-component of the collision-induced dipole moment at an internuclear distance of 2.2 bohr for geometry 15-0-75-30...........................................................................................272  Table 163. The z-component of the collision-induced dipole moment at an internuclear distance of 2.28187 bohr for geometry 15-0-75-30...................................................................................272  Table 164. The z-component of the collision-induced dipole moment at an internuclear distance of 2.296 bohr for geometry 15-0-75-30.......................................................................................273  Table 165. The z-component of the collision-induced dipole moment at an internuclear distance of 2.36 bohr for geometry 15-0-75-30.........................................................................................273  Table 166. The z-component of the collision-induced dipole moment at an internuclear distance of 2.457 bohr for geometry 15-0-75-30.......................................................................................273  Table 167. The z-component of the collision-induced dipole moment at an internuclear distance of 2.646 bohr for geometry 15-0-75-30.......................................................................................274  Table 168. The z-component of the collision-induced dipole moment at an internuclear distance of 2.929 bohr for geometry 15-0-75-30.......................................................................................274  Table 169. The z-component of the collision-induced dipole moment at an internuclear distance of 3.213 bohr for geometry 15-0-75-30.......................................................................................274  Table 170. The z-component of the collision-induced dipole moment at an internuclear distance of 3.3 bohr for geometry 15-0-75-30...........................................................................................275  xv  Table 171. The z-component of the collision-induced dipole moment at an internuclear distance of 2 bohr for geometry 15-0-95-10..............................................................................................275  Table 172. The z-component of the collision-induced dipole moment at an internuclear distance of 2.1 bohr for geometry 15-0-95-10...........................................................................................275  Table 173. The z-component of the collision-induced dipole moment at an internuclear distance of 2.2 bohr for geometry 15-0-95-10...........................................................................................276  Table 174. The z-component of the collision-induced dipole moment at an internuclear distance of 2.28187 bohr for geometry 15-0-95-10...................................................................................276  Table 175. The z-component of the collision-induced dipole moment at an internuclear distance of 2.296 bohr for geometry 15-0-95-10.......................................................................................276  Table 176. The z-component of the collision-induced dipole moment at an internuclear distance of 2.36 bohr for geometry 15-0-95-10.........................................................................................277  Table 177 The z-component of the collision-induced dipole moment at an internuclear distance of 2.457 bohr for geometry 15-0-95-10.......................................................................................277  Table 178. The z-component of the collision-induced dipole moment at an internuclear distance of 2.646 bohr for geometry 15-0-95-10.......................................................................................277  Table 179. The z-component of the collision-induced dipole moment at an internuclear distance of 2.929 bohr for geometry 15-0-95-10.......................................................................................278  Table 180. The z-component of the collision-induced dipole moment at an internuclear distance of 3.213 bohr for geometry 15-0-95-10.......................................................................................278  Table 181. The z-component of the collision-induced dipole moment at an internuclear distance of 3.3 bohr for geometry 15-0-95-10...........................................................................................278  Table 182. The z-component of the collision-induced dipole moment at an internuclear distance of 2 bohr for geometry 10-0-160-15............................................................................................279  Table 183. The z-component of the collision-induced dipole moment at an internuclear distance of 2.1 bohr for geometry 10-0-160-15.........................................................................................279  Table 184. The z-component of the collision-induced dipole moment at an internuclear distance of 2.2 bohr for geometry 10-0-160-15.........................................................................................279  Table 185. The z-component of the collision-induced dipole moment at an internuclear distance of 2.28187 bohr for geometry 10-0-160-15.................................................................................280  xvi  Table 186. The z-component of the collision-induced dipole moment at an internuclear distance of 2.296 bohr for geometry 10-0-160-15.....................................................................................280  Table 187. The z-component of the collision-induced dipole moment at an internuclear distance of 2.36 bohr for geometry 10-0-160-15.......................................................................................280  Table 188. The z-component of the collision-induced dipole moment at an internuclear distance of 2.457 bohr for geometry 10-0-160-15.....................................................................................281  Table 189. The z-component of the collision-induced dipole moment at an internuclear distance of 2.646 bohr for geometry 10-0-160-15.....................................................................................281  Table 190. The z-component of the collision-induced dipole moment at an internuclear distance of 2.929 bohr for geometry 10-0-160-15.....................................................................................281  Table 191. The z-component of the collision-induced dipole moment at an internuclear distance of 3.213 bohr for geometry 10-0-160-15.....................................................................................282  Table 192. The z-component of the collision-induced dipole moment at an internuclear distance of 3.3 bohr for geometry 10-0-160-15.........................................................................................282  Table 193. The z-component of the collision-induced dipole moment at an internuclear distance of 2 bohr for geometry 20-0-145-25............................................................................................282  Table 194. The z-component of the collision-induced dipole moment at an internuclear distance of 2.1 bohr for geometry 20-0-145-25.........................................................................................283  Table 195. The z-component of the collision-induced dipole moment at an internuclear distance of 2.2 bohr for geometry 20-0-145-25.........................................................................................283  Table 196. The z-component of the collision-induced dipole moment at an internuclear distance of 2.28187 bohr for geometry 20-0-145-25.................................................................................283  Table 197. The z-component of the collision-induced dipole moment at an internuclear distance of 2.296 bohr for geometry 20-0-145-25.....................................................................................284  Table 198. The z-component of the collision-induced dipole moment at an internuclear distance of 2.36 bohr for geometry 20-0-145-25.......................................................................................284  Table 199. The z-component of the collision-induced dipole moment at an internuclear distance of 2.457 bohr for geometry 20-0-145-25.....................................................................................284  Table 200. The z-component of the collision-induced dipole moment at an internuclear distance of 2.646 bohr for geometry 20-0-145-25.....................................................................................285  xvii  Table 201. The z-component of the collision-induced dipole moment at an internuclear distance of 2.929 bohr for geometry 20-0-145-25.....................................................................................285  Table 202. The z-component of the collision-induced dipole moment at an internuclear distance of 3.213 bohr for geometry 20-0-145-25.....................................................................................285  Table 203. The z-component of the collision-induced dipole moment at an internuclear distance of 3.3 bohr for geometry 20-0-145-25.........................................................................................286  Table 204. The z-component of the collision-induced dipole moment at an internuclear distance of 2 bohr for geometry 30-0-130-35............................................................................................286  Table 205. The z-component of the collision-induced dipole moment at an internuclear distance of 2.1 bohr for geometry 30-0-130-35.........................................................................................286  Table 206. The z-component of the collision-induced dipole moment at an internuclear distance of 2.2 bohr for geometry 30-0-130-35.........................................................................................287  Table 207. The z-component of the collision-induced dipole moment at an internuclear distance of 2.28187 bohr for geometry 30-0-130-35.................................................................................287  Table 208. The z-component of the collision-induced dipole moment at an internuclear distance of 2.296 bohr for geometry 30-0-130-35.....................................................................................287  Table 209. The z-component of the collision-induced dipole moment at an internuclear distance of 2.36 bohr for geometry 30-0-130-35.......................................................................................288  Table 210. The z-component of the collision-induced dipole moment at an internuclear distance of 2.457 bohr for geometry 30-0-130-35.....................................................................................288  Table 211. The z-component of the collision-induced dipole moment at an internuclear distance of 2.646 bohr for geometry 30-0-130-35.....................................................................................288  Table 212. The z-component of the collision-induced dipole moment at an internuclear distance of 2.929 bohr for geometry 30-0-130-35.....................................................................................289  Table 213. The z-component of the collision-induced dipole moment at an internuclear distance of 3.213 bohr for geometry 30-0-130-35.....................................................................................289  Table 214. The z-component of the collision-induced dipole moment at an internuclear distance of 3.3 bohr for geometry 30-0-130-35.........................................................................................289  Table 215. The z-component of the collision-induced dipole moment at an internuclear distance of 2 bohr for geometry 30-0-60-0................................................................................................290  xviii  Table 216. The z-component of the collision-induced dipole moment at an internuclear distance of 2.1 bohr for geometry 30-0-60-0.............................................................................................290  Table 217. The z-component of the collision-induced dipole moment at an internuclear distance of 2.2 bohr for geometry 30-0-60-0.............................................................................................290  Table 218. The z-component of the collision-induced dipole moment at an internuclear distance of 2.28187 bohr for geometry 30-0-60-0.....................................................................................291  Table 219. The z-component of the collision-induced dipole moment at an internuclear distance of 2.36 bohr for geometry 30-0-60-0...........................................................................................291  Table 220.The z-component of the collision-induced dipole moment at an internuclear distance of 2.457 bohr for geometry 30-0-60-0.........................................................................................291  Table 221. The z-component of the collision-induced dipole moment at an internuclear distance of 2.646 bohr for geometry 30-0-60-0.........................................................................................292  Table 222. The z-component of the collision-induced dipole moment at an internuclear distance of 2.929 bohr for geometry 30-0-60-0.........................................................................................292  Table 223. The z-component of the collision-induced dipole moment at an internuclear distance of 3.213 bohr for geometry 30-0-60-0.........................................................................................292  Table 224. The z-component of the collision-induced dipole moment at an internuclear distance of 3.3 bohr for geometry 30-0-60-0.............................................................................................293  Table 225. The z-component of the collision-induced dipole moment at an internuclear distance of 2 bohr for geometry 30-0-90-0................................................................................................293  Table 226. The z-component of the collision-induced dipole moment at an internuclear distance of 2.1 bohr for geometry 30-0-90-0.............................................................................................293  Table 227. The z-component of the collision-induced dipole moment at an internuclear distance of 2.2 bohr for geometry 30-0-90-0.............................................................................................294  Table 228. The z-component of the collision-induced dipole moment at an internuclear distance of 2.28187 bohr for geometry 30-0-90-0.....................................................................................294  Table 229. The z-component of the collision-induced dipole moment at an internuclear distance of 2.296 bohr for geometry 30-0-90-0........................................................................................294  Table 230. The z-component of the collision-induced dipole moment at an internuclear distance of 2.36 bohr for geometry 30-0-90-0...........................................................................................295  xix  Table 231. The z-component of the collision-induced dipole moment at an internuclear distance of 2.457 bohr for geometry 30-0-90-0.........................................................................................295  Table 232. The z-component of the collision-induced dipole moment at an internuclear distance of 2.646 bohr for geometry 30-0-90-0.........................................................................................295  Table 233. The z-component of the collision-induced dipole moment at an internuclear distance of 2.929 bohr for geometry 30-0-90-0.........................................................................................296  Table 234. The z-component of the collision-induced dipole moment at an internuclear distance of 3.213 bohr for geometry 30-0-90-0.........................................................................................296  Table 235. The z-component of the collision-induced dipole moment at an internuclear distance of 3. 3 bohr for geometry 30-0-90-0............................................................................................296  Table 236. The z-component of the collision-induced dipole moment at an internuclear distance of 2 bohr for geometry 45-0-30-0................................................................................................297  Table 237. The z-component of the collision-induced dipole moment at an internuclear distance of 2.1 bohr for geometry 45-0-30-0.............................................................................................297  Table 238. The z-component of the collision-induced dipole moment at an internuclear distance of 2.2 bohr for geometry 45-0-30-0.............................................................................................297  Table 239. The z-component of the collision-induced dipole moment at an internuclear distance of 2.28187 bohr for geometry 45-0-30-0.....................................................................................298  Table 240. The z-component of the collision-induced dipole moment at an internuclear distance of 2.296 bohr for geometry 45-0-30-0.........................................................................................298  Table 241. The z-component of the collision-induced dipole moment at an internuclear distance of 2.36 bohr for geometry 45-0-30-0...........................................................................................298  Table 242. The z-component of the collision-induced dipole moment at an internuclear distance of 2.457 bohr for geometry 45-0-30-0.........................................................................................299  Table 243. The z-component of the collision-induced dipole moment at an internuclear distance of 2.646 bohr for geometry 45-0-30-0.........................................................................................299  Table 244. The z-component of the collision-induced dipole moment at an internuclear distance of 2.929 bohr for geometry 45-0-30-0.........................................................................................299  Table 245. The z-component of the collision-induced dipole moment at an internuclear distance of 3.213 bohr for geometry 45-0-30-0.........................................................................................300  xx  Table 246. The z-component of the collision-induced dipole moment at an internuclear distance of 3.3 bohr for geometry 45-0-30-0.............................................................................................300  Table 247. The z-component of the collision-induced dipole moment at an internuclear distance of 2 bohr for geometry 45-0-60-0................................................................................................300  Table 248. The z-component of the collision-induced dipole moment at an internuclear distance of 2.1 bohr for geometry 45-0-60-0.............................................................................................301  Table 249. The z-component of the collision-induced dipole moment at an internuclear distance of 2.2 bohr for geometry 45-0-60-0.............................................................................................301  Table 250. The z-component of the collision-induced dipole moment at an internuclear distance of 2.28187 bohr for geometry 45-0-60-0.....................................................................................301  Table 251. The z-component of the collision-induced dipole moment at an internuclear distance of 2.296 bohr for geometry 45-0-60-0.........................................................................................302  Table 252. The z-component of the collision-induced dipole moment at an internuclear distance of 2.36 bohr for geometry 45-0-60-0...........................................................................................302  Table 253. The z-component of the collision-induced dipole moment at an internuclear distance of 2.457 bohr for geometry 45-0-60-0.........................................................................................302  Table 254. The z-component of the collision-induced dipole moment at an internuclear distance of 2.646 bohr for geometry 45-0-60-0.........................................................................................303  Table 255. The z-component of the collision-induced dipole moment at an internuclear distance of 2.929 bohr for geometry 45-0-60-0.........................................................................................303  Table 256. The z-component of the collision-induced dipole moment at an internuclear distance of 3.213 bohr for geometry 45-0-60-0.........................................................................................303  Table 257. The z-component of the collision-induced dipole moment at an internuclear distance of 3.3 bohr for geometry 45-0-60-0.............................................................................................304  Table 258. The z-component of the collision-induced dipole moment at an internuclear distance of 2 bohr for geometry 45-0-90-0................................................................................................304  Table 259. The z-component of the collision-induced dipole moment at an internuclear distance of 2.1 bohr for geometry 45-0-90-0.............................................................................................304  Table 260. The z-component of the collision-induced dipole moment at an internuclear distance of 2.2 bohr for geometry 45-0-90-0.............................................................................................305  xxi  Table 261. The z-component of the collision-induced dipole moment at an internuclear distance of 2.28187 bohr for geometry 45-0-90-0.....................................................................................305  Table 262. The z-component of the collision-induced dipole moment at an internuclear distance of 2.296 bohr for geometry 45-0-90-0.........................................................................................305  Table 263. The z-component of the collision-induced dipole moment at an internuclear distance of 2.36 bohr for geometry 45-0-90-0...........................................................................................306  Table 264. The z-component of the collision-induced dipole moment at an internuclear distance of 2.457 bohr for geometry 45-0-90-0.........................................................................................306  Table 265. The z-component of the collision-induced dipole moment at an internuclear distance of 2.646 bohr for geometry 45-0-90-0.........................................................................................306  Table 266. The z-component of the collision-induced dipole moment at an internuclear distance of 2.929 bohr for geometry 45-0-90-0.........................................................................................307  Table 267. The z-component of the collision-induced dipole moment at an internuclear distance of 3.213 bohr for geometry 45-0-90-0.........................................................................................307  Table 268. The z-component of the collision-induced dipole moment at an internuclear distance of 3.3 bohr for geometry 45-0-90-0.............................................................................................307  Table 269. The z-component of the collision-induced dipole moment at an internuclear distance of 2 bohr for geometry 35-0-65-80..............................................................................................308  Table 270. The z-component of the collision-induced dipole moment at an internuclear distance of 2.1 bohr for geometry 35-0-65-80...........................................................................................308  Table 271. The z-component of the collision-induced dipole moment at an internuclear distance of 2.2 bohr for geometry 35-0-65-80...........................................................................................308  Table 272. The z-component of the collision-induced dipole moment at an internuclear distance of 2.28187 bohr for geometry 35-0-65-80...................................................................................309  Table 273. The z-component of the collision-induced dipole moment at an internuclear distance of 2.296 bohr for geometry 35-0-65-80.......................................................................................309  Table 274. The z-component of the collision-induced dipole moment at an internuclear distance of 2.36 bohr for geometry 35-0-65-80.........................................................................................309  Table 275. The z-component of the collision-induced dipole moment at an internuclear distance of 2.457 bohr for geometry 35-0-65-80.......................................................................................310  xxii  Table 276. The z-component of the collision-induced dipole moment at an internuclear distance of 2.646 bohr for geometry 35-0-65-80.......................................................................................310  Table 277. The z-component of the collision-induced dipole moment at an internuclear distance of 2.929 bohr for geometry 35-0-65-80.......................................................................................310  Table 278. The z-component of the collision-induced dipole moment at an internuclear distance of 3.213 bohr for geometry 35-0-65-80.......................................................................................311  Table 279. The z-component of the collision-induced dipole moment at an internuclear distance of 3.3 bohr for geometry 35-0-65-80...........................................................................................311  Table 280. The z-component of the collision-induced dipole moment at an internuclear distance of 2 bohr for geometry 60-0-80-65..............................................................................................311  Table 281. The z-component of the collision-induced dipole moment at an internuclear distance of 2.1 bohr for geometry 60-0-80-65...........................................................................................312  Table 282. The z-component of the collision-induced dipole moment at an internuclear distance of 2.2 bohr for geometry 60-0-80-65...........................................................................................312  Table 283. The z-component of the collision-induced dipole moment at an internuclear distance of 2.28187 bohr for geometry 60-0-80-65...................................................................................312  Table 284. The z-component of the collision-induced dipole moment at an internuclear distance of 2.296 bohr for geometry 60-0-80-65.......................................................................................313  Table 285. The z-component of the collision-induced dipole moment at an internuclear distance of 2.36 bohr for geometry 60-0-80-65.........................................................................................313  Table 286. The z-component of the collision-induced dipole moment at an internuclear distance of 2.457 bohr for geometry 60-0-80-65.......................................................................................313  Table 287. The z-component of the collision-induced dipole moment at an internuclear distance of 2.646 bohr for geometry 60-0-80-65.......................................................................................314  Table 288. The z-component of the collision-induced dipole moment at an internuclear distance of 2.929 bohr for geometry 60-0-80-65.......................................................................................314  Table 289. The z-component of the collision-induced dipole moment at an internuclear distance of 3.213 bohr for geometry 60-0-80-65.......................................................................................314  Table 290. The z-component of the collision-induced dipole moment at an internuclear distance of 3.3 bohr for geometry 60-0-80-65...........................................................................................315  xxiii  Table 291. The z-component of the collision-induced dipole moment at an internuclear distance of 2 bohr for geometry 70-0-65-75..............................................................................................315  Table 292. The z-component of the collision-induced dipole moment at an internuclear distance of 2.1 bohr for geometry 70-0-65-75...........................................................................................315  Table 293. The z-component of the collision-induced dipole moment at an internuclear distance of 2.2 bohr for geometry 70-0-65-75...........................................................................................316  Table 294. The z-component of the collision-induced dipole moment at an internuclear distance of 2.28187 bohr for geometry 70-0-65-75...................................................................................316  Table 295. The z-component of the collision-induced dipole moment at an internuclear distance of 2.296 bohr for geometry 70-0-65-75.......................................................................................316  Table 296. The z-component of the collision-induced dipole moment at an internuclear distance of 2.36 bohr for geometry 70-0-65-75.........................................................................................317  Table 297. The z-component of the collision-induced dipole moment at an internuclear distance of 2.457 bohr for geometry 70-0-65-75.......................................................................................317  Table 298. The z-component of the collision-induced dipole moment at an internuclear distance of 2.646 bohr for geometry 70-0-65-75.......................................................................................317  Table 299. The z-component of the collision-induced dipole moment at an internuclear distance of 2.929 bohr for geometry 70-0-65-75.......................................................................................318  Table 300. The z-component of the collision-induced dipole moment at an internuclear distance  of 3.213 bohr for geometry 70-0-65-75.......................................................................................318  Table 301. The z-component of the collision-induced dipole moment at an internuclear distance of 3.3 bohr for geometry 70-0-65-75...........................................................................................318  Table 302. The z-component of the collision-induced dipole moment at an internuclear distance of 2 bohr for geometry 50-0-95-55..............................................................................................319  Table 303. The z-component of the collision-induced dipole moment at an internuclear distance of 2.1 bohr for geometry 50-0-95-55...........................................................................................319  Table 304. The z-component of the collision-induced dipole moment at an internuclear distance of 2.2 bohr for geometry 50-0-95-55...........................................................................................319  Table 305. The z-component of the collision-induced dipole moment at an internuclear distance of 2.28187 bohr for geometry 50-0-95-55...................................................................................320  xxiv  Table 306. The z-component of the collision-induced dipole moment at an internuclear distance of 2.296 bohr for geometry 50-0-95-55.......................................................................................320  Table 307. The z-component of the collision-induced dipole moment at an internuclear distance of 2.36 bohr for geometry 50-0-95-55.........................................................................................320  Table 308. The z-component of the collision-induced dipole moment at an internuclear distance of 2.457 bohr for geometry 50-0-95-55.......................................................................................321  Table 309. The z-component of the collision-induced dipole moment at an internuclear distance of 2.646 bohr for geometry 50-0-95-55.......................................................................................321  Table 310. The z-component of the collision-induced dipole moment at an internuclear distance of 2.929 bohr for geometry 50-0-95-55.......................................................................................321  Table 311. The z-component of the collision-induced dipole moment at an internuclear distance of 3.213 bohr for geometry 50-0-95-55.......................................................................................322  Table 312. The z-component of the collision-induced dipole moment at an internuclear distance of 3.3 bohr for geometry 50-0-95-55...........................................................................................322                        xxv  LIST OF FIGURES   Figure 1. Comparison of the observed and computed O2-O2 fundamental band given by McKellar, Rich and Welsh...............................................................................................................8  Figure 2. Four main geometries stuAB, and = A- B),  bond distance rA and rB, and the intermolecular separation R80....................................................16  Figure 3. Wavefunctions for the first six vibrational levels (=0-5) for the ground rotational state (J=0) of the O2 molecule................................................................................................................67  Figure 4. The dipole polarizability (zz) in atomic units versus bond length for the CASPT2 and MRCI methods, using the doubly augmented basis sets................................................................70  Figure 5. The dipole polarizability (zz)  in atomic units versus bond length obtained with the CASPT2 method, using the singly and doubly augmented basis sets...........................................71  Figure 6. The dipole polarizability (zz)  in atomic units versus bond length, obtained using the MRCI method and the singly and doubly augmented basis sets...................................................72  Figure 7. The dipole polarizability (xx) in atomic units versus bond length, obtained with the MRCI and CASPT2 methods and the doubly augmented basis sets.............................................73  Figure 8. The dipole polarizability (xx) in atomic units versus bond length, obtained with the CASPT2 method and the singly and doubly augmented basis sets...............................................74  Figure 9. The dipole polarizability (xx) in atomic units versus bond length, obtained with the MRCI method and the singly and doubly augmented basis sets...................................................75  Figure 10. The quadrupole moment (zz) in atomic units versus bond length obtained with the MRCI and CASPT2 methods and the doubly augmented basis sets.............................................78  Figure 11. The shape of the quadrupole moment as determined by the cos2 function82..............79  Figure 12. The quadrupole moment (zz) in atomic units  versus bond length obtained using the CASPT2 method and the doubly and singly augmented basis sets...............................................80   Figure 13. The quadrupole moment (zz) in atomic units versus bond length obtained using the MRCI method and the doubly and singly augmented basis sets...................................................81   Figure 14. The hexadecapole moment  (zzzz) in atomic units versus bond length obtained using the MRCI and CASPT2 methods, and the doubly augmented basis sets......................................82  xxvi  Figure 15. The hexadecapole moment  (zzzz)  in atomic units versus bond length obtained using the CASPT2 method and the singly and doubly augmented basis sets.........................................83  Figure 16. The hexadecapole moment  (zzzz)  in atomic units versus bond length obtained using the MRCI method and the singly and doubly augmented basis sets..............................................84  Figure 17. The dipole-octopole polarizability (Ex,xxx) in atomic units versus the bond length obtained using the MRCI and CASPT2 methods, with the doubly augmented basis sets.............87  Figure 18. The dipole-octopole polarizability (Ex,xxx) in atomic units obtained using the CASPT2 method and the singly and doubly augmented basis sets...............................................................88  Figure 19. The dipole-octopole polarizability (Ex,xxx) in atomic units obtained using the MRCI method and the singly and doubly augmented basis sets...............................................................89  Figure 20. The dipole-octopole polarizability (Ez,zzz) in atomic units obtained using the MRCI and CASPT2 methods, and the doubly augmented basis sets.......................................................90  Figure 21. The dipole-octopole polarizability (Ez,zzz) in atomic units obtained using the CASPT2 method and the singly and doubly augmented basis sets...............................................................91  Figure 22. The dipole-octopole polarizability (Ez,zzz) in atomic units obtained using the MRCI method and the singly and doubly augmented basis sets. .............................................................92  Figure 23. The dipole-dipole-quadrupole polarizability (Bx,z,xz) in atomic units versus bond length obtained using the MRCI and CASPT2 methods and the doubly augmented basis sets ...93  Figure 24. The dipole-dipole-quadrupole polarizability (Bx,z,xz) in atomic units versus bond length obtained using the CASPT2 method and the singly and doubly augmented basis sets......94  Figure 25. The dipole-dipole-quadrupole polarizability (Bx,z,xz) in atomic units versus bond length obtained using the MRCI method and the singly and doubly augmented basis sets. ........95  Figure 26. The dipole-dipole-quadrupole polarizability (Bx,x,zz ) in atomic units versus bond length obtained using MRCI and CASPT2 methods using the doubly augmented basis sets.......97  Figure 27. The dipole-dipole-quadrupole polarizability (Bx,x,z,z) in atomic units obtained using the CASPT2 method and the singly and doubly augmented basis sets ........................................98  Figure 28. The dipole-dipole-quadrupole polarizability( Bx,x,z,z) in atomic units obtained using the MRCI method and the singly and doubly augmented basis sets ..................................................99  Figure 29. The dipole-dipole-quadrupole polarizability (Bx,x,xx) in atomic units versus bond length, obtained using the MRCI and CASPT2 methods, and the doubly augmented basis  sets................................................................................................................................................101  xxvii  Figure 30. The dipole-dipole-quadrupole polarizability (Bx,x,xx) in atomic units versus bond length, obtained using the CASPT2 method and the singly and doubly augmented basis sets...102  Figure 31. The dipole-dipole-quadrupole polarizability( Bx,x,xx) in atomic units versus bond length, obtained with the MRCI method and the singly and doubly augmented basis sets.........103  Figure 32. The dipole-dipole-quadrupole polarizability (Bz,z,zz) in atomic units versus bond length obtained using the MRCI and CASPT2 methods, with the doubly augmented basis sets ..........105  Figure 33. The dipole-dipole-quadrupole polarizability (Bz,z,zz) in atomic units versus bond length obtained using the CASPT2 method and the singly and doubly augmented basis sets...............106  Figure 34. The dipole-dipole-quadrupole polarizability (Bz,z,zz) in atomic units versus bond length obtained using the MRCI method and the singly and doubly augmented basis sets...................107  Figure 35. The quadrupole-quadrupole polarizability (Cxx,xx) in atomic units versus bond distance obtained using the MRCI and CASPT2 methods and the doubly augmented basis sets.............112  Figure 36. The quadrupole-quadrupole polarizability (Cxx,xx) in atomic units versus bond length obtained using the CASPT2 method and the singly and doubly augmented basis sets...............113  Figure 37. The quadrupole-quadrupole polarizability (Cxx,xx) in atomic units obtained using the MRCI method and the singly and doubly augmented basis sets.................................................114  Figure 38. The quadrupole-quadrupole polarizability (Czz,zz) in atomic units versus bond length obtained using the MRCI and CASPT2 methods and the doubly augmented basis sets.............115  Figure 39. The quadrupole-quadrupole polarizability (Czz,zz) in atomic units versus bond length obtained using the CASPT2 method and the singly and doubly augmented basis sets...............116  Figure 40. The quadrupole-quadrupole polarizability (Czz,zz)  in atomic units versus bond length obtained using the CASPT2 method and the singly and doubly augmented basis sets..............117  Figure 41. The quadrupole-quadrupole polarizability (Cxz,xz) in atomic units versus bond distance obtained using the MRCI and CASPT2 basis sets and the doubly augmented basis sets...........118   Figure 42. The quadrupole-quadrupole polarizability (Cxz,xz) in atomic units versus bond distance obtained using the CASPT2 basis set and the singly and doubly augmented basis sets.............119  Figure 43. The quadrupole-quadrupole polarizability (Cxz,xz) in atomic units versus bond distance obtained using the MRCI method and the singly and doubly augmented basis sets...................120   Figure 44. The second hyperpolarizability (xxxx) in atomic units versus bond distance obtained using the MRCI and CASPT2 methods and the doubly augmented basis sets............................121   xxviii  Figure 45. The second hyperpolarizability (xxxx) in atomic units versus bond distance obtained using the CASPT2 method and the singly and doubly augmented basis sets..............................122   Figure 46. The second hyperpolarizability (xxxx) in atomic units versus bond distance obtained using the MRCI method and the singly and doubly augmented basis sets..................................123  Figure 47. The second hyperpolarizability (xxzz) in atomic units versus bond distance obtained using the MRCI and CASPT2 methods and the  doubly augmented basis sets...........................124   Figure 48. The second hyperpolarizability (xxzz) in atomic units versus bond distance obtained using the CASPT2 method and the singly and doubly augmented basis sets.............................125  Figure 49. The second hyperpolarizability (xxzz) in atomic units versus bond distance obtained using the MRCI method and the singly and doubly augmented basis sets..................................126  Figure 50. The second hyperpolarizability (zzzz) in atomic units versus bond distance obtained using the MRCI and CASPT2 methods and the doubly augmented basis sets............................127   Figure 51. The second hyperpolarizability (zzzz) in atomic units versus bond distance obtained using the CASPT2 method and the singly and doubly augmented basis sets..............................128  Figure 52. The second hyperpolarizability (zzzz) in atomic units versus bond distance obtained using the MRCI method and the singly and doubly augmented basis sets..................................129  Figure 53. CBS limit for the dipole-dipole-quadrupole tensor (Bx,z,xz), with assigned error.......131  Figure 54. Convergence behavior of aug-cc-pVXZ and d-aug-cc-pVXZ basis sets, as illustrated  for values of the dipole-dipole-quadrupole tensor, Bx,z,xz at the equilibrium O2 bond distance.  The x-axis has values of the cardinal number (X), given by X-1................................................132   Figure 55. The direct quadrupolar induction contribution to the dipole for the four main geometries, at an intermolecular distance of  R=10 bohr............................................................173  Figure 56. E-tensor induction contribution to the collision-induced dipole moment for the four main geometries studied, at an intermolecular distance of 10 bohr.............................................174   Figure 57. Direct hexadecapolar induction contribution to the collision-induced dipole moment, for the four main geometries studied, at an intermolecular distance of R=10 bohr.....................176  Figure 58. The back-induction contribution to the collision-induced dipole for the four main geometries studied, at an intermolecular distance of R=10 bohr.................................................177   Figure 59. The dispersion contribution to the collision-induced dipole moment, for the four main geometries considered, at an intermolecular distance of 10 bohr................................................178   xxix  Figure 60. Comparison of dipole moments as a function of intermolecular distance (R) for the linear geometry, with r=2 bohr for . The dipole moment has atomic units, ea0.....................183   Figure 61. Comparison of dipole moments as a function of intermolecular distance (R) for the linear geometry, with r=3.3 bohr for  . The dipole moment has atomic units, ea0.................184  Figure 62a. Comparison of dipole moments as a function of intermolecular distance (R) for the T-shaped geometry, with r=2 bohr for . The dipole moment has atomic units, ea0................185  Figure 62b. Close-up view of dipole moments as a function of intermolecular distance (R) for the T-shaped geometry, with r=2 bohr for . The dipole moment has atomic units, ea0...............186   Figure 63. Comparison of dipole moments as a function of intermolecular distance (R) for the T-shaped geometry, with r=req for  . The dipole moment has atomic units, ea0........................186  Figure 64. Comparison of dipole moments as a function of intermolecular distance (R) for the T-shaped geometry, with r=3.3 bohr for . The dipole moment has atomic units, ea0................187  Figure 65a. Comparison of dipole moments as a function of intermolecular distance (R) for the H-shaped geometry, with r=2 bohr for . The dipole moment has atomic units, ea0...............188   Figure 65b. Close-up view of dipole moments as a function of intermolecular distance (R) for the T-shaped geometry, with r=2 bohr for . The dipole moment has atomic units, ea0...............189   Figure 66. Comparison of dipole moments as a function of intermolecular distance (R) for the H- shaped geometry, with r=3.3 bohr for . The dipole moment has atomic units, ea0................189  Figure 67 . Comparison of dipole moments as a function of intermolecular distance (R) for the H-shaped geometry, with r=2 bohr for . The dipole moment has atomic units, ea0...............190   Figure 68. Comparison of dipole moments as a function of intermolecular distance (R) for the H-shaped geometry, with r=3.3 bohr for . The dipole moment has atomic units, ea0................191  Figure 69. Comparison of dipole moments as a function of intermolecular distance (R) for  the 30-0-0-0 geometry, with r=2.1 bohr for . The dipole moment has atomic units, ea0.............192.  Figure 70. Comparison of dipole moments as a function of intermolecular distance (R) for  the 30-0-0-0 geometry, with r=2.296 bohr for . The dipole moment has atomic units, ea0.........193  Figure 71. Comparison of dipole moments as a function of intermolecular distance (R) for the 30-0-0-0 geometry, with r=3.3 bohr for . The dipole moment has atomic units, ea0.............194  Figure 72. Comparison of dipole moments as a function of intermolecular distance (R) for the 60-0-0-0 geometry, with r=2 bohr for . The dipole moment has atomic units, ea0................195  xxx  Figure 73. Comparison of dipole moments as a function of intermolecular distance (R) for the 60-0-0-0 geometry, with r=2.296 bohr for . The dipole moment has atomic units, ea0.........196  Figure 74. Comparison of dipole moments as a function of intermolecular distance (R) for the 60-0-0-0 geometry, with r=3.3 bohr for . The dipole moment has atomic units, ea0.............196  Figure 75. Comparison of dipole moments as a function of intermolecular distance (R) for the 105-0-60-60 geometry, with r=2 bohr for . The dipole moment has atomic units, ea0..........197  Figure 76. Comparison of dipole moments as a function of intermolecular distance (R) for the 105-0-60-60 geometry, with r=req for . The dipole moment has atomic units, ea0................198  Figure 77. Comparison of dipole moments as a function of intermolecular distance (R) for the 105-0-60-60 geometry, with r=3.3 bohr for . The dipole moment has atomic units, ea0.......199  Figure 78 . Comparison of dipole moments as a function of intermolecular distance (R) for the 40-0-115-45 geometry, with r=2 bohr for . The dipole moment has atomic units, ea0..........200  Figure 79. Comparison of dipole moments as a function of intermolecular distance (R) for the 40-0-115-45 geometry, with r=req for . The dipole moment has atomic units, ea0................201  Figure 80. Comparison of dipole moments as a function of intermolecular distance (R) for the 40-0-115-45 geometry, with r=3.3 bohr for . The dipole moment has atomic units, ea0.......202  Figure 81. Comparison of dipole moments as a function of intermolecular distance (R) for the 20-0-115-45 geometry, with r=2 bohr for . The dipole moment has atomic units, ea0..........202  Figure 82 . Comparison of dipole moments as a function of intermolecular distance (R) for the 20-0-115-45 geometry, with r=req for . The dipole moment has atomic units, ea0.................203  Figure 83 . Comparison of dipole moments as a function of intermolecular distance (R) for the 20-0-115-45 geometry, with r=3.3 bohr for . The dipole moment has atomic units, ea0.......204  Figure 84. Comparison of dipole moments as a function of intermolecular distance (R) for the 15-0-75-30 geometry, with r=2 bohr for . The dipole moment has atomic units, ea0............205  Figure 85.  Comparison of dipole moments as a function of intermolecular distance (R) for the 15-0-75-30 geometry, with r=req  for . The dipole moment has atomic units, ea0..................206  Figure 86. Comparison of dipole moments as a function of intermolecular distance (R) for the 15-0-75-30 geometry, with r=3.3 bohr for . The dipole moment has atomic units, ea0.........206  Figure 87. Comparison of dipole moments as a function of intermolecular distance (R) for the 15-0-95-10 geometry, with r=2 bohr for . The dipole moment has atomic units, ea0............207  xxxi  Figure 88. Comparison of dipole moments as a function of intermolecular distance (R) for the 15-0-95-10 geometry, with r=req for . The dipole moment has atomic units, ea0...................208  Figure 89. Comparison of dipole moments as a function of intermolecular distance (R) for the 15-0-95-10 geometry, with r=3.3 bohr for . The dipole moment has atomic units, ea0.........209  Figure (90). Comparison of dipole moments as a function of intermolecular distance (R) for the 10-0-160-15 geometry, with r=2 bohr for . The dipole moment has atomic units, ea0..........209  Figure (91). Comparison of dipole moments as a function of intermolecular distance (R) for the 10-0-160-15 geometry, with r=req for . The dipole moment has atomic units, ea0.................210  Figure 92. Comparison of dipole moments as a function of intermolecular distance (R) for the 10-0-160-15 geometry, with r=3.3 bohr for . The dipole moment has atomic units, ea0.......211  Figure 93. Comparison of dipole moments as a function of intermolecular distance (R) for the 20-0-145-25 geometry, with r=2 bohr for . The dipole moment has atomic units, ea0..........212  Figure 94. Comparison of dipole moments as a function of intermolecular distance (R) for the 20-0-145-25 geometry, with r=req for . The dipole moment has atomic units, ea0.................213  Figure 95. Comparison of dipole moments as a function of intermolecular distance (R) for the 20-0-145-25 geometry, with r=3.3 bohr for . The dipole moment has atomic units, ea0.......214  Figure 96. Comparison of dipole moments as a function of intermolecular distance (R) for the 30-0-135-35 geometry, with r=2 bohr for . The dipole moment has atomic units, ea0..........215  Figure 97. Comparison of dipole moments as a function of intermolecular distance (R) for the 30-0-135-35 geometry, with r=req for . The dipole moment has atomic units, ea0.................215  Figure 98. Comparison of dipole moments as a function of intermolecular distance (R) for the 30-0-135-35 geometry, with r=3.3 bohr for . The dipole moment has atomic units, ea0.......216  Figure 99. Comparison of dipole moments as a function of intermolecular distance (R) for the 30-0-60-0 geometry, with r=2 bohr for . The dipole moment has atomic units, ea0..............217  Figure 100. Comparison of dipole moments as a function of intermolecular distance (R) for the 30-0-60-0 geometry, with r=2.296 bohr for . The dipole moment has atomic units, ea0.......217  Figure 101. Comparison of dipole moments as a function of intermolecular distance (R) for the 30-0-60-0 geometry, with r=3.3 bohr for . The dipole moment has atomic units, ea0...........218  Figure 102.  Comparison of dipole moments as a function of intermolecular distance (R) for the 30-0-90-0 geometry, with r=2 bohr for . The dipole moment has atomic units, ea0..............219  xxxii  Figure 103. Comparison of dipole moments as a function of intermolecular distance (R) for the 30-0-90-0 geometry, with r=req for . The dipole moment has atomic units, ea0.....................219  Figure 104. Comparison of dipole moments as a function of intermolecular distance (R) for the 30-0-90-0 geometry, with r=3.3 bohr for . The dipole moment has atomic units, ea0...........220  Figure 105. Comparison of dipole moments as a function of intermolecular distance (R) for the 35-0-65-80 geometry, with r=2 bohr for . The dipole moment has atomic units, ea0............221  Figure 106. Comparison of dipole moments as a function of intermolecular distance (R) for the 35-0-65-80 geometry, with r=req for . The dipole moment has atomic units, ea0...................221  Figure 107. Comparison of dipole moments as a function of intermolecular distance (R) for the 35-0-65-80 geometry, with r=3.3 bohr for . The dipole moment has atomic units, ea0.........222  Figure 108. Comparison of dipole moments as a function of intermolecular distance (R) for the 45-0-30-0 geometry, with r=2 bohr for . The dipole moment has atomic units, ea0..............223  Figure 109. Comparison of dipole moments as a function of intermolecular distance (R) for the 45-0-30-0 geometry, with r=req for . The dipole moment has atomic units, ea0.....................224  Figure 110. Comparison of dipole moments as a function of intermolecular distance (R) for the 45-0-30-0 geometry, with r=3.3 bohr for . The dipole moment has atomic units, ea0...........225  Figure 111. Comparison of dipole moments as a function of intermolecular distance (R) for the 45-0-60-0 geometry, with r=2 bohr for . The dipole moment has atomic units, ea0..............226  Figure 112. Comparison of dipole moments as a function of intermolecular distance (R) for the 45-0-60-0 geometry, with r=req for . The dipole moment has atomic units, ea0....................226  Figure 113. Comparison of dipole moments as a function of intermolecular distance (R) for the 45-0-60-0 geometry, with r=3.3 bohr for . The dipole moment has atomic units, ea0...........227  Figure 114. Comparison of dipole moments as a function of intermolecular distance (R) for the 45-0-90-0 geometry, with r=2 bohr for . The dipole moment has atomic units, ea0..............228  Figure 115. Comparison of dipole moments as a function of intermolecular distance (R) for the 45-0-90-0 geometry, with r=req for . The dipole moment has atomic units, ea0....................229  Figure 116. Comparison of dipole moments as a function of intermolecular distance (R) for the  45-0-90-0 geometry, with r=3.3 bohr for . The dipole moment has atomic units, ea0...........229  Figure 117. Comparison of dipole moments as a function of intermolecular distance (R) for the 50-0-95-55 geometry, with r=2 bohr for . The dipole moment has atomic units, ea0............230  xxxiii  Figure 118. Comparison of dipole moments as a function of intermolecular distance (R) for the 50-0-95-55 geometry, with r=req bohr for . The dipole moment has atomic units, ea0..........231  Figure 119. Comparison of dipole moments as a function of intermolecular distance (R) for the 50-0-95-55 geometry, with r=3.3 bohr for . The dipole moment has atomic units, ea0.........232  Figure 120. Comparison of dipole moments as a function of intermolecular distance (R) for the 60-0-80-65 geometry, with r=2 bohr for . The dipole moment has atomic units, ea0............233  Figure 121. Comparison of dipole moments as a function of intermolecular distance (R) for the 60-0-80-65 geometry, with r=req for . The dipole moment has atomic units, ea0...................233  Figure 122. Comparison of dipole moments as a function of intermolecular distance (R) for the 60-0-80-65 geometry, with r=3.3 bohr for . The dipole moment has atomic units, ea0.........234  Figure 123. Comparison of dipole moments as a function of intermolecular distance (R) for the 70-0-65-75 geometry, with r=2 bohr for . The dipole moment has atomic units, ea0............235  Figure 124. Comparison of dipole moments as a function of intermolecular distance (R) for the 70-0-65-75 geometry, with r=req for . The dipole moment has atomic units, ea0...................236  Figure 125. Comparison of dipole moments as a function of intermolecular distance (R) for the 70-0-65-75 geometry, with r=3.3 bohr for . The dipole moment has atomic units, ea0.........237           xxxiv   1  CHAPTER 1: Introduction In order to absorb and emit radiation in the infrared region of the electromagnetic spectrum, a molecule must possess a charge separation in the form of a dipole moment1.  The oxygen molecule exists as a linear, centrosymmetric molecule, with no dipole moment, and therefore is infrared inactive. However, during  collisions between oxygen molecules at high pressures, transition probabilities are enhanced as the transitions can acquire some electric dipole character as a result of breakdown of the symmetry during interactions2. This situation is thus useful for investigating intermolecular, as opposed to intramolecular, processes.  These collision-induced interactions were first detected, in fact, in a sample of oxygen, when Crawford, Welsh and Locke3 observed it while studying the O2 fundamental rotation-vibrational band with a Perkin-Elmer spectrometer. The absorption varied as the square of the density, suggesting absorption by pairs of molecules (since absorption by monomers has a linear dependence on the density of the material).   Although occurring in the infrared region of the electromagnetic spectrum, this absorption followed the Raman selection rules of , leading to a Q, O, and S branch, centered at the O2 molecule's fundamental frequency of 1556 cm-1.  In both Raman scattering and collision-induced absorption an electric dipole is induced; in the case of Raman scattering the dipole is induced by the field of a light wave, while for collision-induced absorption the electric dipole is produced by an intermolecular force field4.  Further measurements5 of the infrared absorption of samples of O2 at high densities showed  a local maximum in regions of twice the fundamental frequency of the oxygen molecule. These findings supported the hypothesis that the absorption is caused by  distortion of the charge distribution of the absorbing molecule during a two-body collision.   2  In 1966, Shapiro and Gush6 also used a  Perkin-Elmer spectrometer to make absorption measurements of a sample of oxygen molecules at room temperature,  at a pressure of a few atmospheres, where only binary collisions could be observed (ternary collisions are not observed until pressures of 85 atm7). The goal of this work was to obtain accurate absorption coefficients for the O2-O2 collision-induced  infrared bands, including the fundamental. Higher pressures were necessary to obtain the  absorption coefficients for the weaker overtone bands.  This work reinforced the earlier finding of Welsh and Locke3. During a collision, interactions can occur through  classical electrostatic effects, dispersion forces (a long-range interaction), exchange/repulsion forces (a short range interaction), and others. Shapiro and Gush noted the broadening of the absorption profile, and the complete "smearing out" of the rotational bands, observed because "collisions occur on the duration of a "fly-by encounter"  and thus lead to a broad, continuous spectrum over many wavelengths instead of resolved, sharp peaks"6. This is a direct consequence of Heisenberg's uncertainty principle, which says the product of line width and lifetime is never smaller than a number on the order of unity. Typical line widths for collision-induced 13 Hz, corresponding to a radiative lifetime of -12 s1.1         Shapiro and Gush6 concluded from their band-profile analysis that  the collision-induced dipole moments responsible for the absorption had two sources: overlap induction and quadrupolar induction. The quadrupole-induced dipole moment is orientation-dependent and follows the before-mentioned Raman selection rules, while the overlap-induced dipole moment is less orientation-dependent and therefore contributes mostly to the J=0, or Q branch.  3  This collision-induced dipole moment has been modeled in early work by Van Kranendonk8 by the following:  (1.1) where r is the intermolecular separation and ,In equation (1.1), the first factor is due to the quadrupolar interaction (long range) and is strongly orientationally dependent, while the second factor is the short range overlap induction factor and is accordingly less orientationally dependent as stated above.      Shapiro and Gush6 further attribute the quadrupolar induction to two types of molecular transitions. The first is a single transition, where one molecule of the colliding pair makes a ernal energy of the other molecule remains unchanged.  The integrated intensity of the absorption due 2, where Q' is the rate of change of the quadrupole the polarizability of the molecule. The second transition type that contributes to the quadrupole-induced absorption spectrum is a double transition, which can occur in two different ways. The first is when one molecule of the colliding pair makes a vibrational transition only, while the other molecule makes a rotational transition according to the Raman selection rules. The intensity of this type of transition is proportional to 2, where  is the rate of change of the polarizability with respect to the internuclear distance, and Q is the quadrupole moment . The second is a double transition during which the first molecule makes a rotation-vibration transition, while the other molecule makes a rotational transition. The spectrum for this case is the most complex, and is proportional to the square of the directionally dependent component of the polarizability. 4  A year after the work of Shapiro and Gush6, Rettschnick & Hoytink9 published a theoretical study of  electronic transitions from collision-induced absorption. In this work, they discussed the following very weak magnetic and forbidden electronic  transitions that had been found to occur during collisions of oxygen molecules (where for example, the notation  refers to a transition from the ground state to the first accessible excited energy state,  a transition from the ground electronic state to the second accessible energy state, ect.): 31; weak magnetic dipole transition9,10,11,12,13,14,15 (the "infrared system")        (1.2)   31; weak magnetic dipole transition16 (the "red system")                                  (1.3)   33; weak orbitally forbidden electric dipole transition17                                                       (1.4) 33; weak ± forbidden electric dipole transition (Herzberg bands)18                   (1.5) ,31; weak spin-forbidden electric dipole transition17                                           (1.6) Most importantly, they noted that these transitions occur due to exchange interaction between the two oxygen molecules, which leads to these weak transitions borrowing intensity from the Schumann-Runge transition, which is the strongest allowed electronic transition: ,33; strongly allowed electric dipole transition (Schumann-Runge bands)19       (1.7)   Not long after the theoretical work of  Rettschnick & Hoytink9, Tabisz, Allin, and Welsh4 also studied the electronic transitions for compressed oxygen using absorption spectra. Their primary goal in doing so was to attempt to gain evidence for bound or at least "quasi -stable" (O2)2 complexes, separated from the collision-induced or van der Waals spectra.  They also studied the 5  "infrared system"9-15 and the "red system"16 described by Rettschnick & Hoytink9, as well as what are known as the combination systems20,21  3 + 31+1                                                                                                                                                           (1.8) 3 + 31+1                                                                                                                 (1.9) 3 + 31+ 1                                                                                                              (1.10) They concluded from these studies that a bound complex could have a shortening of its lifetime because of predissociation when the rotational energy exceeded the dissociation energy of the pair during a collision. They further stated that their experimental investigation demonstrated that the electronic band systems of oxygen observed in the compressed gas in the near IR and visible regions conform to all known criteria for collision-induced spectra. They assumed a Boltzmann relation known to hold for pressure-induced absorption bands in modeling their spectra, and the fit of their absorption coefficients to this model was their primary evidence for the observed bands being assigned as collision-induced electronic spectra. In the same year, Blickensderfer and Ewing22 studied just the 31  transition, or "infrared system" at very long path lengths and low pressures, in order to limit their measurements to binary collisions. They also reported finding no evidence of bound (O2)2 complexes, and found that their absorption coefficients fit the Boltzmann distribution for translationally broadened, or collision-induced, spectra. In 1972, McKellar, Rich and Welsh23 published their study of the infrared fundamental and the five strongest near-infrared and visible electronic bands of gaseous oxygen (Table 1), using long path lengths as was done by Blickensderfer and Ewing9, but extending to lower temperatures (90-115K) than the experiments done by Tabisz et al.4 (205-297 K) .  6    Table 1. Electronic bands of gaseous oxygen studied by McKellar, Rich, and Welsh23  Additionally, low temperature experiments were done on the fundamental band in order to supplement the high-temperature observations of Shapiro and Gush6. McKellar et al23. defined the binary absorption coefficient as                                                    (1.11)  is the path length in centimeters,  is the frequency in cm-1,  is the intensity which is transmitted by the gas, and  is the intensity transmitted by the evacuated cell. The integrated absorption is then defined as                                                           (1.12) 7  They noted a marked narrowing of the fundamental band at the lower temperatures used when compared to the work of Shapiro and Gush6. This was attributed to two causes: the higher rotational populations tend towards lower energy levels, and the lower kinetic energies of collision pairs give smaller translation broadening of the individual induced transitions.   McKellar, Rich and Welsh23 used a computed profile that assumed that only the quadrupole induction is important in producing the collision-induced spectrum for the . They compared this calculated spectrum to their experimental spectrum obtained at 102 K and found very little difference between the two. They thus concluded that the induction in the fundamental band is due almost entirely to anisotropic interactions operative during collisions (Figure (1)). 8   Figure 1. Comparison of the observed and computed O2-O2 fundamental band given by McKellar, Rich and Welsh23   For the near-IR and visible electronic bands,  McKellar, Rich and Welsh23 fit their data to the following Boltzmann relation:                                                    (1.13) They noted that the application of this relation to an entire band with an unknown rotational structure rests on the assumption that the translation broadening is much greater than the extent of the structure, as is the case for collision-induced absorption.  They found that the bands at 1.26 , 6290 Å , and 5770 Å satisfied the relation well. However, they concluded that the low-9  temperature profile for the band at 4770 Å did not satisfy the relation satisfactorily, unlike the high-temperature data for this band that did fit the Boltzmann equation used in the work by Tabisz et al.4 The Boltzma This band showed two clearly resolved maxima in the low temperature spectra. These were explained by McKellar, Rich and Welsh23 and in earlier work by Cho14 et al. as arising from two distinct transitions of different energies. In one transition, the vibrational 1-0 transition takes place in the same molecule as the electronic transition. In the second, the vibrational transition takes place in the other of the two colliding molecules.   Perner and Platt24 carried out the first atmospheric observation of collision-induced absorption of O2 pairs.  They obtained absorption data for the "infrared", "red" and combination systems. Continued atmospheric  studies focused on the fundamental vibration-rotation band (see Orlando25 et al. for a review). Thibault26 et al. noted that the first overtone band is approximately 100 times less intense6, and concluded that the first and therefore sequential overtone bands were thus negligible for modeling the absorption of O2 in the atmosphere. They studied  collision-induced absorption in the fundamental vibration-rotation band with a view to application for atmospheric physics, and thus the acquisition of more accurate absorption coefficients. They performed absorption measurements using an infrared Fourier spectrometer, on gaseous samples of oxygen in the 0-20 atm pressure and 193-293 K temperature ranges.  Rinsland et al.27 studied the collision-induced absorption of the fundamental vibration-rotation band by recording high resolution stratospheric absorption spectra from a balloon-borne interferometer. They noted that for spectroscopic experiments of the stratosphere, such as the limb infrared monitor of the stratosphere (LIMS) experiment aboard the Nimbus 728, it is important to verify the O2 10  laboratory absorption coefficients3,4,5 by direct comparison to O2 absorption measured in the atmosphere.  Research on the collision-induced absorption in O2 pairs has in more recent years focused on its impact in the Earth's global solar short wave (SW) radiation budget. The global average of SW radiation transmitted to the Earth by the Sun has been estimated to be 342 W/m2. For clear sky conditions, roughly 16% of this radiation is absorbed in the atmosphere by the gases O2, H2O, O3, CO2, CH4, and others, with the remaining 84% transmitted to the surface of the Earth29-31.      Absorption measurements done in both clear32 and cloudy29,32skies using Differential Optical Absorption Spectrometry (DOAS)33 showed that atmospheric absorption by O2 collision pairs contribute a minor but non-negligible role for both clear and cloudy sky SW heating, with an estimated 0.57 W/m2, or about 1% of the globally averaged SW heating. This model took the 1.06 m and the visible bands into account.  Solomon et al.34 extended the work of Pfeilsticker et al.29, showing that the 1.27 m band of O2-O2, and the same band for O2-N2 collision pairs, should also be considered in the Earth's radiation budget. To model the collision-induced band at 1.27 m, Solomon et al.32 stretched the 1.06 m band shape to fit relationships between the integrated intensity of the two bands taken from older low resolution measurements. As a consequence of using data from different studies20,23,35, they presented their SW estimates as a range of values.   For clear sky conditions alone, they estimated that the absorption in the 1.27 m band from both complexes is about 0.64-1.55 W/m2. The corresponding globally averaged, all sky condition values are then 0.93-1.32 W/m2, which is roughly double of the estimate put forth by Pfeilsticker et al.29 More recent work36 that used improved data for the band at 1.27 m 11  estimated the overall globally averaged impact of collision-induced absorption in mixtures of oxygen and nitrogen to be about 1 W/m2.   Much work has been done on considerations of O2-O2 collisions contributing to the formation/dissociation of ozone, both experimentally (Miller37, Park and Slanger38, Stranges39, Toumi40) and theoretically (Hernandez41, Lauvergnat42). Ozone absorbs UV radiation in what is known as the Hartley band38,43,44between 200-300 nm. When a photon in this region is absorbed, the ozone molecule dissociates. A mechanism has been proposed in which this dissociation of ozone produces vibrationally excited O2 (), which then reacts with a ground state O2 to produce ozone45. This mechanism arose from two observations. The first was a bimodal O(3P) translational energy distribution following photodissociation of ozone at 226 nm, implying the coincident production of triplet O2, which was identified as O2 ( in very high vibrational levels in subsequent experiments. The second was a large cross section for the reaction of vibrationally excited ground electronic O2 with O2 to yield ozone, based on measurements of relaxation rates of vibrationally excited O246,39.  Miller37 et al. used photofragment imaging and pump/probe experiments to explore the photodissociation of ozone at 226 nm by the following mechanism:                                               (1.14)                                                   (1.15) where hproblem"45,46,47,48, or why previously predicted stratospheric  ozone concentrations were less than observed. From quantum-yield calculations, Miller et al. concluded that  highly vibrationally excited  is in fact produced by the dissociation of ozone in the atmosphere at wavelengths of 226 nm (ultraviolet). They determined the yield of this excited   to be about 12  1%.  Using steady-state calculations, they estimated a 10% upper stratospheric ozone increase as a result of this yield. They therefore concluded that the mechanism in () could help account for the deficit. Later work confirmed the production of  from the dissociation of ozone following absorption of UV light49,50,51,52, and additional work done on the photodissociation of ozone by high-resolution photofragment translational spectroscopy by the Stranges group39 estimated the excited O2 yield to be even higher, at 7%.   Park and Slanger38 also addressed the ozone deficit problem. It had been hypothesized that the  vibrationally excited oxygen molecules produced in the Hartley band by ozone photodissociation could then undergo subsequent photoexcitation in the Schumann-Runge bands53.This photoexcitation could lead to predissociation, and subsequent autocatalytic production of ozone, by the following mechanism:  (3                                                    (1.16) (3(3                                                           (1.17) (3]                                                    (1.18)  In (1.17), the vibrationally excited oxygen molecule is in the =7 vibrational state because absorption at that step must occur at the photolysis wavelength of 248 nm, which is the location of the (2,7) band  in the Schumann-Runge system.    This new source of ozone could then help alleviate the discrepancies between modeled and observed ozone profiles. In order for this mechanism to generate enough ozone to help account for the discrepancy, the lifetime of the vibrationally excited oxygen molecules in (1.17) would have to be long enough to then absorb a photon in the Schumann-Runge band. Toumi54  13  previously modeled atmospheric ozone production by the autocatalytic mechanism in (1.16)-(1.18), assuming that O2  would be the only important quencher of the excited oxygen molecules produced in the Hartley bands. In this model, the quenching was slow enough to allow the autocatalytic mechanism to generate significant amounts of ozone.  In order to evaluate this possibility, Park and Slanger38 studied the nascent distribution of vibrationally excited oxygen at 248 nm, near the peak of the Hartley band. They also obtained their rate coefficients for relaxation by O2 and N2.  Their results were similar to that of Toumi54 for O2 as a quencher of the excited O2 molecules. However, Park and Slanger also considered N2 as a quencher. They found that O2 collisions with N2 quenched the excited O2 molecules five times faster than when O2 was the collisional partner. They concluded that this precludes a significant role for the vibrationally excited oxygen photodissociation as a source of stratospheric ozone.   Geiser55 et al. later used resonance-enhanced multiphoton ionization techniques, coupled with time-of-flight product imaging to determine the yield of  from photodissociation of ozone between 226 and 240 and at 260 nm. The yields at these wavelengths were between 0.6±0.1% and 11.8±1.9%. They additionally stated that it now seemed unlikely that the production of ozone from (1.14) and (1.15) occurs, given that there had been no direct evidence found for step (1.15). Specifically, theoretical work41,42,56-66 involving this reaction using ab initio surfaces60,62 and semi-empirical potentials57,59,60,62-65 found no evidence to support the proposed mechanism.  As mentioned at the beginning of this chapter, it is through the induced dipoles that collision-induced absorption is able to occur between otherwise spectroscopically inactive molecules.  If the induced dipole moment and the interaction potential are known as functions of the intermolecular separation and molecular orientation, an absorption spectrum can be 14  obtained1. For the purpose of computing spectral line shapes and spectral moments, the induced es1,67,68,69 as follows:                 (1.19) In (1.19), and  are unit vectors along the symmetry axes of molecules A and B,  is a unit vector along the intermolecular axis R, and  is a Clebsch-Gordan coefficient. The functions  are expansion coefficients.  To model the collision-induced spectra of binary or pure mixtures of diatomic gases, we first define an absorption coefficient66,70 as follows:                                             (1.20) where . By considering only binary collisions, we can then write the function  as                                                       (1.21) for a pure gas such as O2-O2. The function  consists of a linear superposition of individual components . For pure rotational-translational spectra, vibrational quantum numbers are zero,  and therefore n is specified by the set of integers:                                            (1.22) where  and  are the initial and final rotational quantum numbers, and the other four integers characterize the induced dipole mechanism, as in the expansions coefficients in (1.19).  Therefore, the primary goal of this work will be to calculate the expansion coefficients  in (1.19), and subsequently the collision-induced dipole moments, for use in spectroscopic line shape analysis. 15   CHAPTER 2: The Interaction Energy of O2-O2 Collisional Complexes  2.1 Introduction  Knowledge of the interaction potential energy for colliding oxygen molecules is relevant for  atmospheric chemistry and physics71,72 (including the airglow phenomenon), in the chemical oxygen iodine laser (COIL)73, in ultracold collisions74, and in condensed phase studies 75,82.  Diatomic oxygen has an unusual ground electronic state of 3. This electron configuration has two unpaired electrons in antibonding, degenerate molecular orbitals, and therefore the oxygen molecule is paramagnetic. Interaction between these two open-shelled molecules leads to non-dynamical correlation effects and requires a multireference method for calculation of the energies involved. In general, non-dynamical or static electron correlation effects arise when the wavefunction of an electronic state is well described only with more than one Slater determinant. This effect is dominant in bond cleavage cases, such as at the long O-O distances of the stretched O2 molecule. Therefore the inclusion of the static correlation via the multi-reference self-consistent field (MCSCF) is essential. MCSCF allows the simultaneous optimization of the coefficients and the molecular orbitals pertaining to the contributing determinants.      When two diatomic oxygen molecules in their ground states interact through a collision, three asymptotic states can be formed, leading to a quintet, triplet, and singlet spin state. The potential energy curves for these states have been investigated in the current work with an aug-cc-PVQZ basis set76, using the CASPT277,78 method. The CASPT2 method was chosen because the singlet and triplet asymptotic states require a multireference method.  For each oxygen monomer, we correlate 16 electrons to 12 orbitals at the MCSCF level, namely the 2p orbitals of all oxygens, and then we allow excitations out of the 2s orbitals at the CASPT2, and so we correlate 24 16  electrons to 16 orbitals. All calculations have been done using the MOLPRO package79, unless otherwise noted.        Four main geometries for the O2-O2 collisional complex were examined. These are the H-shape geometry, the T-shape geometry, the X-shape geometry, and the linear geometry. These are represented in Figure (2)80, in Jacobi coordinates.   Figure 2. Four main geometries studied, as a function AB, and = A- B), bond distance rA and rB, and the intermolecular separation R80  The linear and H-shape geometries possess D2h point group symmetry, while the X-shape and  T-shape  geometries possess C2v symmetry.   17  2.2 Basis Set Superposition Error (BSSE) The supermolecular method is used to calculate the interaction energy by the following equation: (2.1) where EAB is the energy of the complex, and EA, EB are the energies of the isolated molecules A and B, respectively (all as a function of the intermolecular distance R). One of the problems this method suffers from is Basis Set Superposition Error (BSSE), which is an artificial lowering of the energy of the pair relative to the monomers, caused by the overlapping basis sets of the monomers in the complex.  In order to correct this error, the Counterpoise Method, as proposed by Boys and Bernardi81, has been used. The BSSE for each of the monomers A, B in the complex is given by: (2.2) (2.3) where  and  are the energies of molecule A and B, respectively, using the basis set of the complex AB.  Subtracting these equations from equation (2.1), we obtain                                            (2.4) There has been some disagreement as to the merits of using this correction, as discussed by A. J. Stone82 in his book "Theory of Intermolecular Forces".       Looking at Tables (1)-(3), it is apparent that the BSSE is largest for the smallest intermolecular distances (R) between monomers, because the artifiial reduction of the dimer energy is largest when each molecule can take advantage of the basis functions of the other 18  molecule. This is true regardless of the geometry formed by the interacting monomers. One point worth mentioning is that the BSSE has the same values regardless of whether we are discussing the singlet, triplet, or quintet spin state within a particular geometry. However, we can compare the magnitude of the BSSE between different geometries. Out of the four main geometries, the linear geometry shows the largest magnitude of BSSE, with a value of 1.87 mHar (10000 mHar=1 Hartree) at a intermolecular separation of 5 bohr (Tables 4-6). The second largest error is in the T-shaped geometry, with an error of 0.815 mHar at the same intermolecular distance. Next is the X-shaped geometry, with an error of 0.557 mHar at 5 bohr, and finally, the H-shape geometry has the smallest error at 5 bohr of 0.393 mHar. Table 2. BSSE and interaction energy for the triplet spin state of the T-shape geometry Intermolecular Distance (bohr) Uncorrected Interaction Energy (mHar) BSSE (mHar) Corrected Interaction Energy (mHar) 5 19.306 0.8154 20.121 6 -1.329 0.3919 -0.937 8 -3.092 0.1026 -0.299 10 -2.808 0.0365 -2.771 12 -2.738 0.0204 -2.718 15 -2.714 0.0131 -2.701 20 -2.708 0.0118 -2.696  Table 3. BSSE and interaction energy for the singlet spin state of the T-shape geometry Intermolecular Distance (bohr) Uncorrected Interaction Energy (mHar) BSSE(mHar) Corrected Interaction Energy (mHar) 5 16.777 0.8154 17.592 6 -1.840 0.3919 -1.448 8 -3.167 0.1026 -3.064 10 -2.872 0.0365 -2.836 12 -2.803 0.0204 -2.783 15 -2.778 0.0131 -2.765 20 -2.772 0.0118 -2.761  19  Table 4. BSSE and interaction energy for the quintet spin state of the T-shape geometry Intermolecular Distance (bohr) Uncorrected Interaction Energy (mHar) BSSE (mHar) Corrected Interaction Energy (mHar) 5 26.232 0.8154 27.048 6 -0.651 0.3919 -0.259 8 -3.123 0.1026 -3.020 10 -2.851 0.0365 -2.814 12 -2.782 0.0204 -2.761 15 -2.757 0.0131 -2.744 20 -2.751 0.0118 -2.739  Table 5. BSSE and interaction energy for the triplet spin state of the linear geometry Intermolecular Distance (bohr) Uncorrected Interaction Energy (mHar) BSSE (mHar) Corrected Interaction Energy (mHar) 5 195.639 1.8710 197.510 6 20.640 0.8181 21.458 8 -3.254 0.1904 -3.064 10 -2.927 0.0593 -2.868 12 -2.800 0.0163 -2.780 15 -2.757 0.0038 -2.754 20 -2.748 0.0002 -2.747  Table 6. BSSE and interaction energy for the quintet spin state of the linear geometry Intermolecular Distance (bohr) Uncorrected Interaction Energy (mHar) BSSE (mHar) Corrected Interaction Energy (mHar) 5 230.225 1.8710 232.097 6 24.365 0.8181 25.183 8 -3.202 0.1904 -3.011 10 -2.928 0.0593 -2.869 12 -2.799 0.0163 -2.782 15 -2.760 0.0038 -2.756 20 -2.750 0.0002 -2.750     20  Table 7. BSSE and interaction energy for the singlet spin state of the linear geometry Intermolecular Distance (bohr) Uncorrected Interaction Energy (mHar) BSSE (mHar) Corrected Interaction Energy (mHar) 5 166.721 1.8710 168.592 6 18.452 0.8181 19.270 8 -3.306 0.1904 -3.115 10 -2.951 0.0593 -2.892 12 -2.820 0.0163 -2.803 15 -2.781 0.0038 -2.777 20 -2.486 0.0002 -2.486  Table 8. BSSE and interaction energy for the singlet spin state of the X-shape geometry Intermolecular Distance (bohr) Uncorrected Interaction Energy (mHar) BSSE (mHar) Corrected Interaction Energy (mHar) 5 -0.614 0.5574 -0.057 6 -3.353 0.2648 -3.088 8 -3.003 0.0578 -2.945 10 -2.520 0.0176 -2.502 12 -2.476 0.0058 -2.470 15 -2.461 0.0008 -2.460 20 -2.457 0.0002 -2.457  Table 9. BSSE and interaction energy for the quintet spin state of the X-shape geometry Intermolecular Distance  (bohr) Uncorrected Interaction Energy (mHar) BSSE (mHar) Corrected Interaction Energy (mHar) 5 -0.677 0.5574 -0.120 6 -3.324 0.2648 -3.060 8 -2.981 0.0578 -2.923 10 -2.814 0.0176 -2.797 12 -2.771 0.0058 -2.765 15 -2.756 0.0008 -2.755 20 -2.752 0.0002 -2.752     21  Table 10. BSSE and interaction energy for the triplet spin state of the X-shape geometry Intermolecular Distance (bohr) Uncorrected Interaction Energy (mHar) BSSE (mHar) Corrected Interaction Energy (mHar) 5 -33.873 0.5574 -33.316 6 -36.694 0.2648 -36.430 8 -36.359 0.0578 -36.301 10 -36.180 0.0176 -36.162 12 -36.132 0.0058 -36.127 15 -36.115 0.0008 -36.115 20 -36.111 0.0002 -36.111  Table 11. BSSE and interaction energy for the triplet spin state of the H-shape geometry Intermolecular Distance (bohr) Uncorrected Interaction Energy (mHar) BSSE (mHar) Corrected Interaction Energy (mHar) 5 -1.767 0.3928 -1.374 6 -3.500 0.1941 -3.305 8 -2.985 0.0402 -2.945 10 -2.811 0.0128 -2.798 12 -2.770 0.0035 -2.763 15 -2.752 0.0004 -2.751 20 -2.748 0.0002 -2.748  Table 12. BSSE and interaction energy for the singlet spin state of the H-shape geometry Intermolecular Distance (bohr) Uncorrected Interaction Energy (mHar) BSSE (mHar) Corrected Interaction Energy (mHar) 5 -3.178 0.3928 -2.785 6 -3.726 0.1941 -3.532 8 -3.013 0.0402 -2.973 10 -2.834 0.0128 -2.821 12 -2.790 0.0035 -2.787 15 -2.775 0.0004 -2.775 20 -2.771 0.0002 -2.771     22  Table 13. BSSE and interaction energy for the quintet spin state of the H-shape geometry Intermolecular Distance (bohr) Uncorrected Interaction Energy (mHar) BSSE (mHar) Corrected Interaction Energy (mHar) 5 1.457 0.3928 1.850 6 -3.087 0.1941 -2.893 8 -2.978 0.0402 -2.938 10 -2.811 0.0128 -2.799 12 -2.768 0.0035 -2.765 15 -2.753 0.0004 -2.752 20 -2.749 0.0002 -2.749   By looking at the uncorrected energies at the minimum of the potential energy curves for all four geometries, and comparing them to the energies at the minimum of the BSSE corrected curves, we can determine if the BSSE affects the position of the minimum of the potential energy wells.  From Table (13), we can see that the singlet, uncorrected T-shaped potential energy curve minimum occurs at an intermolecular separation of 7 bohr. After applying the correction for BSSE, the minimum occurs at an intermolecular separation of 7.1 bohr . Table (14) shows the corresponding data for the triplet spin state in the same geometry. The uncorrected minimum occurs at an intermolecular separation of 7.1 bohr. The corrected minimum occurs at an intermolecular separation of 7.2 bohr. Finally, Table (15) shows the data acquired for the quintet spin state in the T-shape geometry. The minimum in the uncorrected PEC occurs at 7.2 bohr. This spin state experiences the largest displacement of the minimum with BSSE correction for the T-shape, with a corrected minimum position of 7.4 bohr.       Table (16) contains the data for the singlet spin state within the linear geometry. For the uncorrected PEC, the minimum occurs at 8 bohr. After correcting for the BSSE, the minimum occurs at 8.1 bohr. Table (17) shows the corresponding data for the triplet spin state in the linear 23  geometry. The minimum for the triplet PEC occurs at 8.1 bohr while the corrected well depth minimum occurs at 8.2 bohr.  Finally, for the quintet spin state of the linear geometry, Table (18),  the PEC minimum occurs at an intermolecular distance of 8.2 bohr . The minimum of the corrected PEC occurs at 8.3 bohr.       Table (19) contains the data for the singlet spin state of the H-shape geometry. The minimum for the uncorrected PEC occurs at 5.6 bohr. After applying the correction for the BSSE, the minimum is located at 5.8 bohr.  The corresponding data for the triplet spin state is in Table (20). The uncorrected minimum in the PEC occurs at 6 bohr.  The corrected minimum is located at 6.1 bohr. Lastly, the data for the quintet spin state of the H-shape geometry is in Table (21). The location of the uncorrected minimum is at 6.5 bohr. After the BSSE correction, the minimum is located at 6.6 bohr.       Table (22) shows the data acquired for the singlet spin state of the X-shape geometry. The minimum for the uncorrected PEC occurs at 6.2 bohr and has a well depth of -3.204 mHar. After correction of the BSSE, the minimum of the PEC occurs at 6.3 bohr. Table (23) shows the data for the triplet spin state of the X-shape geometry. The uncorrected PEC minimum is at 6.2 bohr and the corrected minimum is located at 6.3 bohr. Finally, the uncorrected PEC for the quintet spin state is located at 6.3 bohr, and the corrected PEC has a minimum at 6.4 bohr.        24  Table 14. Position of the PEC minimum for the singlet state and T-shape geometry Intermolecular Distance (bohr) Uncorrected Interaction Energy (mHar) BSSE (mHar) Corrected Interaction Energy (mHar) 6.5 -3.169 0.2662 -2.903 6.6 -3.264 0.2484 -3.015 6.7 -3.327 0.2319 -3.095 6.8 -3.366 0.2166 -3.149 6.9 -3.385 0.2022 -3.183 7.0 -3.391 0.1888 -3.202 7.1 -3.385 0.1762 -3.209 7.2 -3.372 0.1643 -3.208 7.3 -3.353 0.1531 -3.200 7.4 -3.330 0.1425 -3.187 7.5 -3.304 0.1325 -3.171  Table 15. Position of the PEC minimum for the triplet state and T-shape geometry Intermolecular Distance (bohr) Uncorrected Interaction Energy (mHar) BSSE (mHar) Corrected Interaction Energy (mHar) 6.5 -3.004 0.2662 -2.738 6.6 -3.124 0.2484 -2.876 6.7 -3.208 0.2319 -2.976 6.8 -3.264 0.2166 -3.047 6.9 -3.298 0.2022 -3.095 7.0 -3.314 0.1888 -3.126 7.1 -3.318 0.1762 -3.142 7.2 -3.313 0.1643 -3.148 7.3 -3.300 0.1531 -3.147 7.4 -3.282 0.1425 -3.140 7.5 -3.261 0.1325 -3.128        25  Table 16. Position of the PEC minimum for the quintet state and T-shape geometry Intermolecular Distance (bohr) Uncorrected Interaction Energy (mHar) BSSE (mHar) Corrected Interaction Energy (mHar) 6.5 -2.721 0.2662 -2.454 6.6 -2.892 0.2484 -2.644 6.7 -3.019 0.2319 -2.787 6.8 -3.109 0.2166 -2.892 6.9 -3.171 0.2022 -2.969 7.0 -3.211 0.1888 -3.022 7.1 -3.234 0.1762 -3.058 7.2 -3.244 0.1643 -3.079 7.3 -3.243 0.1531 -3.090 7.4 -3.236 0.1425 -3.093 7.5 -3.223 0.1325 -3.091  Table 17. Position of the PEC minimum for the singlet state and linear geometry Intermolecular Distance (bohr) Uncorrected Interaction Energy (mHar) BSSE (mHar) Corrected Interaction Energy (mHar) 7.7 -3.246 0.2320 -3.014 7.8 -3.282 0.2172 -3.065 7.9 -3.300 0.2034 -3.097 8.0 -3.306 0.1904 -3.115 8.1 -3.301 0.1783 -3.123 8.2 -3.289 0.1669 -3.122 8.3 -3.272 0.1564 -3.116 8.4 -3.252 0.1465 -3.106 8.5 -3.230 0.1374 -3.093 8.6 -3.207 0.1290 -3.078 8.7 -3.183 0.1211 -3.062        26  Table 18. Position of the PEC minimum for the triplet state and linear geometry Intermolecular Distance (bohr) Uncorrected Interaction Energy(mHar) BSSE (mHar) Corrected Interaction Energy(mHar) 7.7 -3.171 0.2320 -2.939 7.8 -3.217 0.2172 -3.000 7.9 -3.243 0.2034 -3.040 8.0 -3.254 0.1904 -3.064 8.1 -3.255 0.1783 -3.076 8.2 -3.247 0.1669 -3.080 8.3 -3.234 0.1564 -3.077 8.4 -3.216 0.1465 -3.070 8.5 -3.196 0.1374 -3.059 8.6 -3.175 0.1290 -3.046 8.7 -3.153 0.1211 -3.032  Table 19. Position of the PEC minimum for the quintet state and linear geometry Intermolecular Distance (bohr) Uncorrected Interaction Energy(mHar) BSSE (mHar) Corrected Interaction Energy(mHar) 7.7 -3.071 0.2320 -2.839 7.8 -3.136 0.2172 -2.919 7.9 -3.178 0.2034 -2.974 8.0 -3.202 0.1904 -3.011 8.1 -3.212 0.1783 -3.034 8.2 -3.213 0.1669 -3.046 8.3 -3.206 0.1564 -3.050 8.4 -3.194 0.1465 -3.048 8.5 -3.179 0.1374 -3.042 8.6 -3.161 0.1290 -3.032 8.7 -3.142 0.1211 -3.021        27  Table 20. Position of the PEC minimum for the singlet state and H-shape geometry Intermolecular Distance (bohr) Uncorrected Interaction Energy(mHar) BSSE (mHar) Corrected Interaction Energy(mHar) 5.2 -3.554 0.3380 -3.216 5.3 -3.666 0.3141 -3.352 5.4 -3.740 0.2924 -3.447 5.5 -3.783 0.2726 -3.510 5.6 -3.801 0.2546 -3.547 5.7 -3.801 0.2380 -3.563 5.8 -3.786 0.2225 -3.564 5.9 -3.760 0.2080 -3.552 6.0 -3.726 0.1941 -3.532 6.1 -3.686 0.1809 -3.505 6.2 -3.642 0.1682 -3.474  Table 21. Position of the PEC minimum for the triplet state and H-shape geometry Intermolecular Distance (bohr) Uncorrected Interaction Energy(mHar) BSSE (mHar) Corrected Interaction Energy(mHar) 5.5 -3.223 0.2726 -2.950 5.6 -3.336 0.2546 -3.082 5.7 -3.414 0.2380 -3.176 5.8 -3.463 0.2225 -3.241 5.9 -3.490 0.2080 -3.282 6.0 -3.500 0.1941 -3.305 6.1 -3.495 0.1809 -3.315 6.2 -3.481 0.1682 -3.313 6.3 -3.459 0.1559 -3.303 6.4 -3.431 0.1441 -3.287 6.5 -3.400 0.1328 -3.268        28  Table 22. Position of the PEC minimum for the quintet state and H-shape geometry Intermolecular Distance (bohr) Uncorrected Interaction Energy(mHar) BSSE (mHar) Corrected Interaction Energy(mHar) 6.0 -3.087 0.1941 -2.893 6.1 -3.157 0.1809 -2.977 6.2 -3.204 0.1682 -3.036 6.3 -3.232 0.1559 -3.076 6.4 -3.245 0.1441 -3.101 6.5 -3.247 0.1328 -3.115 6.6 -3.241 0.1220 -3.119 6.7 -3.230 0.1118 -3.118 6.8 -3.213 0.1023 -3.111 6.9 -3.194 0.0936 -3.101 7.0 -3.173 0.0856 -3.088  Table 23. Position of the PEC minimum for the singlet state and X-shape geometry Intermolecular Distance (bohr) Uncorrected Interaction Energy(mHar) BSSE (mHar) Corrected Interaction Energy(mHar) 5.7 -36.537 0.1808 -36.357 5.8 -36.642 0.1684 -36.473 5.9 -36.714 0.1569 -36.557 6.0 -36.760 0.1463 -36.614 6.1 -36.787 0.1366 -36.650 6.2 -36.797 0.1277 -36.669 6.3 -36.796 0.1196 -36.676 6.4 -36.785 0.1120 -36.673 6.5 -36.768 0.1050 -36.663 6.6 -36.746 0.0984 -36.648 6.7 -36.721 0.0922 -36.629        29  Table 24. Position of the PEC minimum for the triplet state and X-shape geometry Intermolecular Distance (bohr) Uncorrected Interaction Energy(mHar) BSSE (mHar) Corrected Interaction Energy(mHar) 5.7 -36.469 0.1808 -36.289 5.8 -36.574 0.1684 -36.406 5.9 -36.647 0.1569 -36.490 6.0 -36.695 0.1463 -36.548 6.1 -36.722 0.1366 -36.585 6.2 -36.733 0.1277 -36.606 6.3 -36.733 0.1196 -36.614 6.4 -36.724 0.1120 -36.612 6.5 -36.708 0.1050 -36.603 6.6 -36.687 0.0984 -36.589 6.7 -36.663 0.0922 -36.571  Table 25. Position of the PEC minimum for the quintet state and X-shape geometry Intermolecular Distance (bohr) Uncorrected Interaction Energy(mHar) BSSE (mHar) Corrected Interaction Energy(mHar) 5.7 -36.433 0.1808 -36.252 5.8 -36.539 0.1684 -36.371 5.9 -36.614 0.1569 -36.457 6.0 -36.663 0.1463 -36.517 6.1 -36.693 0.1366 -36.556 6.2 -36.707 0.1277 -36.579 6.3 -36.709 0.1196 -36.589 6.4 -36.702 0.1120 -36.590 6.5 -36.688 0.1050 -36.583 6.6 -36.670 0.0984 -36.571 6.7 36.648 0.0922 -36.555  Table 26. Positions of BSSE corrected minima for the four main geometries studied  H-shape X-shape T-shape Linear singlet 5.8 bohr 6.3 bohr 7.1 bohr 8.1 bohr triplet 6.1 bohr 6.3 bohr 7.2 bohr 8.2 bohr quintet 6.6 bohr 6.4 bohr 7.4 bohr 8.3 bohr   30  2.3 Comparison of Interaction Energy to Literature Much work has been done to determine the interaction energy of the O2-O2 collisional complex.73,76-96,60,66  In this section, we therefore compare our interaction energy curves to literature values that were determined both theoretically 73,86,92,60,94,95 and experimentally75,78,99 in order to validate their accuracy.        The first reliable potential energy surface for the exchange-interaction driven singlet, triplet, and quintet splittings of the O2(3)-O2(3) interaction surface was obtained by Bussery and Wormer92. The PESs were built by combining ab initio calculations by Wormer and van der Avoird13a using first order exchange perturbation theory at the HartreeFock level and a second-order polarization energy evaluated semiempirically, circumventing in this way the difficulty inherent to an ab initio determination of the dispersion terms93,96.      Tables (26)-(28) shows our results for the position and depth of the attractive interaction potential curves.  Bussery and Wormer92 studied the same four geometries that we have investigated, and paid special attention to the H-shape geometry, noting that it is the structure found in the solid oxygen crystal104, and appeared from their PES to be the most stable geometry. They also discussed the coupling given by a Heisenberg spin Hamiltonian described by a single parameter, JAB for the  pair. This anisotropic term is used to determine if the interaction is ferro-magnetic or antiferromagnetic. The sign of this term is dictated by the overlap between the  orbitals. Their H-shape geometry yields an antiferromagnetic coupling, with a negative value of JAB. Their X-shape potential gives a ferromagnetic coupling, with a positive value for JAB. This is used to explain the stabilization of the H-shape geometry relative to the X-shape geometry for their singlet and triplet states. While we can see from Tables (28) and (29) that our H-shape singlet spin state is indeed more stable, both in position and well depth, than the X-31  shape singlet state, we find that the triplet state for the X-shape has a slightly deeper potential well (). However, the position of the H-shape is slightly more favorable for the triplet state (6.1 bohr) than that of the X-shape triplet (6.3 bohr). Their data shows that the most favorable geometry of the two for the quintet splitting is the X-shape, and our data shows this as well, both for the position and depth of the potential minimum. Their T-shape and linear geometries are more repulsive and less favored than the H-shape and X-shape, and our data agrees. Their conclusion of the H-shape as the most stable geometry helped validate the experiments done by Goodman et al. 105, in which electronic spectra in the visible region was used to make the same conclusion.   We next compare our data for the position and depth of the interaction energy curves to the experimental work of Aquilanti et al.80 They explored they nature of the bond in the O2-O2 "dimer" using molecular beam experiments for collisions between oxygen molecules. The resulting cross sections were used to extract information on the position and energy of the minima of the singlet, triplet, and quintet splittings. This work is significant as it allowed the first complete experimental characterization of the interaction in such a system. Experiments of this type must be carried out with sufficiently high angular and energy resolution in order to allow the measurement of quantum interference effects for an accurate description of the interaction potential.  Earlier attempts106 made with hot effusive beams of fast rotating molecules emerging from a microwave discharge source had yielded the interaction potential that was averaged over all molecular orientations. Considerable progress in the production and characterization of beams emerging from supersonic sources107 in work done by R.B.  allowed for collisional studies of cold molecules, with slow rotations and more control over 32  molecular alignment, in order to more fully understand the anisotropy of the collisions.  From the "glory pattern" produced by quantum interference, it is possible to extract information on the potential well features100 while the absolute value of the cross section contains information about the long-range attraction.101       In generating their potential energy surface, Aquilanti10 et al. considered oxygen molecule A and oxygen molecule B in the pair  to be at their fixed equilibrium bond distances, and therefore behaving as rigid rotors. They separate the interaction potential into the radial and angular components as follows:                       (2.5) with La, Lb=0,1,2...and |La-Lba+Lb. The functions  are coupled spherical harmonics. The radial components contain the interaction information, whether from induction, dispersion, electrostatic repulsion, etcetera. Because the oxygen molecule is a homonuclear diatomic, only even moments contribute to the sum in equation (2.5). After inserting the proper values into equation (2.5) for the three angles , and , the interaction potentials (truncated after the first four terms) for the four basic configurations H,X,T, and linear are given by:                    (2.6)                  (2.7) (2.8)                     (2.9)      Aquilanti80 et al. compared their experimental potential and glory pattern to that of the semi ab initio potential and subsequent glory pattern in [16] and found they were dephased with 33  respect to each other, and had a different frequency. They concluded that the marked difference suggests that the isotropic term for the potential   needs to be much larger (more negative than) that calculated in [Ref. 16].  When comparing their potential energy curves to that of Bussery21 et al., they concur that the lowest energy minimum for the singlet state occurs when the oxygen molecules are in the planar parallel (H-shape) geometry. Indeed, this is the same result we have found in the present work, and our ab initio curves estimate an even deeper singlet well depth for the minimum in the H-shape geometry, at 23.5 eV, compared to 19.0 eV from Bussery92 and 17 eV from the work of Aquilanti80 and coworkers (Table (28)). In addition, both Bussery and Aquilanti find that the lowest potential energy for the quintet state occurs for the X-shape geometry, and again, our results confirm this (Table (29)).      We next compare our work to the ab initio study done by Minaev et al.89. They performed ab initio complete active space configuration interaction (CASCI) calculations on the singlet, triplet, and quintet state of , using a 6-31G* basis set108. They performed calculations only for the parallel planar (H-shape ) geometry. In addition, they used restricted open-shell Hartree Fock quintet state orbitals in all of their calculations.  This was presumably done because the quintet splitting is a single reference state, while the singlet and triplet state contain multireference character. Though the dependence on the intermolecular distance R and subsequent positions of the van der Waals minima are similar in value to our work and that of all other literature values in Table (28), the well depths are significantly smaller.    Hernández60 and coworkers have performed  ab initio calculations of the singlet splitting for all four main geometries, at the CASSCF level. With the exception of the linear geometry, where we have calculated a deeper potential well for the singlet state, their results match ours almost exactly, both in position and magnitude of the interaction potential minima.  34    Dayou98 and coworkers used the CASPT2 method, as we have, to obtain their two-dimensional potential energy curves, with an atomic natural orbital basis set with s,p,d and f functions. They calculated the van der Waals minima for the singlet state of the H-shape geometry, and is in good agreement with our results, with our potential giving a slightly deeper interaction well.       More recently, Bartolomei100 et al.  performed a detailed study on the intermolecular potentials for the three lowest multiplet states.  They used the CASPT2 method and an ANO basis set to obtain the quintet state's minima and well depth for all four main geometries. In addition they obtained the potential energy curve for the quintet state using the CCSD(T) method109. Because the triplet and singlet multiplet states are multireference, CCSD(T) is not an appropriate method to use in calculating their potential energy curves. Therefore, the potential energy for these states was obtained as follows:                                                      (2.9)                                                    (2.10) where  and  are the potentials for the singlet state and triplet state, respectively. The quintet energy obtained using the CCSD(T) method is given by , and , are the singlet-quintet and triplet-quintet energy differences obtained using the CASPT2 method. In this way they obtained a lower and upper limit to the position of the singlet and triplet states. These are given in the CCSD(T) column in Tables (26)-(29). Comparing to our data, we can see that our results reproduce theirs for the quintet at the CASPT2 level well. Our T,X, and H-shape results are slightly more attractive in the van der Waals well, perhaps because of the use of additional functions in our basis set. Examining their CCSD(T) results for the linear geometry in Table (26), we see that our singlet state is more attractive than that found by Bartolomei100 at the 35  CCSD(T) level. Our data for the triplet state shows a well depth that is exactly the same as the lower limit calculated by Bartolomei100 at the CCSD(T) level. Finally, our quintet state for the linear geometry is somewhat more repulsive than the CCSD(T) one.      The same analysis is applied to Table (27), corresponding to our data for the T-shape geometry. Our singlet state is more repulsive than that obtained by Bartolomei29 et al. using the CCSD(T) method, with a minimum that is about 4 meV more shallow than the CCSD(T) lower limit. Our triplet state is 2.6 meV shallower than the CCSD(T) lower limit, and our quintet state is 3.4 meV shallower than the corresponding CCSD(T) quintet state.        Now turning to Table (28), we see that our singlet state is near the CCSD(T) upper limit of the curve obtained by Bartolomei100 et al. for the H-shape geometry. Our triplet state is about 4 meV shallower than the upper limit of the CCSD(T) curve. Finally, our quintet state is 3.6 meV shallower than the CCSD(T) value. All of the positions of these minima are comsistent between methods.       Finally we compare our position and depth of our X-shape minima to that obtained by Bartolomei100 et al. using the CCSD(T) method.  Our singlet state is only 0.6 meV from their upper limit of the CCSD(T) results. The same is true for the triplet state. It is worth noting that while previous works80,92 determined the quintet state to be more attractive than either the singlet or triplet for the X-shape, our results agree with that of Bartolomei that this is not necessarily the case.         In conclusion, from comparison with the results in the literature, we find that our choice of method and basis set give  reliable results for the interaction of the  system for the four main geometries. These methods will therefore be employed, with changes as appropriate, in subsequent chapters, particularly when calculating the ab initio dipole moments. 36    Table 27. The position and depth of the interaction potential energy curve for the linear geometry of O2-O2 Linear (D2h)  X1Ag 3B1u 5Ag Reference Method/Basis Re(a0)  Re(a0)  Re(a0)  present work CASPT2/ aug-cc-pVQZ 8.1 17.4 8.2 9.1 8.3 8.2 Bussery et al.92 SCF/ GTO[5s3p2d] 8.6 5.8 8.6 5.7 8.8 5.6 Aquilanti et al.80 Experiment 8.0 9.1±0.8 8.1 8.6±0.8 8.3 7.7±0.8 Hernández et al.60 CASSCF/ 5s3p2d 8.3 8.5 - - - - Bartolomei100 CCSD(T)/ 5s4p3d2f 8.01/7.97 14.0/14.6 8.05/8.06 9.1/13.4 8.16 12.3 Bartolomei100 CASPT2/5s4p3d2f - - - - 8.25 8.59   Table 28. The position and depth of the interaction potential energy curve for the T-shape geometry of O2-O2 T-shape (C2v)  X1B1  3B1  5B1  Reference  Method/Basis  Re(a0)    Re(a0)    Re(a0)    present work  CASPT2/ aug-cc-pVQZ(6s5p4d3f2g)  7.1  12.2  7.2  12.2  7.4  9.7  Bussery et al.92  SCF/GTO[5s3p2d] 7.6  8.0  7.6  7.9  7.6  7.6  Aquilanti et al.80  Experiment  7.1  16.0±0.8   7.2  14.7±0.8   7.3  12.9 ±0.8  Hernández et al.60  CASSCF/5s3p2d  7.2  10.9  -  -  -  -  Bartolomei100 CCSD(T)/5s4p3d2f 7.02/6.95 16.0/17.2 7.12/7.07 14.8/15.5 7.26 13.1 Bartolomei100 CASPT2/5s4p3d2f - - - - 7.5 9.08  37    Table 29. The position and depth of the interaction potential energy curve for the H-shape geometry of O2-O2  H-shape (D2h)  X1Ag  3B1u  5Ag  Reference  Method/Basis  Re(a0)    Re(a0)    Re(a0)    present work  CASPT2/ aug-cc-pVQZ (6s5p4d3f2g) 5.8  23.5  6.1  15.4  6.6  10.1  Bussery et al.92 SCF/GTO[5s3p2d] 6.1  19.0  6.2  17.3  6.4  14.8  Aquilanti et al.2  Experimental  6.71  17.0 ±0.8  6.82  15.9 ±0.8  6.95  14.3 ±0.8  Minaev et al.3  CASCI/6-31G*  6.33  6.96  6.46  5.67  6.69  3.81  Dayou et al.98  CASPT2/ 5s4p3d2f (ANO) 5.85  19.9      Hernández et al.60  CASSCF/5s3p2d  5.8  24.4  -  -  -  -  Bartolomei100 CCSD(T)/ 5s4p3d2f 5.77/5.56 25.4/32.3 6.07/5.94 19.5/22.2 6.48 13.7 Bartolomei100 CASPT2/5s4p3d2f - - - - 6.5 8.72                  38   Table 30. The position and depth of the interaction potential energy curve for the X-shape geometry of O2-O2 X-Shape (C2v)  X1A2  3A2  5A2  Reference  Method/Basis  Re(a0)    Re(a0)    Re(a0)  eV)  Present work *spin-averaged CASPT2/ aug-cc-pVQZ(6s5p4d3f2g) 6.3 15.6 6.3 15.6 6.4 13.7 Bussery et al.92 *spin-averaged SCF/GTO[5s3p2d] 6.2  14.7  5.8  15.0  6.8  15.6  Hernández et al.60 CASSCF/5s3p2d  6.2  15.7  -  -  -  -  Aquilanti et al.80  Experimental  6.86  15.3±0.8    6.84  15.5 ±0.8   6.8  16 ±0.8   Bartolomei100 CCSD(T)/5s4p3d2f 6.24/6.18 16.2/17.3 6.23/6.19 16.2/17.0 6.22 16.3 Bartolomei100 CASPT2/5s4p3d2f - - - - 6.5 10.67             39   CHAPTER 3: Multipole Moments and (hyper)polarizabilites of the O2 Molecule 3.1 Introduction  The elucidation of the energy transfer processes between oxygen molecules can provide a better understanding of the energy balance occurring in the Earth's atmosphere.112 Although the oxygen molecule does not possess a permanent dipole moment, collision-induced electric dipoles (in addition to the magnetic dipole) facilitate the absorption of radiation in the roto-translational region. This effect can potentially change the role of the molecular oxygen in the atmospheric thermodynamics. It is therefore crucial to understand the underlying physics of the O2-O2 interaction and how each molecule is polarized in the electric field of its neighbor.      The collision-induced dipole moment is dependent on the dipole polarizability (),  dipole-octupole (), quadrupole-quadrupole (), and second hyperpolarizabilities (), as well as the quadrupole (zz ) and hexadecapole (zzzz) moments. These properties have been reported in the literature for the O2 molecule at its equilibrium bond length113-118,120-126. In particular, we focus on comparing our data to that of  Y. N. Kalugina and V. N. Cherepanov113, who have calculated the dipole polarizabilities, quadrupole moment, hexadecapole moment, dipole-octupole polarizabilities, and quadrupole-quadrupole polarizabilities using finite field techniques and the R-CCSD(T) method, with an aug-cc-pVQZ basis set. We also compare our data to that of Bartolomei114 et al., who has calculated these same properties using again the aug-cc-pVQZ basis set but the MRCI method. In particular, we compare our data to their value obtained using what they label as the "CAS1" active space, as it is identical to ours. In this active space, u,g orbitals closed, 40  u,g u,g orbitals. However, there are few available values for these properties at non-equilibrium internuclear bond distances. In addition, there are no values in the literature for the dipole-dipole-quadrupole polarizability, as far as we are aware.       We have calculated these properties at several bond distances, for use in the calculation of average rovibrational properties for the =0-5, J=0 energy levels. In order to calculate energies at bond distances that are shorter or longer than the equilibrium bond distance, we require a multireference method. This is because as the bond distance is shortened/stretched, the occupied orbitals and the nearest unoccupied orbitals can become quasi-degenerate (near each other in energy). This is especially important for the oxygen molecule, because its open shell nature leads to many low-lying excited electronic energy states128,129. One of the methods we will therefore employ is the Complete Active Space Second Order Perturbation Theory method (CASPT2). In the CASPT2 method, a multi-configurational CASSCF wave function is used to obtain the dynamic electron correlation, which allows the construction of appropriate molecular orbitals at distances longer than the equilibrium bond length. Single, double, and all higher excitations to virtual orbitals (non-dynamic correlation) are considered through the second order perturbation theory scheme. Our second method used to calculate the electronic properties is the MRCI method, also starting from a CASSCF wavefunction. In this method, excitations into the quasi-degenerate orbitals encountered away from equilibrium are explicitly included in the reference wavefunction. We have considered only double and single excitations in our MRCI calculations. There are advantages and diadvantages inherent in both methods. The MRCI method is variational, which means that we approach the true energy for our system from above; that is, the energy obtained using this method will always be larger than the exact energy. However, the CASPT2 method is not variational, and therefore can give an energy that is lower than the exact 41  energy. In this way the CASPT2 method can be seen as somewhat less trustworthy than the MRCI method. The advantage of the CASPT2 method is that it is faster and less computationally demanding than the MRCI method, because the coefficients in the wavefunction are obtained through the second order perturbation theory formulas (instead of variationally).       In addition to the use in calculating the vibrational averages, these properties have been used to obtain the long-range approximate collision-induced dipole moments (chapter 4).       This chapter is organized as follows: in Section 3.2 we will explain our methodology and computational details. Section 3.3 begins our results & discussion, beginning with the electronic properties obtained at the equilibrium bond distance, and using the singly augmented basis sets aug-cc-pVXZ. Section 3.4 continues our results with the properties obtained at the equilibrium bond distance with the doubly augmented basis sets d-aug-cc-pVXZ. These sections also include a comparison to selected literature values. Section 3.5 contains our results for properties obtained away from the equilibrium internuclear distance. Section 3.6 gives our results for the estimate of the complete basis set (CBS) limit. Section 3.7 gives our results for the rovibrational averages of the properties obtained using the Numerov-Cooley Method128,129. Finally, we give our conclusions in Section 3.8.   3.2 Methodology Dipole, quadrupole, octopole, and hexadecapole fields must be applied in order to obtain the multipole moments and polarizabilities.  Our Hamiltonian includes the dipole (), quadrupole (), octopole (), and hexadecapole () moment operators:                                                        (3.1) 42  In equation (3.1),  is the unperturbed electronic Hamiltonian, and the Greek subscripts run through the x,y,z coordinates. The Einstein convention of summation over Greek indices is being followed throughout this study. The , , ,  amplitudes define the strength of the corresponding fields (and field gradients).  Applying perturbation theory, the most important terms of the energy expansion are:          (3.2) In equation (3.2),  is the energy of the unperturbed system, and  and are the quadrupole and hexadecapole moments. The dipole moment and octopole moment,   and , respectively, vanish for linear centrosymmetric molecules. This is also true for the dipole-quadrupole polarizability ( ) and the dipole hyperpolarizability 130. If we place O2 molecule along the z-axis, then by symmetry, and using equation(3.2), we require the following tensors only:                                                                              (3.3)                                                                                   (3.4)                                                                              (3.5)                                                                        (3.6)                                                                            (3.7) 43      (3.8)                                                                              (3.9) Practically, the derivatives in equations 3-9 are calculated via the central differences formulae as follows:                                                      (3.10)                                              (3.11)                        (3.12)                             (3.13)                                                                               (3.14)                                                                        44                                                                                                                      (3.15) The field amplitudes f, g, and h have to be selected wisely. They should be big enough to generate numerically significant energy differences but not too big to render the above equations invalid. In addition, using different field strengths will yield slightly different results, and this is still under investigation. We finally chose f = 10-3 a.u, g = 10-3 a.u., and h = 10-4 a.u. The above equations yield errors on the order of the second power of f, g, or h. Thirty-four energy calculations were performed for each internuclear distance to fulfill the above calculations. It was necessary to impose symmetry constraints on the wavefunction according to the symmetry of the fields applied to obtain accurate values. For example, we considered the D2h point group for zero field and quadrupole field calculations. The C2v and Cs symmetry was also applied for various other combinations of fields required. All computational work was performed using the MOLPRO79 package.   3.3. Properties at the Equilibrium Bond Length, Obtained using the Singly Augmented Basis Sets  All electronic properties calculated for the oxygen molecule that are discussed in this section and the following sections are given in atomic units. The units for the quadrupole moment are ea02, where e is the charge on the electron, and a0 is the bohr radius. The units for the hexadecapole moment are ea04, and the units of the dipole-octupole polarizability E are e2a04/Eh , where Eh is the energy in atomic units (Hartrees). The dipole-dipole-quadrupole polarizability B has units of e3a04/Eh2, and the units of the quadrupole-quadrupole polarizability C are e2a04/Eh. Finally, the e4a04/Eh3.  Referring to Table (31), we examine the 45  MRCI and CASPT2 values of the dipole polarizability (),  dipole-octupole (), quadrupole-quadrupole (), and dipole-dipole-quadrupole () polarizabilities and second hyperpolarizabilities (), as well as the quadrupole (zz ) and hexadecapole (zzzz) moments, at the equilibrium bond distance of r=2.28187 bohr. We compare these values with some in the literature113-115the singly augmented correlation consistent  polarized valence (aug-cc-pVXZ) basis sets of Dunning76 . The numbers in these tables are replicable numbers, as opposed to significant digits. That is, they are numbers the method will reliably generate on different machines, as opposed to being numbers that are correct to these many digits. In Table (32), we will discuss the same values obtained using the doubly augmented (d-aug-cc-pVXZ) basis sets. This analysis of basis set sensitivity is useful for determining what basis set to use when trying to balance the accuracy of the result using that basis set with the basis set cost. In general, the addition of more functions of angular momentum l to a given basis set increases it's cost and is more computationally demanding. However, in some cases, a certain number of functions is necessary in order to obtain accurate results. Increasing the cardinal number increases the number of functions in the basis set, as does adding one (in the aug-cc-PVXZ case) or two (d-aug-cc-pVXZ)  diffuse functions for every angular momentum present in the basis. Diffuse functions allow greater flexibility in the wavefunction by allowing the electrons to be held further away from the nucleus. These functions are known to be necessary for weakly bound systems, especially when obtaining the polarization for a system.  Looking first at the x-component of the polarizability (xx), we see that using the MRCI method and our largest basis set ((5Z)=aug-cc-pV5Z), there is about a 5% difference between our value and that calculated by Yu N. Kalugina and V. N. Cherepanov113, who used a slightly smaller 46  aug-cc-pVQZ basis set and the R-CCSD(T) method . Our value of (xx) obtained using the CASPT2 method is closer, with a 2.6% difference. Our value for the z-component of the zz) using the MRCI method is only just over 1% different from that of Kalugina and Cherepanov, and again the CASPT2 value is even closer, with only a 0.6% difference from the literature. As far as basis set sensitivity, the x-component of the polarizability (xx) is -cc-pVDZ to the aug-cc-pV5Z basis set for the CASPT2 method, and a 6.5% difference between the same levels for the MRCI method. The z-zz) is even more valabout 0.8% difference between the same values for MRCI. 47  Table 31. Electronic properties (in a.u.) of O2 at the equilibrium bond length, r(O-O)=2.28187  bohr, using the MRCI and CASPT2/aug-cc-pVXZ levels of theory. Method MRCI MRCI MRCI MRCI CASPT2 CASPT2 CASPT2 CASPT2 Literature Basis set X = D X = T X = Q X = 5 X = D X = T X = Q X = 5   -E0 150.0007 150.111 150.146 150.1575 149.9954 150.1135 150.1535 150.1683 - xx 7.303 7.729 7.825 7.797 7.332 7.853 7.987 7.99 8.20a zz 14.788 14.888 14.897 14.906 14.95 15.143 15.182 15.178 15.09a zz -0.1589 -0.2393 -0.2508 -0.2597 -0.1594 -0.2331 -0.2422 -0.25 -0.21b, -0.22d,-0.24c zzzz 5.3889 4.5787 4.5058 4.3609 5.3273 4.5794 4.5162 4.3703 4.67b, 4.70d,4.48c Ex,xxx -15.9879 -17.0159 -17.1905 -17.1309 -16.0291 -17.31 -17.5451 -17.5232 20.95a,18.81e Ez,zzz 14.532 18.6798 19.7699 20.2751 14.3343 18.5936 19.8435 20.3474 -17.93b,-18.06e Bx,z,xz -70.8515 -85.3055 -95.1795 -98.73 -72.0735 -90.854 -102.702 -107.307 - Bx,x,zz 12.556 6.246 23.480 30.88 24.470 32.157 38.685 42.836 - Bx,x,xx -34.831 -42.927 -60.291 -68.536 -42.229 -58.444 -71.814 -79.270 - Bz,z,zz -77.566 -101.017 -118.399 -124.483 -71.68 -98.349 -115.116 -122.036 - Cxx,xx 8.699 10.863 12.073 12.798 8.722 11.069 12.373 13.162 12.84b,13.64e Czz,zz 17.902 20.320 21.159 21.709 17.974 20.698 21.660 22.277 22.30b,23.00e Cxz,xz 16.575 18.000 18.648 18.935 18.006 19.737 20.501 20.833 19.75b,20.48e xxxx 187.977 272.1 374.406 439.931 190.68 286.435 402.458 471.133 490f xxzz 198.926 243.534 290.740 312.823 219.849 285.485 350.420 - 358f zzzz 431.872 500.444 612.824 652.564 367.111 461.353 602.010 663.836 776f  a. Yu N. Kalugina and V. N. Cherepanov113      b Yu N. Kalugina and V. N. Cherepanov113, using R-CCSD(T), with an aug-cc-pVQZ basis set c Bartolomei et al.114, using aug-cc-pVQZ basis set and MRCI method, and active space CAS1 d Bartolomei et al.114, using aug-cc-pVQZ basis set and MRCI method, and active space CAS2 e Values taken from Bartolomei et al.114, as cited in Yu N. Kalugina and V. N. Cherepanov113 f  Neogrády  et al.118, using the CCSD Method and aug-cc-pV5Z basis set  48  Now moving to the quadrupole moment (zz), we see that our value calculated using our largest basis set (5Z), and using the MRCI method, is about 8%  different from the nearest literature value, that is, that of Bartolomei et al.114, who have also used the MRCI method, with a similar active space to ours (labeled CAS1). For our value calculated with the (5Z) basis set but with the CASPT2 method, the nearest literature value is again that of Bartolomei et al.114 also with the CAS1 active space, with only a 4% difference. Examining now the dependence of the more sensitive than the polarizabilities. For the MRCI method, there is a 48% difference in going from the DZ to 5Z level. The majority of the difference is found in going from the DZ to TZ levels (40%). The situation is similar for the CASPT2 method values; there is a 44% difference in values from the DZ to 5Z levels, with 37% of that difference occurring between the DZ and TZ levels.       If we now examine the hexadecapole moment (zzzz),  we find that our value at the highest singly augmented basis set we considered (5Z), is again closest to that of Bartolomei et al.114,  for both our MRCI and CASPT2 value, with about a 2.5% difference. Looking at the basis set dependence, we find that increasing the basis set from the DZ to 5Z in the singly augmented correlation consistent series decreases the value of the MRCI  hexadecapole by 21%; 16% of that difference is between the DZ and TZ levels.  The CASPT2 value decreases by almost 20% in going from the DZ to 5Z levels, with 15% of that difference between the DZ and TZ levels.       We now turn to the higher-order polarizabilities. Our value of the Ex,xxx tensor (dipole-octopole polarizability) using the MRCI method and aug-cc-pV5Z basis set is of opposite sign to that found in the literature113. We find that this is true also for the Ez,zzz tensor. For comparison, we have found that their values for these tensors are also opposite in sign from others we have 49  calculated for H2131.We have reexamined the formulas we have used and found them to be correct. If we assume that the indices were somehow switched, then our Ex,xxx= their Ez,zzz and vice versa. In this case, our value for Ex,xxx using the aug-cc-pV5Z basis and MRCI method is -17.1309 au, which is 4.55% different from their value of Ez,zzz. Our CASPT2 values for Ex,xxx is just over 2% different from their value for Ez,zzz. Using the same analysis for our Ez,zzz values, both the MRCI and CASPT2 values with the 5Z basis set are around 3% different from their reported values for Ex,xxx. Looking at basis set sensitivity, our MRCI value of Ex,xxx is less with only a 7% decrease when increasing the basis set size from the DZ to 5Z level, with 6% of the decrease occurring from the DZ to TZ levels. Our CASPT2 value is slightly more sensitive, with a total decrease of almost 9% when increasing the basis set from the DZ to the 5Z level, with almost 8% of that difference again occurring between the DZ and TZ levels. Our Ez,zzz values are more sensitive: there is a levels, and a 35% difference in going from the DZ level to the 5Z level for our CASPT2 values, with a 26% difference occurring between the DZ and TZ levels.      We now examine the dipole-dipole quadrupole polarizabilities Bx,z,xz. We are unable to find literature values to compare to, but we can do a basis set sensitivity analysis. In going from the DZ to 5Z basis set level using the MRCI method, there is an almost 33% decrease. A large portion of that difference is between the DZ and TZ levels (almost 19%). However, there is also an appreciable difference when increasing the basis set size from the TZ to QZ levels (over 10%). Therefore, the remaining percent difference in going from the QZ to 5Z levels makes up the smallest difference, of only about 4%. The situation is similar for the corresponding CASPT2 50  values, with a 23% difference between the DZ and TZ levels, an appreciable difference of about 12% between the TZ and QZ levels, and again a small difference of 4% between the QZ and 5Z levels. Applying the same analysis to the Bx,x,zz values of the polarizability, we observe some strange behavior with respect to basis set size. In increasing the basis set size from the DZ to TZ levels, there is a 67% decrease in the value of the polarizability. However, when we increase the basis set size from the TZ to QZ levels, there is a 115% increase in the value of the polarizability. Finally, when going from the QZ to 5Z levels, there is a 27% increase in the value of the dipole-dipole-quadrupole polarizability. Thus, we notice two main things from this analysis. The first is that we cannot consider the value of this polarizability to be monotonic with the increase in basis set size. Second, there is a large difference between basis sets for all smallest, it is larger than previous cases considered. In addition, we do not observe this non-monotonic behavior for the same values obtained at the CASPT2 level. Instead, the dipole-dipole-quadrupole polarizability Bx,x,zz increases in a monotonic manner, with a total of almost 55% difference in increasing the basis set size from the DZ to 5Z levels. About half of this difference is found between the DZ and TZ levels. There is a large difference between the TZ and QZ levels as well (about 18%), and the smallest difference is again found between the QZ and 5Z levels, with about a 10% increase in going from the QZ to 5Z level. Because these values increases as we increase the basis set size, they are probably more accurate than those obtained using the MRCI method, at least for the singly augmented basis set series.        If we now examine the MRCI values for the dipole-dipole-quadrupole polarizability Bx,x,xx, we see that in going from the DZ to 5Z levels, there is a decrease of 65%. While there is an almost 51  21% difference between the DZ and TZ levels, the bulk of the percent difference is seen when increasing the basis set from the TZ to QZ levels, with a difference of  almost 33%. There is only a 12% difference when increasing the basis set from the QZ to 5Z level. In contrast, for the Bx,x,xx polarizability obtained using the CASPT2 method, the largest  difference occurs between the DZ and TZ levels (32%), with almost 20% difference between the TZ and QZ levels, and finally an additional 9% difference between the QZ and 5Z levels. This gives a total of nearly 61% difference when increasing the basis set size from the DZ to 5Z level.         For the fourth and final component of the dipole-dipole-quadrupole polarizability, Bz,z,zz, we see a 46% decrease when increasing the basis set from the DZ to 5Z level using the MRCI method. Most of this difference is found between the DZ and TZ levels (26%), with almost 16% difference between the TZ and QZ levels, and less than 5% difference between the QZ and 5Z levels. For the same values obtained using the CASPT2 method, there is a 52% decrease when increasing the basis set from the DZ to 5Z level. Again, most of this difference is between  the DZ and TZ levels (31%), with about 15% of the difference between the TZ and QZ levels, and less than 6% difference between the QZ and 5Z levels.       From Table (31), we see that there are literature values113 to compare to our values of the quadrupole-quadrupole polarizability (). We first discuss the quadrupole-quadrupole polarizability Cxx,xx. Our value obtained using the MRCI method, and the aug-cc-pV5Z basis set, is only 0.33% different from the value of Kalugina and  Cherepanov113, who used a QZ basis. Our CASPT2 value is farther from their value, but is still less than 3% different. Examining basis set sensitivity, we see that there is a 38%  increase in the MRCI value of Cxx,xx when increasing 52  the basis set from the DZ to 5Z levels. Over 22% of this difference is found between the DZ and TZ levels, with a 10% difference between the TZ and QZ levels, and only about 6% difference between the QZ and 5Z levels. The situation with the CASPT2 values is essentially the same, with almost a 41% increase in the value of Cxx,xx when increasing the basis set size from the DZ to 5Z levels; the large portion of the difference (~24%) again occurring between the DZ and TZ levels, about 11% difference between the TZ and QZ, and finally only about 6% difference between the QZ and 5Z levels. The next component of the quadrupole-quadrupole polarizability in Table (31), Czz,zz-cc-pVXZ series. The total difference found when increasing the basis set size from the DZ to 5Z level for the MRCI method is about 19%, with 12% of that difference occurring between the DZ and TZ levels, and only 3-4% difference between additional levels. The values obtained using CASPT2 have a total of a 21% increase between the DZ and 5Z levels, with 14% of the increase occurring between the DZ and TZ levels, with again only 3-4% of an increase for additional increases in the cardinal number. Comparing to the available literature data, we see that there is very good agreement between our values and that of  Kalugina and  Cherepanov113, with less than 3% of a difference between our MRCI value and their value, and our CASPT2 value is identical with their value to the first decimal place. Our best (5Z level) MRCI value for the third component of the quadrupole-quadrupole polarizability in Table  (26), Cxz,xz, is about 4% smaller than that of Kalugina and Cherepanov113. Our CASPT2 value at the same level is about 5% larger than the value calculated by Kalugina and Cherepanov113, but less than 2% larger than the value calculated by Bartolomei et al.114, who used a QZ basis set, and the MRCI method. This tensor is the least sensitive of the three quadrupole-quadrupole polarizabilities to cardinal number with 53  between 13% and 15% total difference in going from the DZ to 5Z basis sets, for the MRCI and CASPT2 methods, respectively.       Our final quantities to be discussed are the second hyperpolarizabilities (). We begin by examining the component xxxx. Our value obtained using the MRCI method and aug-cc-pV5Z basis set, is about 11% smaller than the value reported by P. Neogràdy et al.118 (who used the CCSD119 method and aug-cc-pVQZ basis set).  The same quantity obtained with the CASPT2 method is less than 4% smaller than the literature value. However, in both cases, this quantity the 5Z level is more than 80% larger than the value obtained at the DZ level. Nearly 37% of this difference occurs between the DZ and TZ levels, but there is almost as much difference between the TZ and QZ levels (~30%). The difference between the QZ and 5Z levels is not insignificant either, at 14%. There is an even larger difference between basis set levels for the values obtained using the CASPT2 method, with an increase of almost 85% between the DZ and 5Z levels. Over 40% of this difference is between the DZ and TZ levels, with over 31% difference between the TZ and QZ levels, and an appreciable 13% difference between the QZ and 5Z levels.        The next second hyperpolarizability we examine is xxzz. Though less so than the previous quantity, it is still quite sensitive to the choice of basis set. For our value using the MRCI method, there is an almost 45% increase in the value when increasing the basis set from the DZ to 5Z levels. A large portion of this increase happens between the DZ to TZ levels (20%), but there is almost as much difference between the TZ and QZ levels (17%). The difference is even more pronounced for the CASPT2 values, with almost a 26% increase in the quantity when 54  increasing the basis set from the DZ to TZ level, and about 20% difference between the TZ and QZ levels. Comparing to the literature value for this quantity, our value using the MRCI method and aug-cc-pV5Z basis set is less than 14% smaller than that of the literature. Since we do not have a value at the same basis set level for our CASPT2 series, we compare our value using the aug-cc-pVQZ basis set to the literature value, and find they are in good agreement, with a difference of  about 2%.        zzzz. Our value using the MRCI method and our largest basis set is about 17% smaller than that of P. Neogràdy et al.118.  Our CASPT2 value using our largest basis set is about 16% smaller than the literature value. As far as basis set sensitivity, the MRCI value increases by 40% when we increase the basis set from the DZ to TZ level, with the largest part of the difference (~20%) occurring between the TZ and QZ levels, as opposed to the largest portion of the difference usually being found between the DZ and TZ levels. This is also true for the CASPT2 values, where about 26% of the nearly 58% total difference being found between the TZ and QZ levels. The difference between the DZ and TZ levels are nearly as large for both the MRCI and CASPT2 values, at ~15% and ~23% respectively.    3.4  Properties at the Equilibrium Bond Length, Obtained using the Doubly Augmented Basis Sets   Table (32) shows the same quantities as Table (31), but now using the doubly augmented correlation-consistent polarized valence double-, triple-, quadruple- and quintuple-zeta  basis sets. We abbreviate these basis sets in the following discussions and figures as "d-DZ",  "d-TZ", "d-QZ, and "d-5Z", respectively. We first discuss the dipole polarizability xx. Within the values 55  obtained using the  MRCI method there is only about a 2% decrease in the value of the polarizability when the basis set is increased from the d-DZ to d-5Z levels. Most of the difference is between the d- DZ and d-TZ levels (1.4%), with only about 0.4% difference between subsequent levels. Our values calculated using the CASPT2 method are even more stable amongst basis sets considered, with only a 0.5% decrease in the value when increasing the basis set size from the d-DZ to d-5Z levels. This is significant, because calculations done with a -aug-ccpVXZ series can be performed, at least for this polarizability, without much loss in accuracy. These calculations take significantly less computational time. We know that this stabilization across cardinal number comes from the addition of the second set of diffuse functions in the basis set. We know this because when we compare to the values obtained using the singly augmented series in Table (31), we see that there was a much higher % difference between values when the basis set was increased from the double-zeta to quintuple-zeta levels for both the MRCI and CASPT2 methods (~7% and ~9%, respectively). The values calculated using the d-5Z basis sets and the MRCI and CASPT2 methods are ~5% and ~2% smaller than the value given by Kalugina and  Cherepanov113, respectively. This is about the same difference we saw for the same values obtained using our 5Z basis set and MRCI and CASPT2 methods (about 5% and 3%, respectively). 56   Table 32. Electronic properties (in a.u.) of O2 at the equilibrium bond length, r(O-O)=2.28187 bohr, using the MRCI and CASPT2/d-aug-cc-pVXZ levels of theory. Method MRCI MRCI MRCI MRCI CASPT2 CASPT2 CASPT2 CASPT2 Literature Basis set X = D X = T X = Q X = 5 X = D X = T X = Q X = 5  -E0 150.002067 150.112012 150.146552 150.157681 149.996735 150.114578 150.154184 150.168632 - xx 7.998 7.887 7.849 7.826 8.041 8.034 8.020 8.003 8.20a zz 15.367 15.056 14.948 14.920 15.390 15.309 15.227 15.197 15.09a zz -0.1899  -0.2641  -0.2602  -0.2617  -0.1903 -0.2570 -0.2512 -0.2520 -0.21b, -0.22d,-0.24c zzzz 5.4824 4.1968 4.2921 4.2895 5.4831 4.2134 4.3019 4.2947 4.67b, 4.70d,4.48c Ex,xxx -18.3036 -17.6089 -17.2863 -17.2669 -18.4462 -17.9592 -17.6627 -17.6392 20.95a,18.81d Ez,zzz 19.2409 21.2591 20.7043 20.8193 18.9992 21.2922 20.8353 20.9305 -17.93b,-18.06e Bx,z,xz -110.2305 -112.8545 -107.0655 -105.5565 -112.5295 -114.7905 -114.3285 -113.8950 - Bx,x,zz 36.4501 45.2970 44.7550 67.4910 42.1400 44.4930 45.6490 45.4370 - Bx,x,xx -67.9500 -79.8010 -79.4530 -84.9220 -72.8190 -85.3200 -87.4300 -87.4390 - Bz,z,zz -115.8030 -150.0030 -143.8510 -125.1810 -118.2050 -130.9940 -129.2510 -129.228 - Cxx,xx 9.6903  13.3216 13.3907 13.3873 10.1833 13.6528 13.8109 13.8259 12.84b,13.64e Czz,zz 20.5033 22.2236 22.0262 21.9659 21.0910 22.7200 22.6312 22.5809 22.30b,23.00e Cxz,xz 18.9835 19.4538 19.3547 19.3038 20.4782 21.2897 21.2808 21.2444 19.75b,20.48e xxxx 518.962 526.344 526.528 517.456 526.479 567.99 582.53 580.125 651f xxzz 396.679 370.176 362.012 - 481.705 481.447 486.052 500.751 434g zzzz 1006.78 773.587 771.715 763.38 734.93 781.737 775.59 768.883 906f a Yu N. Kalugina and V. N. Cherepanov113, using CCSD(T), with an aug-cc-pVQZ basis set  b Yu N. Kalugina and V. N. Cherepanov113, using R-CCSD(T), with an aug-cc-pVQZ basis set c Bartolomei et al.114, using aug-cc-pVQZ basis set and MRCI method, and active space CAS1 d Bartolomei et al.114, using aug-cc-pVQZ basis set and MRCI method, and active space CAS2 e Values taken from Bartolomei et al.114, as cited in Yu N. Kalugina and V. N. Cherepanov113 f 118, using the CCSD method and d- aug-cc-pV5Z basis set g Urban118, using the CCSD method and d- aug-cc-pQ5Z basis set  57   zz, we see that there is about a 3% decrease in the MRCI value when increasing the basis set from the d-DZ to the d-5Z level. Again, the CASPT2 series is more stable, with only about a 1% decrease in the value when the basis set size is increased the same way.  This difference for the CASPT2 levels is about the same as was seen for the singly augmented series (1.5%), while the 3% decrease for the MRCI values is actually about three times larger of a difference than was seen for the singly augmented series. However, the differences are still small, especially when compared to the other moments and polarizabilities we have calculated. The difference between the value reported by Kalugina and Cherepanov113 and our MRCI and CASPT2 values obtained with the d-5Z basis set are around 1% in both cases. As with the xx-component of the polarizability, this difference is about the same as that observed between the values of Kalugina and Cherepanov113 and our 5Z MRCI/CASPT2 values.   We will now discuss the quadrupole moment, zz. The values obtained using the MRCI method do not show a monotonic decrease. There is first a substantial decrease of almost 33% when we increase the basis set size from the d-DZ to d-TZ level. There is then about 1.5% increase in the value when we increase the basis set from the d-TZ to d-QZ level, and then finally 0.3% decrease between the d-QZ and d-5Z levels. Therefore, when compared to the large difference between the d-DZ and d-TZ levels, there is essentially no change in the value with a subsequent increase of the basis set size.  The same is true for the CASPT2 values. There is a large decrease between the d-DZ and d-TZ levels (almost 30%), then an increase of ~2% when we increase the basis set from the d-TZ to d-QZ levels, and only about 0.3%  decrease between the d-QZ and d-5Z levels. There is a stabilization between cardinal numbers compared to the singly augmented 58  series here as well. The singly augmented values had a difference of about 40% and 38%  between the DZ/TZ levels for the MRCI and CASPT2 values, respectively. Therefore, there is almost a 10% reduction in this difference when adding a second set of diffuse functions. The difference between the triple -zeta/quadruple zeta and quadruple-zeta/quintuple-zeta basis sets is reduced when adding a second set of diffuse functions as well, from 7-8% to 1.5% at most.  Both our MRCI and CASPT2 d-5Z values for the quadrupole moment are closest to the literature value of Bartolomei et al114., with about 4% and 5% difference, respectively.   Moving to the hexadecapole moment, (zzzz), calculated with the "d-XZ" series, we find that it follows the same non-monotonic behavior as the quadrupole moment just discussed. Focusing on the MRCI values, there is first a nearly 27% decrease when the basis set size is increased from the d-DZ to d-TZ level. Subsequent increases of the basis set, from d-TZ to d-QZ and from d-QZ to d-5Z, result in about a 2% increase in the value of the hexadecapole, and then a 0.06% decrease in the value, respectively. Therefore the value is very stable after the initial 27% decrease. The CASPT2 values follow the same trend, including the large difference in the value between the d-DZ and d-TZ basis sets, which in this case is 26%. The values obtained with subsequent increases of the cardinal number  each other than those at the MRCI level, with only a 0.21% increase in increasing the basis set from the d-TZ level to the d-QZ level, and only a 0.17% decrease when increasing the basis set size further to the d-5Z level. When we compare these differences to the differences between values obtained in the "XZ" (singly-augmented) series, we see there is about a 10%  larger separation between the doubly augmented double-zeta and doubly augmented triplet-zeta basis sets than between the singly augmented double-zeta and triplet-zeta values. The difference between the TZ/QZ and d-TZ/d-59  QZ MRCI values are about the same. However, the difference between the TZ/QZ for the CASPT2 values is an order magnitude larger than between the d-TZ/d-QZ values. The differences between the QZ/5Z levels are significantly larger for both the MRCI and CASPT2 values than between the values obtained using the d-QZ/d-5Z levels. Therefore there is altogether a stabilization in values when using the doubly augmented series as opposed to the singly augmented one.   The values of the hexadecapole obtained using the d-5Z basis set and both the MRCI and CASPT2 methods are about 4% smaller than the literature value from Bartolomei et al.114   The next set of values in Table (32) are the dipole-octupole polarizabilities Ex,xxx. These values do follow a monotonic trend, a decrease in absolute magnitude, when increasing the basis set size. As with most of the properties we have examined, the largest difference is seen between the d-DZ and d-TZ levels. There is a nearly 4% difference between these values obtained using the MRCI method. The difference between the d-TZ and d-QZ is of the same magnitude, at almost 2%. The difference between the d-QZ and d-5Z levels is an order smaller in magnitude, at only about 0.3%. The separations are about the same in the CASPT2 case (same magnitude), but even smaller. Therefore this property is rather insensitive altogether to a change in cardinal number. Both the MRCI and CASPT2 values obtained using the d-5Z basis set are closest to the literature value of Kalugina and Cherepanov113, at only about a 4% and 2% difference, respectively.   When we now examine the dipole-quadrupole polarizability Ez,zzz, we see that this property once again deviates from monotonic behavior. Looking at the MRCI(CASPT2) values, we see there is about a 10% (12%) increase in the dipole-octopole polarizability when we increase the basis set 60  from the d-DZ to d-TZ level. This is followed by a 3% (2%) decrease, when increasing the basis set size from the d-TZ to d-QZ levels, and then a less than 1% (for both MRCI and CASPT2)  increase in the value. The MRCI value using the d-5Z basis set is only 0.63% smaller than the value reported by Kalugina and Cherepanov113, and the CASPT2 value is only ~0.1% different, nearly identical to their value.   We now move on to the dipole-dipole-quadrupole polarizability Bx,z,xz  that was obtained using the doubly augmented basis sets. This tensor does not show monotonic behavior with the size of the basis set. However, the differences between the values obtained at the different cardinal numbers   are an order of magnitude smaller than the differences between the singly augmented basis set levels. There is a ~2% decrease in the MRCI value of Bx,z,xz  when the basis set is increased from the d-DZ to d-TZ levels. When the basis set is increased from the d-TZ to d-QZ levels, there is over a 5% increase. So in this case the largest difference is between the d-TZ/d-QZ levels (this was not true for Bx,z,xz when we considered the singly augmented series, where the largest difference was between the DZ and TZ levels). An additional increase of the cardinal number to the d-5Z level causes an almost 1.5% increase in the value. Looking at the CASPT2 values, we see that there is also about a 2% decrease when we increase the basis set from the d-DZ to d-TZ level, as there was between the same MRCI values. However, in the CASPT2 case, there is a much smaller difference in between the d-TZ and d-QZ levels (0.4%). So in the CASPT2 the largest difference is between the d-DZ and d-TZ levels. The difference between the d-QZ and d-5Z levels is 0.4% as well.   61  The next dipole-dipole-quadrupole polarizability in Table (32) is Bx,x,zz. Again this tensor does not increase monotonically with the basis set. When the basis set is increased from the d-DZ to the d-TZ level using the MRCI method, there is an almost 22% increase in the value of the polarizability. There is then about a 1% decrease when the basis set is further enlarged from the d-TZ to d-QZ level. From the d-QZ to d-5Z level, there is over a 40% increase in the value of the polarizability. Comparing these values to the CASPT2 situation, we see a 5% increase in the value of the polarizability when we increase the basis set from the d-DZ to the d-TZ level. There is then about a 3% increase when the basis set is increased in size from the d-TZ to the d-QZ level, and then finally about a 0.5% difference between the values at the d-QZ and d-5Z level. Given the stability in the CASPT2 values, especially the very small difference between the two largest basis sets, it is likely that the CASPT2 values are the more accurate between the two methods.   We know turn to an examination of the dipole-dipole-quadrupole polarizability Bx,x,xx calculated using the doubly augmented basis sets. Not surprisingly at this point, these values also do not follow a monotonic pattern. There is a 16% decrease in the MRCI value when the basis set size is increased from the d-DZ to d-TZ level. There is then a ~0.5% increase between the d-TZ and d-QZ values (so these two values for the polarizability are essentially the same). Finally, there is a ~7% decrease in the value of the dipole-dipole polarizability when the basis set is further increased to the d-5Z level. The situation for the CASPT2 values is similar. There is about a 16%  decrease in the value of the polarizability when the basis set is increased from the d-DZ to d-TZ level. There is a larger difference between the d-TZ and d-QZ values obtained with the CASPT2 62  method than those obtained with the MRCI  method (about 2% instead of 0.5%). However, the values at the d-QZ and d-5Z are the same to the second decimal place in the CASPT2 case.   The fourth and final non-redundant component of the dipole-dipole-quadrupole polarizability is Bz,z,zz. There is the same decrease seen in the MRCI value of this polarizability when increasing the basis set size from the d-DZ to d-TZ levels as was seen in the last three components of the B-tensor. This decrease is a difference of ~26%. There is then an increase of about 4% when the basis set is increased to the d-QZ level, and then finally an increase of about 13% between the d-QZ and d-5Z levels. The CASPT2 values for this tensor also display non-monotonic behavior, but as seen previously, the values display greater stability against a change in the cardinal number than the MRCI values. There is only a 10% decrease in between the d-DZ and d-TZ levels, and only about a 1% and 0.01% increase between the d-TZ/d-QZ and d-QZ/d-5Z levels, respectively.   We will now discuss the quadrupole-quadrupole polarizabilities (). The first in Table (32) is Cxx,xx. The MRCI values of this tensor increase with basis set size until the d-5Z basis set. There is a 32% difference between the d-DZ and d-TZ levels, and the values at the d-TZ and d-QZ levels are nearly identical, with only about a 0.5% difference. The difference between the d-QZ and d-5Z levels is an extremely slight decrease, with only a 0.03% difference between values. The CASPT2 values increase in value with an increase in basis set size. There is a total of a 30% increase in going from the d-DZ to d-5Z basis set, and a whopping 29% of this difference occurs between the d-DZ and d-TZ levels. Comparing to the literature values, our MRCI and 63  CASPT2 values are nearest to that of Bartolomei et al114., with only between 1-2% difference in both cases.    The next quadrupole-quadrupole polarizability is Czz,zz. These values do not vary monotonically with the basis set size. We instead observe that there is first an increase of 8% in the MRCI value of the polarizability when we increase the basis set size from the d-DZ to the d-TZ level, but then the value decreases when the basis set is increased to the d-QZ level. However, the decrease is only by less than 1%. There is then an additional decrease in the polarizability when the basis set size is increased to the d-5Z level. However, this difference is even smaller.  Therefore, the value of Czz,zz is essentially constant after the increase from the d-DZ to the d-TZ basis set size. The same is seen for the CASPT2 values. There is an initial increase of around 7%, but then the differences in increasing the cardinal number any further are far under 1%. Altogether this is a significant stabilization for both methods compared to the singly augmented series, in which the total percent difference in increasing the basis set size from double-zeta to quintuple-zeta was around 20%. Comparing to the literature values, our values are consistent with the value calculated by Kalugina and Cherepanov113, with a difference of less than 2% for both our MRCI and CASPT2 values. Our values are near that of Bartolomei et al.114 as well, with less than 5% difference between them for both methods.   The third and final component of the quadrupole-quadrupole polarizability is Cxz,xz. For both the MRCI and CASPT2 values, the largest difference is between the d-DZ and d-TZ basis set levels, at less than 3% and less than 4%, respectively. Any additional increase in the cardinal number produces a value of the  polarizability that is under 1% different than the d-TZ level value (and in 64  most cases far under even 0.5%). The MRCI value obtained using our best basis set is closest to the value of Kalugina and Cherepanov113, with only about a 2% difference, but it is still less than 6% different than the Bartolomei et al.114 value. Our CASPT2 value however, is closer to the value calculated by Bartolomei et al.114, with a difference of about 4%, and over 7% different from the value calculated by Kalugina and Cherepanov113.   We now discuss our last set of values, the second (hyper)polarizabilities (). We first look at the values for xxxx. The MRCI values increase with increased basis set size until the d-5Z level, where there is a very slight decrease (around 0.02 %).  In fact, the value for the hyperpolarizability at the d-5Z level is only about 0.3% different than that obtained at the d-DZ level. There is more separation in the CASPT2 levels, with about a 10% difference between the smallest (d-DZ) and largest (d-5Z) basis sets. These differences are dramatically smaller than those between the basis set sizes for the singly augmented series, where the total difference between the DZ/5Z levels was over 80% for both methods. Comparison with the literature value shows that our MRCI value at the d-5Z level is about 22% different than that provided by P. Neogràdy et al.118 and our CASPT2 value is about 12% different.   The second hyper-polarizability in table (2) is xxzz. We were able to obtain values at the d-DZ, d-TZ, and d-QZ level with the MRCI method. There is about a 9% reduction in the value of this hyper-polarizability when we increase the basis set from the d-DZ to the d-5Z level.  The CASPT2 values are not monotonic with the increase in basis set size. However, our value at the d-5Z level is the farthest from that at the d-DZ level, and the difference is still less than 4%. 65   P. Neogràdy et al.118  did not obtain a value using the d-aug-cc-pV5Z basis set, but they did obtain a value using the d-aug-cc-pVQZ basis set that we can use to compare to our values using the same basis set.  Our value at the d-QZ level and using the MRCI method is about 18% different from the literature value, while our CASPT2 value using the d-5Z basis set is about 11% different from the literature value.       Finally, our last hyperpolarizability to discuss is zzzz. The MRCI values for this quantity decrease monotonically with an increase in basis set size. The separation between basis set levels is larger than for the previous hyperpolarizabilities discussed. There is a 27.5% reduction in the MRCI value of this tensor when we increase the basis set size to the d-5Z level. Over 26% of this difference is between the d-DZ and d-TZ levels, however, so subsequent increases of the cardinal number only cause small decreases. The CASPT2 values, in contrast, do not decrease monotonically with the basis set size, and in fact the value of zzzz is larger at our highest basis set level than at our smallest. However, the difference between the d-DZ and d-5Z levels is only 4.5%, with differences of less than 1% between the d-TZ/d-QZ and d-QZ/d-5Z levels. So in general the CASPT2 values show less change with basis set size in this case than the MRCI values. Our MRCI value using the d-5Z basis set is less than 6% different from the literature value in Table (32), and our CASPT2 value is less than 7% different.     66  3.5 Properties Obtained at non-Equilibrium Bond Lengths In order to calculate vibrational averages of the multipole moments and (hyper)polarizabilities discussed, we must have values of these quantities away from equilibrium. In particular, we wish to obtain these quantities for the first six vibrational levels. In order to decide what bond distances we wish to obtain these properties at, two things were done. First, a calculation done at the MCSCF level that included all nine triplet states formed from two triplet oxygen atoms showed that the ground state is well separated from the other eight excited states for O-O bond lengths under about 3.78 bohr. Most calculations have unphysical results at bond lengths over 3.6 bohr however, so this was used as the upper limit of the bond length. Indeed, a majority of the calculations attempted for bond lengths larger than this either did not converge or gave unphysical results, blowing up to extremely large numbers. Secondly, using our potential energy curve obtained with the MRCI method and the d-aug-cc-pVQZ basis set, we used the Numerov-Cooley129,130 technique to numerically solve the rovibrational Schr˘0dinger equation. The vibrational wavefunctions in Figure (3) were thus obtained. This figure shows that the vibrational wavefunctions are well embedded in the region of O-O=1.8-3 bohr. Therefore, the lower limit for our calculations was a bond distance of 1.8 bohr.  67    Figure 3. Wavefunctions for the first six vibrational levels (=0-5) for the ground rotational state (J=0) of the O2 molecule  68  Figure (4) shows the dipole polarizability (zz) using the "d-XZ" basis sets, using both the MRCI and CASPT2 methods, as a function of the O-O bond length. The MRCI and CASPT2 values agree within 2% for the d-DZ levels, within 3% for the d-TZ levels, and within about 7% for the d-QZ levels. Once we reach the d-5Z basis set level, differences are again within 3% for all bond distances considered. For all levels, the greatest difference between the MRCI and CASPT2 values occurs at the bond distances of 2.646 bohr and 2.929 bohr.   We now observe the basis set sensitivity of the dipole polarizability (zz) within the CASPT2 method for the bond distances studied (Figure  (5)). Although the figure shows all levels of the cardinal number considered, we will restrict our discussions within a particular method to the effect of adding a second set of diffuse functions to our largest basis set, that is, X=5. For this component of the dipole polarizability, the effect of adding a second set of these functions is negligible, with the largest difference between the 5Z and d-5Z basis sets occurring at the shortest bond distance (1.8 bohr), where it is only 0.5%. Figure (6) shows the same quantity and basis sets for the MRCI method. From this figure, we can see that the DZ/d-DZ basis sets are separated from the values calculated using higher order basis sets. As with the CASPT2 values, the differences between the singly and doubly augmented basis sets 5Z/d-5Z are very small, at 0.4% or less.    Our next figure shows the dipole polarizability (xx) using the "d-XZ" basis sets and the MRCI and CASPT2 methods versus the O-O bond length. There is improved agreement between methods for the d-DZ basis set, with less than 1% difference between methods. The largest differences between the methods using the d-TZ basis set is around 2%, and then there is a 69  difference of almost 9% between methods at a bond distance of 2.9 bohr when using the d-QZ basis set. As with the polarizability (zz), the difference between methods using the d-5Z basis set is within 3%.   Figure (8) shows the dipole polarizability (xx) obtained with the CASPT2 method and all basis sets. The difference between the singly and doubly augmented basis sets is over 13% at some points between the DZ/d-DZ basis sets,  but this difference grows smaller between sets as the cardinal number is increased, until it is under 1% between the 5Z/d-5Z basis sets. This is also the case for the values obtained using the MRCI method and the singly and doubly augmented basis sets (Figure (9)).    70    Figure 4. The dipole polarizability (zz) in atomic units versus  bond length for the CASPT2 and MRCI methods, using the doubly augmented basis sets. 71   Figure 5. The dipole polarizability (zz) in atomic units versus  bond length obtained with the CASPT2 method, using the singly and doubly augmented basis sets.  72   Figure 6. The dipole polarizability (zz) in atomic units versus  bond length, obtained using the MRCI method and the singly and doubly augmented basis sets.  73   Figure 7. The dipole polarizability (xx) in atomic units versus  bond length, obtained with the MRCI and CASPT2 methods and the doubly augmented basis sets.   74   Figure 8. The dipole polarizability (xx) in atomic units versus  bond length, obtained with the CASPT2 method and the singly and doubly augmented basis sets.  75   Figure 9. The dipole polarizability (xx) in atomic units versus  bond length, obtained with the MRCI method and the singly and doubly augmented basis sets.    76  zz, shown in Figure  (10). Because the quadrupole moment is very small and positive for the CASPT2 method and very small and negative for the MRCI method at 2.457 bohr, the difference between them blows up to 407% at this point when looking at the d-5Z basis set values. All other bond distances have differences of 0.01-9%, with the higher end of the difference again occurring for bond distances longer than equilibrium, as was seen for the dipole polarizabilities. Also seen clearly is the separation of the d-DZ basis set from all other basis sets, as was seen for many of the properties we examined at equilibrium. It is interesting to note the sign change of the quadrupole moment, that occurs as the bond distance is increased past equilibrium. Figure (11) shows a schematic diagram for why this occurs82. The quadrupole moment behaves as a cos2()  function. Therefore, when the bond distance is lengthened in the O2 molecule, the electron density is shifted from a region where the cosine squared function is negative to one that is positive.       zz versus bond distance for the singly and doubly augmented basis sets within the CASPT2 method.  The DZ/d-DZ levels are again clearly separated from the larger basis set. The values have a large difference when they are very small, near the point where the quadrupole moment changes sign, as was seen in the previous figure (here observed in the region between 2.36-2.46 bohr). Otherwise, the effect of adding a second diffuse function is very small (less than 1% between the 5Z/d-5Z basis sets). Figure (13) shows the quadrupole moment versus both length obtained using the MRCI method. The difference between the singly and doubly augmented basis set values at the point 2.46 bohr are even larger 77  for the MRCI method, as the value of the quadrupole at this point is even smaller, on the order of 10-2 a.u. Otherwise, the differences are again very small.        We now discuss the hexadecapole moment(zzzz)  for both methods as a function of bond distance, as shown in Figure  (14). Right away we can see the separation of the d-DZ level values from all other basis set values. The two methods are in good agreement for this property as well, with around 1% difference at most between the d-DZ values, less than 3% difference between the values using the d-TZ basis set, around 3% difference at most between the d-QZ values, and finally around 3% at most difference between the methods when using the d-5Z basis set.  Unlike the previous properties we examined, the largest differences occur for the shortest bond distance in this case, of 1.8 bohr.       The next figure shows the hexadecapole moment obtained using the CASPT2 method, and all basis sets. We can see that in this case both the singly and doubly augmented double zeta basis sets are separated from all other basis sets. As with the quadrupole moment, adding a second diffuse function has little effect at the quintuple zeta level, with less than a 2% difference. Figure (16) shows the same quantities for MRCI, which show the same behavior as those obtained using the CASPT2 method.    78    Figure 10. The quadrupole moment (zz) in atomic units versus  bond length obtained with the MRCI and CASPT2 methods and the doubly augmented basis sets.    79   Figure 11. The shape of the quadrupole moment as determined by the cos2() function82   80   Figure 12. The quadrupole moment (zz) in atomic units versus  bond length obtained using the CASPT2 method and the doubly and singly augmented basis sets.  81   Figure 13. The quadrupole moment (zz) in atomic units versus  bond length obtained using the MRCI method and the doubly and singly augmented basis sets.   82   Figure 14. The hexadecapole moment (zzzz) in atomic units versus  bond length obtained using the MRCI and CASPT2 methods, and the doubly augmented basis sets.    83    Figure 15. The hexadecapole moment (zzzz) in atomic units versus  bond length obtained using the CASPT2 method and the singly and doubly augmented basis sets.  84    Figure 16. The hexadecapole moment (zzzz) in atomic units versus  bond length obtained using the MRCI method and the singly and doubly augmented basis sets.  85  We will now discuss the dipole-octopole polarizability, Ex,xxx. As was the case for the hexadecapole moment, the largest differences occur at the shortest distance and grow smaller as the bond is lengthened. This is seen clearly in Figure  (17). There is good agreement between methods however, with even the largest differences being under 5%.      We now look at the dipole-octopole polarizability Ex,xxx as obtained within the CASPT2 method. Figure (18) shows these values. We can see clearly that as the cardinal number is increased, the difference between the singly and doubly augmented curves grows smaller. At the shortest distance of 1.8 bohr, the difference between the 5Z/d-5Z basis sets is almost 4%, but the difference quickly drops off as the bond is lengthened, and is under 1% for most bond distances. The same is true for the values obtained using the MRCI method (Figure  (19)).        Figure (20) shows a comparison of methods for the dipole-octopole polarizability Ez,zzz. Again, the largest difference occurs at the shortest bond distance, and in this case is as much as 13% (between the d-QZ levels). However, the differences between methods at all other bond lengths considered are smaller than those for Ex,xxx.        Figure (21) shows the values for this tensor obtained using the CASPT2 method, and all singly and doubly augmented basis sets studied. As observed for the dipole-octopole polarizability Ex,xxx, the difference between the singly and doubly augmented basis sets decreases with increasing cardinal number. At a bond distance of 1.8 bohr, the largest difference occurs, at 33%, with a values of 3.05 a.u. at the 5Z level and a value of 4.30 at the d-5Z level. This difference quickly drops off with increasing bond length and is between 1-3% for most bond lengths. Figure (22) shows the MRCI values for this tensor, which behaves very similarly to the CASPT2 values, with the largest difference occurring for 1.8 bohr, at 27%. 86   Figure (23) shows the dipole-dipole-quadrupole polarizability, Bx,z,xz versus the bond length. The differences between methods grow larger with the basis set size for this tensor, growing from around  2% between the d-DZ values to around 9% for the d-5Z values. The points at 1.8 bohr show this trend well, as this is where the largest differences are found. For the d-QZ levels, there is also around an 8% difference after 3 bohr, which occurs because the MRCI values have some unreliability in that region.       Figure (24) shows the values of  Bx,z,xz obtained using just the CASPT2 method, and all singly and doubly augmented basis sets. Like previous properties discussed, the differences between the singly and doubly augmented basis sets do grow smaller with increased cardinal number. However, even between the 5Z/d-5Z basis sets, there is an appreciable difference between values. The largest difference is at shorter distances of the bond length, with ~17% difference at 1.8 bohr, but there is at least 4% difference at all bond lengths studied. The situation is the same for the MRCI values (Figure  (25)).   87  Figure 17. The dipole-octopole polarizability (Ex,xxx) in atomic units versus  the bond length obtained using the MRCI and CASPT2 methods, with the doubly augmented basis sets.  88   Figure 18. The dipole-octopole polarizability (Ex,xxx) in atomic units obtained using the CASPT2 method and the singly and doubly augmented basis sets 89   Figure 19. The dipole-octopole polarizability (Ex,xxx) in atomic units obtained using the MRCI method and the singly and doubly augmented basis sets  90   Figure 20. The dipole-octopole polarizability (Ez,zzz) in atomic units obtained using the MRCI and CASPT2 methods, and the doubly augmented basis sets 91   Figure 21. The dipole-octopole polarizability (Ez,zzz) in atomic units obtained using the CASPT2 method and the singly and doubly augmented basis sets.  92   Figure 22. The dipole-octopole polarizability (Ez,zzz) in atomic units obtained using the MRCI method and the singly and doubly augmented basis sets.  93   Figure 23. The dipole-dipole-quadrupole polarizability (Bx,z,xz) in atomic units versus  bond length obtained using the MRCI and CASPT2 methods and the doubly augmented basis sets  94   Figure 24. The dipole-dipole-quadrupole polarizability (Bx,z,xz) in atomic units versus  bond length obtained using the CASPT2 method and the singly and doubly augmented basis sets 95   Figure 25. The dipole-dipole-quadrupole polarizability (Bx,z,xz) in atomic units versus  bond length obtained using the MRCI method and the singly and doubly augmented basis sets.   96  The next property we examine as a function of the bond length is the dipole-dipole-quadrupole polarizability Bx,x,zz, Figure  (26). As with the last property, we see that the MRCI values do not converge smoothly. This results in some very large differences between methods. The points at a bond distance of 1.8 bohr show this separation well. At this point, there is a 54% difference between the MRCI and CASPT2 values here for the d-DZ basis set, a 49% difference between values for the d-TZ basis set, a 40% difference in values for the d-QZ basis set, and an 83% difference in values for the d-5Z basis set. Around a bond distance of 3 bohr, these differences become much smaller, and are on the order of 1% in some cases.        We now look at the values of Bx,x,zz obtained using just the CASPT2 method. As we can see in Figure  (27), there is an appreciable difference between even the 5Z/d-5Z basis sets. At the shortest distance of 1.8  bohr, the difference is largest, at ~28%. But even at the longest bond lengths studied, there is a difference of 10-12%. The difference is even larger for the values obtained using the MRCI method (Figure  (28)). This is partially because the doubly augmented curves are not smooth, and probably contain some convergence errors. For the 5Z/d-5Z basis sets, the difference at the shortest distance of 1.8 bohr is 160%. As the bond length is lengthened, the difference slowly decreases, until it reaches a minimum at the longest distance of 3.3 bohr, with a difference between basis sets of 11.4%. 97   Figure 26. The dipole-dipole-quadrupole polarizability (Bx,x,zz )in atomic units versus  bond length obtained using MRCI and CASPT2 methods using the doubly augmented basis sets  98   Figure 27. The dipole-dipole-quadrupole polarizability (Bx,x,z,z) in atomic units obtained using the CASPT2 method and the singly and doubly augmented basis sets  99   Figure 28. The dipole-dipole-quadrupole polarizability( Bx,x,z,z) in atomic units obtained using the MRCI method and the singly and doubly augmented basis sets  100  Figure (29) shows the third dipole-dipole-quadrupole polarizability, Bx,x,xx, versus the bond length. We can see that the MRCI values again do not converge smoothly, especially for the d-QZ and d-5Z levels. Because of this, the largest differences between the two methods occur for the longer distances of the bond length, at up to 17%.        We now examine Bx,x,xx within the CASPT2 method (Figure  (30)). These curves converged smoothly, and it is apparent that there is a basis set sensitivity even at the 5Z/d-5Z levels, with a difference between 10-19% for all bond lengths.  Figure (31) shows the same MRCI values, and as the curves are not as smooth as the CASPT2 curves, the difference between the singly and doubly augmented values is larger, between 11-33%.  101   Figure 29. The dipole-dipole-quadrupole polarizability (Bx,x,xx) in atomic units versus  bond length, obtained using the MRCI and CASPT2 methods, and the doubly augmented basis sets.  102   Figure 30. The dipole-dipole-quadrupole polarizability (Bx,x,xx) in atomic units versus  bond length, obtained using the CASPT2 method and the singly and doubly augmented basis sets 103   Figure 31. The dipole-dipole-quadrupole polarizability( Bx,x,xx) in atomic units versus  bond length, obtained with the MRCI method and the singly and doubly augmented basis sets 104   Figure (32) shows the fourth and final dipole-dipole-quadrupole polarizability, Bz,z,zz. In the region from 2.5 to 3 bohr, there is good agreement between the MRCI and CASPT2 methods for all four doubly augmented basis sets, on the order of 1%.  The agreement at shorter distances is the worst, with differences as large as 14%.        The next graph, Figure  (33), shows Bz,z,zz versus bond length obtained using the CASPT2 method and all singly and doubly augmented basis sets. There is almost a 20% difference between the 5Z/d-5Z basis sets for the shortest bond length, and at least 5% difference for all bond lengths studied. The same quantities obtained using the MRCI method (Figure  (34))  show much smaller differences between the 5Z/d-5Z basis set, with only a 2.7% difference at the shortest bond length, and less than one percent difference in the region between 2.1-2.3 bohr. The largest difference between these basis sets for the MRCI values actually occurs at the longest bond distance (3.3 bohr) with an 8% difference at this point.  105   Figure 32. The dipole-dipole-quadrupole polarizability (Bz,z,zz) in atomic units versus  bond length obtained using the MRCI and CASPT2 methods, with the doubly augmented basis sets  106   Figure 33. The dipole-dipole-quadrupole polarizability (Bz,z,zz) in atomic units versus  bond length obtained using the CASPT2 method and the singly and doubly augmented basis sets 107   Figure 34. The dipole-dipole-quadrupole polarizability (Bz,z,zz) in atomic units versus  bond length obtained using the MRCI method and the singly and doubly augmented basis sets  108  We now turn to the quadrupole-quadrupole polarizabilities, starting with Cxx,xx (Figure  (35)). For this property it is easily seen again that the d-DZ values of both methods are well separated from the values obtained with the larger basis sets. It is easy to see where the differences are between methods using that basis set as well.  The points in the region from 2.2-2.5 bohr show the largest differences between methods, with around 5% difference, while all other points obtained with the d-DZ basis set show less than a 1% difference. For the values obtained using the d-TZ basis set, the difference between methods is fairly uniform for all distances considered, with a difference between 2-3%. For the d-QZ basis set, we again see the largest difference between methods in the values from 2.2-2.5 bohr, where the difference is as much as 11%. Finally, for the d-5Z values the differences are once again uniform, and between 3-4%.      In Figure  (36), we consider just the values of Cxx,xx obtained within the CASPT2 method with the singly and doubly augmented basis sets. The difference between the 5Z/d-5Z basis sets is pretty uniform throughout, between about 5-6%.  This is also true for the MRCI values (Figure (37)).       The second quadrupole-quadrupole polarizability we have calculated values for is Czz,zz (Figure  (38)). For the values obtained using the d-DZ basis set, there is again the same increased difference between values obtained with the two methods from 2.2-2.5 bohr, with the largest difference being about 5% at 2.2 bohr. For all other basis sets, the difference is uniformly between 2-3% for all distances. When we view just the values obtained using the CASPT2 method (Figure  (39)), we see that there is a very small difference between the values obtained at the 5Z and d-5Z level. amounting to between 1-2% for all bond lengths. The same is true for the values obtained with the singly and doubly augmented basis sets within the MRCI method (Figure  (40)). 109  The third and last quadrupole-quadrupole polarizability is Cxz,xz (Figure (41)). This component of the quadrupole-quadrupole polarizability exhibits larger differences between the methods in general, though the curves are qualitatively similar in behavior. For the d-DZ basis set, there again is the largest differences between methods in the region from 2.2-2.5 bohr, though the increase in difference in this region is not as large as in the previous two components examined. The largest difference between methods at this level of basis set is between 8-9%. For the d-TZ basis set, the difference between methods increases with the bond length, and is almost 10% at 2.6 bohr. The same is true for the d-5Z basis set. Note that for the MRCI values, there is no point at 3.6, as the finite field calculations done for this quantity gave unphysical results after 3.5 bohr. The singly and doubly augmented basis sets for just the CASPT2 method (Figure  (42)) show a 7% difference between the 5Z/d-5Z basis sets at the shortest distance, but the difference drops off quickly as the bond is lengthened and is between 1-3% for most bond lengths. Once again, the MRCI values are very similar (Figure  (43)).       The last property to compare for the performance of the two methods is the second hyperxxxx (Figure  (44)). The separation between the values obtained using both methods at the d-DZ basis set level from the larger basis sets is apparent in this figure. The MRCI and CASPT2 values at this level show good agreement, with the largest difference being about 3% and most differences under 2%. For the d-TZ values, there is a clear outlier at 2.9 bohr in the MRCI values. Other than this point, the d-TZ values are within 10% of each other between methods. The largest difference is found between methods at the d-QZ level, where the difference reaches 16% at 3.3 bohr. The values at the d-5Z level have their largest difference at 2.2 bohr, where there is a 12.4% difference. 110   Looking now at just the CASPT2 values for xxxx in Figure  (45),  we can see that even at the 5Z/d-5Z level, there is a 20-27% difference between the singly and doubly augmented levels (the DZ/d-DZ values for this property using the CASPT2 method are nearly 100% different for all bond distances). The MRCI singly and doubly augmented values at the 5Z/d-5Z level are slightly closer together (Figure  (44)), but still 16-25% different for all bond lengths.       The next component of the second hyperpolarizability is xxzz. Figure (47) shows the MRCI and CASPT2 values versus bond distance for the d-DZ, d-TZ, and d-QZ basis sets. The calculations for this quantity attempted with the d-aug-cc-pv5Z basis set did not converge. As far as this property is concerned, the MRCI method seems to do better, and we were able to obtain more values and a smoother curve with this method than with the CASPT2 method. However, for the curves we do have, we note that there are large differences between the two methods. At the d-DZ basis set level, the differences range between 19-45%. There is as much as a 70% difference between methods at the d-TZ basis set level, and 58% between values at the d-QZ level.        Figure (48) shows just the CASPT2 values for xxzz. Because we do not have values at the X=5 level, we will examine the singly and doubly augmented values at the QZ/d-QZ level. This tensor shows sensitivity to the addition of a second diffuse function, as the difference between the QZ/d-QZ basis set values are 23-33% for all bond distances considered.  The MRCI values show similar differences between the QZ and d-QZ levels, with differences ranging from 14-41% (Figure (49)).  111  The last component of the second hyperpolarizability is zzzz . At the d-DZ level, Figure  (50) shows that the MRCI value deviates sharply from that obtained using the same basis set and the CASPT2 method at 2.46 bohr. Therefore the difference at this point between methods at this basis set level is 80%. Many of the points are vary at anywhere from 10-32% difference at this basis set level, but at a distance of 2.65 bohr the difference is only about 2.5%. The values at the d-TZ level differ by up to 15% for some of the distances considered, but the difference is very small again at 2.65 bohr, at only 0.2%. Very similar behavior in the differences between methods are observed for the two larger basis sets.       Figure (51) shows the CASPT2 values of zzzz for all basis sets. This tensor is sensitive to the addition of a second set of diffuse functions. The difference is largest for shorter distances, with a difference of nearly 32% between the 5Z and d-5Z basis sets at this point. Even at the longest distances of the bond length, we see a 14-17% difference between the singly and doubly augmented quintuple zeta basis sets. The MRCI values (Figure  (52)) are similar, starting with a 23% difference at the shortest bond length, but the difference drops off with increased bond length and is only 2-4% at the longer bond lengths.   112   Figure 35. The quadrupole-quadrupole polarizability (Cxx,xx) in atomic units versus  bond distance obtained using the MRCI and CASPT2 methods and the doubly augmented basis sets 113   Figure 36. The quadrupole-quadrupole polarizability (Cxx,xx) in atomic units versus  bond length obtained using the CASPT2 method and the singly and doubly augmented basis sets 114   Figure 37. The quadrupole-quadrupole polarizability (Cxx,xx) in atomic units obtained using the MRCI method and the singly and doubly augmented basis sets 115   Figure 38. The quadrupole-quadrupole polarizability (Czz,zz) in atomic units versus  bond length obtained using the MRCI and CASPT2 methods and the doubly augmented basis sets    116  Figure 39. The quadrupole-quadrupole polarizability (Czz,zz) in atomic units versus  bond length obtained using the CASPT2 method and the singly and doubly augmented basis sets 117  Figure 40. The quadrupole-quadrupole polarizability (Czz,zz)  in atomic units versus  bond length obtained using the CASPT2 method and the singly and doubly augmented basis sets  118   Figure 41. The quadrupole-quadrupole polarizability (Cxz,xz) in atomic units versus  bond distance obtained using the MRCI and CASPT2 basis sets and the doubly augmented basis sets  119   Figure 42. The quadrupole-quadrupole polarizability (Cxz,xz) in atomic units versus  bond distance obtained using the CASPT2 basis set and the singly and doubly augmented basis sets 120   Figure 43. The quadrupole-quadrupole polarizability (Cxz,xz) in atomic units versus  bond distance obtained using the MRCI method and the singly and doubly augmented basis sets  121   Figure 44. The second hyperpolarizability (xxxx) in atomic units versus  bond distance obtained using the MRCI and CASPT2 methods and the doubly augmented basis sets  122   Figure 45. The second hyperpolarizability (xxxx) in atomic units versus  bond distance obtained using the CASPT2 method and the singly and doubly augmented basis sets   123   Figure 46. The second hyperpolarizability (xxxx) in atomic units versus  bond distance obtained using the MRCI method and the singly and doubly augmented basis sets  124    Figure 47. The second hyperpolarizability (xxzz) in atomic units versus  bond distance obtained using the MRCI and CASPT2 methods and the  doubly augmented basis sets  125     Figure 48. The second hyperpolarizability (xxzz) in atomic units versus  bond distance obtained using the CASPT2 method and the singly and doubly augmented basis sets   126   Figure 49. The second hyperpolarizability (xxzz) in atomic units versus  bond distance obtained using the MRCI method and the singly and doubly augmented basis sets  127   Figure 50. The second hyperpolarizability (zzzz) in atomic units versus bond distance obtained using the MRCI and CASPT2 methods and the doubly augmented basis sets  128   Figure 51. The second hyperpolarizability (zzzz) in atomic units versus  bond distance obtained using the CASPT2 method and the singly and doubly augmented basis sets  129   Figure 52. The second hyperpolarizability (zzzz) in atomic units versus  bond distance obtained using the MRCI method and the singly and doubly augmented basis sets  130  3.6 Estimate of Complete Basis Set Limit  From the data gathered for each electronic property, we have estimated the CBS limit as follows:                            (3.16)  In this equation, we have found the average of the property calculated with the aug-cc-pV5Z basis set and the corresponding value calculated with the d-aug-cc-pV5Z basis set within a given method. We then find the difference between the values in order to assign an error to this value. This method of calculating the CBS limit operates on the assumption that the two basis sets should theoretically give the same limit within a given method.  Therefore, the CBS limit should lie somewhere between the two values calculated with each basis set. Figure (53) shows the values calculated for the CBS limit for the Bx,z,xz tensor, using the MRCI method. For points where the error between the two basis sets is larger, such as for the value at the bond length of 1.8 bohr in Figure (53), the value calculated using the d-aug-ccc-pV5Z basis set is probably the better one. We base this assumption on the behavior of these basis sets given in Figure (54). 131    Figure 53. CBS limit for the dipole-dipole-quadrupole tensor (Bx,z,xz), with assigned error  -200.000 -180.000 -160.000 -140.000 -120.000 -100.000 -80.000 -60.000 -40.000 -20.000 0.000 1.5 1.7 1.9 2.1 2.3 2.5 2.7 2.9 3.1 3.3 3.5 O-O bond length (bohr) Bx,z,xz 132   Figure 54. Convergence behavior of aug-cc-pVXZ and d-aug-cc-pVXZ basis sets, as illustrated for values of the dipole-dipole-quadrupole tensor, Bx,z,xz at the equilibrium O2 bond distance. The x-axis has values of the cardinal number (X), given by X-1.   In Figure (54), it is apparent that the d-aug-cc-pVXZ basis set series converges to the X=5 level much more quickly than the aug-cc-pVXZ basis set. Indeed, there is very little change between the d-aug-cc-pVDZ and the d-aug-cc-pV5Z basis sets. Therefore, the value calculated with the d-aug-cc-pV5Z basis set should be used whenever larger differences are observed between the singly and doubly augmented basis sets.        Tables (33) and (34) give the CBS limit estimated values for the MRCI method, along with their calculated errors. Tables (35) and (36) give the same, calculated using the CASPT2 values.  -120 -110 -100 -90 -80 -70 0 1 2 3 4 X-1 Bx,z,xz CASPT2 XZ CASPT2 d-XZ  133  Table 33.  CBS Limit Estimated Values (B-tensors and C-tensors) Obtained with the MRCI Method O-O (bohr) -Bx,z,xz Bx,x,zz -Bx,x,xx -Bz,z,zz Cxx,xx Czz,zz Cxz,xz 1.8  167.109±14.174 17.880±14.219 73.075±11.856 116.452±1.552 11.700±0.342 17.078±0.169 19.195±0.646 1.9 129.605±8.993 25.801±13.637 69.693±10.026 109.354±1.859 11.650±0.289 17.665±0.132 18.142±0.435 2.0 111.245±6.487 33.265±15.890 67.976±8.801 110.171±0.691 11.849±0.268 18.510±0.118 17.863±0.319 2.1 102.483±5.631 38.645±17.490 69.021±8.294 113.829±0.292 12.203±0.267 19.549±0.115 18.051±0.249 2.2 100.719±4.159 44.910±17.832 73.444±8.535 119.256±0.030 12.661±0.279 20.748±0.120 18.547±0.206 2.28187 102.143±3.413 49.185±18.305 76.729±8.193 124.832±0.349 13.093±0.294 21.837±0.129 19.119±0.184 2.296 102.552±3.353 49.536±18.682 77.010±8.537 125.882±0.410 13.171±0.298 22.035±0.130 19.231±0.181 2.36 105.002±3.203 53.309±18.545 80.568±8.052 130.844±0.705 13.541±0.314 22.960±0.139 19.775±0.171 2.457 110.439±3.287 57.664±18.376 85.861±8.217 138.820±1.099 14.135±0.340 24.461±0.155 20.711±0.161 2.646 124.812±4.006 63.424±15.831 96.206±8.522 154.219±1.421 15.376±0.399 27.696±0.194 22.842±0.158 2.929 148.070±4.711 69.972±11.427 109.991±8.732 176.311±1.285 17.332±0.484 33.207±0.261 26.592±0.175 3.213 161.305±7.418 69.756±4.008 117.431±6.919 181.094±4.973 19.281±0.549 39.377±0.320 30.822±0.208 3.3 165.074±5.917 70.929±4.126 119.439±6.836 178.633±7.143 19.858±0.562 41.365±0.337 32.187±0.218  Table 34. CBS Limit Estimated Values (polarizabilities, quadrupole moments, hexadecapole moments and E-tensors) obtained with the MRCI method O-O (bohr) xx zz zz zzzz -Ex,xxx Ez,zzz 1.8 7.302±0.027 11.399±0.025 -0.983±0.001 5.345±0.013 18.541±0.307 4.178±0.560 1.9 7.266±0.022 11.706±0.014 -0.834±0.001 4.649±0.018 17.068±0.204 8.526±0.420 2.0 7.340±0.019 12.316±0.011 -0.684±0.001 4.265±0.024 16.448±0.149 11.909±0.346 2.1 7.477±0.017 13.126±0.008 -0.534±0.001 4.118±0.029 16.367±0.117 14.949±0.305 2.2 7.654±0.014 14.070±0.007 -0.384±0.001 4.164±0.033 16.687±0.100 17.966±0.282 2.28187 7.812±0.015 14.913±0.007 -0.261±0.001 4.325±0.036 17.199±0.068 20.547±0.272 2.296 7.840±0.015 15.063±0.007 -0.240±0.001 4.363±0.036 17.306±0.063 21.008±0.271 2.36 7.970±0.014 15.757±0.007 -0.145±0.001 4.572±0.038 17.858±0.044 23.164±0.267 2.457 8.167±0.013 16.842±0.007 0.003±0.001 4.991±0.040 18.879±0.020 26.675±0.265 2.646 8.537±0.013 18.989±0.007 0.259±0.001 6.122±0.044 21.476±0.008 34.526±0.271 2.929 9.010±0.016 21.837±0.009 0.598±0.001 8.447±0.047 26.749±0.031 49.529±0.289 3.213 9.350±0.018 23.189±0.016 0.853±0.001 11.360±0.046 33.594±0.019 69.923±0.314 3.3 9.425±0.018 23.140±0.004 0.911±0.001 12.346±0.045 35.996±0.008 77.267±0.355 134   Table 35. CBS Limit Estimated Values (B-tensors and C-tensors) obtained with the CASPT2 method O-O (bohr) -Bx,z,xz Bx,x,zz -Bx,x,xx -Bz,z,zz Cxx,xx Czz,zz Cxz,xz 1.8 183.195±15.756 11.664±1.680 77.356±7.178 125.469±12.397 12.112±0.382 17.633±0.194 20.486±0.735 1.9 140.849±9.400 19.973±1.131 72.853±5.235 115.827±7.772 12.027±0.323 18.192±0.153 19.428±0.491 2.0 120.579±6.316 27.346±0.989 72.512±4.094 113.153±5.525 12.215±0.300 19.035±0.137 19.250±0.355 2.1 111.691±4.662 34.067±0.997 75.119±3.776 115.050±4.379 12.573±0.300 20.087±0.135 19.602±0.277 2.2 109.176±3.735 39.543±1.145 79.047±3.796 120.102±3.812 13.046±0.314 21.312±0.142 20.296±0.229 2.28187 110.601±3.294 44.137±1.301 83.355±4.085 125.643±3.607 13.494±0.332 22.429±0.152 21.039±0.206 2.296 111.028±3.245 44.823±1.196 84.161±4.137 126.698±3.584 13.576±0.336 22.631±0.154 21.179±0.202 2.36 113.624±3.061 48.254±1.484 87.901±4.288 131.855±3.581 13.961±0.354 23.583±0.165 21.853±0.192 2.457 119.279±2.921 52.850±1.774 94.491±5.658 140.374±3.682 14.582±0.384 25.127±0.184 22.971±0.182 2.646 133.903±2.974 60.566±2.471 107.164±5.905 159.213±4.047 15.879±0.451 28.460±0.229 25.374±0.181 2.929 158.381±3.460 70.103±3.564 121.561±7.676 178.153±5.104 17.917±0.548 34.131±0.307 29.260±0.204 3.213 173.121±4.095 75.920±4.130 130.935±9.083 179.067±6.324 19.933±0.620 40.463±0.380 33.319±0.242 3.3 174.079±4.165 76.022±4.433 132.262±9.381 174.362±6.114 20.526±0.635 42.499±0.400 34.587±0.256                 135  Table 36. CBS Limit Estimated Values  (polarizabilities, quadrupole moments, hexadecapole moments and E-tensors) obtained with the CASPT2 method O-O (bohr) xx zz zz zzzz -Ex,xxx Ez,zzz 1.8 7.500±0.025 11.664±0.033 -0.990±0.001 5.509±0.015 19.406±0.363 3.677±0.624 1.9 7.446±0.015 11.929±0.021 -0.837±0.001 4.750±0.020 17.713±0.237 8.349±0.464 2.0 7.515±0.010 12.527±0.015 -0.684±0.001 4.325±0.026 16.976±0.161 11.908±0.377 2.1 7.655±0.008 13.345±0.012 -0.531±0.001 4.151±0.031 16.838±0.112 15.035±0.329 2.2 7.832±0.007 14.314±0.010 -0.377±0.001 4.179±0.035 17.107±0.078 18.076±0.303 2.28187 7.996±0.006 15.188±0.010 -0.251±0.001 4.333±0.038 17.581±0.058 20.639±0.292 2.296 8.026±0.006 15.344±0.010 -0.229±0.001 4.370±0.038 17.684±0.055 21.093±0.290 2.36 8.163±0.007 16.072±0.010 -0.132±0.001 4.576±0.040 18.229±0.042 23.208±0.287 2.457 8.368±0.007 17.215±0.009 0.013±0.001 4.997±0.042 19.230±0.027 26.621±0.286 2.646 8.743±0.010 19.495±0.009 0.282±0.001 6.146±0.046 21.865±0.004 34.173±0.297 2.929 9.251±0.015 22.480±0.010 0.627±0.001 8.527±0.049 27.297±0.015 48.832±0.328 3.213 9.607±0.018 23.768±0.010 0.884±0.001 11.515±0.047 34.384±0.012 69.616±0.363 3.3 9.686±0.019 23.672±0.010 0.941±0.001 12.525±0.046 36.880±0.007 77.327±0.374 136  3.7 Rovibrational Averages  As we stated at the beginning of the last section, the purpose of obtaining the electric properties of the O2 molecule at non-equilibrium values is for the use in calculating the averages of the properties at the first six vibrational levels(=0-5), in the rotational ground state (J=0). We used our MRCI potential energy curve and the Numerov-Cooley129,130 method to first obtain the rovibrational wavefunction, (R). Our grid spanned the region of 1.5 to 5.5 bohr and contained 3000 points. We then integrated numerically over the internuclear distance R to obtain each property P as  (R) |P(R)| (R) . The averages thus obtained are shown in the following tables.   137   Table 37. Vibrational averages obtained for the MRCI properties, at the d-aug-cc-pV5Z basis set level, unless otherwise noted.    xx zz -zz zzzz -Ex,xxx Ez,zzz xxzz* xxxx zzzz r=r0 7.8451 15.0188 0.2478 4.3140 17.3338 21.1222 361.1860 519.8290 763.3750 0 7.8473 15.0335 0.2483 4.3494 17.4011 21.1820 365.5860 523.8240 774.8550 1 7.8885 15.2640 0.2211 4.4721 17.6725 21.9396 373.6440 538.5700 804.6570 2 7.9295 15.4958 0.1940 4.5998 17.9557 22.7229 381.7650 552.8010 838.4130 3 7.9704 15.7288 0.1669 4.7327 18.2513 23.5354 389.6740 566.5900 877.0130 4 8.0113 15.9639 0.1397 4.8713 18.5604 24.3831 397.1410 580.0360 921.4350 5 8.0522 16.2009 0.1122 5.0156 18.8835 25.2692 404.0280 593.2320 972.6930 *Using properties obtained with the d-aug-cc-pVQZ basis set Table 38. Vibrational averages obtained for the MRCI properties, at the d-aug-cc-pV5Z basis set level   Bx,z,xz Bx,x,zz Bx,x,xx Bz,z,zz Cxx,xx Czz,zz Cxz,xz r=r0 -105.7740 68.0230 -85.7960 -125.9130 13.4413 22.0970 19.3750 0 -106.5965 68.0267 -85.4824 -126.3420 13.4563 22.1296 19.4201 1 -108.5077 68.4101 -86.7799 -128.0340 13.5951 22.4641 19.6518 2 -110.5015 68.7584 -88.0485 -129.7910 13.7360 22.8066 19.8898 3 -112.5631 69.0784 -89.2946 -131.6020 13.8792 23.1578 20.1344 4 -114.6779 69.3803 -90.5280 -133.4630 14.0253 23.5196 20.3863 5 -116.8290 69.6675 -91.7536 -135.3590 14.1743 23.8925 20.6458            138  Table 39. Vibrational averages obtained for the CASPT2 properties, at the d-aug-cc-pV5Z basis set level, unless otherwise noted.    xx zz zz zzzz Ex,xxx Ez,zzz xxzz* xxxx zzzz r=r0 7.6278 15.4064 -0.2232 4.3444 -17.7727 21.5352 -351.760 -585.777 -769.817 0 8.0265 15.3188 -0.2382 4.3558 -17.7796 21.2784 -356.507 -588.686 -777.03 1 8.0692 15.5649 -0.2103 4.4807 -18.0554 22.0029 -366.168 -604.538 -800.22 2 8.1117 15.8124 -0.1824 4.6110 -18.3442 22.7516 -375.824 -619.604 -829.849 3 8.1540 16.0612 -0.1546 4.7466 -18.6463 23.5282 -385.481 -634.017 -867.172 4 8.1965 16.3121 -0.1267 4.8881 -18.9627 24.3394 -395.148 -647.944 -913.489 5 8.2390 16.5646 -0.0986 5.0357 -19.2941 25.1888 -404.847 -661.55 -970.149 *Obtained using the aug-cc-pVQZ basis set  Table 40. Vibrational averages obtained for the CASPT2 properties, at the d-aug-cc-pV5Z basis set level   Bx,z,xz Bx,x,zz Bx,x,xx Bz,z,zz Cxx,xx Czz,zz Cxz,xz r=r0 -114.3830 46.5130 -88.5480 -130.6210 13.9404 22.8544 21.4273 0 -114.9325 45.8690 -88.3302 -130.2810 13.8987 22.7504 21.3782 1 -116.8292 46.7178 -89.9921 -132.5340 14.0450 23.0961 21.6446 2 -118.8062 47.5639 -91.6356 -134.7980 14.1934 23.4501 21.9156 3 -120.8533 48.4096 -93.2589 -137.0600 14.3441 23.8129 22.1913 4 -122.9601 49.2609 -94.8642 -139.3110 14.4978 24.1867 22.4728 5 -125.1133 50.1181 -96.4479 -141.5370 14.6546 24.5720 22.7602      139  Often, the vibrationally averaged bond distance (r0) is used in ab initio calculations to obtain the expectation value of a given property at the =0 vibrational energy level. The Numerov-Cooley method is a good method for evaluating how good of an approximation this procedure is.  We have calculated the properties in the above tables ab initio at the vibrationally averaged  bond distance (r=r0)for each method, and compared them with the averages at =0 obtained with the Numerov-Cooley method. The average properties for the MRCI method at =0 (Tables 3 and 4)  are less than 1% different from the ab initio properties obtained at r0=2.2912 bohr. The properties for the CASPT2 method at =0 (tables 5 and 6) are less than 1.4% different than the properties obtained at r0=2.3007 bohr. Therefore, at least for the O2 molecule and the first five vibrational levels, using the vibrationally averaged bond distance to obtain these properties is an excellent approximation.  3.8 Conclusion We have obtained values for the dipole polarizability (), quadrupole moment (zz ), hexadecapole (zzzz) moment,  dipole-octopole (), quadrupole-quadrupole (), and dipole-dipole-quadrupole () polarizabilities and second hyper-polarizabilities () for the oxygen molecule O2 using both the MRCI and CASPT2 methods. We have examined the effect on these properties of changing the cardinal number () in the aug-cc-PVXZ basis set, with  have also examined the effect of adding a second diffuse function to the aug-cc-pVXZ basis sets to result in the d-aug-cc-pVXZ basis sets.       We have also compared our properties at the equilibrium bond length with properties in the literature, and find good agreement. We have been unable to find values for comparison to our dipole-dipole-quadrupole () polarizabilities in the literature.  140       We have exhaustively examined the properties for their basis set sensitivity, and find that for the singly augmented basis set aug-cc-pVXZ, the components of the dipole polarizability () are quite insensitive to the choice of cardinal number X. However, the quadrupole moment (zz ), hexadecapole (zzzz) moment,  dipole-octopole (), quadrupole-quadrupole (), and dipole-dipole-quadrupole () polarizabilities and second hyper-polarizabilities () all displayed sensitivity to a change in cardinal number, with the second hyper-polarizability showing the greatest change in value when increasing the basis set size from the aug-cc-pVDZ to aug-cc-PV5Z level.       For the properties obtained with the doubly augmented basis set d-aug-cc-pVXZ we find that there is a large stabilization in values amongst cardinal number; that is, many of the properties do not vary much when the cardinal number is increased. Notably, although the second hyperpolarizability  was amongst the most sensitive to a change in cardinal number for the singly augmented series, for the doubly augmented series we see virtually no change in this property when the cardinal number is increased from X=D to X=5. We therefore agree with Bartolomei et al.114 when they state that "the use of polarized functions of high zeta order in the basis sets is important to describe accurately multipole moments especially those of high order..".       The quadrupole moment, hexadecapole moment, dipole-dipole-quadrupole polarizability, and some components of the quadrupole-quadrupole polarizability and second hyperpolarizability still show sensitivity to the cardinal number chosen for the doubly augmented basis set. Most often, the majority of the difference is found when the basis set is increased from the d-aug-cc-pVDZ to the d-aug-cc-pVTZ level. This separation was seen in many of the singly augmented properties as well. This strong effect of f  functions in the basis set has been attributed to the fact 141  orbitals are occupied in the reference state133.       We also compared the properties obtained at the MRCI and CASPT2 methods and examined them for any differences. We found that the dipole polarizabilities, quadrupole-quadrupole polarizabilities, quadrupole moment, and hexadecapole moment were in excellent agreement. Larger differences between methods were found for the dipole-dipole-quadrupole and dipole-octopole polarizabilities, and for the second hyperpolarizabilities.       In addition, we examined the properties for sensitivity to adding a second set of diffuse functions to the aug-cc-pV5Z basis set to form the d-aug-cc-pV5Z basis set. We conclude that the dipole polarizabilities, quadrupole-quadrupole polarizabilities, quadrupole moment and hexadecapole moment are essentially unchanged when adding a diffuse function. The largest changes are seen in the dipole-dipole-quadrupole polarizabilities, dipole-octupole polarizabilities, and second hyperpolarizabilities.142  CHAPTER 4: Long-Range Approximation for the Interaction Dipole Moments of the O2-O2 Supermolecule, for Collision-Induced Absorption Applications 4.1 Introduction: The Use of Electric Properties in the Long-Range Approximation In order to be useful for spectroscopic rototranslational line-shape analysis134-136, it is customary to cast collision-induced dipole moments into a symmetry-adapted spherical harmonic expansion as follows137-139,68,69:                                                     (4.1)  where  and  are unit vectors along the symmetry axes of molecules A and B,  is a unit vector along the intermolecular vector R between molecules A and B and  is a Clebsch-Gordan coefficient. The first step in calculating the dipole moments using the expansion is to determine the expansion coefficients . To calculate these expansion coefficients we first assume that the diatomic oxygen molecules are interacting at a sufficiently long distance apart as to consider overlap, exchange, and charge transfer effects to be negligible; therefore this model is only valid at long-range distances of the intermolecular separation R. The equations used include induction by permanent multipolar fields and their gradients137,138,140-143, dispersion144-147, and back-induction148 and are complete through order R-7 in the intermolecular separation R.       To calculate the expansion coefficients for the direct induction contribution to the collision-induced dipole moment, we have used the following expressions69,146,149:                                                                                                                          (4.2) 143                                                                                                               (4.3)       (4.4)                                                    (4.5)                                                                                                                   (4.6)                                                                                                                              (4.7) whereis the isotropic dipole polarizability of molecule B in the A-B collision pair, given by                                                                                                         (4.8)                  and  .  The coefficient   contains additionally the permanent quadrupole moment () and thus describes the quadrupolar induction through order R-4; and the coefficient  contains the permanent hexadecapole moment  and therefore describes the hexadecapolar induction through order R-6. Equation (4.4) contains the permanent quadrupole moment, and also the dipole-octopole polarizability . Similarly, equation (4.5) contains the permanent hexadecapole moment and again the dipole-octopole polarizability . The coefficients ,,,and  are given in the Ref. [69]. In this work, the superscripts A and B refer to molecules A and B, which in this work are an oxygen molecule at its equilibrium bond length, and an oxygen molecule with a bond length of 2-3.3 bohr, respectively. In equations (4.6) and (4.7), the operator interchanges molecules A and B, but does not change the direction of the intermolecular vector R.        Back-induction effects appear at order R-7. This is a classical effect that occurs when the quadrupolar field of molecule A polarizes its neighbor (molecule B), setting up a static reaction 144  field that acts back onto molecule A, producing a dipole69,150. The back-induction expansion coefficients for the collision-B=0 are calculated using the following:                                                             (4.9)                   (4.10)              (4.11)                                                   (4.12)  A=0 are given by the analog of equation (4.6).  All remaining back-induction terms can be obtained using the following general formula:               (4.13)  Coefficients a and b are given in Ref. [69].       Dispersion is the last effect that contributes to the collision-induced dipole at long range. This effect is purely quantum mechanical, and arises from the constantly fluctuating charge distributions of the interacting molecules82.  To evaluate the dispersion contribution to the dipole, the dipole-quadrupole hyperpolarizabilites () are required. as  functions of the imaginary frequency. However, the dispersion coefficients can be evaluated using static response tensors and dispersion energy coefficients , as demonstrated by Bohr and Hunt69, using the Constant Ratio approximation. The isotropic dispersion coefficient is given by: 145                                                    (4.14)     where                              (4.15)  The anisotropic dispersion coefficients are given approximately by                                                                      (4.16)                                                                              (4.17)                                                                                      (4.18)                                                   (4.19)                                                                          (4.20)   The formulas for ,,, ,and are calculated using components of the dipole-dipole-quadrupole, quadrupole-quadrupole, and dipole-octopole polarizabilities and are given explicitly in Ref [69].  The dispersion coefficients of Bartolomei et al 114 have been used in these expressions: , ,,, and. These coefficients were calculated by use of static properties obtained using  the MRCI method111 with a CASSCF151,152 reference wavefunction and formulas contained within reference [150].  Our results for the direct multipolar induction, back induction, and dispersion terms in the 146  collision-induced dipole coefficients    for O2-O2 are shown in Tables (41)-(52). We take molecule A to be an oxygen molecule with an equilibrium bond length of 2.28187 bohr, and molecule B with a non-equilibrium bond length ranging from 2-3.3 bohr. For the calculation of these terms we have used our static  multipole moments and polarizabilities obtained with the CASPT277,78 method, using the doubly augmented correlation consistent polarized valence quadruple zeta (d-aug-cc-pVQZ) basis set of Dunning76.                    147  Table 41. Direct multipolar induction, back-induction, and dispersion terms in the collision-induced dipole coefficients  for rB=2 bohr, at the CASPT2 level, with the d-aug-cc-pVQZ basis set     D0001 0 -22.9 217 D2021 0 -74.2 237 D2023 4.010 31.1 195 D4043 0 -11.4 -101 D2201 0 -0.801 0.273 D2211 0 -1.30 -0.683 D2221 0 -6.33 0.167 D2223 0.5065 3.24 -0.021 D2233 -2.261 20.9 -1.50 D2243 2.632 10.0 -1.07 D2245 -51.27 -36.1 1.83 D4221 0 -0.236 -0.0805 D4223 0 -0.032 -0.0110 D4233 0 0.150 0.0512 D4243 0 -0.326 -0.111 D4245 4.126 -0.265 -0.0904 D4255 16.27 1.34 0.458 D4265 1.572 -4.61 -1.57 D2421 0 0.642 0.110 D2423 0 0.087 0.0150 D2433 0 0.408 0.0702 D2443 0 0.887 0.153 D2445 -6.853 0.721 0.124 D2455 19.02 3.65 0.628 D2465 -17.76 12.6 2.16 D0221 0 197 -459 D0223 -12.35 -94.1 -93.0 D0443 0 24.5 137 D0445 100.9 0 0 D4045 -88.66 0 0       148  Table 42. Direct multipolar induction, back-induction, and dispersion terms in the collision-induced dipole coefficients  for rB=2.1, at the CASPT2 level, with the d-aug-cc-pVQZ basis set     D0001 0 -17.0 85.4 D2021 0 -77.3 243 D2023 4.168 33.0 173 D4043 0 -11.9 -101 D2201 0 -0.587 0.133 D2211 0 -1.23 -0.614 D2221 0 -4.24 0.0779 D2223 0.3305  2.16 -0.00808 D2233 -1.920 18.4 -1.23 D2243 1.717 6.87 -0.517 D2245 -33.50 -24.4 0.876 D4221 0 -0.268 -0.0805 D4223 0 -0.0364 -0.0110 D4233 0 0.170 0.0512 D4243 0 -0.370 -0.111 D4245 5.049 -0.301 -0.0904 D4255 16.64 1.52 0.458 D4265 8.113 -5.24 -1.57 D2421 0 0.565 0.0950 D2423 0 0.077 0.0129 D2433 0 0.360 0.0605 D2443 0 0.781 0.131 D2445 -6.616 0.635 0.107 D2455 17.84 3.22 0.541 D2465 -18.11 11.1 1.86 D0221 0 161 -350 D0223 -9.584 -75.7 -128 D0443 0 21.6 118 D0445 96.48 0 0 D4045 -92.16 0 0     149  Table 43. Direct multipolar induction, back-induction, and dispersion terms in the collision-induced dipole coefficients  for rB=2.2, at the CASPT2 level, with the d-aug-cc-pVQZ basis set     D0001 0 -8.86 19.1 D2021 0 -81.2 247 D2023 4.359 35.5 162 D4043 0 -12.4 -101 D2201 0 -0.301 0.0472 D2211 0 -1.13 -0.570 D2221 0 -1.99 0.0291 D2223 0.1498 1.00 -0.00370 D2233 -1.588 15.7 -1.07 D2243 0.7784 3.30 -0.185 D2245 -15.24 -11.5 0.318 D4221 0 -0.304 -0.0805 D4223 0 -0.0415 -0.0110 D4233 0 0.194 0.0512 D4243 0 -0.421 -0.111 D4245 6.090 -0.343 -0.0904 D4255 17.25 1.74 0.458 D4265 15.11 -5.97 -1.57 D2421 0 0.457 0.0850 D2423 0 0.0623 0.0116 D2433 0 0.291 0.0541 D2443 0 0.632 0.117 D2445 -6.7105 0.514 0.0955 D2455 17.57 2.60 0.484 D2465 -19.38 8.95 1.66 D0221 0 122 -284 D0223 -6.805 -55.7 -148 D0443 0 17.5 106 D0445 96.86 0 0 D4045 -96.39 0 0      150  Table 44. Direct multipolar induction, back-induction, and dispersion terms in the collision-induced dipole coefficients  for rB=2.28187, at the CASPT2 level, with the d-aug-cc-pVQZ basis set     D0001 0 0 0 D2021 0 -84.7 248 D2023 4.534 37.9 159 D4043 0 -12.9 -101 D2201 0 0 0 D2211 0 -1.00 -0.547 D2221 0 0 0 D2223 0 0 0 D2233 -1.322 13.4 -0.981 D2243 0 0 0 D2245 0 0 0 D4221 0 -0.338 -0.0805 D4223 0 -0.0460 -0.0110 D4233 0 0.215 0.0512 D4243 0 -0.467 -0.111 D4245 7.003 -0.380 -0.0904 D4255 17.89 1.93 0.458 D4265 21.06 -6.62 -1.57 D2421 0 0.338 0.0805 D2423 0 0.0460 0.0110 D2433 0 0.215 0.0512 D2443 0 0.467 0.111 D2445 -7.003 0.380 0.0904 D2455 17.89 1.93 0.458 D2465 -21.06 6.62 1.57 D0221 0 84.7 -248 D0223 -4.534 -37.9 -159 D0443 0 12.9 101 D0445 100.3 0 0 D4045 -100.3 0 0     151  Table 45. Direct multipolar induction, back-induction, and dispersion terms in the collision-induced dipole coefficients  for rB=2.296, at the CASPT2 level, with the d-aug-cc-pVQZ basis set     D0001 0 1.74 -1.66 D2021 0 -85.4 248 D2023 4.565 38.3 159 D4043 0 -13.0 -101 D2201 0 0.0584 -0.00610 D2211 0 -0.974 -0.544 D2221 0 0.357 -0.00245 D2223 -0.0259 -0.179 -0.000240 D2233 -1.277 13.0 -0.972 D2243 -0.1348 -0.606 0.0223 D2245 2.648 2.08 -0.0342 D4221 0 -0.344 -0.0805 D4223 0 -0.0468 -0.0110 D4233 0 0.219 0.0512 D4243 0 -0.476 -0.111 D4245 7.166 -0.387 -0.0904 D4255 18.01 1.96 0.458 D4265 22.10 -6.74 -1.57 D2421 0 0.315 0.0804 D2423 0 0.0428 0.0109 D2433 0 0.200 0.0511 D2443 0 0.435 0.111 D2445 -7.072 0.353 0.0903 D2455 17.99 1.79 0.457 D2465 -21.41 6.16 1.57 D0221 0 78.0 -243 D0223 -4.144 -34.6 -160 D0443 0 12.0 101 D0445 101.1 0 -1.66 D4045 -100.9 0 248     152  Table 46. Direct multipolar induction, back-induction, and dispersion terms in the collision-induced dipole coefficients  for rB=2.36, at the CASPT2 level, with the d-aug-cc-pVQZ basis set     D0001 0 10.4 -5.81 D2021 0 -88.4 248 D2023 4.710 40.5 158 D4043 0 -13.4 -101 D2201 0 0.348 -0.0196 D2211 0 -0.840 -0.537 D2221 0 2.02 -0.0283 D2223 -0.1434 -1.01 0.0104 D2233 -1.075 11.0 -0.915 D2243 -0.7449 -3.49 0.0959 D2245 14.70 11.9 -0.218 D4221 0 -0.372 -0.0805 D4223 0 -0.0506 -0.0110 D4233 0 0.237 0.0512 D4243 0 -0.514 -0.111 D4245 7.916 -0.418 -0.0904 D4255= 18.60 2.12 0.458 D4265 26.87 -7.28 -1.57 D2421 0 0.196 0.0783 D2423 0 0.0266 0.0107 D2433 0 0.125 0.0498 D2443 0 0.270 0.108 D2445 -7.445 0.220 0.0880 D2455 18.61 1.11 0.446 D2465 -23.17 3.83 1.53 D0221 0 46.4 -227 D0223 -2.385 -19.4 -167 D0443 0 7.49 98.6 D0445 105.8 0 0 D4045 -104.1 0 0     153  Table 47. Direct multipolar induction, back-induction, and dispersion terms in the collision-induced dipole coefficients  for rB=2.457, at the CASPT2 level, with the d-aug-cc-pVQZ basis set     D0001 0 26.1 13.3 D2021 0 -93.1 247 D2023 4.936 44.0 161 D4043 0 -14.1 -101 D2201 0 0.863 -0.0719 D2211 0 -0.583 -0.511 D2221 0 4.68 -0.0281 D2223 -0.3203 -2.32 -0.00318 D2233 -0.7790 7.87 -0.874 D2243 -1.664 -8.26 0.262 D2245 32.96 27.7 -0.399 D4221 0 -0.416 -0.0805 D4223 0 -0.0566 -0.0110 D4233 0 0.265 0.0512 D4243 0 -0.575 -0.111 D4245 9.091 -0.467 -0.0904 D4255 19.60 2.37 0.458 D4265 34.19 -8.14 -1.57 D2421 0 -0.0216 0.0791 D2423 0 -0.00293 0.0108 D2433 0 -0.0137 0.0504 D2443  -0.0298 0.109 D2445 -8.195 -0.0242 0.0889 D2455 19.96 -0.123 0.450 D2465 -26.48 -0.422 1.55 D0221 0 -5.84 -197 D0223 0.2349 5.26 -174 D0443 0 -0.824 100 D0445 115.5 0 0 D4045 -109.1 0 0     154  Table 48. Direct multipolar induction, back-induction, and dispersion terms in the collision-induced dipole coefficients  for rB=2.646, at the CASPT2 level, with the d-aug-cc-pVQZ basis set     D0001 0 64.8 57.3 D2021 0 -103 245 D2023 5.377 51.8 169 D4043 0 -15.3 -101 D2201 0 2.11 -0.0959 D2211 0 0.0901 -0.499 D2221 0 10.2 -0.0377 D2223 -0.6535 -5.01 -0.00414 D2233 -0.2474 1.18 -0.838 D2243 -3.395 -18.8 0.350 D2245 68.08 61.2 -0.533 D4221 0 -0.506 -0.0805 D4223 0 -0.0688 -0.0110 D4233 0 0.322 0.0512 D4243 0 -0.699 -0.111 D4245 11.44 -0.568 -0.0904 D4255 21.83 2.88 0.458 D4265 48.38 -9.89 -1.57 D2421 0 -0.566 0.0886 D2423 0 -0.077 0.0121 D2433 0 -0.361 0.0564 D2443 0 -0.783 0.122 D2445 -10.22 -0.636 0.0996 D2455 23.92 -3.22 0.504 D2465 -34.85 -11.1 1.73 D0221 0 -120 -180 D0223 5.081 58.0 -178 D0443 0 -21.7 113 D0445 142.2 0 0 D4045 -118.9 0 0     155  Table 49. Direct multipolar induction, back-induction, and dispersion terms in the collision-induced dipole coefficients  for rB=2.929, at the CASPT2 level, with the d-aug-cc-pVQZ basis set     D0001 0 134 124 D2021 0 -116 241 D2023 5.959 64.0 180 D4043 0 -17.0 -101 D2201 0 4.29 -0.103 D2211 0 1.38 -0.495 D2221 0 18.4 -0.0631 D2223 -1.083 -8.90 0.00775 D2233 0.4367 -9.46 -0.787 D2243 -5.629 -35.3 0.403 D2245 117.5 112 -0.693 D4221 0 -0.621 -0.0805 D4223 0 -0.0846 -0.0110 D4233 0 0.396 0.0512 D4243 0 -0.859 -0.111 D4245 14.48 -0.698 -0.0904 D4255 24.73 3.54 0.458 D4265 66.72 -12.2 -1.57 D2421 0 -1.55 0.129 D2423 0 -0.211 0.0176 D2433 0 -0.988 0.0822 D2443 0 -2.15 0.179 D2445 -14.40 -1.74 0.145 D2455 32.29 -8.84 0.736 D2465 -51.81 -30.4 2.53 D0221 0 -303 -167 D0223 11.33 134 -184 D0443 0 -59.4 168 D0445 197.7 0 0 D4045 -131.8 0 0     156  Table 50. Direct multipolar induction, back-induction, and dispersion terms in the collision-induced dipole coefficients  for rB=3.123, at the CASPT2 level, with the d-aug-cc-pVQZ basis set     D0001 0 188 168 D2021 0 -123 239 D2023 6.249 72.2 187 D4043 0 -17.8 -101 D2201 0 5.92 -0.114 D2211 0 2.48 -0.490 D2221 0 24.4 -0.0925 D2223 -1.370 -11.8 0.0210 D2233 1.028 -19.0 -0.725 D2243 -7.120 -47.6 0.473 D2245 161.6 149 -0.886 D4221 0 -0.665 -0.0805 D4223 0 -0.0904 -0.0110 D4233 0 0.423 0.0512 D4243 0 -0.918 -0.111 D4245 15.80 -0.747 -0.0904 D4255 24.85 3.78 0.458 D4265 76.82 -13.0 -1.57 D2421 0 -2.34 0.203 D2423 0 -0.318 0.0276 D2433 0 -1.49 0.129 D2443 0 -3.23 0.280 D2445 -19.72 -2.62 0.228 D2455 42.61 -13.3 1.15 D2465 -74.00 -45.8 3.97 D0221 0 -449 -151 D0223 15.96 205 -191 D0443 0 -89.5 275 D0445 267.7 0 0 D4045 -138.2 0 0     157  Table 51. Direct multipolar induction, back-induction, and dispersion terms in the collision-induced dipole coefficients  for rB=3.3, at the CASPT2 level, with the d-aug-cc-pVQZ basis set     D0001 0 196 172 D2021 0 -123 239 D2023 6.258 73.1 188 D4043 0 -17.9 -101 D2201 0 6.14 -0.124 D2211 0 2.67 -0.485 D2221 0 25.5 -0.104 D2223 -1.422 -12.3 0.0246 D2233 1.196 -21.3 -0.697 D2243 -7.387 -49.6 0.518 D2245 173.7 156 -0.980 D4221 0 -0.656 -0.0805 D4223 0 -0.089 -0.0110 D4233 0 0.418 0.0512 D4243 0 -0.907 -0.111 D4245 15.70 -0.737 -0.0904 D4255 24.02 3.74 0.458 D4265 77.62 -12.8 -1.57 D2421 0 -2.46 0.239 D2423 0 -0.335 0.0325 D2433 0 -1.57 0.152 D2443 0 -3.40 0.330 D2445 -21.54 -2.76 0.268 D2455 46.05 -14.0 1.36 D2465 -81.73 -48.1 4.68 D0221 0 -477 -141 D0223 17.00 218 -195 D0443 0 -94.1 332 D0445 291.5 0 0 D4045 -138.4 0 0    158  Bohr and Hunt69 have derived the following expression for the collision-induced dipole for two linear, centrosymmetric molecules in a T-shape configuration with , , , in terms of the coefficients                                                                                       (4.21)      We have derived corresponding expressions for two linear, centrosymmetric molecules for the X-shape, linear, and H-shape configurations as well, given by equations (4.22), (4.23), and (4.24), respectively:                                                                                          (4.22)                                                                                                                               (4.23)  159                                                                                                                                             (4.24)      Using these equations for the different geometries, we can calculate the contributions to the dipole due to quadrupolar induction, hexadecapolar induction, induction via the E-tensor mechanism, back-induction, and dispersion. For example, from equation (4.22), the quadrupolar induction contribution to the dipole moment through order R-4 for O2-O2 in the X-shape geometry depends on the coefficients   and  and is given by the following:                                                          (4.25)     The hexadecapolar induction contribution for the same geometry is calculated the following way:                                                                                                      (4.26)  In this equation,  is given by:                                                                                                    (4.27) where  and  are the parallel and perpendicular components of the polarizability for molecule B (O2 with non-equilibrium bond distance) and  is the hexadecapolar moment for molecule A (O2 with equilibrium bond distance), and  is given by:                                                                                                                               (4.28)         160  To find the induction contribution to the dipole moment from the E-tensor mechanism for O2-O2 in the X-shape geometry, equation (22) is used to find:                                                                                                                                  (4.29)   where  is given by                                                                                  (4.30)  and  and  are dipole-octupole polarizabilities for molecule A (O2 with an equilibrium bond distance), and  is the quadrupole moment for molecule B (O2 with an equilibrium or non- equilibrium bond distance).  The coefficient is given by the following:                                                                             (4.31)          and the coefficient  is found using:                                 (4.32)     The back-induction contribution to the dipole moment using equation (22) for O2-O2 in the X-shape geometry is found by:                                                                                                                              (4.33)                                                                                                                                                                                                                                                                       161  The dispersion contribution is calculated using the above equation and replacing the back-induction coefficients by dispersion coefficients.       From equation (4.23), the quadrupolar induction contribution through order R-4  to the dipole moment for O2-O2 in the linear geometry is given by:                                                             (4.34) and the hexadecapolar induction contribution is calculated using:                                                                                                             (4.35)                        Induction via the E-tensor is calculated using:                                                                                                                              (4.36) Also from equation (4.23), the back-induction contribution to the dipole moment for the linear geometry is:                                                                                                                  (4.37)          Again, the dispersion contribution for the linear geometry is found by using equation (4.23) and substituting the dispersion coefficients for the back-induction coefficients.        From equation (4.24), the quadrupolar induction contribution through order R-4  to the dipole moment for O2-O2 in the H-shape geometry is  162                                                         (4.38) and the hexadecapolar induction contribution is given by                                                                                                      (4.39)      The contribution to the dipole from  induction via the E-tensor for O2-O2 in the H-shape geometry can be obtained using:                                                                                                                               (4.40)      From equation (4.21), the quadrupolar induction contribution through order R-4 to the dipole moment for O2-O2 in the T-shape geometry is                                                            (4.41)                                 The hexadecapolar induction contribution can be found using:                                                                                                   (4.42) The contribution to the dipole moment via E-tensor induction for O2-O2 in the T-shape geometry is given by:                                                                                                                               (4.43)               163  Using equation (4.21), the back-induction contribution to the dipole moment for O2-O2 in the T-shape geometry is found using:                                                                                                            (4.44)  Again, we can find the dispersion contribution to the dipole moment by replacing the back-induction coefficients in (4.44) by dispersion coefficients.  4.2 Contributions to the Collision-Induced Dipole Moment from Hexadecapolar Induction, Quadrupolar Induction, E-tensor Induction, Back Induction and Dispersion for Four Main Geometries  The direct quadrupolar induction through order R-4,  direct hexadecapolar induction, E-tensor induction, back-induction and dispersion terms calculated using the above formulas are given in Tables (52)-(63). The terms have been calculated for oxygen molecule A at the equilibrium bond distance, interacting with a second oxygen molecule B with a bond distance between 2-3.3 bohr.  Table 52. Contributions to the long-range dipole moment, for four main O2-O2 orientations, with rA=req and rB =2 bohr rB =2 CASPT2 Quadrupolar induction Hexadecapolar induction E-tensor induction Back induction Dispersion X-shape -8.075 R-4 4.283 R-6 -19.436 R-6 -37.066 R-7 132.116 R-7 Linear -14.446 R-4 53.037 R-6 -117.393 R-6 -491.793 R-7 644.298 R-7 H-shape -8.075 R-4 4.283 R-6 67.081 R-6 12.104 R-7 132.958 R-7 T-shape 17.641 R-4 -50.600 R-6 28.175 R-6 373.737 R-7 65.686 R-7  164  From Table (52), we see that the quadrupolar induction contributes a negative term to the dipole moment for all O2-O2 configurations except T-shape when oxygen A is at an equilibrium bond length, and oxygen B has a bond distance of 2 bohr (shorter than equilibrium). The T-shaped configuration quadrupolar induction term is largest in magnitude, followed by the linear quadrupolar induction term, and the X-shape and H-shape terms are identical. The hexadecapolar induction term is positive for all orientations except the T-shape. The linear orientation hexadecapolar induction term is largest in magnitude, followed by the T-shape orientation term, and as with the quadrupolar induction, the hexadecapolar induction is the same for the X-shape and H-shape, and is an order of magnitude smaller than the T-shape and linear terms.       The next term in Table (52) comes from the E-tensor induction. This term is negative for the X-shape and linear orientations, and positive for the H-shape and T-shape orientations. The term for the linear orientation is largest in magnitude, followed by the H-shape term, then the T-shape term, and the X-shape orientation term is the smallest term from induction via the E-tensor.       Also from Table (52), we see that the dipole for the X-shape and linear orientations contains a negative contribution from back-induction as well, and the term is positive for the H-shape and T-shape orientations. The linear orientation receives the largest contribution from back-induction, and the T-shape contribution is the second largest, and is of the same magnitude if not sign. The H-shape and X-shape terms are an order of magnitude smaller.       The last contribution in Table (52) arises from dispersion. This term is positive for all four orientations, and by far largest for the linear orientation.  The X-shape and T-shape terms are less than 1% different, but not identical. The smallest contribution from dispersion occurs for the T-165  shape orientation, and this term is an order of magnitude smaller than that for the linear orientation.   Table 53. Contributions to the long-range dipole moment, for four main O2-O2 orientations, with rA=req and rB =2.1 bohr rB =2.1 CASPT2 Quadrupolar induction Hexadecapolar induction E-tensor Induction Back Induction Dispersion X-shape -5.244 R-4 0.301 R-6 -12.786 R-6 -26.632 R-7 42.035 R-7 Linear -9.380 R-4 23.777 R-6 -75.953 R-6 -338.136 R-7 284.389 R-7 H-shape -5.244 R-4 0.301 R-6 43.564 R-6 6.448 R-7 42.475 R-7 T-shape 15.519 R-4 -59.357 R-6 12.680 R-6 338.679 R-7 11.526 R-7  Table (53) shows the five long-range contributions to the collision-induced dipole moment, for an O2-O2 supermolecular complex with oxygen molecule A at an equilibrium bond distance, and oxygen molecule B  at a bond distance of 2.1 bohr. The terms possess the same signs as discussed for the terms in Table (52). The trends for the quadrupolar induction remain the same amongst the four orientations, but have all reduced slightly in magnitude. The hexadecapolar term is now largest in magnitude for the T-shape orientation, as the term for that orientation has grown in magnitude, while the term for the linear orientation has decreased in magnitude significantly.       The trends seen in Table (52) for the E-tensor contribution continue here. For the back-induction contribute however, the value of the contribution to the dipole in the linear and T-shape orientations are nearly identical, although opposite in sign. The contribution to the dipole via back-induction for the X-shape orientation is an order of magnitude smaller than for the linear and T-shape orientations, and the contribution to the dipole for the H-shape orientation is the smallest, being an order of magnitude smaller than the X-shape orientations term. Finally, the dispersion term is still the largest for the linear orientation.   166   Table 54. Contributions to the long-range dipole moment, for four main O2-O2 orientations, with rA=req and rB =2.2 bohr rB =2.2 CASPT2 Quadrupolar induction Hexadecapolar induction E-tensor induction Back induction Dispersion X-shape -2.36802 R-4 -0.86383 R-6 -5.84119 R-6 -13.3986 R-7 2.94658 R-7 Linear -4.23605 R-4 6.388039 R-6 -34.3306 R-6 -162.975 R-7 88.58555 R-7 H-shape -2.36802 R-4 -0.86383 R-6 19.73887 R-6 2.158335 R-7 3.080553 R-7 T-shape 13.44424 R-4 -63.4552 R-6 -3.94721 R-6 304.2219 R-7 -4.24005 R-7  Table (54) shows the contributions to the dipole for the four main O2-O2 configurations studied when oxygen A is at an equilibrium bond length, and oxygen B has a bond distance of 2.2 bohr. The sign and relative contributions for the quadrupolar induction remain the same as in Tables (52) and (53). However, at this distance, the hexadecapolar contributions to the collision-induced dipole for the X-shape and H-shape have become negative. The hexadecapolar contribution is still largest for the T-shape orientation.  For oxygen molecule B at 2.2 bohr, the induction via the E-tensor now contributes a negative term for the T-shape orientation.  The linear orientation still possesses the largest term for E-tensor induction. At this point, the term for back induction possessed by the T-shape orientation has overcome that of the linear orientation, but dispersion still contributes the most to the linear orientation out of the four orientations.  Table 55. Contributions to the long-range dipole moment, for four main O2-O2 orientations, with rA=req and rB =req rB =2.28187 CASPT2 Quadrupolar induction Hexadecapolar induction E-tensor induction Back induction Dispersion X-shape 0 0 0 0 0 Linear 0 0 0 0 0 H-shape 0 0 0 0 0 T-shape 11.781 R-4 -63.771 R-6 -18.199 R-6 275.929 R-7 10.433 R-7  167  Table (55) is a good check for the accuracy of our dipole equations; all terms for the contribution to the dipole for the  X-shape, linear, and H-shape orientations are zero. This is because these three orientations are centrosymmetric, and thus have no dipole moment when both oxygen molecules are at equilibrium. Only the T-shape orientation contains non-zero contributions. Note that the dispersion term for the T-shape orientation has switched signs here, becoming positive.   Table 56. Contributions to the long-range dipole moment, for four main O2-O2 orientations, with rA=req and rB =2.296 bohr rB =2.296 CASPT2 Quadrupolar induction Hexadecapolar induction E-tensor induction Back induction Dispersion X-shape 0.4078 R-4 0.316389 R-6 1.017649 R-6 2.56186 R-7 0.018696 R-7 Linear 0.729495 R-4 -0.40313 R-6 5.944907 R-6 30.01532 R-7 -10.4399 R-7 H-shape 0.4078 R-4 0.316389 R-6 -3.42286 R-6 -0.2384 R-7 -0.00116 R-7 T-shape 11.49593 R-4 -63.5292 R-6 -20.7022 R-6 271.0092 R-7 12.42856 R-7  Table (56) contains the terms for contributions to the dipole when oxygen B has a bond distance of 2.296 bohr, slightly longer than equilibrium. All contributions from the quadrupolar induction are now positive. The T-shape orientation continues to possess the largest quadrupolar induction term.  Concerning the hexadecapolar induction, the term for the linear orientation has now for the first time become negative, while the term for the H-shape and X-shape orientations have reverted back to positive values. The T-shape orientation continues to possess the largest contribution from hexadecapolar induction of the four geometries.       The terms arising via the E-tensor mechanism have switched sign for the X-shape and Linear orientations. These terms are now positive. The E-tensor contribution term for the H-shape orientation has switched sign as well, now becoming negative. The E-tensor contribution is now largest for the T-shape orientation, as the linear orientation term has decreased in magnitude.  168  The terms for the back induction have switched sign as well; the terms are now positive for the X-shape and linear orientations, and negative for the H-shape orientation. The T-shape contribution from back-induction remains positive and is still the largest of the four.   Concerning the dispersion contribution when oxygen molecule B is at 2.296 bohr, the contribution to the linear orientation has now become negative, as has the H-shape term. The T-shape term has now become larger in magnitude than the linear term.  Table 57. Contributions to the long-range dipole moment, for four main O2-O2 orientations, with rA=req and rB =2.36 bohr rB =2.36 CASPT2 Quadrupolar induction Hexadecapolar induction E-tensor Induction Back Induction Dispersion X-shape 2.25119 R-4 2.247441 R-6 5.655948 R-6 15.10757 R-7 2.263756 R-7 Linear 4.027051 R-4 0.018957 R-6 32.92205 R-6 172.979 R-7 -47.2249 R-7 H-shape 2.25119 R-4 2.247441 R-6 -18.9711 R-6 -0.78169 R-7 2.324716 R-7 T-shape 10.22455 R-4 -61.6178 R-6 -32.1975 R-6 248.9452 R-7 26.43155 R-7  Table (57) shows the contributions to the dipole for the four main O2-O2 orientations when oxygen molecule B has a bond distance of 2.36 bohr. The signs and trends for the quadrupolar induction are the same as for the data in Table (56). For the hexadecapolar induction, the term for the contribution to the linear orientation dipole has reverted to a positive value, having been negative only when oxygen molecule B had a bond length of 2.296 bohr. It will remain positive for the over the range of the rest of the bond lengths considered.   The hexadecapolar contribution to the T-shape orientation remains larger than the contribution for the other three orientations.  The largest contribution to the dipole via the E-tensor is now the term for the linear  orientation, although it is only about 2% larger in magnitude than the T-shape term (but opposite in sign).  169  The signs and trends for the back-induction term in Table (56) remain the same as in Table (55), with the largest term contributing to the T-shape orientation. The largest term from dispersion now contributes to the linear orientation, over-taking the T-shape orientation term. The dispersion contribution term for the H-shape orientation has once again become positive, and will remain so for the rest of the bond distances for oxygen molecule B examined. Table 58. Contributions to the long-range dipole moment, for four main O2-O2 orientations, with rA=req and rB =2.457 bohr rB =2.457 CASPT2 Quadrupolar induction Hexadecapolar induction E-tensor induction Back induction Dispersion X-shape 5.007147 R-4 6.83522 R-6 12.6871 R-6 37.05524 R-7 36.92338 R-7 Linear 8.957057 R-4 7.740126 R-6 73.80978 R-6 410.9558 R-7 -78.1674 R-7 H-shape 5.007147 R-4 6.83522 R-6 -42.5375 R-6 0.108028 R-7 36.68581 R-7 T-shape 8.346778 R-4 -55.7851 R-6 -49.8056 R-6 215.3483 R-7 78.0336 R-7  In Table (58), we can see that the quadrupolar induction term is close in value for the linear and T-shape orientations, with oxygen molecule B at a bond distance of r=2.457 bohr. However, the T-shape term is still about 7% larger.  The largest hexadecapolar induction contribution is still the term for the T-shape orientation.  The linear E-tensor contribution term has now over-taken that of the T-shape orientation in magnitude.       Concerning the back-induction contribution, the H-shape term has now once again become positive, and will remain so for all longer bond distances of oxygen molecule B considered. As with the E-tensor induction contribution, the linear back-induction contribution has now becoming larger for the linear orientation than for the T-shape orientation. As for the dispersion contribution, the largest term still contributes to the linear orientation, though it is only slightly larger in magnitude than the T-shape term.  170  Table 59. Contributions to the long-range dipole moment, for four main O2-O2 orientations, with rA=req and rB =2.646 rB =2.646 CASPT2 Quadrupolar induction Hexadecapolar induction E-tensor induction Back induction Dispersion X-shape 10.1259 R-4 20.99384 R-6 26.18792 R-6 89.09999 R-7 83.80728 R-7 Linear 18.11376 R-4 45.67368 R-6 152.6361 R-6 937.2741 R-7 -77.8011 R-7 H-shape 10.1259 R-4 20.99384 R-6 -87.9285 R-6 8.003046 R-7 83.65027 R-7 T-shape 4.913846 R-4 -34.9795 R-6 -84.3609 R-6 150.6621 R-7 144.1964 R-7  In Table (59), the bond distance for oxygen molecule B has been lengthened to 2.646 bohr. The largest quadrupolar induction term still contributes to the linear orientation dipole. In addition, the largest term for the hexadecapolar induction now contributes to the linear orientation dipole as well, replacing the T-shape orientation contribution.  The largest contribution via E-tensor induction continues to be for the linear orientation. As far as back-induction is concerned, the largest term by far is that for the linear orientation, which has more than doubled in going from rB=2.457 bohr to rB=2.646 bohr. The largest dispersion term now contributes to the T-shape orientation.  Table 60. Contributions to the long-range dipole moment, for four main O2-O2 orientations, with rA=req and rB =2.929 rB =2.929 CASPT2 Quadrupolar induction Hexadecapolar induction E-tensor induction Back induction Dispersion X-shape 16.73762 R-4 52.68705 R-6 44.88263 R-6 177.969 R-7 121.1966 R-7 Linear 29.94117 R-4 158.5104 R-6 266.4013 R-6 1780.611 R-7 -128.085 R-7 H-shape 16.73762 R-4 52.68705 R-6 -152.828 R-6 30.88505 R-7 121.833 R-7 T-shape 0.511506 R-4 15.69696 R-6 -134.069 R-6 61.11158 R-7 207.0611 R-7  The trends in Table (59) continue for the quadrupolar induction terms in Table (60), now for an oxygen molecule B bond distance of 2.929 bohr. The largest hexadecapolar induction term, E-171  tensor induction term, and back-induction term all contribute to the linear orientation. The largest contribution from dispersion now contributes to the T-shape orientation dipole.  Table 61. Contributions to the long-range dipole moment, for four main O2-O2 orientations, with rA=req and rB =3.213 rB =3.213 CASPT2 Quadrupolar induction Hexadecapolar induction E-tensor induction Back induction Dispersion X-shape 21.50182 R-4 94.97172 R-6 60.68108 R-6 245.7033 R-7 95.71674 R-7 Linear 38.46362 R-4 347.8871 R-6 374.6841 R-6 2430.554 R-7 -321.211 R-7 H-shape 21.50182 R-4 94.97172 R-6 -213.057 R-6 49.93977 R-7 97.68184 R-7 T-shape -2.9961 R-4 87.92899 R-6 -178.679 R-6 -16.1983 R-7 203.2541 R-7  The trends for the quadrupolar induction have continued in Table (61), with the largest term belonging to the linear orientation. The only difference is that the T-shape term is now negative and will be negative for the last bond distance next considered as well. The largest terms for the hexadecapolar, E-tensor, and back induction continue to be for the linear orientation. The largest dispersion contribution now belongs to the linear orientation as well. These trends continue in Table (62), when the bond distance for oxygen molecule B is extended to the longest bond distance considered, 3.3 bohr.  Table 62. Contributions to the long-range dipole moment, for four main O2-O2 orientations, with rA=req and rB =3.3 rB =3.3 CASPT2 Quadrupolar induction Hexadecapolar induction E-tensor induction Back induction Dispersion X-shape 22.51966 R-4 109.6372 R-6 64.81026 R-6 255.7391 R-7 60.74578 R-7 Linear 40.2844 R-4 422.0736 R-6 406.3436 R-6 2543.824 R-7 -451.641 R-7 H-shape 22.51966 R-4 109.6372 R-6 -230.288 R-6 51.38599 R-7 63.31649 R-7 T-shape -3.88434 R-4 113.7906 R-6 -191.011 R-6 -37.0266 R-7 174.7359 R-7  The above information is summarized in Table (63).   172  Table 63. Orientations with largest contribution from quadrupolar induction, hexadecapolar induction, E-tensor induction, back-induction and dispersion for rB=2-3.3 bohr Oxygen molecule B bond distance (rB) Quadrupolar induction Hexadecapolar induction E-tensor induction Back induction Dispersion 2 bohr T-shape Linear Linear Linear Linear 2.1 bohr T-shape T-shape Linear T-shape Linear 2.2 bohr T-shape T-shape Linear T-shape Linear 2.28187 bohr (eq) T-shape T-shape T-shape T-shape T-shape 2.296 bohr T-shape T-shape T-shape T-shape T-shape 2.36 bohr T-shape T-shape Linear T-shape Linear 2.457 bohr Linear T-shape Linear Linear Linear 2.646 bohr Linear Linear Linear Linear T-shape 2.929 bohr Linear Linear Linear Linear T-shape 3.213 bohr Linear Linear Linear Linear Linear 3.3 bohr Linear Linear Linear Linear Linear  4.3 Contributions to the Dipole Moment at an Intermolecular Distance of 10 bohr The intermolecular distance of R=10 bohr has been substituted into the terms in tables (51)-(61) to illustrate the relative magnitudes of the direct quadrupolar induction, direct hexadecapolar induction, E-tensor induction, back-induction, and dispersion contributions to the collision-induced dipole moment. This corresponds to a distance that is about 4 bohr outside of the H-shape and X-shape potential energy minimum, about 3 bohr outside the potential energy minimum for the T-shape orientation, and about 2 bohr outside of the potential energy minimum for the linear orientation of the O2-O2 supermolecule. Figure (55) shows the direct quadrupolar induction contribution for the four main geometries (T-shaped, Linear, H-shaped and X-shaped), versus the O2 bond length for the non-equilibrium oxygen molecule in the O2-O2 supermolecule. This contribution is by far the largest to the collision-induced dipole moment, accounting for  ~80% or more of the total dipole for all geometries and all at bond distances considered.  173   Figure 55. The direct quadrupolar induction contribution to the dipole for the four main geometries, at an intermolecular distance of  R=10 bohr.  From the figure, we can see that at distances less than equilibrium, the contribution to the dipole moment for O2-O2 in the T-shape is positive in value, with a negative slope, while the opposite is true for the other three geometries. The contribution to the dipole becomes zero at equilibrium for the X-shape, H-shape, and linear geometries, due to the fact that these arrangements are centrosymmetric. After this point the contribution becomes positive for these three geometries. The contribution for the T-shaped geometry is also still positive at this point, and does not become negative until we reach the two longest distances we considered. It is clear from the figure that the magnitude of this component is generally largest for the linear geometry. Also note that that quadrupolar induction contribution to the collision-induced dipole has the same value for the X-shape and the H-shape geometries.       The second largest contribution to the collision-induced dipole moment for the X-shaped orientation is the E-tensor induction, at ~2-3% of the dipole, except for at the three longest bond distances of oxygen B considered, in which the hexadecapolar induction becomes larger. For the 174  H-shaped orientation of the two oxygen molecules, the E-tensor is the second largest contribution for all bond distances, contributing 9-11% to the collision-induced dipole.  For the linear orientation, the E-tensor induction is the second largest contribution for all but the longest bond length of 3.3 bohr, where hexadecapolar induction contributes slightly more. The contribution to the dipole from E-tensor induction in this case is 7-8%.  For the T-shaped geometry, E-tensor induction is the second largest contribution as well, contributing ~3-7% to the dipole, until the non-equilibrium oxygen molecule in the collision pair becomes stretched past 2.5 bohr, at which point the hexadecapolar induction becomes larger. The E-tensor induction contribution to the dipole at an intermolecular distance of 10 bohr is shown in Figure (56). From the figure, we can see that the linear geometry displays the largest magnitude of this component, as was seen for quadrupolar induction.  Figure 56. E-tensor induction contribution to the collision-induced dipole moment for the four main geometries studied, at an intermolecular distance of 10 bohr.   175  Figure (57) shows the direct hexadecapolar contribution to the collision-induced dipole moment. For the linear orientation, hexadecapolar induction contributes between 0.004-4.4% to the dipole moment when the bond distance of oxygen molecule B is between 2 and 2.929 bohr. This makes it the fourth and fifth largest out of the five contributions to the dipole for the linear orientation. We can see clearly in the figure that as the bond is stretched to the longest two distances, the hexadecapolar contribution increases significantly. At these distances, the hexadecapolar induction contributes 7-8% to the dipole moment, making it the third largest contribution to the dipole at a bond length of 3.213 bohr, and the second largest contribution at the bond length of 3.3 bohr. In regards to the T-shaped geometry, the hexadecapolar induction is second only to the quadrupolar induction contribution for bond distances of rB=2-2.5 bohr. At longer distances, it decreases to become the third largest contribution.       For the H-shape orientation, hexadecapolar induction is the third largest contribution to the dipole, after quadrupolar induction and E-tensor induction, being 4-48% that of the E-tensor contribution in the range from 2.2-3.3 bohr. At distances shorter than 2.2 bohr, the hexadecapolar term is smaller, as seen in Figure (57), and the dispersion replaces hexadecapolar induction as the third largest contribution.      Note that as was seen with quadrupolar induction, the hexadecapolar contribution to the dipole for the X-shape and H-shape geometries has the same value. Therefore the hexadecapolar induction is overcome by dispersion at shorter distances for this orientation as well. As was also seen for the H-shape orientation, it is the third largest contribution for rB>2.2 bohr. However, at bond distances>2.646 bohr, the hexadecapolar induction becomes the second largest contribution 176  to the dipole for the X-shape orientation; this is because the E-tensor induction term does not become as large in magnitude at long distances as was seen for the H-shape (See Figure 56).     Figure 57. Direct hexadecapolar induction contribution to the collision-induced dipole moment, for the four main geometries studied, at an intermolecular distance of R=10 bohr.  Figure (58) shows the back-induction contribution to the dipole moment. It is apparent that again the magnitude of this contribution to the dipole moment is much larger for the linear geometry than for the other three geometries. For the T-shaped geometry, this contribution accounts for ~0.5-4% of the collision-induced dipole. For the linear, H-shaped, and X-shaped geometries, back-induction contributes ~3-5%, less than 1%, and ~0.5-1% to the dipole, respectively.  177   Figure 58. The back-induction contribution to the collision-induced dipole for the four main geometries studied, at an intermolecular distance of R=10 bohr.   The fifth and final contribution to the collision-induced dipole moment in the long-range approximation comes from the dispersion, Figure (59). It is interesting to note that although almost all of the terms in the collision-induced dipole equations for the H-shaped and X-shaped geometries have different coefficients, this contribution ends up being so similar in value for the two geometries. As with the previous four contributions, we also note that the magnitude of this contribution is largest for the linear geometry.  For the T-shaped geometry, dispersion accounts for up to ~5% of the collision-induced dipole, but for most bond distances it contributes far less than 1%. For the linear geometry, dispersion only accounts for ~0.4-3% for all bond distances studied, while for the H-shaped geometry, dispersion accounts for only 2% or less of the total dipole. The situation is very similar for the X-shaped geometry, with dispersion only accounting for at most ~2% of the dipole, though for most distances the contribution is under 1%.  178   Figure 59. The dispersion contribution to the collision-induced dipole moment, for the four main geometries considered, at an intermolecular distance of 10 bohr.   4.4 Discussion and Summary We have calculated the long-range z-component of the collision-induced dipole moments, using a previously derived equation for the T-shape orientation of two linear, centrosymmetric molecules8. In addition, we have now derived equations for the collision-induced dipole moment for two linear molecules in an H-shape, X-shape, and linear orientation. Using our static multipole moments and (hyper)polarizabilities that were obtained with the CASPT2 method and d-aug-cc-pVQZ basis set, we have calculated the necessary coefficients  for use in the dipole expressions. We have thus calculated the z-component of the collision-induced dipole moment, for comparison with our ab initio values of the dipole moment in the next chapter.    We have separated the dipole moment equations into contributions from direct quadrupolar induction through order R-4  in the intermolecular separation R, contributions from direct 179  hexadecapolar induction and induction via non-uniformities of the local field acting on each molecule (via the E-tensor mechanism) through order R-6, and contributions from back-induction and dispersion through order R-7. Table (72) shows that the largest terms from all five contributions to the collision-induced dipole occur for the linear and T-shaped orientations.       We have also evaluated these contributions to the collision-induced dipole numerically, for an intermolecular distance of 10 bohr. From this analysis, we can see that the direct quadrupolar induction contributes the most to the collision-induced dipole moment, accounting for 80% or more of the total dipole for all four orientations studied. The second largest contribution to the dipole moment comes from the E-tensor induction for all orientations except the T-shape, contributing about 2-11% of the dipole over the range of non-equilibrium bond distances for oxygen molecule B. Some exceptions occur as the bond of oxygen molecule B is stretched to the longest distances considered, where hexadecapolar induction becomes slightly larger than the E-tensor induction contribution.        For the H-shape and X-shape orientations, the hexadecapolar induction term is the third largest for most bond distances, except for at the same longer bond lengths mentioned above, where it becomes the second largest contribution, and at some distances shorter than equilibrium, where it is overcome by the dispersion terms.       For the linear orientation, the third largest contribution to the dipole at most bond lengths is the back-induction term, with 3-5% of the total dipole. Depending on the bond distance of oxygen molecule B, the smallest contributions come from the hexadecapolar induction and dispersion contribution.   180       For the T-shape orientation, hexadecapolar induction makes the second largest contribution, at ~2-7% of the total dipole, except for the longest three bond distances, at which the E-tensor term takes over, at 21-42% of the total collision-induced dipole. The fourth largest contribution comes from back-induction for rB=2-2.646 bohr, with the dispersion contributing the least in this region, at less than 1% of the total dipole in most cases. The situation is reversed for the bond distances in the region from 2.929-3.3 bohr, with back induction contributing the least there.                   181  CHAPTER 5: Ab Initio Versus Long-Range Approximation from the Dipole Moments 5.1 Introduction: Methodology Used in Obtaining Ab Initio Dipole Moments  The long-range approximation discussed in the last chapter has one large advantage. That is, the moments and polarizabilities required are found for the O2 monomer, and therefore complex and costly supermolecular calculations are avoided. However, the long-range method only takes into account the long-range effects acting on the molecule(s), as its name suggests. These include electrostatic, induction, and dispersion effects, which all vary with the intermolecular distance as . Short-range effects, which vary with the intermolecular distance as , are neglected. These effects include resonance interactions, exchange-repulsion effects, charge transfer effects, and penetration and damping effects82. Therefore, it would be helpful to know at what intermolecular distances the long-range method is valid, and at what intermolecular distances it begins to fail due to the neglect of the short-range effects. In this section, we will compare the z-components of the dipole moment calculated using ab initio methods, and those found using the long-range method. As noted in the beginning of the last chapter, electric multiple moments can be calculated by using the appropriate derivative of the energy E as follows:                     (5.1) As  was done in chapter 3, the finite field approach can be used to find the dipole moment, using the following derivative113:                                                         (5.2) 182  We will use a simple two-point model, which has an associated error of , as follows:                                                 (5.3) 5.2 Comparison of Ab Initio and Long-Range Dipole  Moments Over 4,380 points have been obtained ab initio, using the CASPT2 method and aug-cc-pVQZ basis set. Table (64) shows the geometries and symmetry point groups used to obtain this data.  Table 64.  Geometries Used in the Calculation of Ab Initio Dipole Moments --- Symmetry Point Group 0-0-0-0 D2h 90-0-0-0 C2v 90-0-90-0 D2h 90-0-90-90 C2v 30-0-0-0 Cs 60-0-0-0 Cs 105-0-60-60 C1 40-0-115-45 C1 20-0-115-45 C1 15-0-75-30 C1 15-0-95-10 C1 10-0-160-15 C1 20-0-145-25 C1 30-0-130-35 C1 30-0-60-0 Cs 30-0-90-0 Cs 35-0-65-80 C1 45-0-30-0 Cs 45-0-60-0 Cs 45-0-90-0 Cs 50-0-95-55 C1 60-0-80-65 C1 70-0-65-75 C1  We will now compare the dipole moments from the long-range and ab initio methods, beginning with the linear (0-0-0-0) geometry and proceeding through the above geometries, ending with the 183  70-0-65-75 conformation. In the interest of brevity, we will compare the dipole moments with monomer A having a bond distance fixed at equilibrium (2.28187 bohr), and monomer B having a bond distance of 2 bohr (the shortest bond length considered), 2.28187 bohr (equilibrium bond distance), and 3.3 bohr (the longest bond distance considered). Figure (60) details the long-range and CASPT2 dipole moments for the linear geometry, with monomer B fixed at a bond distance of 2 bohr.   Figure 60. Comparison of dipole moments as a function of intermolecular distance (R) for the linear geometry, with r=2 bohr for . The dipole moment has atomic units, ea0.   From Figure (60), we see that the dipole moments for the ab initio triplet, singlet, and quintet spin states, as well as the dipole calculated using the long-range method, are in good agreement for the four longest intermolecular distances (8,10,12, and 15 bohr). At an intermolecular distance of of 6 bohr, the four values begin to separate, with the ground spin state (the singlet) having the largest magnitude. The difference is even more pronounced as the molecules draw nearer to each other, at R=5 bohr. Neglect of the short range effects in the long-range dipole 184  gives this dipole the smallest magnitude at this distance. We also note that the long-range dipole most closely replicates the ab initio quintet spin state. Figure (61) makes the same comparison for the linear geometry's dipole moments, but with monomer B fixed at a bond distance of 3.3 bohr.   Figure 61. Comparison of dipole moments as a function of intermolecular distance (R) for the linear geometry, with r=3.3 bohr for  . The dipole moment has atomic units, ea0.  We note immediately that the z-component of the dipole moment has switched direction, becoming negative for all points save the quintet spin state dipole at R=5 bohr. As in Figure (60), the four values we compare are in good agreement until the molecules are at R=6 bohr, at which point we clearly see separation amongst them of a large magnitude, with differences of over 100%. With monomer B at this longest bond distance, the separation is more pronounced than for the case of monomer B at the shortest bond distance (Figure (60)). Again, the singlet state ab initio value is largest in magnitude, and  the long-range method value is the smallest. The long-185  range values most closely replicate the quintet spin state values, until a distance of R=5 bohr, where the triplet state is closer to the long-range value. We do not discuss values of the dipole moment for both monomers fixed at equilibrium, as in the linear geometry, the supermolecular complex is centrosymmetric, and therefore gives a dipole moment of zero.       Figure (62a) depicts the dipole moment as a function of intermolecular distance for the T-shaped (90-0-0-0) geometry.   Figure 62a. Comparison of dipole moments as a function of intermolecular distance (R) for the T-shaped geometry, with r=2 bohr for . The dipole moment has atomic units, ea0.   As for the linear geometry, Figure (62a) shows that there is good agreement between the four values until R=6 bohr. The singlet state again has the largest magnitude dipole moment, and the long-range values are closest to those of the quintet state. In Figure (62b), we see a close up of the dipole moment values from 8-15 bohr, where the long-range method can be seen to be valid.  186   Figure 62b. Close-up view of dipole moments as a function of intermolecular distance (R) for the T-shaped geometry, with r=2 bohr for . The dipole moment has atomic units, ea0.        Figure (63) shows the dipole moments obtained with both monomers fixed at equilibrium.   Figure 63. Comparison of dipole moments as a function of intermolecular distance (R) for the T-shaped geometry, with r=req for  . The dipole moment has atomic units, ea0. 187   We note that when both molecules are at the equilibrium bond distance,. the quintet state dipole moments are now largest in magnitude, and the long-range values again are closest to those of the quintet spin state dipole moments.  In Figure (64), we see the dipole moments for monomer B fixed at 3.3 bohr for the T-shaped geometry.   Figure 64. Comparison of dipole moments as a function of intermolecular distance (R) for the T-shaped geometry, with r=3.3 bohr for . The dipole moment has atomic units, ea0.        In Figure (64), the values of the dipole moment have again become mostly negative, as was the case for monomer B fixed at the shortest distance (r=2 bohr). The singlet state dipole moment is the largest in magnitude, and the long-range values are again closest to that of the quintet spin state values.  188       The next geometry considered is the H-shape geometry, with angles 90-0-90-0, and D2h point group symmetry. Figure (65a) shows the z-component of the dipole moment for this geometry when monomer B has its bond length constricted to a length of r=2 bohr. Here, the z-component for the ab initio singlet spin state is the largest in magnitude. The long-range dipole is much closer in magnitude to the triplet and quintet dpiole, which are nearly identical.    Figure 65a. Comparison of dipole moments as a function of intermolecular distance (R) for the H-shaped geometry, with r=2 bohr for . The dipole moment has atomic units, ea0.   Figure (65b) shows a zoomed in view of the dipole moments in Figure (65a),  from 8-15 bohr, where the long-range method is valid.  189   Figure 65b. Close-up view of dipole moments as a function of intermolecular distance (R) for the T-shaped geometry, with r=2 bohr for . The dipole moment has atomic units, ea0.  Figure (66) shows the z-component of the dipole moment for the H-shape geometry, with monomer B having a bond distance of r=3.3 bohr.   Figure 66. Comparison of dipole moments as a function of intermolecular distance (R) for the H-shaped geometry, with r=3.3 bohr for . The dipole moment has atomic units, ea0.  190  The values are in good agreement until an intermolecular distance of R=6 bohr. The singlet spin state again has the largest magnitude, and the long-range values are closest to the quintet spin state ab initio values.      Figure (67) compares the z-component of the dipole moments for the X-shape geometry, with the bond distance of monomer B fixed at 2 bohr.    Figure 67 . Comparison of dipole moments as a function of intermolecular distance (R) for the X-shaped geometry, with r=2 bohr for . The dipole moment has atomic units, ea0.   This geometry is different from the last three that we considered in that the largest magnitude dipole moment is the long-range value (apparent at 6 bohr and 5 bohr). However, the long-range values again most closely replicate the quintet spin state values, and  the values are in good agreement until R=6 bohr.  191       In Figure (68), we compare values for the same geometry but at a monomer B bond distance of 3.3 bohr. As in Figure (67), the long-range dipole has the largest magnitude, and is somewhat closer to the quintet spin state values than the other two multiplet states; however, the difference between the three ab initio spin state values is much less than the difference between the ab initio and long-range values.   Figure 68. Comparison of dipole moments as a function of intermolecular distance (R) for the X-shaped geometry, with r=3.3 bohr for . The dipole moment has atomic units, ea0.  The next geometry we will consider is that with angles 30-0-0-0. This is a similar orientation to the T-shape, but because of the reduction of the first angle to 30 degrees, the only symmetry plane it contains is the reflection plane. In Figure (69), we compare the dipole moments with monomer B at the shortest bond distance, r=2.1 bohr.  192   Figure 69. Comparison of dipole moments as a function of intermolecular distance (R) for  the 30-0-0-0 geometry, with r=2.1 bohr for . The dipole moment has atomic units, ea0.  The ab initio dipole moment calculated for the triplet spin state is the largest in magnitude at the intermolecular distance of nearest approach of the two monomers studied in this work (R=5 bohr).  Figure (70) shows the dipole moments for this geometry when monomer A is at the equilibrium bond length (r=2.28187 bohr), and the bond length of monomer B is very near this, at r=2.296 bohr. We see that at this geometry and with the monomers at/near equilibrium, the dipole of the quintet state is the largest in magnitude. The singlet state has the smallest dipoles of the three spin states, and thus the long-range dipole moments most closely model the singlet state data.   193   Figure 70. Comparison of dipole moments as a function of intermolecular distance (R) for  the 30-0-0-0 geometry, with r=2.296 bohr for . The dipole moment has atomic units, ea0.          194       Finally, Figure (71) compares the dipole moments with monomer B at the longest bond length considered.  Figure 71. Comparison of dipole moments as a function of intermolecular distance (R) for the 30-0-0-0 geometry, with r=3.3 bohr for . The dipole moment has atomic units, ea0.  The dipoles are in good agreement until the distance of R=6 bohr. All other previous trends hold here, with the long-range dipoles closest to those of the quintet spin state ab initio values, and the singlet state ab initio dipoles largest in magnitude.  In Figure (72), we compare the dipole moments for the geometry with angles 60-0-0-0 ("halfway" between the T-shaped and 30-0-0-0 geometries), at the shortest monomer B bond distance.  195   Figure 72. Comparison of dipole moments as a function of intermolecular distance (R) for the 60-0-0-0 geometry, with r=2 bohr for . The dipole moment has atomic units, ea0.  The previous trends in the data hold here again. We note that at this geometry, the dipoles have the same overall trends as those for the T-shape geometry, but at least for the ab initio values, are almost twice as large as the T-shape values.       In Figure (73), the z-component of the dipole moment versus the intermolecular distance is given for the 60-0-0-0 geometry with monomer A at the O2 equilibrium bond length (r=2.28187 bohr), and monomer B very near equilibrium, at r=2.296 bohr.  196   Figure 73. Comparison of dipole moments as a function of intermolecular distance (R) for the 60-0-0-0 geometry, with r=2.296 bohr for . The dipole moment has atomic units, ea0.    Figure 74. Comparison of dipole moments as a function of intermolecular distance (R) for the 60-0-0-0 geometry, with r=3.3 bohr for . The dipole moment has atomic units, ea0. 197    Figure (74) shows the dipole moments for the 60-0-0-0 conformation but with monomer B at the longest bond distance considered, 3.3 bohr. The dipoles are about five times larger than they were when monomer B was constricted below the equilibrium bond length at r=2 bohr.       We now turn to our first geometry that was obtained ab initio with a C1 symmetry point group (the point group containing only the identity element, E). The angles for this geometry are 105-0-60-60 and no higher symmetry can be imposed. Figure (75) shows the dipole moments obtained for this geometry with the bond distance of monomer B constricted to r=2 bohr.   Figure 75. Comparison of dipole moments as a function of intermolecular distance (R) for the 105-0-60-60 geometry, with r=2 bohr for . The dipole moment has atomic units, ea0.  For this geometry and bond distance, the previously observed trends hold, except the long-range value at R=6 bohr is closer to the value for the singlet and triplet states than the value for the 198  quintet state. The separation in values starts at R=8 bohr here, with more separation at 6 bohr and then the largest differences at the shortest distance (R=5), compared with  all previous data.     Figure 76. Comparison of dipole moments as a function of intermolecular distance (R) for the 105-0-60-60 geometry, with r=req for . The dipole moment has atomic units, ea0.  Figure (76) shows the dipole moments obtained for the geometry with angles 105-0-60-60 when both monomers are at the equilibrium bond length of r=2.28187 bohr. Here, the long-range dipole is actually larger in magnitude than the ab initio dipoles at near range. Out of the three dipoles calculated ab initio at R=5 bohr, the quintet state dipole is largest in magnitude.       In Figure (77), we compare the z-components of the dipole moment for the 105-0-60-60 geometry, but with monomer B at r=3.3 bohr. For this distance, the values at R=8 bohr are in better agreement than they were with monomer B constricted to r=2 bohr, and there is not much 199  separation until R=6 bohr. One difference from previous trends here is that the ab initio values are closer to that of the triplet state values than to the quintet state values, and this is most clearly seen at R=5 bohr.    Figure 77. Comparison of dipole moments as a function of intermolecular distance (R) for the 105-0-60-60 geometry, with r=3.3 bohr for . The dipole moment has atomic units, ea0.  The next geometry we examine is that with angles 40-0-115-45. This again is a C1 point group symmetry arrangement. Figure (78) shows the dipoles at this geometry with monomer B at the most constricted bond length.  200   Figure 78 . Comparison of dipole moments as a function of intermolecular distance (R) for the 40-0-115-45 geometry, with r=2 bohr for . The dipole moment has atomic units, ea0.  In Figure (78), we see that all previous observed trends are seen here again.       Figure (79) shows the data for this geometry with both monomers at equilibrium.  In this case, all three ab initio values are in good agreement for the entire curve, and the ab initio as well as long-range values show the same trends, curving downward (becoming more negative) the shorter R becomes, though the long-range values become less negative than the ab initio values. This is presumably due to the neglect of short range effects in the interaction.   201   Figure 79. Comparison of dipole moments as a function of intermolecular distance (R) for the 40-0-115-45 geometry, with r=req for . The dipole moment has atomic units, ea0.  Figure (80) shows the values for the dipoles at this geometry when the bond distance for monomer B is longest. Previous trends are seen here again, and there is no strange behavior at R=15 bohr as there was for the data with the constricted monomer B bond distance.   202   Figure 80. Comparison of dipole moments as a function of intermolecular distance (R) for the 40-0-115-45 geometry, with r=3.3 bohr for . The dipole moment has atomic units, ea0. The next geometry we compare values for has angles 20-0-115-45, similar to our last geometry. Figure (81) shows the dipoles obtained when monomer B is constricted to r=2 bohr.   Figure 81. Comparison of dipole moments as a function of intermolecular distance (R) for the 20-0-115-45 geometry, with r=2 bohr for . The dipole moment has atomic units, ea0.  203  For this geometry and these bond distances, the quintet state is largest in magnitude, and is about twice as large as the value for the dipole for the singlet state at R=5 bohr. The long-range dipole component is actually closer to that of the singlet and triplet state at intermolecular distances of 5 and 6 bohr, respectively.  The data for the dipole moment for this geometry with both monomers at their equilibrium bond lengths is seen in Figure (82).  Figure 82 . Comparison of dipole moments as a function of intermolecular distance (R) for the 20-0-115-45 geometry, with r=req for . The dipole moment has atomic units, ea0.  Here there is very little difference between the three ab initio values until an intermolecular distance of R=5 bohr. As seen for most of the previous data, the ab initio and long-range data are in good agreement for intermolecular distances of R=8,10,12, and 15 bohr.      The last figure for this geometry, with monomer B at the longest bond distance, is Figure (83). 204   Figure 83 . Comparison of dipole moments as a function of intermolecular distance (R) for the 20-0-115-45 geometry, with r=3.3 bohr for . The dipole moment has atomic units, ea0.  It can be seen from this figure that the ab initio values become steeply negative when the molecules are five bohr apart, while the long-range value changes very little.  The next geometry discussed is again of C1 symmetry, with angles 15-0-75-30. Figure (84) shows the dipole moments when monomer B is most constricted.   205   Figure 84. Comparison of dipole moments as a function of intermolecular distance (R) for the 15-0-75-30 geometry, with r=2 bohr for . The dipole moment has atomic units, ea0.  These data show no marked deviation from the overwhelming trends, with the singlet spin state having the largest dipole moment.       In Figure (85), we view the dipole moments for this geometry with both monomers at equilibrium. Again, there is no deviation from previous trends.  206   Figure 85.  Comparison of dipole moments as a function of intermolecular distance (R) for the 15-0-75-30 geometry, with r=req  for . The dipole moment has atomic units, ea0.  Finally, we discuss the data for this geometry with monomer B at r=3.3 bohr in Figure (86).  Figure 86. Comparison of dipole moments as a function of intermolecular distance (R) for the 15-0-75-30 geometry, with r=3.3 bohr for . The dipole moment has atomic units, ea0. 207  Here, there is a significant difference from past behavior seen for this geometry or any others. The ab initio triplet state data, though displaying the same overall behavior as the data for the other two spin states, is significantly separated from the other ab initio data and the long-range dipole, and is additionally largest in magnitude.  The next geometry keeps monomer A at the same angles (=15°, =0°), but changes the orientation of monomer B, for an overall geometry with angles 15-0-95-10. The data for monomer B at the most constricted bond length are shown in Figure (87).  Figure 87. Comparison of dipole moments as a function of intermolecular distance (R) for the 15-0-95-10 geometry, with r=2 bohr for . The dipole moment has atomic units, ea0.  This figure shows very little difference from previously observed trends.   Figure (88) shows the z-component of the dipole moment for this geometry when both monomers are at equilibrium.  208   Figure 88. Comparison of dipole moments as a function of intermolecular distance (R) for the 15-0-95-10 geometry, with r=req for . The dipole moment has atomic units, ea0.  This figure shows that the long-range dipole is closer to the singlet and triplet state dipoles at R=6 bohr than to the quintet state dipole. When the molecules become closer at R=5 bohr, the long-range dipole reverts to the commonly observed behavior of most closely replicating the quintet state dipole moment. In Figure (89), we observe the dipole moment when monomer B is at r=3.3 bohr. These data show the same trends as seen previously for other geometries.  209   Figure 89. Comparison of dipole moments as a function of intermolecular distance (R) for the 15-0-95-10 geometry, with r=3.3 bohr for . The dipole moment has atomic units, ea0. The next geometry we discuss is another C1 symmetry point group conformation, with angles 10-0-160-15. Figure (90) shows the z-components of the dipole moment for this geometry, with monomer B constricted shorter than equilibrium to r=2 bohr.  Figure (90). Comparison of dipole moments as a function of intermolecular distance (R) for the 10-0-160-15 geometry, with r=2 bohr for . The dipole moment has atomic units, ea0.  210  This geometry possesses dipole moments that are larger than those for the last few geometries discussed.  Figure (91) shows the dipoles for the same geometry but with both monomers at their equilibrium bond distances.    Figure (91). Comparison of dipole moments as a function of intermolecular distance (R) for the 10-0-160-15 geometry, with r=req for . The dipole moment has atomic units, ea0.  Unlike most of the previous cases we have examined, for this geometry the quintet spin state possesses the largest magnitude of the z-component of the dipole moment. In Figure (92), we examine the z-component of the dipole moment for the geometry with angles 10-0-160-15, with the bond distance of monomer B at r=3.3 bohr.  211   Figure 92. Comparison of dipole moments as a function of intermolecular distance (R) for the 10-0-160-15 geometry, with r=3.3 bohr for . The dipole moment has atomic units, ea0.  As with Figure (86), which displays the dipole moment for a geometry with angles 15-0-75-30, and for monomer B at the bond distance of r=3.3 bohr, the triplet spin state dipole moment is well separated from the singlet and quintet spin state ab initio dipoles, as well as from the long-range dipole.  Figure (93) shows the z-component of the dipole moment for the geometry given by the angles 20-0-145-25, with monomer B's bond length at r=2 bohr.  212   Figure 93. Comparison of dipole moments as a function of intermolecular distance (R) for the 20-0-145-25 geometry, with r=2 bohr for . The dipole moment has atomic units, ea0.  Previous trends are observed here, with the singlet state having the largest magnitude of dipole moment, and the long-range data most closely resembling the quintet state ab initio dipole.       Figure (94) shows the same data but for both O2 molecules at the equilibrium bond distance of 2.28187 bohr. In this figure, we see that the ab initio quintet spin state dipole moment is now largest in magnitude. In addition, the long-range dipole moment most closely resembles the singlet state, both in magnitude and behavior.   213   Figure 94. Comparison of dipole moments as a function of intermolecular distance (R) for the 20-0-145-25 geometry, with r=req for . The dipole moment has atomic units, ea0.  The dipole moments for this geometry and the situation in which monomer A is at its equilibrium bond distance and monomer B is at a bond distance of 3.3 bohr are shown in Figure (95). As the two molecules approach, the difference between the ab initio and long-range data becomes more stark than in some previous geometries, with differences of over 100% between the dipole moments obtained from the two methods at an intermolecular distance of R=6 bohr.  214   Figure 95. Comparison of dipole moments as a function of intermolecular distance (R) for the 20-0-145-25 geometry, with r=3.3 bohr for . The dipole moment has atomic units, ea0.  Figure (96) shows the dipole moments calculated for the geometry with angles 30-0-130-35. Monomer B has been fixed at the constricted bond distance of 2 bohr. The ab initio dipoles calculated for the singlet spin state are largest in magnitude for this geometry and these bond lengths. As with most previous cases, the long-range approximation is valid up to an intermolecular separation of 8 bohr.       The next figure shows the z-component of the dipole moment for the same geometry, but with both monomers at the equilibrium bond distance, r=2.28187 bohr. When both monomers are at equilibrium, the ab initio dipole moment calculated for the quintet state is largest in magnitude.   215   Figure 96. Comparison of dipole moments as a function of intermolecular distance (R) for the 30-0-135-35 geometry, with r=2 bohr for . The dipole moment has atomic units, ea0.   Figure 97. Comparison of dipole moments as a function of intermolecular distance (R) for the 30-0-135-35 geometry, with r=req for . The dipole moment has atomic units, ea0.  216  Figure (98) shows the dipole moments for this geometry when monomer B has been stretched past equilibrium to the bond length of 3.3 bohr. The dipole moment calculated for the quintet spin state is slightly larger in magnitude than the magnitude of the dipole for the singlet and triplet states, though all three ab initio values are close in magnitude for this geometry and set of bond lengths.   Figure 98. Comparison of dipole moments as a function of intermolecular distance (R) for the 30-0-135-35 geometry, with r=3.3 bohr for . The dipole moment has atomic units, ea0.  We know turn to a geometry that contains one symmetry plane (the reflection plane) and thus has Cs point group symmetry. Figure (99) shows the z-components of the dipoles calculated for this geometry, which has angles of 30-0-60-0. In Figure (99), monomer B is constricted to 2 bohr. Previous trends are observed, with the ab initio dipoles calculated for the singlet spin state largest in magnitude.  217   Figure 99. Comparison of dipole moments as a function of intermolecular distance (R) for the 30-0-60-0 geometry, with r=2 bohr for . The dipole moment has atomic units, ea0.  Figure (100) shows the z-component of the dipole moment for the same geometry, but with both monomers at their equilibrium bond distances.   Figure 100. Comparison of dipole moments as a function of intermolecular distance (R) for the 30-0-60-0 geometry, with r=2.296 bohr for . The dipole moment has atomic units, ea0. 218  In Figure (101), we examine the dipole moments for the  geometry with angles 30-0-60-0, and for monomer B at the bond distance of 3.3 bohr. Previous trends are shown here as well, with the singlet state dipole largest in magnitude. However, the ab initio dipole moment for the triplet sin state is nearly as large as the singlet state dipole.   Figure 101. Comparison of dipole moments as a function of intermolecular distance (R) for the 30-0-60-0 geometry, with r=3.3 bohr for . The dipole moment has atomic units, ea0.  The next geometry examined is similar to that we just considered, and has the same point group symmetry. The angles of this geometry are 30-0-90-0. Figure (102) shows the z-component of the dipole moment for this geometry, with the bond distance of monomer B contracted to r=2 bohr. The dipole moments here are quite a bit smaller (about 80%) than the dipole moments for the last geometry.   219  Figure (103) shows the z-component of the dipole moment for this geometry when each monomer is fixed at its equilibrium bond distance.   Figure 102.  Comparison of dipole moments as a function of intermolecular distance (R) for the 30-0-90-0 geometry, with r=2 bohr for . The dipole moment has atomic units, ea0.   Figure 103. Comparison of dipole moments as a function of intermolecular distance (R) for the 30-0-90-0 geometry, with r=req for . The dipole moment has atomic units, ea0. 220  The final figure for this geometry is Figure (104), where we see the z-component of the dipole moments when monomer B is stretched past equilibrium to r=3.3 bohr. The dipole component here displays the same type of behavior as for the last geometry in Figure (101), but again the magnitude of this component is an order of magnitude smaller, due to monomer B being at a right angle to the intermolecular axis.   Figure 104. Comparison of dipole moments as a function of intermolecular distance (R) for the 30-0-90-0 geometry, with r=3.3 bohr for . The dipole moment has atomic units, ea0.  Our next geometry belongs to the C1 symmetry point group, containing only the identity element E. In Figure (105), the z-component of the dipole moment is displayed for this geometry, with monomer B fixed at a bond distance of r=2 bohr. This geometry displays the same trends we have observed previously, with the dipole component for the ab initio singlet state the largest in magnitude.   221   Figure 105. Comparison of dipole moments as a function of intermolecular distance (R) for the 35-0-65-80 geometry, with r=2 bohr for . The dipole moment has atomic units, ea0.  Figure (106) now examines this component of the dipole at this geometry with both monomers at equilibrium. The ab initio dipole is largest in magnitude for the quintet state.  Figure 106. Comparison of dipole moments as a function of intermolecular distance (R) for the 35-0-65-80 geometry, with r=req for . The dipole moment has atomic units, ea0. 222  Figure (107) displays the z-components of the dipole for this geometry, with the bond length of monomer B now stretched past equilibrium to 3.3 bohr.   Figure 107. Comparison of dipole moments as a function of intermolecular distance (R) for the 35-0-65-80 geometry, with r=3.3 bohr for . The dipole moment has atomic units, ea0.  The z-component of the dipole moment has now become about an order of magnitude larger than it was when both monomers were at equilibrium bond lengths.       The next three geometries we will consider are all of Cs symmetry, containing only the identity element E and one reflection plane. Figure (108) shows the z-components of the dipole for a geometry with angles 45-0-30-0, and with monomer B at the shortest contracted bond distance considered.  223   Figure 108. Comparison of dipole moments as a function of intermolecular distance (R) for the 45-0-30-0 geometry, with r=2 bohr for . The dipole moment has atomic units, ea0.  Previous trends are displayed in this figure. We also note that the long-range method underestimates the magnitude of the dipole moment at the smallest intermolecular distance of R=5 bohr by about a factor of four.       The z-component for this geometry when both monomers are fixed at the equilibrium bond lengths is shown in Figure (109). Here, the ab initio data predicts that the dipole is largest in magnitude for the quintet spin state. The long-range dipole is almost always smallest in magnitude, and therefore it is closer to the ab initio singlet state dipole here.  Again, the long-range dipole is about one fourth the value of the largest ab initio dipole here.   224   Figure 109. Comparison of dipole moments as a function of intermolecular distance (R) for the 45-0-30-0 geometry, with r=req for . The dipole moment has atomic units, ea0.  Finally, Figure (110) shows the z-component of the dipole when monomer B is stretched to a bond distance of r=3.3 bohr. Previous trends are observed for this set of dipoles, and their magnitude is about twice of those in the last two figures, for monomer B contracted or at equilibrium.  225   Figure 110. Comparison of dipole moments as a function of intermolecular distance (R) for the 45-0-30-0 geometry, with r=3.3 bohr for . The dipole moment has atomic units, ea0.  Figure (111) displays the behavior of the z-component of the dipole moment when the angle of the second monomer, with respect to the intermolecular axis, has been increased by 30 degrees, resulting in a geometry with angles 45-0-60-0. The data in the figure is for this geometry when monomer B is contracted to r=2 bohr.  226   Figure 111. Comparison of dipole moments as a function of intermolecular distance (R) for the 45-0-60-0 geometry, with r=2 bohr for . The dipole moment has atomic units, ea0.  From Figure (111), we see that increasing the angle theta of the second monomer has caused the z-component of the dipole moment to decrease by about a factor of four.  Figure (112) shows the dipole moments for this geometry when both monomers are at their equilibrium bond lengths.   Figure 112. Comparison of dipole moments as a function of intermolecular distance (R) for the 45-0-60-0 geometry, with r=req for . The dipole moment has atomic units, ea0. 227  When both monomers are at equilibrium, the z-component of the dipole moment for the quintet and triplet spin state have become smaller, while the z-component of the dipole moment for the singlet state has more than doubled in size when the distance between the monomers is R=5 bohr. Additionally, the singlet dipole is inconsistent with the data for the long-range dipole at an intermolecular separation of 12 bohr here.       In Figure (113), the z-component of the dipole for this geometry and with monomer B with a bond length of r=3.3 bohr is examined. Again, the singlet spin state dipole is the largest, but the triplet dipole has also increased by an order of magnitude and is nearly as large as the singlet dipole.    Figure 113. Comparison of dipole moments as a function of intermolecular distance (R) for the 45-0-60-0 geometry, with r=3.3 bohr for . The dipole moment has atomic units, ea0.  228  In constructing the next geometry, the angle of the second monomer with respect to the intermolecular axis has again been increased by 30 degrees, so that the angles for this geometry are 45-0-90-0 (the second monomer is now at a right angle with respect to the intermolecular axis). Figure (114) shows the z-component of the dipole moment when monomer B is contracted to its shortest distance for this geometry.   Figure 114. Comparison of dipole moments as a function of intermolecular distance (R) for the 45-0-90-0 geometry, with r=2 bohr for . The dipole moment has atomic units, ea0.  When the dipoles are at their largest in Figure (116), they are about 4 times as large as the dipoles were with a geometry of 45-0-60-0, when monomer B was at this contracted bond distance.       Figure (115) shows the z-components of the dipole for this geometry when the bond distance of both monomers is at equilibrium. It is unusual, when compared to previous geometries, that the long-range z-component of the dipole is actually the largest here in magnitude. That is followed by the quintet and triplet dipoles, with the singlet dipole the smallest in magnitude.  229   Figure 115. Comparison of dipole moments as a function of intermolecular distance (R) for the 45-0-90-0 geometry, with r=req for . The dipole moment has atomic units, ea0.  Finally, the z-component of the dipole moment is shown in Figure (116) for this geometry when the bond of monomer B is stretched to r=3.3 bohr.   Figure 116. Comparison of dipole moments as a function of intermolecular distance (R) for the 45-0-90-0 geometry, with r=3.3 bohr for . The dipole moment has atomic units, ea0. 230   In Figure (116), the dipole calculated with the long-range method and the ab initio dipole for the singlet spin state are roughly equal in magnitude, but opposite in sign.       The next geometry for which we compare the ab initio and long-range dipoles is the geometry with angles 50-0-95-55 (of the C1 symmetry point group). Figure (117) shows the z-component of the dipole when monomer B is contracted to r=2 bohr.    Figure 117. Comparison of dipole moments as a function of intermolecular distance (R) for the 50-0-95-55 geometry, with r=2 bohr for . The dipole moment has atomic units, ea0.  Here again, the long-range dipole is actually the largest in magnitude, then followed by the singlet ab initio dipole, which is opposite in sign.   231  Figure (118) shows the z-components of the dipole when both monomers are at equilibrium bond distances for this geometry.   Figure 118. Comparison of dipole moments as a function of intermolecular distance (R) for the 50-0-95-55 geometry, with r=req bohr for . The dipole moment has atomic units, ea0.  The long-range dipole is still the largest in magnitude in Figure (120), followed by the quintet, triplet, and then singlet dipoles (the quintet and singlet dipoles have now reversed order here).      In Figure (119), the corresponding dipoles for monomer B stretched to r=3.3 bohr are shown. When monomer B is stretched to this distance, the quintet state displays the largest z-component dipole. 232   Figure 119. Comparison of dipole moments as a function of intermolecular distance (R) for the 50-0-95-55 geometry, with r=3.3 bohr for . The dipole moment has atomic units, ea0.  The next to last geometry for which dipoles have been obtained is again a geometry belonging to the C1 symmetry point group, with angles 60-0-80-65. Figure (120) shows this data for this geometry when the bond length of monomer B is fixed at r=2 bohr. The long-range dipole moment is the largest in magnitude for this geometry.        Figure (121) shows the corresponding dipoles for this geometry when the bond length of monomer B is at equilibrium.    233   Figure 120. Comparison of dipole moments as a function of intermolecular distance (R) for the 60-0-80-65 geometry, with r=2 bohr for . The dipole moment has atomic units, ea0.   Figure 121. Comparison of dipole moments as a function of intermolecular distance (R) for the 60-0-80-65 geometry, with r=req for . The dipole moment has atomic units, ea0.  234  As for the last geometry considered (Figure (118)), the singlet state dipole and quintet state dipole have reversed in their orders of magnitude, so that now the quintet state dipole is the largest, when both monomers are at equilibrium.       Figure (122) displays the behavior of the z-component of the dipole moment for the 60-0-80-65 geometry when the bond length of monomer B is r=3.3 bohr. The long-range dipole is the largest in magnitude, followed by the ab initio quintet, triplet and singlet dipole.   Figure 122. Comparison of dipole moments as a function of intermolecular distance (R) for the 60-0-80-65 geometry, with r=3.3 bohr for . The dipole moment has atomic units, ea0.  Finally, the last geometry for which dipole moments have been calculated for is that with angles 70-0-65-75. Figure (123) shows the z-component for the dipoles for this geometry when the bond distance of monomer B is at r=2 bohr.  235   Figure 123. Comparison of dipole moments as a function of intermolecular distance (R) for the 70-0-65-75 geometry, with r=2 bohr for . The dipole moment has atomic units, ea0.  This figure follows the overriding previous trends, with the singlet state dipole the largest in magnitude.       In Figure (124), the z-component of the dipole moment for this geometry, with both monomers at their equilibrium bond length, is shown. The triplet spin state z-component shows some strange behavior, deviating away from the other two ab initio values as the molecules approach each other. At the shortest intermolecular distance of 5 bohr, the triplet state dipole becomes large and positive, and is the largest in magnitude of the four dipoles.   236   Figure 124. Comparison of dipole moments as a function of intermolecular distance (R) for the 70-0-65-75 geometry, with r=req for . The dipole moment has atomic units, ea0.  Figure (125) is the last for this chapter. In it, we see the dipoles for the 70-0-65-75 geometry, with the bond distance of monomer B stretched to r=3.3 bohr. In this figure, the long-range dipole is the largest in magnitude, shown clearly at the intermolecular distance of R=5 bohr.      The figures analyzed in this section are only a small portion of the dipole moment data acquired. The z-components of the dipole moment calculated using ab initio energy calculations for the singlet, triplet, and quintet spin states are tabulated in the appendix, in Tables (65)-(312).  237   Figure 125. Comparison of dipole moments as a function of intermolecular distance (R) for the 70-0-65-75 geometry, with r=3.3 bohr for . The dipole moment has atomic units, ea0.  5.3 Conclusions and Future Directions We have obtained the z-component of the collision-induced dipole moment for 23 angular geometries for the O2-O2 system, for use in modeling spectroscopic line profiles. Over 4,000 energies were calculated for this purpose, while applying a perturbative field with a strength of 0.001 a.u.   The geometries that result in the largest magnitude (0.1-1.0 ea0, or 0.25-2.5 Debye) of dipole moment include the linear, T-shaped, 30-0-0-0, 60-0-0-0, and 10-0-160-15 geometries. In all of these cases, at least one of the monomers in the O2-O2 system is approaching the second monomer in a linear, or near linear, head-on orientation. This results in the largest distortion of the electronic charge around the oxygen atoms. For the same reason, for all geometries, the distortion and thus resulting dipole moment is larger when monomer B is at a non-equilibrium bond length, whether contracted shorter or stretched longer than the equilibrium bond length. In 238  most cases, the singlet spin state, which is the ground state of the three asymptotic spin states for the O2-O2 system, displays the largest magnitude of the dipole moment.        We have compared the ab initio dipole moments to the same component that was calculated using multipole moments and polarizabilities for the monomers, and making use of the long-range approximation. In general, the long-range expansion is a good approximation for intermolecular distances of 8 bohr or larger.       The ab initio calculations of the dipole moment require a large amount of CPU time and are costly to perform. For this reason, we have only obtained the z-component of the dipole moment.  If the x and y components of the dipole are calculated ab initio, the total dipole moment can be obtained, and then the expansion coefficients in equation (1.19) can be calculated and compared directly to those that were obtained using the long-range approximation. This is a next step in future directions of this project.           239              APPENDIX               240  APPENDIX  All dipole moments in the appendix are in atomic units (ea0). Table 65. The z-component of the collision-induced dipole moment at an internuclear distance of 2 bohr for geometry 0-0-0-0 (linear) Intermolecular Distance (bohr)  z (singlet)  z (triplet)  z (quintet) 5 0.377021 0.248095 0.082433 6 0.071249 0.053053 0.02546 8 0.006778 0.006448 0.005829 10 0.00217 0.002172 0.002148 12 0.001022 0.001023 0.001022 15 0.000431 0.000431 0.000431 20 0.000137 0.000137 0.000137  Table 66. The z-component of the collision-induced dipole moment at an internuclear distance of 2.1 bohr for geometry 0-0-0-0 (linear) Intermolecular Distance (bohr)   z (singlet)  z (triplet)  z (quintet) 5 0.239599 0.163078 0.062996 6 0.045774 0.034755 0.018044 8 0.004439 0.004265 0.003902 10 0.001434 0.001433 0.001422 12 0.000686 0.000676 0.000673 15 0.000281 0.000282 0.000281 20 8.91E-05 8.91E-05 8.91E-05  Table 67. The z-component of the collision-induced dipole moment at an internuclear distance of 2.2 bohr for geometry 0-0-0-0 (linear) Intermolecular Distance (bohr)   z (singlet)  z (triplet)  z (quintet) 5 0.107269 0.075721 0.033086 6 0.020979 0.016145 0.008867 8 0.002035 0.001979 0.001813 10 0.000676 0.070126 0.000653 12 0.000306 0.000307 0.000307 15 0.000128 0.000128 0.000128 20 4.03E-05 4.03E-05 4.03E-05   241  Table 68. The z-component of the collision-induced dipole moment at an internuclear distance of 2.296 bohr for geometry 0-0-0-0 (linear) Intermolecular Distance (bohr)   z (singlet)  z (triplet)  z (quintet) 5 -0.01845 -0.01349 -0.00651 6 -0.00373 -0.0029 -0.00165 8 -0.00036 -0.00038 -0.00032 10 -0.00011 -0.00012 -0.00011 12 -5.2E-05 -5.4E-05 -5.3E-05 15 -2.2E-05 -2.2E-05 -2.2E-05 20 -7E-06 -7E-06 -7E-06  Table 69. The z-component of the collision-induced dipole moment at an internuclear distance of 2.36 bohr for geometry 0-0-0-0 (linear) Intermolecular Distance (bohr)   z (singlet)  z (triplet)  z (quintet) 5 -0.10158 -0.07606 -0.02247 6 -0.02118 -0.01652 -0.01728 8 -0.00198 -0.00193 -0.00973 10 -0.00064 -0.10682 -0.00774 12 -0.0003 -0.0003 -0.00714 15 -0.00012 -0.00012 -0.00682 20 -3.9E-05 -3.9E-05 -0.00666  Table 70. The z-component of the collision-induced dipole moment at an internuclear distance of 2.457 bohr for geometry 0-0-0-0 (linear) Intermolecular Distance (bohr)   z (singlet)  z (triplet)  z (quintet) 5 -0.22472 -0.17359 -0.09542 6 -0.04962 -0.03886 -0.02325 8 -0.00454 -0.00439 -0.00412 10 -0.00144 -0.00143 -0.00143 12 -0.00067 -0.00066 -0.00067 15 -0.00027 -0.00027 -0.00027 20 -8.6E-05 -8.6E-05 -8.6E-05      242  Table 71. The z-component of the collision-induced dipole moment at an internuclear distance of 2.646 bohr for geometry 0-0-0-0 (linear) Intermolecular Distance (bohr)   z (singlet)  z (triplet)  z (quintet) 5 -0.44061 -0.35034 -0.2037 6 -0.11541 -0.08914 -0.05523 8 -0.01855 -0.00939 -0.00879 10 -0.01216 -0.00298 -0.00297 12 -0.0106 -0.00137 -0.00137 15 -0.00979 -0.00056 -0.00056 20 -0.00939 -0.00017 -0.00017  Table 72. The z-component of the collision-induced dipole moment at an internuclear distance of 2.929 bohr for geometry 0-0-0-0 (linear) Intermolecular Distance (bohr)   z (singlet)  z (triplet)  z (quintet) 5 -0.65431 -0.53422 -0.13891 6 -0.23243 -0.30686 -0.11817 8 -0.01752 -0.02427 -0.01589 10 -0.00512 -0.00686 -0.00509 12 -0.00231 -0.00303 -0.00231 15 -0.00096 -0.00128 -0.00093 20 -0.00029 -0.00037 -0.00029  Table 73. The z-component of the collision-induced dipole moment at an internuclear distance of 3.213 bohr for geometry 0-0-0-0 (linear) Intermolecular Distance (bohr)   z (singlet)  z (triplet)  z (quintet) 5 -0.6678 -0.4887 0.091813 6 -0.37965 -0.16174 -0.20466 8 -0.02514 -0.00543 -0.02266 10 -0.00687 0.005506 -0.00683 12 -0.00305 0.00806 -0.00304 15 -0.00122 0.009313 -0.00122 20 -0.00038 0.009879 -0.00038      243  Table 74. The z-component of the collision-induced dipole moment at an internuclear distance of 3.3 bohr for geometry 0-0-0-0 (linear) Intermolecular Distance (bohr)   z (singlet)  z (triplet)  z (quintet) 5 -0.67239 -0.08092 0.086246 6 -0.42854 -0.35174 -0.23611 8 -0.02735 -0.02629 -0.0246 10 -0.0073 -0.00728 -0.00726 12 -0.00322 -0.00321 -0.00322 15 -0.0013 -0.00124 -0.00128 20 -0.0004 -0.0004 -0.0004  Table 75. The z-component of the collision-induced dipole moment at an internuclear distance of 2 bohr for geometry 90-0-0-0 (T-shaped) Intermolecular Distance (bohr)   z (singlet)  z (triplet)  z (quintet) 5 0.080953 0.072238 0.038309 6 0.026797 0.024538 0.020391 8 0.005055 0.004996 0.005134 10 0.001717 0.001723 0.001806 12 0.000816 0.00082 0.00085 15 0.000377 0.000377 0.000386 20 0.000124 0.000124 0.000128  Table 76. The z-component of the collision-induced dipole moment at an internuclear distance of 2.1 bohr for geometry 90-0-0-0 (T-shaped) Intermolecular Distance (bohr)   z (singlet)  z (triplet)  z (quintet) 5 0.051272 0.047376 0.034422 6 0.021035 0.020084 0.018246 8 0.004287 0.004263 0.004213 10 0.001435 0.001432 0.001431 12 0.000674 0.000675 0.000674 15 0.000311 0.00031 0.00031 20 0.000102 0.000102 0.000102      244  Table 77. The z-component of the collision-induced dipole moment at an internuclear distance of 2.2 bohr for geometry 90-0-0-0 (T-shaped) Intermolecular Distance (bohr)   z (singlet)  z (triplet)  z (quintet) 5 0.028411 0.027541 0.029298 6 0.01656 0.016551 0.016027 8 0.003884 0.003886 0.003506 10 0.001269 0.001273 0.001134 12 0.000573 0.000576 0.000524 15 0.00026 0.000261 0.000243 20 8.71E-05 8.7E-05 8.04E-05  Table 78. The z-component of the collision-induced dipole moment at an internuclear distance of 2.296 bohr for geometry 90-0-0-0 (T-shaped) Intermolecular Distance (bohr)   z (singlet)  z (triplet)  z (quintet) 5 0.00023 0.002376 0.022787 6 0.011155 0.012126 0.014366 8 0.003176 0.003206 0.003256 10 0.000996 0.000997 0.000999 12 0.000436 0.000438 0.000436 15 0.000198 0.000202 0.0002 20 6.77E-05 6.77E-05 6.77E-05  Table 79. The z-component of the collision-induced dipole moment at an internuclear distance of 2.36 bohr for geometry 90-0-0-0 (T-shaped) Intermolecular Distance (bohr)   z (singlet)  z (triplet)  z (quintet) 5 -0.01988 -0.01605 0.019561 6 0.007349 0.008959 0.013032 8 0.002698 0.002738 0.002607 10 0.000815 0.000822 0.000692 12 0.000344 0.000344 0.000282 15 0.00016 0.00016 0.000135 20 5.5E-05 5.5E-05 4.69E-05      245  Table 80. The z-component of the collision-induced dipole moment at an internuclear distance of 2.457 bohr for geometry 90-0-0-0 (T-shaped) Intermolecular Distance (bohr)   z (singlet)  z (triplet)  z (quintet) 5 -0.05309 -0.04727 0.007684 6 0.001156 0.003714 0.009556 8 0.001963 0.002027 0.002152 10 0.000537 0.000543 0.000545 12 0.000211 0.000209 0.000209 15 0.000102 0.000102 0.000102 20 3.62E-05 3.62E-05 3.62E-05  Table 81. The z-component of the collision-induced dipole moment at an internuclear distance of 2.646 bohr for geometry 90-0-0-0 (T-shaped) Intermolecular Distance (bohr)   z (singlet)  z (triplet)  z (quintet) 5 -0.12932 -0.12225 -0.01669 6 -0.01272 -0.00804 0.002473 8 0.000522 0.000629 0.000837 10 1.78E-05 2.74E-05 3.18E-05 12 -4.5E-05 -4.5E-05 -4.5E-05 15 -6.9E-06 -6.9E-06 -6.9E-06 20 1.79E-06 1.75E-06 1.81E-06  Table 82. The z-component of the collision-induced dipole moment at an internuclear distance of 2.929 bohr for geometry 90-0-0-0 (T-shaped) Intermolecular Distance (bohr)   z (singlet)  z (triplet)  z (quintet) 5 -0.27528 -0.27403 -0.06827 6 -0.0388 -0.03078 -0.01136 8 -0.00161 -0.00145 -0.00111 10 -0.00067 -0.00067 -0.00066 12 -0.00038 -0.00038 -0.00038 15 -0.00015 -0.00015 -0.00015 20 -4.3E-05 -4.3E-05 -4.3E-05      246  Table 83. The z-component of the collision-induced dipole moment at an internuclear distance of 3.213 bohr for geometry 90-0-0-0 (T-shaped) Intermolecular Distance (bohr)   z (singlet)  z (triplet)  z (quintet) 5 -0.45121 -0.45764 -0.14135 6 -0.07186 -0.06104 -0.03015 8 -0.00376 -0.00351 -0.00301 10 -0.00127 -0.00128 -0.00126 12 -0.00066 -0.00066 -0.00066 15 -0.00026 -0.00031 -0.00026 20 -7.8E-05 -7.8E-05 -7.8E-05  Table 84. The z-component of the collision-induced dipole moment at an internuclear distance of 3.3 bohr for geometry 90-0-0-0 (T-shaped) Intermolecular Distance (bohr)   z (singlet)  z (triplet)  z (quintet) 5 -0.51662 -0.52613 -0.16807 6 -0.08294 -0.07188 -0.03709 8 -0.00439 -0.00411 -0.00357 10 -0.00145 -0.00149 -0.00141 12 -0.00074 -0.00074 -0.00074 15 -0.00029 -0.00029 -0.00029 20 -8.7E-05 -8.7E-05 -8.7E-05  Table 85. The z-component of the collision-induced dipole moment at an internuclear distance of 2 bohr for geometry 90-0-90-0 (H-shaped) Intermolecular Distance (bohr)   z (singlet)  z (triplet)  z (quintet) 5 0.032664174 -0.001807933 -0.001492126 6 0.003142046 -0.002699614 -0.002317612 8 -0.001098531 -0.001334431 -0.001257441 10 -0.000572854 -0.000571589 -0.000579044 12 -0.000270651 -0.000252792 -0.000270573 15 -0.000103089 -9.932E-05 -0.000103051 20 -3.28355E-05 -3.16545E-05 -3.283E-05      247  Table 86. The z-component of the collision-induced dipole moment at an internuclear distance of 2.1 bohr for geometry 90-0-90-0 (H-shaped) Intermolecular Distance (bohr)   z (singlet)  z (triplet)  z (quintet) 5 0.018571174 0.013151735 -0.000555847 6 0.001625818 0.000618447 -0.001352498 8 -0.000714349 -0.000742677 -0.000794234 10 -0.000358792 -0.000364055 -0.000365078 12 -0.000171416 -0.000170589 -0.000170799 15 -6.6299E-05 -6.6257E-05 -6.628E-05 20 -2.12235E-05 -2.1222E-05 -2.1221E-05  Table 87. The z-component of the collision-induced dipole moment at an internuclear distance of 2.2 bohr for geometry 90-0-90-0 (H-shaped) Intermolecular Distance (bohr)   z (singlet)  z (triplet)  z (quintet) 5 0.007604828 0.005418126 -0.000112935 6 0.000617271 0.000219947 -0.000557593 8 -0.000322302 -0.000330525 -0.000350138 10 -0.000159499 -0.000156803 -0.00016064 12 -7.62935E-05 -7.44485E-05 -7.52515E-05 15 -2.9693E-05 -2.9787E-05 -2.9669E-05 20 -9.56149E-06 -9.57502E-06 -9.54702E-06  Table 88. The z-component of the collision-induced dipole moment at an internuclear distance of 2.296 bohr for geometry 90-0-90-0 (H-shaped) Intermolecular Distance (bohr)   z (singlet)  z (triplet)  z (quintet) 5 -0.001223966 -0.000877103 1.41802E-06 6 -9.3438E-05 -3.2527E-05 8.93875E-05 8 4.3606E-05 5.2751E-05 5.91645E-05 10 2.615E-05 2.75745E-05 2.7101E-05 12 -8.504E-06 1.2762E-05 1.2713E-05 15 5.10451E-06 5.03351E-06 5.07998E-06 20 1.63902E-06 1.6355E-06 1.6415E-06      248  Table 89. The z-component of the collision-induced dipole moment at an internuclear distance of 2.36 bohr for geometry 90-0-90-0 (H-shaped) Intermolecular Distance (bohr)   z (singlet)  z (triplet)  z (quintet) 5 -0.006509539 -0.00468223 -3.58225E-05 6 -0.000483635 -0.000162563 0.000465138 8 0.000300857 0.000307024 0.000321828 10 0.000146554 0.000146137 0.000147383 12 6.8986E-05 6.92375E-05 6.9215E-05 15 2.78425E-05 2.78505E-05 2.78855E-05 20 9.02651E-06 9.03151E-06 9.02651E-06  Table 90. The z-component of the collision-induced dipole moment at an internuclear distance of 2.457 bohr for geometry 90-0-90-0 (H-shaped) Intermolecular Distance (bohr)   z (singlet)  z (triplet)  z (quintet) 5 -0.013873828 -0.010036202 -0.00021081 6 -0.000997227 -0.000328662 0.000980201 8 0.000658445 0.000674754 0.000702408 10 0.000320664 0.0003207 0.000321407 12 0.000149483 0.000151378 0.000151181 15 6.1616E-05 6.16005E-05 6.1604E-05 20 1.99725E-05 1.9969E-05 1.99705E-05  Table 91. The z-component of the collision-induced dipole moment at an internuclear distance of 2.646 bohr for geometry 90-0-90-0 (H-shaped) Intermolecular Distance (bohr)   z (singlet)  z (triplet)  z (quintet) 5 -0.026612266 -0.019446753 -0.000760019 6 -0.001853874 -0.000619212 0.001794119 8 0.001293608 0.001320693 0.001372668 10 0.000625663 0.000626417 0.000627532 12 0.000296343 0.000296496 0.000296457 15 0.00012324 0.00012322 0.000123233 20 4.0204E-05 4.01945E-05 4.01965E-05      249  Table 92. The z-component of the collision-induced dipole moment at an internuclear distance of 2.929 bohr for geometry 90-0-90-0 (H-shaped) Intermolecular Distance (bohr)   z (singlet) z (triplet) z (quintet) 5 -0.042138557 -0.03124864 -0.001614153 6 -0.002910275 -0.001022437 0.002654339 8 0.002052387 0.002089907 0.002163685 10 0.000990094 0.000990965 0.000991066 12 0.000473282 0.00047049 0.000471987 15 0.000201017 0.000196354 0.000200756 20 6.62395E-05 6.57925E-05 6.5907E-05  Table 93. The z-component of the collision-induced dipole moment at an internuclear distance of 3.213 bohr for geometry 90-0-90-0 (H-shaped) Intermolecular Distance (bohr)   z (singlet)  z (triplet) z (quintet) 5 -0.051844839 -0.039338699 -0.002131642 6 -0.003545545 -0.001271731 0.003200946 8 0.002540543 0.002584656 0.002678435 10 0.001229398 0.00124104 0.001234128 12 0.000592768 0.000601174 0.000593133 15 0.000255038 0.000267267 0.000256205 20 8.47495E-05 8.40285E-05 8.47255E-05  Table 94. The z-component of the collision-induced dipole moment at an internuclear distance of 3.3 bohr for geometry 90-0-90-0 (H-shaped) Intermolecular Distance (bohr)   z (singlet)  z (triplet) z (quintet) 5 -0.053182254 -0.040804785 -0.002213646 6 -0.00359987 -0.001287265 0.003309089 8 0.002634653 0.002658444 0.002778047 10 0.001280079 0.001281476 0.001283648 12 0.00061584 0.00058639 0.000618762 15 0.000267023 0.000268167 0.000268026 20 8.8727E-05 8.8729E-05 8.8685E-05      250  Table 95. The z-component of the collision-induced dipole moment at an internuclear distance of 2 bohr for geometry 90-0-90-90 (X-shaped) Intermolecular Distance (bohr)   z (singlet)  z (triplet)  z (quintet) 5 -0.0007545 -0.00180793 -0.00206339 6 -0.0026053 -0.00269961 -0.00265887 8 -0.001399 -0.00133443 -0.00138858 10 -0.0006038 -0.00057159 -0.00060324 12 -0.0002676 -0.00025279 -0.00026763 15 -0.000105 -9.932E-05 -0.00010507 20 -3.344E-05 -3.1654E-05 -3.3443E-05  Table 96. The z-component of the collision-induced dipole moment at an internuclear distance of 2.1 bohr for geometry 90-0-90-90 (X-shaped) Intermolecular Distance (bohr)   z (singlet)  z (triplet)  z (quintet) 5 -0.00042 -0.00067559 -0.00116044 6 -0.00162 -0.00162971 -0.00164827 8 -0.00089 -0.00089175 -0.0008848 10 -0.00038 -0.00038309 -0.00038339 12 -0.00017 -0.00016838 -0.00017076 15 -6.8E-05 -6.7356E-05 -6.7766E-05 20 -2.2E-05 -3.2982E-05 -2.162E-05  Table 97. The z-component of the collision-induced dipole moment at an internuclear distance of 2.2 bohr for geometry 90-0-90-90 (X-shaped) Intermolecular Distance (bohr)   z (singlet)  z (triplet)  z (quintet) 5 -0.00016 -0.00026874 -0.0004672 6 -0.0007 -0.00070769 -0.00071758 8 -0.00039 -0.00039727 -0.00039266 10 -0.00017 -0.00017303 -0.00016967 12 -7.1E-05 -7.5556E-05 -7.5801E-05 15 -3E-05 -3.0227E-05 -3.0393E-05 20 -9.7E-06 -9.7595E-06 -9.7315E-06       251  Table 98. The z-component of the collision-induced dipole moment at an internuclear distance of 2.296 bohr for geometry 90-0-90-90 (X-shaped) Intermolecular Distance (bohr)   z (singlet)  z (triplet)  z (quintet) 5 2.5382E-05 4.11145E-05 7.40345E-05 6 0.00011535 0.000119454 0.000123286 8 6.6981E-05 6.68835E-05 6.6877E-05 10 2.9545E-05 5.8636E-05 2.8712E-05 12 1.4514E-05 1.3342E-05 1.28525E-05 15 5.533E-06 7.37748E-06 5.20399E-06 20 1.675E-06 1.68748E-06 1.67E-06  Table 99. The z-component of the collision-induced dipole moment at an internuclear distance of 2.36 bohr for geometry 90-0-90-90 (X-shaped) Intermolecular Distance (bohr)   z (singlet)  z (triplet)  z (quintet) 5 0.000131 0.000219081 -0.00069105 6 0.000633 0.000637981 0.000530715 8 0.000366 0.000349621 0.000486198 10 0.000153 0.000149927 0.000220337 12 7.17E-05 6.9705E-05 9.8668E-05 15 2.88E-05 2.59445E-05 3.94215E-05 20 9.2E-06 9.20201E-06 1.26465E-05  Table 100. The z-component of the collision-induced dipole moment at an internuclear distance of 2.457 bohr for geometry 90-0-90-90 (X-shaped) Intermolecular Distance (bohr)   z (singlet)  z (triplet)  z (quintet) 5 0.00027 0.001352359 0.001883054 6 0.001369 0.001476301 0.001519432 8 0.000801 0.000680756 0.000677524 10 0.000342 0.000280084 0.000279552 12 0.000154 0.000128763 0.000126312 15 6.32E-05 5.30885E-05 5.29005E-05 20 2.04E-05 1.703E-05 1.70115E-05      252  Table 101. The z-component of the collision-induced dipole moment at an internuclear distance of 2.646 bohr for geometry 90-0-90-90 (X-shaped) Intermolecular Distance (bohr)   z (singlet)  z (triplet)  z (quintet) 5 0.000512 0.000882771 0.001575633 6 0.002646 0.002667611 0.002708654 8 0.001572 0.001569746 0.001564577 10 0.000671 0.000670603 0.000670138 12 0.000303 0.000303638 0.000303081 15 0.000127 0.000127965 0.000126856 20 4.1E-05 4.0997E-05 4.09955E-05  Table 102. The z-component of the collision-induced dipole moment at an internuclear distance of 2.929 bohr for geometry 90-0-90-90 (X-shaped) Intermolecular Distance (bohr)   z (singlet)  z (triplet)  z (quintet) 5 0.000891 0.001412741 0.002542371 6 0.004133 0.004196042 0.004275948 8 0.002498 0.002497056 0.002485307 10 0.001064 0.00106795 0.001062597 12 0.000483 0.000485822 0.000484465 15 0.000207 0.000208773 0.000207034 20 6.73E-05 6.7289E-05 6.72385E-05  Table 103. The z-component of the collision-induced dipole moment at an internuclear distance of 3.213 bohr for geometry 90-0-90-90 (X-shaped) Intermolecular Distance (bohr)   z (singlet)  z (triplet)  z (quintet) 5 0.001181 0.001755721 0.00320051 6 0.00516 0.005211106 0.005333383 8 0.003102 0.003109162 0.003094536 10 0.001325 0.001333343 0.001327877 12 0.000609 0.000638792 0.000611505 15 0.000264 0.000254868 0.000264949 20 8.65E-05 0.000132413 8.6535E-05      253  Table 104. The z-component of the collision-induced dipole moment at an internuclear distance of 3.3 bohr for geometry 90-0-90-90 (X-shaped) Intermolecular Distance (bohr)   z (singlet)  z (triplet)  z (quintet) 5  0.001223 0.001781882 0.003291267 6 0.00536 0.005386253 0.005546102 8 0.003223 0.003215423 0.003217491 10 0.001383 0.0013798 0.001383636 12 0.000667 0.000641964 0.000639178 15 0.000276 0.000237446 0.000277544 20 9.07E-05 9.06905E-05 9.07895E-05  Table 105. The z-component of the collision-induced dipole moment at an internuclear distance of 2 bohr for geometry 30-0-0-0  Intermolecular Distance (bohr)   z (singlet)  z (triplet)  z (quintet) 5  0.24899 - 0.210709 6 0.053624 - 0.043862 8 0.006148 - 0.005758 10 0.001881 - 0.001862 12 0.000927 - 0.000927 15 0.000411 - 0.000411 20 0.000132 - 0.000132  Table 106. The z-component of the collision-induced dipole moment at an internuclear distance of 2.1 bohr for geometry 30-0-0-0  Intermolecular Distance (bohr)   z (singlet)  z (triplet)  z (quintet) 5  0.183603 0.202742 0.198831 6 0.038029 0.003678 0.035441 8 0.003696 0.037085 0.003538 10 0.001095 0.001089 0.001081 12 0.000557 0.000528 0.000556 15 0.000257 0.000256 0.000256 20 8.38E-05 8.3E-05 8.37E-05      254  Table 107. The z-component of the collision-induced dipole moment at an internuclear distance of 2.2 bohr for geometry 30-0-0-0  Intermolecular Distance (bohr)   z (singlet)  z (triplet)  z (quintet) 5  0.143861 0.155759 0.190949 6 0.028882 0.001642 0.030602 8 0.002327 0.026233 0.002179 10 0.000546 0.000483 0.000532 12 0.0003 0.000214 0.000299 15 0.000147 0.000113 0.000147 20 4.81E-05 3.93E-05 4.81E-05  Table 108. The z-component of the collision-induced dipole moment at an internuclear distance of 2.28187 bohr for geometry 30-0-0-0  Intermolecular Distance (bohr)   z (singlet)  z (triplet)  z (quintet) 5  0.09276 - 0.173306 6 0.017546 - 0.023367 8 0.000642 - 0.000567 10 -3.1E-05 - -3.1E-05 12 4.13E-05 - 4.96E-05 15 -1.5E-05 - 4.15E-05 20 1.88E-05 - 2.79E-05  Table 109. The z-component of the collision-induced dipole moment at an internuclear distance of 2.296 bohr for geometry 30-0-0-0  Intermolecular Distance (bohr)   z (singlet)  z (triplet)  z (quintet) 5  0.083189 0.116187 0.169583 6 0.015481 0.000319 0.022003 8 0.000347 0.017648 0.00028 10 -0.00014 -0.00014 -0.00015 12 -1.7E-05 -1.8E-05 -1.8E-05 15 1.67E-05 1.69E-05 1.67E-05 20 7.35E-06 7.35E-06 7.34E-06      255  Table 110. The z-component of the collision-induced dipole moment at an internuclear distance of 2.36 bohr for geometry 30-0-0-0  Intermolecular Distance (bohr)   z (singlet)  z (triplet)  z (quintet) 5  0.036699 0.072648 0.150166 6 0.005628 -0.00103 0.015348 8 -0.00102 0.008839 -0.00103 10 -0.0006 -0.0006 -0.00061 12 -0.00023 -0.00023 -0.00023 15 -7E-05 -7E-05 -7E-05 20 -2E-05 -2E-05 -2E-05  Table 111. The z-component of the collision-induced dipole moment at an internuclear distance of 2.457 bohr for geometry 30-0-0-0  Intermolecular Distance (bohr)   z (singlet)  z (triplet)  z (quintet) 5  -0.04409 -0.00398 0.11258 6 -0.01101 -0.00315 0.003708 8 -0.00317 -0.00617 -0.00312 10 -0.00131 -0.00131 -0.00132 12 -0.00056 -0.00056 -0.00056 15 -0.0002 -0.0002 -0.0002 20 -6.1E-05 -6.1E-05 -6.1E-05  Table 112. The z-component of the collision-induced dipole moment at an internuclear distance of 2.646 bohr for geometry 30-0-0-0  Intermolecular Distance (bohr)   z (singlet)  z (triplet)  z (quintet) 5  -0.21342 -0.17384 0.019225 6 -0.05054 -0.00747 -0.02507 8 -0.00753 -0.04235 -0.00735 10 -0.00269 -0.00269 -0.0027 12 -0.00117 -0.00117 -0.00117 15 -0.00045 -0.00045 -0.00045 20 -0.00014 -0.00014 -0.00014      256  Table 113. The z-component of the collision-induced dipole moment at an internuclear distance of 2.929 bohr for geometry 30-0-0-0  Intermolecular Distance (bohr)   z (singlet)  z (triplet)  z (quintet) 5  -0.87639 -0.34226 -0.08526 6 -0.13188 -0.01417 -0.08735 8 -0.01435 -0.11878 -0.0139 10 -0.00459 -0.00459 -0.0046 12 -0.002 -0.002 -0.002 15 -0.00077 -0.00077 -0.00077 20 -0.00024 -0.00024 -0.00024  Table 114. The z-component of the collision-induced dipole moment at an internuclear distance of 3.213 bohr for geometry 30-0-0-0  Intermolecular Distance (bohr)   z (singlet)  z (triplet)  z (quintet) 5  -0.65283 -0.32931 -0.22403 6 -0.24602 -0.02098 -0.18259 8 -0.02113 -0.23348 -0.0204 10 -0.00616 -0.00616 -0.00617 12 -0.00265 -0.00258 -0.00265 15 -0.00102 -0.00102 -0.00102 20 -0.00031 -0.00031 -0.00031  Table 115. The z-component of the collision-induced dipole moment at an internuclear distance of 3.3 bohr for geometry 30-0-0-0  Intermolecular Distance (bohr)   z (singlet)  z (triplet)  z (quintet) 5  -0.5214 -0.29269 -0.25393 6 -0.28675 -0.02288 -0.21935 8 -0.02317 -0.27451 -0.02236 10 -0.00656 -0.00658 -0.00654 12 -0.0028 -0.0028 -0.0028 15 -0.00108 -0.00108 -0.00107 20 -0.00033 -0.00033 -0.00033      257  Table 116. The z-component of the collision-induced dipole moment at an internuclear distance of 2 bohr for geometry 60-0-0-0  Intermolecular Distance (bohr)   z (singlet)  z (triplet)  z (quintet) 5  0.12791 0.114761 0.107634 6 0.035354 0.033962 0.033405 8 0.005064 0.005008 0.00524 10 0.001679 0.00163 0.001718 12 0.00084 0.000798 0.000838 15 0.000373 0.000373 0.000388 20 0.000123 0.000123 0.000127  Table 117. The z-component of the collision-induced dipole moment at an internuclear distance of 2.1 bohr for geometry 60-0-0-0  Intermolecular Distance (bohr)   z (singlet)  z (triplet)  z (quintet) 5  0.094175 0.093327 0.101247 6 0.029115 0.028873 0.028918 8 0.003883 0.003868 0.003799 10 0.001208 0.001189 0.00119 12 0.000587 0.000588 0.000588 15 0.000282 0.000281 0.000281 20 9.38E-05 9.3E-05 9.37E-05  Table 118. The z-component of the collision-induced dipole moment at an internuclear distance of 2.2 bohr for geometry 60-0-0-0  Intermolecular Distance (bohr)   z (singlet)  z (triplet)  z (quintet) 5  0.05693 0.068898 0.09651 6 0.022512 0.023316 0.026832 8 0.002652 0.002642 0.003107 10 0.00074 0.00072 0.000876 12 0.000367 0.000365 0.000429 15 0.000188 0.000185 0.000212 20 6.42E-05 6.42E-05 7.16E-05      258  Table 119. The z-component of the collision-induced dipole moment at an internuclear distance of 2.28187 bohr for geometry 60-0-0-0  Intermolecular Distance (bohr)   z (singlet)  z (triplet)  z (quintet) 5  0.033086 - 0.087902 6 0.019092 - 0.023209 8 0.002212 - 0.002151 10 0.000525 - 0.000527 12 0.00026 - 0.00025 15 0.000133 - 0.000139 20 4.86E-05 - 4.89E-05  Table 120. The z-component of the collision-induced dipole moment at an internuclear distance of 2.296 bohr for geometry 60-0-0-0  Intermolecular Distance (bohr)   z (singlet)  z (triplet)  z (quintet) 5  0.026741 0.046878 0.086138 6 0.018029 0.019501 0.022525 8 0.002038 0.002013 0.001985 10 0.000461 0.000453 0.000448 12 0.000218 0.000224 0.000224 15 0.000127 0.000127 0.000127 20 4.48E-05 4.48E-05 4.48E-05  Table 121. The z-component of the collision-induced dipole moment at an internuclear distance of 2.36 bohr for geometry 60-0-0-0  Intermolecular Distance (bohr)   z (singlet)  z (triplet)  z (quintet) 5  0.015424 0.024383 0.07702 6 0.015901 0.015061 0.019189 8 0.001369 0.001209 0.001207 10 0.000127 0.000155 0.000157 12 6.7E-05 8.75E-05 8.69E-05 15 6.07E-05 7.04E-05 7.04E-05 20 2.36E-05 2.71E-05 2.71E-05      259  Table 122. The z-component of the collision-induced dipole moment at an internuclear distance of 2.457 bohr for geometry 60-0-0-0  Intermolecular Distance (bohr)   z (singlet)  z (triplet)  z (quintet) 5  -0.05879 -0.01653 0.073222 6 0.004348 0.007477 0.016077 8 -1.1E-05 -1.2E-05 1.85E-05 10 -0.00028 -0.00028 -0.00034 12 -0.00012 -0.00012 -0.00015 15 -1.5E-05 -1.5E-05 -2.8E-05 20 5.97E-07 -5.8E-06 -4.1E-06  Table 123. The z-component of the collision-induced dipole moment at an internuclear distance of 2.646 bohr for geometry 60-0-0-0  Intermolecular Distance (bohr)   z (singlet)  z (triplet)  z (quintet) 5  -0.22757 -0.12521 0.009187 6 -0.01904 -0.01078 -0.00104 8 -0.00342 -0.00254 -0.00246 10 -0.0014 -0.00112 -0.00113 12 -0.00063 -0.00051 -0.00051 15 -0.00022 -0.00017 -0.00017 20 -6.3E-05 -4.8E-05 -4.8E-05  Table 124. The z-component of the collision-induced dipole moment at an internuclear distance of 2.929 bohr for geometry 60-0-0-0  Intermolecular Distance (bohr)   z (singlet)  z (triplet)  z (quintet) 5  -0.43584 -0.33796 -0.10787 6 -0.05852 -0.04887 -0.03202 8 -0.00643 -0.00635 -0.00623 10 -0.00229 -0.00229 -0.0023 12 -0.00103 -0.00105 -0.00104 15 -0.00038 -0.00038 -0.00038 20 -0.00011 -0.00011 -0.00011      260  Table 125. The z-component of the collision-induced dipole moment at an internuclear distance of 3.213 bohr for geometry 60-0-0-0  Intermolecular Distance (bohr)   z (singlet)  z (triplet)  z (quintet) 5  -0.59612 -0.52856 -0.20986 6 -0.11801 -0.10408 -0.07951 8 -0.01034 -0.01021 -0.01 10 -0.00326 -0.00327 -0.00327 12 -0.00146 -0.00146 -0.00146 15 -0.00054 -0.00054 -0.00054 20 -0.00016 -0.00016 -0.00016  Table 126. The z-component of the collision-induced dipole moment at an internuclear distance of 3.3 bohr for geometry 60-0-0-0  Intermolecular Distance (bohr)   z (singlet)  z (triplet)  z (quintet) 5  -0.59257 -0.49433 -0.20332 6 -0.13962 -0.12521 -0.09841 8 -0.01152 -0.01137 -0.01113 10 -0.00352 -0.00352 -0.00349 12 -0.00156 -0.00156 -0.00155 15 -0.00058 -0.00058 -0.00058 20 -0.00017 -0.00017 -0.00017  Table 127. The z-component of the collision-induced dipole moment at an internuclear distance of 2 bohr for geometry 105-0-60-60  Intermolecular Distance (bohr)   z (singlet)  z (triplet)  z (quintet) 5  0.073382 0.057572 -0.0171 6 0.001165 -0.00248 -0.00986 8 -0.00156 -0.00166 -0.00184 10 -0.00053 -0.00053 -0.00054 12 -0.00026 -0.00026 -0.00026 15 -0.00012 -0.00012 -0.00012 20 -3.5E-05 -3.8E-05 -3.5E-05      261  Table 128. The z-component of the collision-induced dipole moment at an internuclear distance of 2.1 bohr for geometry 105-0-60-60  Intermolecular Distance (bohr)   z (singlet)  z (triplet)  z (quintet) 5  0.035371 0.02461 -0.02489 6 -0.00356 -0.00578 -0.01027 8 -0.00153 -0.00157 -0.00166 10 -0.00047 -0.00047 -0.00047 12 -0.00022 -0.00022 -0.00022 15 0.067422 0.068422 0.068338 20 -3.4E-05 -3.4E-05 -3.4E-05  Table 129. The z-component of the collision-induced dipole moment at an internuclear distance of 2.2 bohr for geometry 105-0-60-60  Intermolecular Distance (bohr)   z (singlet)  z (triplet)  z (quintet) 5  0.00025 -0.00623 -0.03477 6 -0.00833 -0.0093 -0.01141 8 -0.00155 -0.00155 -0.00156 10 -0.00042 -0.00043 -0.00042 12 -0.0002 -0.0002 -0.0002 15 0.116822 0.117736 0.117665 20 -3.3E-05 -3.3E-05 -3.3E-05  Table 130. The z-component of the collision-induced dipole moment at an internuclear distance of 2.28187 bohr for geometry 105-0-60-60  Intermolecular Distance (bohr)   z (singlet)  z (triplet)  z (quintet) 5  -0.001132967 -0.002726543 -0.006380962 6 0.001419402 0.001157685 0.000632795 8 0.000761469 0.00075914 0.000761863 10 0.000237468 0.000251805 0.000250892 12 0.000143739 0.000102293 0.000100046 15 -5.455E-05 3.9957E-05 4.45315E-05 20 1.14425E-05 1.1644E-05 1.379E-05      262  Table 131. The z-component of the collision-induced dipole moment at an internuclear distance of 2.296 bohr for geometry 105-0-60-60  Intermolecular Distance (bohr)   z (singlet)  z (triplet)  z (quintet) 5  -0.03191 -0.03494 -0.04531 6 -0.01275 -0.01282 -0.01292 8 -0.0016 -0.00156 -0.00152 10 -0.00044 -0.00039 -0.00038 12 -0.00015 -0.00017 -0.00017 15 -9.5E-05 -9.5E-05 -9.5E-05 20 -3.2E-05 -3.2E-05 -3.2E-05  Table 132. The z-component of the collision-induced dipole moment at an internuclear distance of 2.36 bohr for geometry 105-0-60-60  Intermolecular Distance (bohr)   z (singlet)  z (triplet)  z (quintet) 5  -0.05232 -0.05337 -0.05224 6 -0.01567 -0.01513 -0.01399 8 -0.0016 -0.00157 -0.0015 10 -0.00036 -0.00036 -0.00035 12 -0.00016 -0.00016 -0.00016 15 -9.1E-05 -9.2E-05 -9.1E-05 20 -3.1E-05 -3.1E-05 -3.1E-05  Table 133. The z-component of the collision-induced dipole moment at an internuclear distance of 2.457 bohr for geometry 105-0-60-60  Intermolecular Distance (bohr)   z (singlet)  z (triplet)  z (quintet) 5  -0.08038 -0.07898 -0.06135 6 -0.01982 -0.0184 -0.01546 8 -0.00164 -0.00158 -0.00145 10 -0.00033 -0.00031 -0.00031 12 -0.00014 -0.00013 -0.00014 15 -8.5E-05 -8.6E-05 -8.5E-05 20 -3E-05 -3E-05 -3E-05      263  Table 134. The z-component of the collision-induced dipole moment at an internuclear distance of 2.646 bohr for geometry 105-0-60-60  Intermolecular Distance (bohr)   z (singlet)  z (triplet)  z (quintet) 5  -0.11795 -0.1132 -0.06928 6 -0.02597 -0.02306 -0.017 8 -0.00161 -0.0015 -0.00128 10 -0.0002 -0.0002 -0.00019 12 -8.6E-05 -8.3E-05 -8.5E-05 15 -6.8E-05 -6.9E-05 -6.8E-05 20 -2.5E-05 -2.4E-05 -2.5E-05  Table 135. The z-component of the collision-induced dipole moment at an internuclear distance of 2.929 bohr for geometry 105-0-60-60  Intermolecular Distance (bohr)   z (singlet)  z (triplet)  z (quintet) 5  -0.16734 -0.15969 -0.07316 6 -0.02852 -0.02436 -0.01477 8 -0.00131 -0.00114 -0.00079 10 2.87E-05 3.91E-05 4.64E-05 12 1.76E-05 1.83E-05 1.83E-05 15 -3.1E-05 -3.1E-05 -3.1E-05 20 -1.3E-05 -1.3E-05 -1.3E-05  Table 136. The z-component of the collision-induced dipole moment at an internuclear distance of 3.213 bohr for geometry 105-0-60-60  Intermolecular Distance (bohr)   z (singlet)  z (triplet)  z (quintet) 5  -0.2682 -0.27247 -0.15093 6 -0.01299 -0.00832 0.0025 8 -0.00077 -0.00061 -0.00012 10 0.000259 0.000261 0.000284 12 0.000118 0.000118 0.000119 15 3.83E-06 3.81E-06 4E-06 20 -3.3E-06 -3.3E-06 -3.2E-06      264  Table 137. The z-component of the collision-induced dipole moment at an internuclear distance of 3.3 bohr for geometry 105-0-60-60  Intermolecular Distance (bohr)   z (singlet)  z (triplet)  z (quintet) 5  -0.28948 0.057572 -0.17492 6 -0.01172 -0.00248 0.010214 8 -0.00031 -0.00166 0.000345 10 0.000337 -0.00053 0.000364 12 0.000149 -0.00026 0.00015 15 1.38E-05 -0.00012 1.42E-05 20 -6.3E-07 -3.8E-05 -5E-07  Table 138. The z-component of the collision-induced dipole moment at an internuclear distance of 2 bohr for geometry 40-0-115-45 Intermolecular Distance (bohr)   z (singlet)  z (triplet)  z (quintet) 5  0.065926 0.050802 -0.00286 6 0.003377 0.000121 -0.00634 8 -0.00165 -0.00176 -0.00198 10 -0.00066 -0.00067 -0.00067 12 -0.00031 -0.00031 -0.00031 15 0.038356 0.0394 0.039311 20 -3.9E-05 -3.9E-05 -3.9E-05  Table 139. The z-component of the collision-induced dipole moment at an internuclear distance of 2.1 bohr for geometry 40-0-115-45 Intermolecular Distance (bohr)   z (singlet)  z (triplet)  z (quintet) 5  0.03431 0.025402 -0.00673 6 -0.00065 -0.00246 -0.00607 8 -0.00155 -0.0016 -0.00171 10 -0.00057 -0.00056 -0.00057 12 -0.00026 -0.00026 -0.00026 15 0.067413 0.068411 0.068327 20 -3.5E-05 -3.5E-05 -3.5E-05      265  Table 140. The z-component of the collision-induced dipole moment at an internuclear distance of 2.2 bohr for geometry 40-0-115-45 Intermolecular Distance (bohr)   z (singlet)  z (triplet)  z (quintet) 5  0.005545 0.001866 -0.01217 6 -0.00449 -0.00514 -0.00638 8 -0.00145 -0.00148 -0.00151 10 -0.00047 -0.00047 -0.00047 12 -0.00021 -0.00021 -0.00021 15 0.116811 0.117725 0.117654 20 -3.1E-05 -3.1E-05 -3.1E-05  Table 141. The z-component of the collision-induced dipole moment at an internuclear distance of 2.28187 bohr for geometry 40-0-115-45 Intermolecular Distance (bohr)   z (singlet)  z (triplet)  z (quintet) 5  -0.01677 -0.01672899 -0.0173 6 -0.00756 -0.00736159 -0.0069 8 -0.0014 -0.00139343 -0.00138 10 -0.00044 -0.00039839 -0.0004 12 -0.00013 -0.00017371 -0.00017 15 0.000193 -8.6111E-05 -8.6E-05 20 -2.8E-05 -2.7504E-05 -2.8E-05  Table 142. The z-component of the collision-induced dipole moment at an internuclear distance of 2.296 bohr for geometry 40-0-115-45 Intermolecular Distance (bohr)   z (singlet)  z (triplet)  z (quintet) 5  -0.02051 -0.01987 -0.01821 6 -0.00808 -0.00774 -0.007 8 -0.0014 -0.00138 -0.00135 10 -0.00039 -0.00038 -0.00039 12 -0.00016 -0.00016 -0.00017 15 -8.4E-05 -8.4E-05 -8.4E-05 20 -2.7E-05 -2.7E-05 -2.7E-05      266  Table 143. The z-component of the collision-induced dipole moment at an internuclear distance of 2.36 bohr for geometry 40-0-115-45 Intermolecular Distance (bohr)   z (singlet)  z (triplet)  z (quintet) 5  -0.03692 -0.03376 -0.02223 6 -0.01039 -0.00943 -0.00748 8 -0.00131 -0.00131 -0.00125 10 -0.00033 -0.00033 -0.00033 12 -0.00014 -0.00014 -0.00014 15 -7.5E-05 -7.5E-05 -7.5E-05 20 -2.4E-05 -2.4E-05 -2.4E-05  Table 144. The z-component of the collision-induced dipole moment at an internuclear distance of 2.457 bohr for geometry 40-0-115-45 Intermolecular Distance (bohr)   z (singlet)  z (triplet)  z (quintet) 5  -0.05954 -0.05315 -0.02753 6 -0.01371 -0.0118 -0.00811 8 -0.00128 -0.00122 -0.0011 10 -0.00027 -0.00025 -0.00025 12 -0.0001 -9.7E-05 -0.0001 15 -6.1E-05 -6.1E-05 -6.1E-05 20 -2E-05 -2E-05 -2E-05  Table 145. The z-component of the collision-induced dipole moment at an internuclear distance of 2.646 bohr for geometry 40-0-115-45 Intermolecular Distance (bohr)   z (singlet)  z (triplet)  z (quintet) 5  -0.09051 -0.01992 -0.03169 6 -0.01836 -0.00378 -0.0085 8 -0.0011 -0.00024 -0.00074 10 -7.2E-05 -1.7E-05 -6.2E-05 12 -1.8E-05 -3.2E-06 -1.7E-05 15 -3E-05 -7.9E-06 -3E-05 20 -1E-05 -2.6E-06 -1E-05      267  Table 146. The z-component of the collision-induced dipole moment at an internuclear distance of 2.929 bohr for geometry 40-0-115-45 Intermolecular Distance (bohr)   z (singlet)  z (triplet)  z (quintet) 5  -0.12162 -0.10790316 -0.02241 6 -0.02035 -0.01593643 -0.00633 8 -0.0006 -0.00043337 -8.6E-05 10 0.000215 0.000173164 0.000231 12 0.000114 0.00011396 0.000114 15 1.95E-05 1.9518E-05 1.94E-05 20 5.26E-06 5.32802E-06 5.28E-06  Table 147. The z-component of the collision-induced dipole moment at an internuclear distance of 3.213 bohr for geometry 40-0-115-45 Intermolecular Distance (bohr)   z (singlet)  z (triplet)  z (quintet) 5  -0.22056 -0.2266882 -0.07412 6 -0.00565 -0.00089932 0.007251 8 -3.4E-05 0.00018111 0.00061 10 0.000463 0.000467813 0.000486 12 0.000225 0.000225785 0.000226 15 6.02E-05 6.02215E-05 6.04E-05 20 1.78E-05 1.78185E-05 1.79E-05  Table 148. The z-component of the collision-induced dipole moment at an internuclear distance of 3.3 bohr for geometry 40-0-115-45 Intermolecular Distance (bohr)   z (singlet)  z (triplet)  z (quintet) 5  -0.24612 -0.25325903 -0.10001 6 0.002133 0.009164444 0.01855 8 0.000353 0.000567235 0.000983 10 0.000538 0.000545603 0.000563 12 0.000256 0.000256543 0.000257 15 7.03E-05 7.13705E-05 7.1E-05 20 2.1E-05 2.0973E-05 2.11E-05      268  Table 149. The z-component of the collision-induced dipole moment at an internuclear distance of 2 bohr for geometry 20-0-115-45 Intermolecular Distance (bohr)   z (singlet)  z (triplet)  z (quintet) 5  0.019952 0.004181 -0.03697 6 -0.01055 -0.01241 -0.01592 8 -0.00297 -0.00298 -0.00295 10 -0.00096 -0.00096 -0.00095 12 -0.00046 -0.00046 -0.00046 15 -0.00021 -0.0002 -0.00021 20 -6.3E-05 -6.2E-05 -6.3E-05  Table 150. The z-component of the collision-induced dipole moment at an internuclear distance of 2.1 bohr for geometry 20-0-115-45 Intermolecular Distance (bohr)   z (singlet)  z (triplet)  z (quintet) 5  -0.00503 -0.01676 -0.04593 6 -0.01274 -0.01397 -0.01626 8 -0.00261 -0.00261 -0.00258 10 -0.00078 -0.00079 -0.00079 12 -0.00038 -0.00038 -0.00038 15 0.067363 -0.00018 0.06828 20 -5.6E-05 -5.6E-05 -5.6E-05  Table 151. The z-component of the collision-induced dipole moment at an internuclear distance of 2.2 bohr for geometry 20-0-115-45 Intermolecular Distance (bohr)   z (singlet)  z (triplet)  z (quintet) 5  -0.02817 -0.03662 -0.05655 6 -0.01514 -0.01586 -0.01723 8 -0.00233 -0.00232 -0.00227 10 -0.00065 -0.00065 -0.00064 12 -0.0003 -0.0003 -0.0003 15 0.116778 0.117691 0.117621 20 -5E-05 -5E-05 -5E-05      269  Table 152. The z-component of the collision-induced dipole moment at an internuclear distance of 2.28187 bohr for geometry 20-0-115-45 Intermolecular Distance (bohr)   z (singlet)  z (triplet)  z (quintet) 5  -0.04591 -0.05232 -0.06581 6 -0.01722 -0.0176 -0.0183 8 -0.00212 -0.00211 -0.00206 10 -0.00054 -0.00053 -0.00053 12 -0.00028 -0.00024 -0.00025 15 -0.00044 -0.00013 -0.00013 20 -4.6E-05 -4.5E-05 -4.5E-05  Table 153. The z-component of the collision-induced dipole moment at an internuclear distance of 2.296 bohr for geometry 20-0-115-45 Intermolecular Distance (bohr)   z (singlet)  z (triplet)  z (quintet) 5  -0.04888 -0.06646 -0.0674 6 -0.01756 -0.01924 -0.01849 8 -0.00209 -0.00193 -0.00203 10 -0.00055 -0.00042 -0.00051 12 -0.00025 -0.0002 -0.00024 15 -0.00013 -0.00011 -0.00013 20 -4.4E-05 -4E-05 -4.4E-05  Table 154. The z-component of the collision-induced dipole moment at an internuclear distance of 2.36 bohr for geometry 20-0-115-45 Intermolecular Distance (bohr)   z (singlet)  z (triplet)  z (quintet) 5  -0.06186 -0.06646 -0.07434 6 -0.01909 -0.01924 -0.01938 8 -0.00196 -0.00193 -0.00187 10 -0.00042 -0.00042 -0.00042 12 -0.0002 -0.0002 -0.0002 15 -0.00011 -0.00011 -0.00011 20 -4E-05 -4E-05 -4E-05      270  Table 155. The z-component of the collision-induced dipole moment at an internuclear distance of 2.457 bohr for geometry 20-0-115-45 Intermolecular Distance (bohr)   z (singlet)  z (triplet)  z (quintet) 5  -0.07941 -0.08191 -0.08318 6 -0.02134 -0.02107 -0.02057 8 -0.00173 -0.00168 -0.00164 10 -0.00028 -0.00029 -0.00029 12 -0.00014 -0.00014 -0.00014 15 -9.3E-05 -9.3E-05 -9.3E-05 20 -3.3E-05 -3.3E-05 -3.3E-05  Table 156. The z-component of the collision-induced dipole moment at an internuclear distance of 2.646 bohr for geometry 20-0-115-45 Intermolecular Distance (bohr)   z (singlet)  z (triplet)  z (quintet) 5  -0.10192 -0.10135 -0.09023 6 -0.02419 -0.02331 -0.02162 8 -0.00128 -0.00123 -0.00115 10 -2.4E-05 -2.2E-05 -1.8E-05 12 -1.3E-05 -1.8E-05 -1.3E-05 15 -4.9E-05 -4.9E-05 -4.9E-05 20 -2E-05 -2E-05 -2E-05  Table 157. The z-component of the collision-induced dipole moment at an internuclear distance of 2.929 bohr for geometry 20-0-115-45 Intermolecular Distance (bohr)   z (singlet)  z (triplet)  z (quintet) 5  -0.111 -0.11094 -0.09181 6 -0.02441 -0.02267 -0.01904 8 -0.00053 -0.00047 -0.00035 10 0.000371 0.000373 0.000378 12 0.000166 0.000183 0.000166 15 1.75E-05 1.74E-05 1.74E-05 20 1.42E-06 1.43E-06 1.42E-06      271  Table 158. The z-component of the collision-induced dipole moment at an internuclear distance of 3.213 bohr for geometry 20-0-115-45 Intermolecular Distance (bohr)   z (singlet)  z (triplet)  z (quintet) 5  -0.16961 -0.15148 -0.13925 6 -0.01227 -0.00982 -0.00368 8 0.000219 0.000311 0.000495 10 0.000701 0.000705 0.000711 12 0.000312 0.00033 0.000313 15 7.19E-05 7.22E-05 7.2E-05 20 1.8E-05 1.8E-05 1.82E-05  Table 159. The z-component of the collision-induced dipole moment at an internuclear distance of 3.3 bohr for geometry 20-0-115-45 Intermolecular Distance (bohr)   z (singlet)  z (triplet)  z (quintet) 5  -0.18937 -0.17686 -0.15273 6 -0.00435 -0.0018 0.003677 8 0.000615 0.000718 0.000944 10 0.000798 0.000778 0.00081 12 0.000353 0.000354 0.000354 15 8.62E-05 8.44E-05 8.65E-05 20 2.23E-05 2.23E-05 2.24E-05  Table 160. The z-component of the collision-induced dipole moment at an internuclear distance of 2 bohr for geometry 15-0-75-30 Intermolecular Distance (bohr)   z (singlet)  z (triplet)  z (quintet) 5  0.086681 0.070393 -0.01745 6 -0.0007 -0.00557 -0.01598 8 -0.00465 -0.00478 -0.00505 10 -0.00184 -0.00185 -0.00185 12 -0.00084 -0.00084 -0.00084 15 -0.00035 -0.00035 -0.00035 20 -0.00011 -0.00011 -0.00011      272  Table 161. The z-component of the collision-induced dipole moment at an internuclear distance of 2.1 bohr for geometry 15-0-75-30 Intermolecular Distance (bohr)   z (singlet)  z (triplet)  z (quintet) 5  0.048914 0.038865 -0.01755 6 -0.00502 -0.00803 -0.01453 8 -0.00398 -0.00405 -0.0042 10 -0.00147 -0.00147 -0.00148 12 -0.00066 -0.00066 -0.00066 15 -0.00028 -0.00028 -0.00028 20 -9.2E-05 -9.2E-05 -9.2E-05  Table 162. The z-component of the collision-induced dipole moment at an internuclear distance of 2.2 bohr for geometry 15-0-75-30 Intermolecular Distance (bohr)   z (singlet)  z (triplet)  z (quintet) 5  0.018333 0.01355 -0.01843 6 -0.00832 -0.00992 -0.01346 8 -0.00329 -0.00334 -0.00342 10 -0.00114 -0.00111 -0.00112 12 -0.00049 -0.00049 -0.00049 15 -0.00022 -0.00022 -0.00022 20 -7.3E-05 -7.3E-05 -7.3E-05  Table 163. The z-component of the collision-induced dipole moment at an internuclear distance of 2.28187 bohr for geometry 15-0-75-30 Intermolecular Distance (bohr)   z (singlet)  z (triplet)  z (quintet) 5  -0.00325 -0.00442 -0.01941 6 -0.01059 -0.01123 -0.01275 8 -0.00275 -0.00277 -0.0028 10 -0.00082 -0.00083 -0.00083 12 -0.00034 -0.00036 -0.00035 15 -0.00029 -0.00017 -0.00017 20 -5.8E-05 -5.8E-05 -5.8E-05      273  Table 164. The z-component of the collision-induced dipole moment at an internuclear distance of 2.296 bohr for geometry 15-0-75-30 Intermolecular Distance (bohr)   z (singlet)  z (triplet)  z (quintet) 5  -0.00679 -0.00733 -0.01959 6 -0.01095 -0.01144 -0.01263 8 -0.00262 -0.00268 -0.0027 10 -0.00079 -0.00078 -0.00078 12 -0.00034 -0.00032 -0.00033 15 0.031621 0.032044 0.031999 20 -5.5E-05 -5.5E-05 -5.5E-05  Table 165. The z-component of the collision-induced dipole moment at an internuclear distance of 2.36 bohr for geometry 15-0-75-30 Intermolecular Distance (bohr)   z (singlet)  z (triplet)  z (quintet) 5  -0.02191 -0.01993 -0.02032 6 -0.01247 -0.01231 -0.01212 8 -0.00228 -0.00225 -0.00224 10 -0.00057 -0.00055 -0.00056 12 -0.00023 -0.00023 -0.00023 15 -0.00012 -0.00012 -0.00012 20 -4.3E-05 -4.3E-05 -4.3E-05  Table 166. The z-component of the collision-induced dipole moment at an internuclear distance of 2.457 bohr for geometry 15-0-75-30 Intermolecular Distance (bohr)   z (singlet)  z (triplet)  z (quintet) 5  -0.04252 -0.03722 -0.02108 6 -0.01449 -0.01343 -0.01136 8 -0.00165 -0.00165 -0.00157 10 -0.00027 -0.00025 -0.00025 12 -8E-05 -7.9E-05 -8.3E-05 15 -6.9E-05 -6.9E-05 -6.9E-05 20 -2.6E-05 -2.6E-05 -2.6E-05       274  Table 167. The z-component of the collision-induced dipole moment at an internuclear distance of 2.646 bohr for geometry 15-0-75-30 Intermolecular Distance (bohr)   z (singlet)  z (triplet)  z (quintet) 5  -0.07394 -0.06404 -0.02037 6 -0.01746 -0.01489 -0.00968 8 -0.00058 -0.00051 -0.00039 10 0.000303 0.000303 0.000307 12 0.000176 0.000174 0.000176 15 2.85E-05 2.65E-05 2.8E-05 20 5.79E-06 5.8E-06 5.78E-06  Table 168. The z-component of the collision-induced dipole moment at an internuclear distance of 2.929 bohr for geometry 15-0-75-30 Intermolecular Distance (bohr)   z (singlet)  z (triplet)  z (quintet) 5  -0.09259 -0.07977 -0.01033 6 -0.01944 -0.01529 -0.00664 8 0.000688 0.000805 0.001015 10 0.000953 0.000954 0.00096 12 0.000479 0.000475 0.000479 15 0.000142 0.000143 0.000143 20 4.38E-05 4.39E-05 4.38E-05  Table 169. The z-component of the collision-induced dipole moment at an internuclear distance of 3.213 bohr for geometry 15-0-75-30 Intermolecular Distance (bohr)   z (singlet)  z (triplet)  z (quintet) 5  -0.52802 -0.12604 0.003248 6 -0.27869 -0.00923 -0.00024 8 -0.20843 0.00164 0.001981 10 -0.20093 0.001402 0.001392 12 -0.19875 0.000682 0.000682 15 -0.19739 0.000219 0.000222 20 -0.19652 7E-05 7E-05      275  Table 170. The z-component of the collision-induced dipole moment at an internuclear distance of 3.3 bohr for geometry 15-0-75-30 Intermolecular Distance (bohr)   z (singlet)  z (triplet)  z (quintet) 5  -0.14985 -0.46116 -0.01022 6 -0.00598 -0.26671 0.00592 8 0.001817 -0.21133 0.00226 10 0.001479 -0.20381 0.001492 12 0.000729 -0.20149 0.000729 15 0.000241 -0.20009 0.00024 20 7.61E-05 -0.19923 7.61E-05  Table 171. The z-component of the collision-induced dipole moment at an internuclear distance of 2 bohr for geometry 15-0-95-10 Intermolecular Distance (bohr)   z (singlet)  z (triplet)  z (quintet) 5  0.034444 0.020291 -0.01609 6 -0.00925 -0.01144 -0.01572 8 -0.00526 -0.00531 -0.0054 10 -0.00204 -0.00204 -0.00204 12 -0.00093 -0.00093 -0.00093 15 -0.00038 -0.00039 -0.00038 20 -0.00012 -0.00012 -0.00012  Table 172. The z-component of the collision-induced dipole moment at an internuclear distance of 2.1 bohr for geometry 15-0-95-10 Intermolecular Distance (bohr)   z (singlet)  z (triplet)  z (quintet) 5  0.015009 0.006193 -0.0169 6 -0.0105 -0.01183 -0.01443 8 -0.00443 -0.00446 -0.0045 10 -0.00162 -0.00162 -0.00163 12 -0.00073 -0.00073 -0.00073 15 -0.00031 -0.00031 -0.00031 20 -0.0001 -0.0001 -0.0001      276  Table 173. The z-component of the collision-induced dipole moment at an internuclear distance of 2.2 bohr for geometry 15-0-95-10 Intermolecular Distance (bohr)   z (singlet)  z (triplet)  z (quintet) 5  -0.00024 -0.00499 -0.01819 6  -0.01131 -0.01199 -0.01338 8 -0.0036 -0.00361 -0.00364 10 -0.00124 -0.00122 -0.00122 12 -0.00054 -0.00054 -0.00054 15 0.047317 -0.00024 -0.00024 20 -7.9E-05 -7.9E-05 -7.9E-05  Table 174. The z-component of the collision-induced dipole moment at an internuclear distance of 2.28187 bohr for geometry 15-0-95-10 Intermolecular Distance (bohr)   z (singlet)  z (triplet)  z (quintet) 5  -0.01093 -0.01284 -0.0194 6  -0.01179 -0.01206 -0.01264 8 -0.00293 -0.00295 -0.00297 10 -0.0009 -0.00091 -0.00091 12 -0.00057 -0.00039 -0.0004 15 -0.00022 -0.00018 -0.00019 20 -6.3E-05 -6.2E-05 -6.2E-05  Table 175. The z-component of the collision-induced dipole moment at an internuclear distance of 2.296 bohr for geometry 15-0-95-10 Intermolecular Distance (bohr)   z (singlet)  z (triplet)  z (quintet) 5  -0.01263 0.019361 -0.01962 6 -0.01199 -0.00637 -0.01252 8 -0.00283 -0.00238 -0.00285 10 -0.00082 -0.00063 -0.00085 12 -0.00037 -0.0003 -0.00037 15 -0.00018 -0.00015 -0.00018 20 -5.9E-05 -6.5E-05 -5.9E-05      277  Table 176. The z-component of the collision-induced dipole moment at an internuclear distance of 2.36 bohr for geometry 15-0-95-10 Intermolecular Distance (bohr)   z (singlet)  z (triplet)  z (quintet) 5  -0.02005 -0.0196 -0.02058 6 -0.01217 -0.01209 -0.012 8 -0.00239 -0.00234 -0.00235 10 -0.00062 -0.00061 -0.00061 12 -0.00026 -0.00025 -0.00026 15 -0.00013 -0.00013 -0.00013 20 -4.6E-05 -4.6E-05 -4.6E-05  Table 177 The z-component of the collision-induced dipole moment at an internuclear distance of 2.457 bohr for geometry 15-0-95-10 Intermolecular Distance (bohr)   z (singlet)  z (triplet)  z (quintet) 5  -0.03033 -0.02721 -0.00877 6 -0.01263 -0.01209 -0.0045 8 -0.00163 -0.00167 -0.00065 10 -0.00027 -0.00026 -0.00011 12 -9.4E-05 -9.3E-05 -3.8E-05 15 -7.3E-05 -7.3E-05 -2.9E-05 20 -2.7E-05 -2.7E-05 -1.1E-05  Table 178. The z-component of the collision-induced dipole moment at an internuclear distance of 2.646 bohr for geometry 15-0-95-10 Intermolecular Distance (bohr)   z (singlet)  z (triplet)  z (quintet) 5  -0.04768 -0.03993 -0.02356 6 -0.01309 -0.012 -0.00993 8 -0.00037 -0.00035 -0.00032 10 0.000353 0.000353 0.000352 12 0.000193 0.000192 0.000193 15 3.49E-05 3.49E-05 3.48E-05 20 8.75E-06 8.78E-06 8.74E-06      278  Table 179. The z-component of the collision-induced dipole moment at an internuclear distance of 2.929 bohr for geometry 15-0-95-10 Intermolecular Distance (bohr)   z (singlet)  z (triplet)  z (quintet) 5  -0.06708 -0.05342 -0.02161 6 -0.01363 -0.01175 -0.00811 8 0.001076 0.001112 0.001172 10 0.001068 0.001074 0.001069 12 0.000528 0.000528 0.000528 15 0.000163 0.000163 0.000163 20 5.16E-05 5.16E-05 5.16E-05  Table 180. The z-component of the collision-induced dipole moment at an internuclear distance of 3.213 bohr for geometry 15-0-95-10 Intermolecular Distance (bohr)   z (singlet)  z (triplet)  z (quintet) 5  -0.07534 -0.05263 -0.01354 6 -0.01353 -0.01085 -0.0053 8 0.001996 0.002053 0.002147 10 0.001531 0.001534 0.001534 12 0.00075 0.00075 0.00075 15 0.00025 0.00025 0.000251 20 8.1E-05 8.1E-05 8.11E-05  Table 181. The z-component of the collision-induced dipole moment at an internuclear distance of 3.3 bohr for geometry 15-0-95-10 Intermolecular Distance (bohr)   z (singlet)  z (triplet)  z (quintet) 5  -0.07207 -0.04574 -0.0169 6 -0.01309 -0.00986 -0.00282 8 0.002209 0.00227 0.00238 10 0.001635 0.001635 0.001639 12 0.0008 0.000801 0.000801 15 0.00027 0.00027 0.000271 20 8.79E-05 8.78E-05 8.79E-05      279  Table 182. The z-component of the collision-induced dipole moment at an internuclear distance of 2 bohr for geometry 10-0-160-15 Intermolecular Distance (bohr)   z (singlet)  z (triplet)  z (quintet) 5  0.365406 0.202099 -0.010841206 6 0.054463 0.03809 0.012776947 8 0.00583 0.005567 0.006065028 10 0.002047 0.002045 0.002651276 12 0.000933 0.000934 0.00130332 15 0.000384 0.000385 0.000556095 20 0.000124 0.000124 0.000179697  Table 183. The z-component of the collision-induced dipole moment at an internuclear distance of 2.1 bohr for geometry 10-0-160-15 Intermolecular Distance (bohr)   z (singlet)  z (triplet)  z (quintet) 5  0.231097 0.111604 -0.046575872 6 0.031765 0.021397 0.004850827 8 0.003772 0.003617 0.003348964 10 0.001366 0.001369 0.001367184 12 0.000617 0.000619 0.000617186 15 0.067721 0.068723 0.068637251 20 7.98E-05 7.97E-05 7.975E-05  Table 184. The z-component of the collision-induced dipole moment at an internuclear distance of 2.2 bohr for geometry 10-0-160-15 Intermolecular Distance (bohr)   z (singlet)  z (triplet)  z (quintet) 5  0.102549 0.020091 -0.096361072 6 0.009832 0.004427 -0.004227737 8 0.001612 0.00154 0.001440999 10 0.000652 0.000647 0.00065081 12 0.000279 0.000279 0.000279186 15 0.11701 0.117924 0.117852914 20 3.39E-05 3.39E-05 3.39265E-05       280  Table 185. The z-component of the collision-induced dipole moment at an internuclear distance of 2.28187 bohr for geometry 10-0-160-15 Intermolecular Distance (bohr)   z (singlet)  z (triplet)  z (quintet) 5  -0.04113 -0.06164 -0.146201016 6 -0.00856 -0.00957 -0.012720113 8 -0.00024 -0.00024 -0.000211757 10 5.25E-05 3.6E-05 4.7051E-05 12 -0.00016 -3.4E-06 -3.79148E-06 15 8.16E-05 -1.5E-05 -9.6405E-06 20 0.000162 -6.6E-06 -3.7785E-06  Table 186. The z-component of the collision-induced dipole moment at an internuclear distance of 2.296 bohr for geometry 10-0-160-15 Intermolecular Distance (bohr)   z (singlet)  z (triplet)  z (quintet) 5  -0.01892 -0.06966 -0.155560463 6 -0.01175 -0.01255 -0.014269185 8 -0.00064 -0.00054 -0.000502449 10 -7.3E-05 -6.1E-05 -5.72165E-05 12 -5.9E-05 -5.3E-05 -5.3132E-05 15 -3.1E-05 -3.1E-05 -3.1221E-05 20 -1E-05 -1E-05 -1.0256E-05  Table 187. The z-component of the collision-induced dipole moment at an internuclear distance of 2.36 bohr for geometry 10-0-160-15 Intermolecular Distance (bohr)   z (singlet)  z (triplet)  z (quintet) 5  -0.12903 -0.06966 -0.200464723 6 -0.0243 -0.01255 -0.021580441 8 -0.00194 -0.00054 -0.001819628 10 -0.00053 -6.1E-05 -0.000526257 12 -0.00027 -5.3E-05 -0.000272184 15 -0.00012 -3.1E-05 -0.000121481 20 -3.9E-05 -1E-05 -3.90655E-05      281  Table 188. The z-component of the collision-induced dipole moment at an internuclear distance of 2.457 bohr for geometry 10-0-160-15 Intermolecular Distance (bohr)   z (singlet)  z (triplet)  z (quintet) 5  -0.21439 -0.21414 -0.276151875 6 -0.04974 -0.04287 -0.033622997 8 -0.00416 -0.00403 -0.003789075 10 -0.00125 -0.00122 -0.001208922 12 -0.00059 -0.00058 -0.000588761 15 -0.00025 -0.00025 -0.000251332 20 -8E-05 -8E-05 -8.0317E-05  Table 189. The z-component of the collision-induced dipole moment at an internuclear distance of 2.646 bohr for geometry 10-0-160-15 Intermolecular Distance (bohr)   z (singlet)  z (triplet)  z (quintet) 5  -0.41577 -0.40002 -0.458151182 6 -0.09931 -0.08351 -0.062456951 8 -0.00804 -0.00779 -0.007312326 10 -0.00237 -0.00236 -0.002352078 12 -0.00111 -0.00111 -0.001111405 15 -0.00046 -0.00046 -0.000463808 20 -0.00015 -0.00015 -0.000147402  Table 190. The z-component of the collision-induced dipole moment at an internuclear distance of 2.929 bohr for geometry 10-0-160-15 Intermolecular Distance (bohr)   z (singlet)  z (triplet)  z (quintet) 5  -0.00029 -0.50865 -0.54798761 6 2.15E-06 -0.16958 -0.132947468 8 -3.1E-08 -0.01253 -0.011730657 10 -9.9E-10 -0.00354 -0.00351567 12 5.12E-10 -0.00163 -0.001631469 15 2.98E-09 -0.00067 -0.000674148 20 -4.8E-10 -0.00021 -0.000212946      282  Table 191. The z-component of the collision-induced dipole moment at an internuclear distance of 3.213 bohr for geometry 10-0-160-15 Intermolecular Distance (bohr)   z (singlet)  z (triplet)  z (quintet) 5  -0.18852 0.029837 -1.93008005 6 -0.32304 0.433593 -0.251160581 8 -0.0174 0.732041 -0.015631827 10 -0.00422 0.728828 -0.004185123 12 -0.00193 0.722891 -0.00192752 15 -0.0008 0.718157 -0.000797123 20 -0.00025 0.714536 -0.000251578  Table 192. The z-component of the collision-induced dipole moment at an internuclear distance of 3.3 bohr for geometry 10-0-160-15 Intermolecular Distance (bohr)   z (singlet)  z (triplet)  z (quintet) 5  -0.23929 -0.00366 0.166188261 6 -0.36813 0.384309 -0.305979138 8 -0.01862 0.74055 -0.01665117 10 -0.00435 0.738684 -0.004312118 12 -0.00199 0.732818 -0.001984403 15 -0.00082 0.727945 -0.000821525 20 -0.00026 0.724468 -0.000259672  Table 193. The z-component of the collision-induced dipole moment at an internuclear distance of 2 bohr for geometry 20-0-145-25 Intermolecular Distance (bohr)   z (singlet)  z (triplet)  z (quintet) 5  0.282384 0.161716 -0.03608645 6  0.030805 0.019108 -0.00068699 8 0.003417 0.003215 0.002849082 10 0.001327 0.001323 0.001320641 12 0.000571 0.000573 0.000573525 15 0.000222 0.000221 0.000222171 20 7.45E-05 7.45E-05 7.4466E-05      283  Table 194. The z-component of the collision-induced dipole moment at an internuclear distance of 2.1 bohr for geometry 20-0-145-25 Intermolecular Distance (bohr)   z (singlet)  z (triplet)  z (quintet) 5  0.173167 0.084347 -0.06598896 6  0.014504 0.006983 -0.00594467 8 0.002079 0.001973 0.001787188 10 0.000893 0.00088 0.000881377 12 0.000373 0.000375 0.000373269 15 0.067628 0.068629 0.068543665 20 4.49E-05 4.48E-05 4.48475E-05  Table 195. The z-component of the collision-induced dipole moment at an internuclear distance of 2.2 bohr for geometry 20-0-145-25 Intermolecular Distance (bohr)   z (singlet)  z (triplet)  z (quintet) 5  0.071043 0.008275 -0.10434333 6  -0.00139 -0.00553 -0.01281815 8 0.000399 0.000612 0.000564775 10 0.000406 0.000417 0.000409936 12 0.000154 0.000155 0.000155531 15 0.116954 0.117867 0.117796404 20 1.42E-05 1.42E-05 1.42375E-05  Table 196. The z-component of the collision-induced dipole moment at an internuclear distance of 2.28187 bohr for geometry 20-0-145-25 Intermolecular Distance (bohr)   z (singlet)  z (triplet)  z (quintet) 5  -0.01031 -0.05433845 -0.14083996 6  -0.01453 -0.01617649 -0.01935133 8 -0.00057 -0.00055559 -0.00050836 10 -3.6E-05 9.23802E-06 1.3633E-05 12 0.000191 -3.3215E-05 -2.8163E-05 15 -4.4E-05 -2.7751E-05 -3.4404E-05 20 -8.8E-06 -7.4025E-06 -1.0908E-05      284  Table 197. The z-component of the collision-induced dipole moment at an internuclear distance of 2.296 bohr for geometry 20-0-145-25 Intermolecular Distance (bohr)   z (singlet)  z (triplet)  z (quintet) 5  -0.02417 -0.06512 -0.14747054 6  -0.01688 -0.01805 -0.02054002 8 -0.00086 -0.00076 -0.00069694 10 -8.3E-05 -6.7E-05 -5.4927E-05 12 -5.5E-05 -5.6E-05 -6.0237E-05 15 -4.4E-05 -4.4E-05 -4.4146E-05 20 -1.5E-05 -1.5E-05 -1.522E-05  Table 198. The z-component of the collision-induced dipole moment at an internuclear distance of 2.36 bohr for geometry 20-0-145-25 Intermolecular Distance (bohr)   z (singlet)  z (triplet)  z (quintet) 5  -0.08596 -0.11335 -0.17818552 6  -0.02715 -0.02659 -0.02607067 8 -0.00187 -0.00168 -0.00155102 10 -0.00036 -0.00037 -0.00036072 12 -0.00021 -0.0002 -0.00020226 15 -0.0001 -0.0001 -0.00010316 20 -3.4E-05 -3.4E-05 -3.4368E-05  Table 199. The z-component of the collision-induced dipole moment at an internuclear distance of 2.457 bohr for geometry 20-0-145-25 Intermolecular Distance (bohr)   z (singlet)  z (triplet)  z (quintet) 5  -0.17446 -0.18221 -0.22516202 6  -0.04283 -0.03952 -0.03470479 8 -0.00316 -0.00304 -0.00280974 10 -0.00081 -0.00082 -0.00079983 12 -0.00041 -0.0004 -0.0004058 15 -0.00019 -0.00019 -0.00018701 20 -6.2E-05 -6.2E-05 -6.155E-05      285  Table 200. The z-component of the collision-induced dipole moment at an internuclear distance of 2.646 bohr for geometry 20-0-145-25 Intermolecular Distance (bohr)   z (singlet)  z (triplet)  z (quintet) 5  -0.31147 -0.29181 -0.32161457 6  -0.07215 -0.06396 -0.05154333 8 -0.00553 -0.00533 -0.0049335 10 -0.00153 -0.00152 -0.00150479 12 -0.00073 -0.00073 -0.00073042 15 -0.00032 -0.00032 -0.00032083 20 -0.0001 -0.0001 -0.0001046  Table 201. The z-component of the collision-induced dipole moment at an internuclear distance of 2.929 bohr for geometry 20-0-145-25 Intermolecular Distance (bohr)   z (singlet)  z (triplet)  z (quintet) 5  -0.49833 -0.50683 -0.53987267 6  -0.11257 -0.09905 -0.07718967 8 -0.00813 -0.00779 -0.00716317 10 -0.00217 -0.00216 -0.00214036 12 -0.00102 -0.00102 -0.00102155 15 -0.00044 -0.00044 -0.00044285 20 -0.00014 -0.00014 -0.000144  Table 202. The z-component of the collision-induced dipole moment at an internuclear distance of 3.213 bohr for geometry 20-0-145-25 Intermolecular Distance (bohr)   z (singlet)  z (triplet)  z (quintet) 5  -0.44862 -0.45963 -1.49121727 6  -0.15138 -0.13409 -0.11373172 8 -0.00981 -0.00937 -0.00850087 10 -0.00246 -0.00245 -0.00241743 12 -0.00116 -0.00116 -0.00115631 15 -0.00051 -0.00051 -0.00050522 20 -0.00017 -0.00017 -0.0001655      286  Table 203. The z-component of the collision-induced dipole moment at an internuclear distance of 3.3 bohr for geometry 20-0-145-25 Intermolecular Distance (bohr)   z (singlet)  z (triplet)  z (quintet) 5  -1.35945 -0.4794 -1.45079488 6  -0.1818 -0.16263 -0.14369502 8 -0.00989 -0.00941 -0.00844852 10 -0.00248 -0.00247 -0.00244262 12 -0.00118 -0.00117 -0.00117387 15 -0.00052 -0.00052 -0.00051588 20 -0.00017 -0.00017 -0.00016968  Table 204. The z-component of the collision-induced dipole moment at an internuclear distance of 2 bohr for geometry 30-0-130-35 Intermolecular Distance (bohr)   z (singlet)  z (triplet)  z (quintet) 5  0.148445 0.102646 -0.023732643 6  0.012617 0.006031 -0.006345359 8 0.000708 0.00055 0.000263747 10 0.000314 0.000312 0.000306958 12 0.000113 0.00011 0.000110506 15 3.04E-05 3.53E-05 3.0443E-05 20 1.37E-05 1.38E-05 1.3729E-05  Table 205. The z-component of the collision-induced dipole moment at an internuclear distance of 2.1 bohr for geometry 30-0-130-35 Intermolecular Distance (bohr)   z (singlet)  z (triplet)  z (quintet) 5  0.081429 0.049905 -0.038740809 6  0.003354 -0.00073 -0.008497206 8 0.000163 7.59E-05 -6.6192E-05 10 0.000143 0.000154 0.000152744 12 3.92E-05 5.03E-05 4.8316E-05 15 0.067512 0.068514 0.068428131 20 2.07E-06 2.03E-06 2.05949E-06      287  Table 206. The z-component of the collision-induced dipole moment at an internuclear distance of 2.2 bohr for geometry 30-0-130-35 Intermolecular Distance (bohr)   z (singlet)  z (triplet)  z (quintet) 5  0.019752 -0.00055 -0.057892688 6  -0.00568 -0.00777 -0.011772396 8 -0.00042 -0.00048 -0.000510123 10 -2.5E-05 -3E-05 -2.073E-05 12 -2.7E-05 -2.7E-05 -2.67445E-05 15 0.116879 0.117792 0.117721846 20 -1E-05 -1E-05 -1.0132E-05  Table 207. The z-component of the collision-induced dipole moment at an internuclear distance of 2.28187 bohr for geometry 30-0-130-35 Intermolecular Distance (bohr)   z (singlet)  z (triplet)  z (quintet) 5  -0.02876 -0.04139223 -0.075747379 6  -0.01307 -0.01373987 -0.015005416 8 -0.00106 -0.000967 -0.000920926 10 -0.00021 -0.00017078 -0.000168553 12 8.38E-05 -0.00010076 -9.2773E-05 15 -0.0001 -6.2303E-05 -6.10115E-05 20 -2E-05 -1.7043E-05 -2.01845E-05  Table 208. The z-component of the collision-induced dipole moment at an internuclear distance of 2.296 bohr for geometry 30-0-130-35 Intermolecular Distance (bohr)   z (singlet)  z (triplet)  z (quintet) 5  -0.03693 -0.04837 -0.078938109 6  -0.01437 -0.01478 -0.01559169 8 -0.00102 -0.00104 -0.000993836 10 -0.00021 -0.0002 -0.000194149 12 -0.00012 -1E-04 -0.000104523 15 -6.5E-05 -6.5E-05 -6.45905E-05 20 -2.2E-05 -2.2E-05 -2.18875E-05      288  Table 209. The z-component of the collision-induced dipole moment at an internuclear distance of 2.36 bohr for geometry 30-0-130-35 Intermolecular Distance (bohr)   z (singlet)  z (triplet)  z (quintet) 5  -0.03693 -0.0794 -0.093429824 6  -0.01437 -0.01945 -0.018291343 8 -0.00102 -0.00144 -0.001322688 10 -0.00021 -0.00031 -0.000307067 12 -0.00012 -0.00016 -0.000156059 15 -6.5E-05 -8.7E-05 -8.67525E-05 20 -2.2E-05 -2.9E-05 -2.94165E-05  Table 210. The z-component of the collision-induced dipole moment at an internuclear distance of 2.457 bohr for geometry 30-0-130-35 Intermolecular Distance (bohr)   z (singlet)  z (triplet)  z (quintet) 5  -0.12266 -0.12266 -0.114095128 6  -0.02828 -0.02623 -0.022266435 8 -0.00204 -0.00198 -0.001792772 10 -0.00048 -0.00047 -0.000463352 12 -0.00023 -0.00022 -0.000228053 15 -0.00012 -0.00012 -0.000117711 20 -4E-05 -4E-05 -3.9793E-05  Table 211. The z-component of the collision-induced dipole moment at an internuclear distance of 2.646 bohr for geometry 30-0-130-35 Intermolecular Distance (bohr)   z (singlet)  z (triplet)  z (quintet) 5  -0.18789 -0.1807 -0.143020098 6  -0.04164 -0.03713 -0.028342025 8 -0.00298 -0.00281 -0.002479804 10 -0.0007 -0.00069 -0.000679518 12 -0.00033 -0.00032 -0.000328992 15 -0.00016 -0.00016 -0.000161842 20 -5.5E-05 -5.4E-05 -5.45315E-05       289  Table 212. The z-component of the collision-induced dipole moment at an internuclear distance of 2.929 bohr for geometry 30-0-130-35 Intermolecular Distance (bohr)   z (singlet)  z (triplet)  z (quintet) 5  -0.26254 -0.26849 -0.215726009 6  -0.05157 -0.04507 -0.030981135 8 -0.00365 -0.0034 -0.002891027 10 -0.00081 -0.0008 -0.000780367 12 -0.00038 -0.00038 -0.000380242 15 -0.00019 -0.00019 -0.000188196 20 -6.4E-05 -6.4E-05 -6.39085E-05  Table 213. The z-component of the collision-induced dipole moment at an internuclear distance of 3.213 bohr for geometry 30-0-130-35 Intermolecular Distance (bohr)   z (singlet)  z (triplet)  z (quintet) 5  -0.36629 -0.37101 -0.361711346 6  -0.04616 -0.03717 -0.019628479 8 -0.00366 -0.00335 -0.002686595 10 -0.00076 -0.00075 -0.000722282 12 -0.00037 -0.00037 -0.000365951 15 -0.00019 -0.00019 -0.000190241 20 -6.7E-05 -6.7E-05 -6.6446E-05  Table 214. The z-component of the collision-induced dipole moment at an internuclear distance of 3.3 bohr for geometry 30-0-130-35 Intermolecular Distance (bohr)   z (singlet)  z (triplet)  z (quintet) 5  -0.3923 -0.39421 -0.401692429 6  -0.05801 -0.04512 -0.025381138 8 -0.0032 -0.00286 -0.002215044 10 -0.00071 -0.0007 -0.000673055 12 -0.00035 -0.00035 -0.000351641 15 -0.00019 -0.00019 -0.000187686 20 -6.7E-05 -6.7E-05 -6.64425E-05      290  Table 215. The z-component of the collision-induced dipole moment at an internuclear distance of 2 bohr for geometry 30-0-60-0 Intermolecular Distance (bohr)   z (singlet)  z (triplet)  z (quintet) 5  0.092503 0.077519 -0.01131 6  0.005819 0.001093 -0.01236 8 -0.00258 -0.00278 -0.00325 10 -0.00104 -0.00107 -0.00106 12 -0.00047 -0.00048 -0.00047 15 -0.00019 -0.00019 -0.00019 20 -5.9E-05 -6E-05 -5.7E-05  Table 216. The z-component of the collision-induced dipole moment at an internuclear distance of 2.1 bohr for geometry 30-0-60-0 Intermolecular Distance (bohr)   z (singlet)  z (triplet)  z (quintet) 5  0.042792 0.032481 -0.01224 6  -0.00113 -0.00448 -0.0112 8 -0.00237 -0.00249 -0.00273 10 -0.00084 -0.00085 -0.00085 12 -0.00038 -0.00037 -0.00037 15 -0.00016 -0.00016 -0.00016 20 -4.8E-05 -4.9E-05 -4.8E-05  Table 217. The z-component of the collision-induced dipole moment at an internuclear distance of 2.2 bohr for geometry 30-0-60-0 Intermolecular Distance (bohr)   z (singlet)  z (triplet)  z (quintet) 5  0.017292 0.013124 -0.00764 6  -0.00384 -0.00534 -0.00851 8 -0.00204 -0.00209 -0.00219 10 -0.00068 -0.00069 -0.00069 12 -0.00029 -0.00029 -0.00029 15 -0.00013 -0.00013 -0.00013 20 -4.2E-05 -4.2E-05 -4.2E-05      291  Table 218. The z-component of the collision-induced dipole moment at an internuclear distance of 2.28187 bohr for geometry 30-0-60-0 Intermolecular Distance (bohr)   z (singlet)  z (triplet)  z (quintet) 5  -0.01016 - -0.00879 6  -0.0075 - -0.00796 8 -0.00182 - -0.00183 10 -0.00054 - -0.00056 12 -0.00022 - -0.00023 15 -0.00012 - -0.00011 20 -3.4E-05 - -3.5E-05  Table 219. The z-component of the collision-induced dipole moment at an internuclear distance of 2.36 bohr for geometry 30-0-60-0 Intermolecular Distance (bohr)   z (singlet)  z (triplet)  z (quintet) 5  -0.03508 -0.03015 -0.01025 6  -0.01082 -0.00975 -0.0076 8 -0.00162 -0.00158 -0.00152 10 -0.00041 -0.00041 -0.0004 12 -0.00015 -0.00016 -0.00016 15 -8.4E-05 -8.3E-05 -8.3E-05 20 -2.9E-05 -2.9E-05 -2.9E-05  Table 220.The z-component of the collision-induced dipole moment at an internuclear distance of 2.457 bohr for geometry 30-0-60-0 Intermolecular Distance (bohr)   z (singlet)  z (triplet)  z (quintet) 5  -0.08142 -0.05568 -0.01247 6  -0.01768 -0.01235 -0.0073 8 -0.00142 -0.00129 -0.00114 10 -0.0002 -0.00024 -0.00024 12 -6.5E-05 -8.7E-05 -8.7E-05 15 -4.7E-05 -5.6E-05 -5.6E-05 20 -1.7E-05 -2E-05 -2E-05       292  Table 221. The z-component of the collision-induced dipole moment at an internuclear distance of 2.646 bohr for geometry 30-0-60-0 Intermolecular Distance (bohr)   z (singlet)  z (triplet)  z (quintet) 5  -0.12097 -0.10569 -0.01769 6  -0.02236 -0.01742 -0.00701 8 -0.00093 -0.00077 -0.00047 10 4.91E-05 5.49E-05 6.36E-05 12 4.58E-05 4.59E-05 4.62E-05 15 -7.7E-06 -7.7E-06 -7.7E-06 20 -5.2E-06 -5.2E-06 -5.2E-06  Table 222. The z-component of the collision-induced dipole moment at an internuclear distance of 2.929 bohr for geometry 30-0-60-0 Intermolecular Distance (bohr)   z (singlet)  z (triplet)  z (quintet) 5  -0.1939 -0.17598 -0.02555 6  -0.03261 -0.02443 -0.00657 8 -0.00039 -0.00013 0.000392 10 0.000425 0.000435 0.00045 12 0.000213 0.000197 0.000213 15 5.53E-05 5.75E-05 5.44E-05 20 1.46E-05 1.47E-05 1.47E-05  Table 223. The z-component of the collision-induced dipole moment at an internuclear distance of 3.213 bohr for geometry 30-0-60-0 Intermolecular Distance (bohr)   z (singlet)  z (triplet)  z (quintet) 5  -0.23724 -0.23004 -0.02601 6  -0.03857 -0.02849 -0.00455 8 0.000127 0.000406 0.001128 10 0.000718 0.00073 0.000751 12 0.000339 0.00034 0.000341 15 0.000102 0.000103 0.000102 20 2.97E-05 3.01E-05 2.98E-05      293  Table 224. The z-component of the collision-induced dipole moment at an internuclear distance of 3.3 bohr for geometry 30-0-60-0 Intermolecular Distance (bohr)   z (singlet)  z (triplet)  z (quintet) 5  -0.24585 -0.24547 -0.02452 6  -0.0381 -0.02783 -0.00294 8 0.000248 0.000656 0.001373 10 0.000791 0.000802 0.000825 12 0.000371 0.000373 0.000347 15 0.000113 0.000115 8.72E-05 20 3.33E-05 3.34E-05 2.32E-05  Table 225. The z-component of the collision-induced dipole moment at an internuclear distance of 2 bohr for geometry 30-0-90-0 Intermolecular Distance (bohr)   z (singlet)  z (triplet)  z (quintet) 5  0.006464679 0.005642 0.005368 6  -0.010010896 -0.01015 -0.01034 8 -0.004928792 -0.00496 -0.005 10 -0.001931991 -0.00193 -0.00194 12 -0.000864529 -0.00086 -0.00086 15 -0.000345852 -0.00035 -0.00035 20 -0.000109145 -0.00011 -0.00011  Table 226. The z-component of the collision-induced dipole moment at an internuclear distance of 2.1 bohr for geometry 30-0-90-0 Intermolecular Distance (bohr)   z (singlet)  z (triplet)  z (quintet) 5  0.001886712 0.002995 0.006008 6  -0.009435215 -0.00928 -0.00891 8 -0.004145383 -0.00416 -0.00418 10 -0.001556671 -0.00157 -0.00156 12 -0.000689202 -0.00069 -0.00069 15 -0.000282595 -0.00028 -0.00028 20 -8.9909E-05 -9E-05 -9E-05      294  Table 227. The z-component of the collision-induced dipole moment at an internuclear distance of 2.2 bohr for geometry 30-0-90-0 Intermolecular Distance (bohr)   z (singlet)  z (triplet)  z (quintet) 5  -0.001750832 0.000858 0.008249 6  -0.008816007 -0.00842 -0.00681 8 -0.003371818 -0.00338 -0.00313 10 -0.001192453 -0.00119 -0.00111 12 -0.000520875 -0.00052 -0.00049 15 -0.000220798 -0.00022 -0.00021 20 -7.08165E-05 -7.1E-05 -6.6E-05  Table 228. The z-component of the collision-induced dipole moment at an internuclear distance of 2.28187 bohr for geometry 30-0-90-0 Intermolecular Distance (bohr)   z (singlet)  z (triplet)  z (quintet) 5  -0.004273687 -0.00064 0.006752 6  -0.008305458 -0.00774 -0.00661 8 -0.002756501 -0.00276 -0.00276 10 -0.000903478 -0.0009 -0.00091 12 -0.000386785 -0.00039 -0.00039 15 -0.000172343 -0.00017 -0.00017 20 -5.525E-05 -5.5E-05 -5.5E-05  Table 229. The z-component of the collision-induced dipole moment at an internuclear distance of 2.296 bohr for geometry 30-0-90-0 Intermolecular Distance (bohr)   z (singlet)  z (triplet)  z (quintet) 5  -0.003788305 -0.00088 0.006794 6  -0.007409649 -0.00763 -0.00645 8 -0.002301896 -0.00266 -0.00266 10 -0.000737951 -0.00086 -0.00086 12 -0.000319215 -0.00037 -0.00036 15 -0.000145981 -0.00016 -0.00016 20 -4.65545E-05 -5.3E-05 -5.3E-05      295  Table 230. The z-component of the collision-induced dipole moment at an internuclear distance of 2.36 bohr for geometry 30-0-90-0 Intermolecular Distance (bohr)   z (singlet)  z (triplet)  z (quintet) 5  -0.006408292 0.006962 0.006794 6  -0.007818475 -0.00571 -0.00645 8 -0.002190451 -0.00218 -0.00266 10 -0.000643865 -0.00064 -0.00086 12 -0.000264425 -0.00026 -0.00036 15 -0.000125593 -0.00013 -0.00016 20 -4.0724E-05 -4.1E-05 -5.3E-05  Table 231. The z-component of the collision-induced dipole moment at an internuclear distance of 2.457 bohr for geometry 30-0-90-0 Intermolecular Distance (bohr)   z (singlet)  z (triplet)  z (quintet) 5  -0.008752539 -0.00333 0.007154 6  -0.00721826 -0.00635 -0.00465 8 -0.001521297 -0.0015 -0.00149 10 -0.000324489 -0.00033 -0.00033 12 -0.000118953 -0.00012 -0.00012 15 -7.0579E-05 -7.1E-05 -7.1E-05 20 -2.2991E-05 -2.3E-05 -2.3E-05  Table 232. The z-component of the collision-induced dipole moment at an internuclear distance of 2.646 bohr for geometry 30-0-90-0 Intermolecular Distance (bohr)   z (singlet)  z (triplet)  z (quintet) 5  -0.011382545 -0.00568 0.007265 6  -0.004835466 -0.00495 -0.00275 8 0.000331852 -0.00028 -0.00025 10 0.000456514 0.00024 0.000239 12 0.000218044 0.00014 0.00014 15 6.0548E-05 2.84E-05 2.84E-05 20 2.16635E-05 9.6E-06 9.6E-06      296  Table 233. The z-component of the collision-induced dipole moment at an internuclear distance of 2.929 bohr for geometry 30-0-90-0 Intermolecular Distance (bohr)   z (singlet)  z (triplet)  z (quintet) 5  -0.017371782 -0.0088 0.00648 6  -0.004598694 -0.00317 -0.00042 8 0.001171391 0.001221 0.001283 10 0.000899784 0.00093 0.000929 12 0.000455144 0.00046 0.000454 15 0.000151605 0.00015 0.000152 20 5.1268E-05 5.13E-05 5.13E-05  Table 234. The z-component of the collision-induced dipole moment at an internuclear distance of 3.213 bohr for geometry 30-0-90-0 Intermolecular Distance (bohr)   z (singlet)  z (triplet)  z (quintet) 5  -0.021616919 -0.01206 0.010649 6  -0.003566443 -0.00194 0.002831 8 0.002212485 0.002209 0.003268 10 0.001403526 0.001404 0.001722 12 0.000674036 0.000675 0.000781 15 0.000240207 0.000161 0.000288 20 8.17755E-05 8.21E-05 0.000102  Table 235. The z-component of the collision-induced dipole moment at an internuclear distance of 3. 3 bohr for geometry 30-0-90-0 Intermolecular Distance (bohr)   z (singlet)  z (triplet)  z (quintet) 5  -0.022959014 -0.01318 0.00357 6  -0.003265422 -0.00163 0.001423 8 0.002428736 0.002475 0.002571 10 0.001503007 0.001504 0.001506 12 0.00072073 0.000722 0.000719 15 0.000259939 0.000261 0.000252 20 8.8543E-05 8.87E-05 8.52E-05      297  Table 236. The z-component of the collision-induced dipole moment at an internuclear distance of 2 bohr for geometry 45-0-30-0 Intermolecular Distance (bohr)   z (singlet)  z (triplet)  z (quintet) 5  0.178755 0.145031778 0.030725 6  0.03208 0.024874276 0.008939 8 0.003008 0.002756865 0.001995 10 0.000889 0.000394083 0.000811 12 0.000393 6.2951E-05 0.000368 15 0.000182 0.000183022 0.000173 20 6.3E-05 3.99899E-06 6.12E-05  Table 237. The z-component of the collision-induced dipole moment at an internuclear distance of 2.1 bohr for geometry 45-0-30-0 Intermolecular Distance (bohr)   z (singlet)  z (triplet)  z (quintet) 5  0.093682 0.077568598 0.025297 6  0.018528 0.014727622 0.007656 8 0.00187 0.001732602 0.001475 10 0.000588 0.000574586 0.000569 12 0.000259 0.000264504 0.000259 15 0.000124 0.000123255 0.000124 20 4.37E-05 4.2973E-05 4.36E-05  Table 238. The z-component of the collision-induced dipole moment at an internuclear distance of 2.2 bohr for geometry 45-0-30-0 Intermolecular Distance (bohr)   z (singlet)  z (triplet)  z (quintet) 5  0.05363 0.046814083 0.026599 6  0.013182 0.011639747 0.008751 8 0.001477 0.001409576 0.001291 10 0.000408 0.000410402 0.000398 12 0.000176 0.000181145 0.00018 15 8.78E-05 8.76935E-05 8.79E-05 20 2.99E-05 2.9925E-05 2.99E-05      298  Table 239. The z-component of the collision-induced dipole moment at an internuclear distance of 2.28187 bohr for geometry 45-0-30-0 Intermolecular Distance (bohr)   z (singlet)  z (triplet)  z (quintet) 5  0.00386 0.005388901 0.019562 6  0.005415 0.005846457 0.006859 8 0.000815 0.00083474 0.000807 10 0.000199 0.000204036 0.000206 12 0.000103 8.8184E-05 0.000108 15 2.7E-05 3.43535E-05 3.81E-05 20 2.62E-05 2.9053E-05 3.12E-05  Table 240. The z-component of the collision-induced dipole moment at an internuclear distance of 2.296 bohr for geometry 45-0-30-0 Intermolecular Distance (bohr)   z (singlet)  z (triplet)  z (quintet) 5  -0.00481 -0.001978181 0.006205 6  0.004025 0.004807257 0.002568 8 0.000725 0.000705883 0.000209 10 0.000164 0.000159242 5.08E-05 12 7E-05 7.0032E-05 1.99E-05 15 4.13E-05 4.03675E-05 2.16E-05 20 1.41E-05 1.4084E-05 9.17E-06  Table 241. The z-component of the collision-induced dipole moment at an internuclear distance of 2.36 bohr for geometry 45-0-30-0 Intermolecular Distance (bohr)   z (singlet)  z (triplet)  z (quintet) 5  -0.04449 -0.036251565 0.011286 6  -0.00234 -6.34905E-05 0.004646 8 0.00018 0.000229283 0.000322 10 1.84E-06 -1.53545E-05 1.26E-05 12 -9E-06 -5.41201E-06 -5.8E-06 15 8.51E-06 8.81099E-06 8.69E-06 20 3.63E-06 3.62201E-06 3.6E-06      299  Table 242. The z-component of the collision-induced dipole moment at an internuclear distance of 2.457 bohr for geometry 45-0-30-0 Intermolecular Distance (bohr)   z (singlet)  z (triplet)  z (quintet) 5  -0.10597 -0.091053242 -0.0011 6  -0.01264 -0.008023429 0.001398 8 -0.00063 -0.000515618 -0.00031 10 -0.00025 -0.000248519 -0.00024 12 -0.00012 -0.00011788 -0.00012 15 -3.9E-05 -3.88225E-05 -3.9E-05 20 -1.2E-05 -1.19295E-05 -1.2E-05  Table 243. The z-component of the collision-induced dipole moment at an internuclear distance of 2.646 bohr for geometry 45-0-30-0 Intermolecular Distance (bohr)   z (singlet)  z (triplet)  z (quintet) 5  -0.22711 -0.203606575 -0.01592 6  -0.03513 -0.025815259 -0.00159 8 -0.00224 -0.002007177 -0.00118 10 -0.00071 -0.000698241 -0.00066 12 -0.00033 -0.000332158 -0.00032 15 -0.00013 -0.000127566 -0.00013 20 -4.1E-05 -4.0581E-05 -4.1E-05  Table 244. The z-component of the collision-induced dipole moment at an internuclear distance of 2.929 bohr for geometry 45-0-30-0 Intermolecular Distance (bohr)   z (singlet)  z (triplet)  z (quintet) 5  -0.37964 -0.314454281 -0.09206 6  -0.07482 -0.050449264 -0.02061 8 -0.00463 -0.003738215 -0.00338 10 -0.00131 - -0.00127 12 -0.00063 -0.000613394 -0.00061 15 -0.00024 -0.00024299 -0.00024 20 -7.7E-05 -7.26905E-05 -7.7E-05      300  Table 245. The z-component of the collision-induced dipole moment at an internuclear distance of 3.213 bohr for geometry 45-0-30-0 Intermolecular Distance (bohr)   z (singlet)  z (triplet)  z (quintet) 5  -0.42386 -0.387211069 -0.14453 6  -0.11589 -0.094386138 -0.03577 8 -0.0067 -0.006065344 -0.00486 10 -0.00175 -0.001781796 -0.0017 12 -0.00081 -0.000807321 -0.00081 15 -0.00033 -0.000326461 -0.00033 20 -0.0001 -0.000104431 -0.0001  Table 246. The z-component of the collision-induced dipole moment at an internuclear distance of 3.3 bohr for geometry 45-0-30-0 Intermolecular Distance (bohr)   z (singlet)  z (triplet)  z (quintet) 5  -0.41054 -0.374806757 -0.1889 6  -0.12579 -0.102082999 -0.03943 8 -0.00726 -0.006526824 -0.00521 10 -0.00184 -0.001820741 -0.00174 12 -0.00085 -0.000849412 -0.00083 15 -0.00035 -0.000344841 -0.00032 20 -0.00011 -0.000110314 -0.00013  Table 247. The z-component of the collision-induced dipole moment at an internuclear distance of 2 bohr for geometry 45-0-60-0 Intermolecular Distance (bohr)   z (singlet)  z (triplet)  z (quintet) 5  0.045842 0.03457 0.004436 6  0.003611 0.00052 -0.00556 8 -0.00199 -0.00215 -0.00244 10 -0.00087 -0.00087 -0.00089 12 -0.00039 -0.00039 -0.00039 15 -0.00015 -0.00016 -0.00015 20 -4.5E-05 -4.5E-05 -4.5E-05      301  Table 248. The z-component of the collision-induced dipole moment at an internuclear distance of 2.1 bohr for geometry 45-0-60-0 Intermolecular Distance (bohr)   z (singlet)  z (triplet)  z (quintet) 5  0.021409 0.016003 0.002371 6  -0.00039 -0.00206 -0.00519 8 -0.00169 -0.00179 -0.00195 10 -0.00065 -0.00065 -0.00066 12 -0.00026 -0.0003 -0.00029 15 -0.00011 -0.00011 -0.00011 20 -3.4E-05 -3.4E-05 -3.4E-05  Table 249. The z-component of the collision-induced dipole moment at an internuclear distance of 2.2 bohr for geometry 45-0-60-0 Intermolecular Distance (bohr)   z (singlet)  z (triplet)  z (quintet) 5  0.01148 0.010136 0.00197 6  -0.00097 -0.00155 -0.00419 8 -0.00129 -0.00132 -0.00145 10 -0.00048 -0.00048 -0.00046 12 -0.00021 -0.0002 -0.00019 15 -8.4E-05 -8.3E-05 -7.9E-05 20 -2.6E-05 -2.6E-05 -2.4E-05  Table 250. The z-component of the collision-induced dipole moment at an internuclear distance of 2.28187 bohr for geometry 45-0-60-0 Intermolecular Distance (bohr)   z (singlet)  z (triplet)  z (quintet) 5  0.120412 0.001588 0.006317 6  0.005561 -0.00228 -0.00186 8 0.006482 -0.001 -0.001 10 0.002691 -0.00033 -0.00033 12 0.001256 -0.00013 -0.00013 15 0.000505 -6.4E-05 -6.2E-05 20 0.000159 -1.8E-05 -1.8E-05      302  Table 251. The z-component of the collision-induced dipole moment at an internuclear distance of 2.296 bohr for geometry 45-0-60-0 Intermolecular Distance (bohr)   z (singlet)  z (triplet)  z (quintet) 5  -0.00224 0.000149 0.00629 6  -0.00274 -0.0024 -0.0017 8 -0.00095 -0.00094 -0.00094 10 -0.00029 -0.0003 -0.0003 12 -0.00012 -0.00012 -0.00012 15 -5.4E-05 -5.3E-05 -5.4E-05 20 -1.7E-05 -1.7E-05 -1.7E-05  Table 252. The z-component of the collision-induced dipole moment at an internuclear distance of 2.36 bohr for geometry 45-0-60-0 Intermolecular Distance (bohr)   z (singlet)  z (triplet)  z (quintet) 5  -0.01092 -0.00627 0.006057 6  -0.00385 -0.00295 -0.00114 8 -0.00072 -0.0007 -0.00064 10 -0.00018 -0.00018 -0.00018 12 -7.3E-05 -6.7E-05 -6.6E-05 15 -3.4E-05 -3.4E-05 -3.4E-05 20 -1.1E-05 -1.1E-05 -1.1E-05  Table 253. The z-component of the collision-induced dipole moment at an internuclear distance of 2.457 bohr for geometry 45-0-60-0 Intermolecular Distance (bohr)   z (singlet)  z (triplet)  z (quintet) 5  -0.02362 -0.01582 0.005392 6  -0.00547 -0.00378 -0.00039 8 -0.00038 -0.00033 -0.00021 10 -4.8E-06 2.95E-06 1.01E-06 12 5.79E-06 1.36E-05 1.28E-05 15 -4.7E-06 -4.5E-06 -4.8E-06 20 -2.4E-06 -2.4E-06 -2.4E-06      303  Table 254. The z-component of the collision-induced dipole moment at an internuclear distance of 2.646 bohr for geometry 45-0-60-0 Intermolecular Distance (bohr)   z (singlet)  z (triplet)  z (quintet) 5  -0.04731 -0.03436 0.003167 6  -0.00847 -0.0054 0.000773 8 0.000227 0.000348 0.000571 10 0.000314 0.00032 0.000325 12 0.000155 0.000155 0.000155 15 4.86E-05 4.88E-05 4.88E-05 20 1.44E-05 1.44E-05 1.44E-05  Table 255. The z-component of the collision-induced dipole moment at an internuclear distance of 2.929 bohr for geometry 45-0-60-0 Intermolecular Distance (bohr)   z (singlet)  z (triplet)  z (quintet) 5  -0.07883 -0.06107 -0.0016 6  -0.01231 -0.00751 0.002136 8 0.001022 0.001218 0.001582 10 0.00072 0.000728 0.000739 12 0.000335 0.000335 0.000334 15 0.000119 0.000122 0.000118 20 3.62E-05 3.63E-05 3.63E-05  Table 256. The z-component of the collision-induced dipole moment at an internuclear distance of 3.213 bohr for geometry 45-0-60-0 Intermolecular Distance (bohr)   z (singlet)  z (triplet)  z (quintet) 5  -0.09669 -0.0764 -0.00414 6  -0.01406 -0.00835 0.003689 8 0.001636 0.001854 0.002384 10 0.001022 0.001029 0.001048 12 0.000475 0.000467 0.000466 15 0.000169 0.00017 0.000168 20 5.23E-05 5.24E-05 5.23E-05      304  Table 257. The z-component of the collision-induced dipole moment at an internuclear distance of 3.3 bohr for geometry 45-0-60-0 Intermolecular Distance (bohr)   z (singlet)  z (triplet)  z (quintet) 5  -0.10003 -0.07939 -0.00226 6  -0.01347 -0.00779 0.004501 8 0.001793 0.002061 0.002567 10 0.00109 0.001094 0.001125 12 0.000494 0.000496 0.000494 15 0.00018 0.000181 0.000202 20 5.58E-05 5.64E-05 4E-05  Table 258. The z-component of the collision-induced dipole moment at an internuclear distance of 2 bohr for geometry 45-0-90-0 Intermolecular Distance (bohr)   z (singlet)  z (triplet)  z (quintet) 5  0.003528 0.003724 0.009915 6  -0.00745 -0.00766 -0.00634 8 -0.00395 -0.00393 -0.00398 10 -0.00157 -0.00154 -0.00157 12 -0.0007 -0.00068 -0.0007 15 -0.00027 -0.00026 -0.00027 20 -8.5E-05 -8.3E-05 -8.5E-05  Table 259. The z-component of the collision-induced dipole moment at an internuclear distance of 2.1 bohr for geometry 45-0-90-0 Intermolecular Distance (bohr)   z (singlet)  z (triplet)  z (quintet) 5  -0.00041 0.005118 0.011127 6  -0.00708 -0.00593 -0.00493 8 -0.00319 -0.00325 -0.00327 10 -0.0012 -0.00125 -0.00125 12 -0.00053 -0.00055 -0.00055 15 -0.00021 -0.00022 -0.00022 20 -6.6E-05 -6.9E-05 -6.9E-05      305  Table 260. The z-component of the collision-induced dipole moment at an internuclear distance of 2.2 bohr for geometry 45-0-90-0 Intermolecular Distance (bohr)   z (singlet)  z (triplet)  z (quintet) 5  0.000574 0.004555 0.012104 6  -0.00549 -0.00488 -0.00363 8 -0.00256 -0.00257 -0.00258 10 -0.00092 -0.00093 -0.00094 12 -0.0004 -0.0004 -0.0004 15 -0.00017 -0.00017 -0.00017 20 -5.2E-05 -5.2E-05 -5.2E-05  Table 261. The z-component of the collision-induced dipole moment at an internuclear distance of 2.28187 bohr for geometry 45-0-90-0 Intermolecular Distance (bohr)   z (singlet)  z (triplet)  z (quintet) 5  -0.0005 0.004082 0.012767 6  -0.0048 -0.00408 -0.00264 8 -0.00203 -0.00203 -0.00203 10 -0.00068 -0.00069 -0.00071 12 -0.00032 -0.00028 -0.0003 15 -0.00012 -0.00012 -0.00012 20 -3.9E-05 -4E-05 -3.8E-05  Table 262. The z-component of the collision-induced dipole moment at an internuclear distance of 2.296 bohr for geometry 45-0-90-0 Intermolecular Distance (bohr)   z (singlet)  z (triplet)  z (quintet) 5  -0.00343 0.005967 0.015685 6  -0.00539 -0.00313 -0.00146 8 -0.00187 -0.00168 -0.0017 10 -0.00057 -0.00057 -0.00057 12 -0.00023 -0.00024 -0.00024 15 -0.0001 -0.00011 -0.00011 20 -3.2E-05 -3.2E-05 -3.2E-05      306  Table 263. The z-component of the collision-induced dipole moment at an internuclear distance of 2.36 bohr for geometry 45-0-90-0 Intermolecular Distance (bohr)   z (singlet)  z (triplet)  z (quintet) 5  -0.00148 0.005693 0.013285 6  -0.00418 -0.00245 -0.00174 8 -0.00152 -0.00121 -0.00152 10 -0.00046 -0.00037 -0.00046 12 -0.00018 -0.00016 -0.00018 15 -8.4E-05 -7.2E-05 -8.4E-05 20 -2.6E-05 -2.1E-05 -2.6E-05  Table 264. The z-component of the collision-induced dipole moment at an internuclear distance of 2.457 bohr for geometry 45-0-90-0 Intermolecular Distance (bohr)   z (singlet)  z (triplet)  z (quintet) 5  -0.0026 0.005238 0.013798 6  -0.00345 -0.00148 -0.00069 8 -0.00093 -0.00059 -0.00091 10 -0.00019 -7.7E-05 -0.00019 12 -5.8E-05 1.81E-05 -5.8E-05 15 -3.6E-05 -2.1E-05 -3.6E-05 20 -1E-05 -4.3E-06 -1E-05  Table 265. The z-component of the collision-induced dipole moment at an internuclear distance of 2.646 bohr for geometry 45-0-90-0 Intermolecular Distance (bohr)   z (singlet)  z (triplet)  z (quintet) 5  -0.00455 0.004193 0.014332 6  -0.00217 0.000185 0.001144 8 0.000124 0.000598 0.00018 10 0.000299 0.000438 0.000297 12 0.000164 0.00022 0.000163 15 5.11E-05 7.52E-05 5.11E-05 20 1.83E-05 2.65E-05 1.83E-05      307  Table 266. The z-component of the collision-induced dipole moment at an internuclear distance of 2.929 bohr for geometry 45-0-90-0 Intermolecular Distance (bohr)   z (singlet)  z (triplet)  z (quintet) 5  -0.00725 - 0.013789 6  -0.00063 - 0.003282 8 0.001419 - 0.001514 10 0.000888 - 0.000887 12 0.000433 - 0.000432 15 0.00016 - 0.00016 20 5.47E-05 - 5.47E-05  Table 267. The z-component of the collision-induced dipole moment at an internuclear distance of 3.213 bohr for geometry 45-0-90-0 Intermolecular Distance (bohr)   z (singlet)  z (triplet)  z (quintet) 5  -0.00725 - 0.013789 6  -0.00063 - 0.003282 8 0.001419 - 0.001514 10 0.000888 - 0.000887 12 0.000433 - 0.000432 15 0.00016 - 0.00016 20 5.47E-05 - 5.47E-05  Table 268. The z-component of the collision-induced dipole moment at an internuclear distance of 3.3 bohr for geometry 45-0-90-0 Intermolecular Distance (bohr)   z (singlet)  z (triplet)  z (quintet) 5  -0.00917 -0.00242 0.011359 6  0.000589 0.001995 0.004788 8 0.002479 0.002528 0.002628 10 0.001377 0.001377 0.001382 12 0.000661 0.000662 0.000656 15 0.000255 0.000255 0.000259 20 8.72E-05 8.73E-05 7.31E-05      308  Table 269. The z-component of the collision-induced dipole moment at an internuclear distance of 2 bohr for geometry 35-0-65-80 Intermolecular Distance (bohr)   z (singlet)  z (triplet)  z (quintet) 5  0.083649 0.067614 0.000258 6  0.004315 0.00024 -0.00819 8 -0.00254 -0.00268 -0.00296 10 -0.00104 -0.00103 -0.00104 12 -0.00047 -0.00047 -0.00047 15 -0.00019 -0.00019 -0.00019 20 -5.9E-05 -5.9E-05 -5.9E-05  Table 270. The z-component of the collision-induced dipole moment at an internuclear distance of 2.1 bohr for geometry 35-0-65-80 Intermolecular Distance (bohr)   z (singlet)  z (triplet)  z (quintet) 5  0.049108 0.040076 -0.00064 6  7.29E-05 -0.00227 -0.00714 8 -0.00225 -0.00231 -0.00247 10 -0.00084 -0.00083 -0.00084 12 -0.00037 -0.00037 -0.00037 15 0.067387 0.068387 0.068302 20 -5E-05 -5E-05 -5E-05  Table 271. The z-component of the collision-induced dipole moment at an internuclear distance of 2.2 bohr for geometry 35-0-65-80 Intermolecular Distance (bohr)   z (singlet)  z (triplet)  z (quintet) 5  0.019086 0.01594 -0.00273 6  -0.00371 -0.00461 -0.00659 8 -0.00192 -0.00197 -0.00203 10 -0.00064 -0.00065 -0.00065 12 -0.00028 -0.00028 -0.00028 15 0.116809 0.117729 0.117653 20 -4.1E-05 -4.1E-05 -4.1E-05      309  Table 272. The z-component of the collision-induced dipole moment at an internuclear distance of 2.28187 bohr for geometry 35-0-65-80 Intermolecular Distance (bohr)   z (singlet)  z (triplet)  z (quintet) 5  -0.00415 -0.00244 -0.00489 6  -0.00647 -0.00644 -0.00636 8 -0.0018 -0.0017 -0.0017 10 -0.00058 -0.0005 -0.0005 12 -0.00022 -0.00021 -0.00021 15 -0.0001 -0.0001 -0.0001 20 -3.3E-05 -3.3E-05 -3.3E-05  Table 273. The z-component of the collision-induced dipole moment at an internuclear distance of 2.296 bohr for geometry 35-0-65-80 Intermolecular Distance (bohr)   z (singlet)  z (triplet)  z (quintet) 5  -0.00734 -0.00551 -0.00526 6  -0.00693 -0.00676 -0.00633 8 -0.00167 -0.00165 -0.00165 10 -0.00047 -0.00047 -0.00047 12 -0.00019 -0.00019 -0.0002 15 -9.7E-05 -9.5E-05 -9.7E-05 20 -3.2E-05 -3.2E-05 -3.2E-05  Table 274. The z-component of the collision-induced dipole moment at an internuclear distance of 2.36 bohr for geometry 35-0-65-80 Intermolecular Distance (bohr)   z (singlet)  z (triplet)  z (quintet) 5  -0.0238 -0.01899 -0.0069 6  -0.00905 -0.0081 -0.00621 8 -0.00151 -0.00147 -0.00141 10 -0.00036 -0.00035 -0.00036 12 -0.00014 -0.00014 -0.00014 15 -7.8E-05 -7.8E-05 -7.8E-05 20 -2.6E-05 -2.6E-05 -2.6E-05      310  Table 275. The z-component of the collision-induced dipole moment at an internuclear distance of 2.457 bohr for geometry 35-0-65-80 Intermolecular Distance (bohr)   z (singlet)  z (triplet)  z (quintet) 5  -0.04652 -0.03772 -0.00871 6  -0.01195 -0.00994 -0.00597 8 -0.00124 -0.00117 -0.00105 10 -0.0002 -0.0002 -0.00019 12 -6.7E-05 -6.9E-05 -6.7E-05 15 -5E-05 -5E-05 -5E-05 20 -1.8E-05 -1.8E-05 -1.8E-05  Table 276. The z-component of the collision-induced dipole moment at an internuclear distance of 2.646 bohr for geometry 35-0-65-80 Intermolecular Distance (bohr)   z (singlet)  z (triplet)  z (quintet) 5  -0.07906 -0.06427 -0.00778 6  -0.01609 -0.01247 -0.00492 8 -0.00072 -0.0006 -0.00038 10 0.000102 0.000106 0.000114 12 7.13E-05 7.28E-05 7.14E-05 15 1.71E-06 8.34E-07 1.62E-06 20 -1.2E-06 -1.1E-06 -1.2E-06  Table 277. The z-component of the collision-induced dipole moment at an internuclear distance of 2.929 bohr for geometry 35-0-65-80 Intermolecular Distance (bohr)   z (singlet)  z (triplet)  z (quintet) 5  -0.10555 -0.088 0.009087 6  -0.01812 -0.0129 -0.00157 8 -2.1E-05 0.000161 0.000531 10 0.00049 0.000494 0.000507 12 0.000247 0.000248 0.000247 15 6.8E-05 6.8E-05 6.79E-05 20 1.99E-05 2E-05 1.99E-05      311  Table 278. The z-component of the collision-induced dipole moment at an internuclear distance of 3.213 bohr for geometry 35-0-65-80 Intermolecular Distance (bohr)   z (singlet)  z (triplet)  z (quintet) 5  -0.19626 -0.20172 -0.00353 6  -0.00428 0.001318 0.010245 8 0.000605 0.000781 0.001304 10 0.000778 0.000784 0.000802 12 0.000377 0.000376 0.000377 15 0.000117 0.000117 0.000116 20 3.52E-05 3.52E-05 3.53E-05  Table 279. The z-component of the collision-induced dipole moment at an internuclear distance of 3.3 bohr for geometry 35-0-65-80 Intermolecular Distance (bohr)   z (singlet)  z (triplet)  z (quintet) 5  -0.22297 -0.22983 -0.0286 6  0.004919 0.013395 0.021606 8 0.000952 0.00119 0.00164 10 0.000855 0.000863 0.000881 12 0.00041 0.00041 0.000411 15 0.000129 0.000129 0.000129 20 3.88E-05 3.88E-05 3.89E-05  Table 280. The z-component of the collision-induced dipole moment at an internuclear distance of 2 bohr for geometry 60-0-80-65 Intermolecular Distance (bohr)   z (singlet)  z (triplet)  z (quintet) 5  0.012921 0.01109923 0.006998589 6  -0.00299 -0.0033133 -0.003972012 8 -0.00265 -0.00267022 -0.002716727 10 -0.00109 -0.00109386 -0.001096505 12 -0.00048 -0.00047762 -0.000477698 15 -0.00019 0.03934983 -0.000185307 20 -5.8E-05 -5.802E-05 0.039376189      312  Table 281. The z-component of the collision-induced dipole moment at an internuclear distance of 2.1 bohr for geometry 60-0-80-65 Intermolecular Distance (bohr)   z (singlet)  z (triplet)  z (quintet) 5  0.009577 0.00725665 0.00875939 6  -0.00238 -0.00086487 -0.002517901 8 -0.00203 -0.0009352 -0.002071723 10 -0.00081 -0.00032217 -0.000814438 12 -0.00035 -0.00011516 -0.000350025 15 0.067394 -5.549E-05 0.068313984 20 -4.3E-05 -1.6395E-05 -4.3416E-05  Table 282. The z-component of the collision-induced dipole moment at an internuclear distance of 2.2 bohr for geometry 60-0-80-65 Intermolecular Distance (bohr)   z (singlet)  z (triplet)  z (quintet) 5  0.006935 0.00815793 0.010150585 6  -0.0017 -0.00154388 -0.001200851 8 -0.00142 -0.00142714 -0.00144361 10 -0.00054 -0.00053789 -0.000539831 12 -0.00023 -0.00022491 -0.000225481 15 -9.3E-05 -9.2511E-05 -9.25455E-05 20 -2.9E-05 -2.873E-05 -2.8732E-05  Table 283. The z-component of the collision-induced dipole moment at an internuclear distance of 2.28187 bohr for geometry 60-0-80-65 Intermolecular Distance (bohr)   z (singlet)  z (triplet)  z (quintet) 5  0.005111 0.00724912 0.011086854 6  -0.0012 -0.00086487 -0.00020401 8 -0.00094 -0.0009352 -0.000943633 10 -0.00036 -0.00032217 -0.000320449 12 -0.00016 -0.00011516 -0.000125418 15 -0.00016 -5.549E-05 -5.3899E-05 20 -1.8E-05 -1.6395E-05 -1.6739E-05       313  Table 284. The z-component of the collision-induced dipole moment at an internuclear distance of 2.296 bohr for geometry 60-0-80-65 Intermolecular Distance (bohr)   z (singlet)  z (triplet)  z (quintet) 5  0.004801 0.00710114 0.011234226 6  -0.00111 -0.00075009 -3.88715E-05 8 -0.00092 -0.00085364 -0.000859238 10 -0.00028 -0.00028493 -0.000284609 12 -0.00011 -0.00010785 -0.000109683 15 -4.8E-05 0.03179388 0.031584293 20 -1.5E-05 -1.4657E-05 -1.46725E-05  Table 285. The z-component of the collision-induced dipole moment at an internuclear distance of 2.36 bohr for geometry 60-0-80-65 Intermolecular Distance (bohr)   z (singlet)  z (triplet)  z (quintet) 5  0.003531 0.00645223 0.011843887 6  -0.00073 -0.00024746 0.000684948 8 -0.0005 -0.00048389 -0.000482212 10 -0.00012 -0.00011862 -0.000120201 12 -3.7E-05 -3.4904E-05 -3.50305E-05 15 -1.9E-05 -1.9101E-05 -1.91715E-05 20 -5.4E-06 -5.4395E-06 -5.44699E-06  Table 286. The z-component of the collision-induced dipole moment at an internuclear distance of 2.457 bohr for geometry 60-0-80-65 Intermolecular Distance (bohr)   z (singlet)  z (triplet)  z (quintet) 5  0.001745 0.0055251 0.012619918 6  -0.00016 0.00046692 0.001701971 8 2.75E-05 5.07E-05 6.31225E-05 10 0.000126 0.00011804 0.000117428 12 7.34E-05 7.307E-05 7.29785E-05 15 2.35E-05 2.3152E-05 2.3134E-05 20 8.15E-06 8.1515E-06 8.14501E-06      314  Table 287. The z-component of the collision-induced dipole moment at an internuclear distance of 2.646 bohr for geometry 60-0-80-65 Intermolecular Distance (bohr)   z (singlet)  z (triplet)  z (quintet) 5  -0.00132 0.00378423 0.013595855 6  0.000755 0.00163809 0.003383198 8 0.000957 0.00097443 0.001008493 10 0.00053 0.00053038 0.000529546 12 0.000261 0.00025953 0.000260425 15 9.88E-05 9.2084E-05 9.80295E-05 20 3.24E-05 3.2419E-05 3.2422E-05  Table 288. The z-component of the collision-induced dipole moment at an internuclear distance of 2.929 bohr for geometry 60-0-80-65 Intermolecular Distance (bohr)   z (singlet)  z (triplet)  z (quintet) 5  -0.00519 0.00100098 0.013536288 6  0.001588 0.00275463 0.005043228 8 0.001965 0.00199917 0.00206309 10 0.000993 0.0009936 0.000994514 12 0.000474 0.00047472 0.00047374 15 0.000185 0.00018418 0.000184059 20 6.11E-05 6.1087E-05 6.109E-05  Table 289. The z-component of the collision-induced dipole moment at an internuclear distance of 3.213 bohr for geometry 60-0-80-65 Intermolecular Distance (bohr)   z (singlet)  z (triplet)  z (quintet) 5  -0.00591 -0.00071694 0.013008996 6  0.001893 0.00317577 0.005832928 8 0.002562 0.00260899 0.002703016 10 0.001284 0.00128675 0.001287576 12 0.000612 0.00061259 0.000611612 15 10.63491 0.00023988 0.000242234 20 8.02E-05 8.0188E-05 8.0214E-05      315  Table 290. The z-component of the collision-induced dipole moment at an internuclear distance of 3.3 bohr for geometry 60-0-80-65 Intermolecular Distance (bohr)   z (singlet)  z (triplet)  z (quintet) 5  -0.00189 0.00117656 0.014529242 6  0.00208 0.00328943 0.006056219 8 0.002679 0.00273255 0.002831867 10 0.001346 0.00134915 0.001350375 12 0.000642 0.00064088 0.000642193 15 0.000256 0.00025568 0.000255522 20 8.45E-05 8.4509E-05 8.45285E-05  Table 291. The z-component of the collision-induced dipole moment at an internuclear distance of 2 bohr for geometry 70-0-65-75 Intermolecular Distance (bohr)   z (singlet)  z (triplet)  z (quintet) 5  0.013083117 0.007991191 -0.003043383 6  0.000506736 -0.00042381 8.07945E-05 8 -0.000739704 -0.000762124 0.000264418 10 -0.000361784 -0.000362577 8.78825E-05 12 -0.000169355 -0.000169816 3.6397E-05 15 -5.74475E-05 -5.85785E-05 1.62235E-05 20 -1.72125E-05 -1.7206E-05 4.71749E-06  Table 292. The z-component of the collision-induced dipole moment at an internuclear distance of 2.1 bohr for geometry 70-0-65-75 Intermolecular Distance (bohr)   z (singlet)  z (triplet)  z (quintet) 5  0.007529162 0.004351609 -0.002301459 6  0.00048671 -8.39445E-05 -0.001181748 8 -0.000376704 -0.000396524 -0.000415309 10 -0.000198354 -0.000198254 -0.000198463 12 -9.45115E-05 -9.4468E-05 -9.4583E-05 15 0.067497574 0.068496472 0.068413302 20 -9.61151E-06 -9.60549E-06 -9.611E-06      316  Table 293. The z-component of the collision-induced dipole moment at an internuclear distance of 2.2 bohr for geometry 70-0-65-75 Intermolecular Distance (bohr)   z (singlet)  z (triplet)  z (quintet) 5  0.00248198 0.000844489 -0.002527649 6  0.000432141 0.000143281 -0.000414758 8 -1.6272E-05 -1.88995E-05 -3.5332E-05 10 -4.00295E-05 -3.53835E-05 -3.81875E-05 12 -2.1499E-05 -2.18605E-05 -2.21005E-05 15 0.116906251 0.117827644 0.117752222 20 -1.76499E-06 -1.76649E-06 -1.774E-06  Table 294. The z-component of the collision-induced dipole moment at an internuclear distance of 2.28187 bohr for geometry 70-0-65-75 Intermolecular Distance (bohr)   z (singlet)  z (triplet)  z (quintet) 5  -0.00137783 0.007249122 -0.003043383 6  0.000351797 -0.000864873 8.0794E-05 8 0.000237922 -0.000935198 0.000264418 10 9.5842E-05 -0.00032217 8.7882E-05 12 4.45575E-05 -0.000115156 3.6397E-05 15 0.00018977 -5.54925E-05 1.6224E-05 20 -1.3537E-05 -1.63955E-05 4.718E-06  Table 295. The z-component of the collision-induced dipole moment at an internuclear distance of 2.296 bohr for geometry 70-0-65-75 Intermolecular Distance (bohr)   z (singlet)  z (triplet)  z (quintet) 5  -0.002014441 -0.002448283 -0.003147054 6  0.000339687 0.000276341 0.000155198 8 0.000299545 0.000310868 0.000314701 10 0.00010312 0.000116626 0.000110535 12 4.84905E-05 4.25085E-05 4.4646E-05 15 1.8854E-05 0.031565533 0.031361766 20 5.83699E-06 5.84149E-06 5.83699E-06      317  Table 296. The z-component of the collision-induced dipole moment at an internuclear distance of 2.36 bohr for geometry 70-0-65-75 Intermolecular Distance (bohr)   z (singlet)  z (triplet)  z (quintet) 5  -0.004871741 -0.004554106 -0.003650482 6  0.000259797 0.000336672 0.000479026 8 0.00052111 0.000529194 0.000538358 10 0.000208174 0.000193879 0.000205704 12 8.2854E-05 8.6665E-05 8.7185E-05 15 3.50035E-05 3.4984E-05 3.49945E-05 20 1.07975E-05 1.07995E-05 1.0801E-05  Table 297. The z-component of the collision-induced dipole moment at an internuclear distance of 2.457 bohr for geometry 70-0-65-75 Intermolecular Distance (bohr)   z (singlet)  z (triplet)  z (quintet) 5  -0.008822006 -0.007482709 -0.004355787 6  0.000139774 0.00040702 0.000912761 8 0.000823628 0.000838882 0.000857546 10 0.000330497 0.000340626 0.000341441 12 0.000147486 0.000148496 0.000147607 15 5.81345E-05 5.8125E-05 5.80805E-05 20 1.8021E-05 1.8026E-05 1.802E-05  Table 298. The z-component of the collision-induced dipole moment at an internuclear distance of 2.646 bohr for geometry 70-0-65-75 Intermolecular Distance (bohr)   z (singlet)  z (triplet)  z (quintet) 5  -0.014659778 -0.011735642 -0.004938887 6  -2.3832E-05 0.000561576 0.001672538 8 0.001328498 0.001351673 0.001395216 10 0.000566273 0.000565704 0.000568773 12 0.000247846 0.000249146 0.000247875 15 9.73135E-05 9.2468E-05 9.7262E-05 20 3.03035E-05 3.02085E-05 3.0305E-05      318  Table 299. The z-component of the collision-induced dipole moment at an internuclear distance of 2.929 bohr for geometry 70-0-65-75 Intermolecular Distance (bohr)   z (singlet)  z (triplet)  z (quintet) 5  -0.017443569 -0.013832797 -0.003010278 6  -2.46945E-05 0.000880725 0.002673931 8 0.001872157 0.001890734 0.001984117 10 0.000812579 0.000812668 0.000816658 12 0.000356217 0.000357384 0.000356193 15 0.000141347 0.000138936 0.00014004 20 4.387E-05 4.38655E-05 4.3859E-05  Table 300. The z-component of the collision-induced dipole moment at an internuclear distance of 3.213 bohr for geometry 70-0-65-75 Intermolecular Distance (bohr)   z (singlet)  z (triplet)  z (quintet) 5  -0.004200936 -0.003648094 0.010353625 6  0.001561565 0.002405413 0.004516636 8 0.00221801 0.0022705 0.002369121 10 0.000976945 0.000982136 0.000982964 12 0.000429469 0.000449846 0.000429416 15 0.000168266 0.000167092 0.000168397 20 5.2616E-05 5.2605E-05 5.2616E-05  Table 301. The z-component of the collision-induced dipole moment at an internuclear distance of 3.3 bohr for geometry 70-0-65-75 Intermolecular Distance (bohr)   z (singlet)  z (triplet)  z (quintet) 5  -0.002078351 0.001341718 0.016234705 6  0.004296294 0.004771711 0.006783943 8 0.002289715 0.0023472 0.002450876 10 0.001013749 0.001014869 0.001020262 12 0.000446352 0.000436642 0.000446545 15 0.000174767 0.000173814 0.000174893 20 5.4558E-05 5.45635E-05 5.45845E-05      319  Table 302. The z-component of the collision-induced dipole moment at an internuclear distance of 2 bohr for geometry 50-0-95-55 Intermolecular Distance (bohr)   z (singlet)  z (triplet)  z (quintet) 5  0.024205 0.02042 0.01064 6  -0.00303 -0.00374 -0.00515 8 -0.00336 -0.0034 -0.00348 10 -0.00136 -0.00136 -0.00137 12 -0.0006 -0.0006 -0.0006 15 -0.00024 -0.00024 -0.00024 20 -7.4E-05 -7.4E-05 -7.4E-05  Table 303. The z-component of the collision-induced dipole moment at an internuclear distance of 2.1 bohr for geometry 50-0-95-55 Intermolecular Distance (bohr)   z (singlet)  z (triplet)  z (quintet) 5  0.015874 0.015132 0.012215 6  -0.00309 -0.00333 -0.00368 8 -0.00272 -0.00276 -0.0028 10 -0.00106 -0.00106 -0.00107 12 -0.00046 -0.00046 -0.00046 15 0.067352 -0.00019 -0.00019 20 -5.8E-05 -5.8E-05 -5.8E-05  Table 304. The z-component of the collision-induced dipole moment at an internuclear distance of 2.2 bohr for geometry 50-0-95-55 Intermolecular Distance (bohr)   z (singlet)  z (triplet)  z (quintet) 5  0.008992 0.010718 0.01064 6  -0.00313 -0.00288 -0.00515 8 -0.00211 -0.00212 -0.00348 10 -0.00077 -0.00077 -0.00137 12 -0.00033 -0.00033 -0.0006 15 -0.00014 -0.00014 -0.00024 20 -4.3E-05 -4.3E-05 -7.4E-05      320  Table 305. The z-component of the collision-induced dipole moment at an internuclear distance of 2.28187 bohr for geometry 50-0-95-55 Intermolecular Distance (bohr)   z (singlet)  z (triplet)  z (quintet) 5  0.003998 0.007475 0.014051 6  -0.0031 -0.00253 -0.00144 8 -0.00162 -0.00163 -0.00163 10 -0.00061 -0.00054 -0.00054 12 -0.00022 -0.00022 -0.00022 15 -0.00027 -9.8E-05 -9.5E-05 20 -3E-05 -3.6E-05 -3E-05  Table 306. The z-component of the collision-induced dipole moment at an internuclear distance of 2.296 bohr for geometry 50-0-95-55 Intermolecular Distance (bohr)   z (singlet)  z (triplet)  z (quintet) 5  0.003177 0.00694 0.01416 6  -0.00311 -0.00247 -0.00128 8 -0.00154 -0.00154 -0.00155 10 -0.00049 -0.0005 -0.0005 12 -0.0002 -0.0002 -0.0002 15 0.031024 -9E-05 0.031496 20 -2.8E-05 -2.8E-05 -2.8E-05  Table 307. The z-component of the collision-induced dipole moment at an internuclear distance of 2.36 bohr for geometry 50-0-95-55 Intermolecular Distance (bohr)   z (singlet)  z (triplet)  z (quintet) 5  -0.00039 0.004602 0.014598 6  -0.00302 -0.00221 -0.00059 8 -0.00118 -0.00117 -0.00116 10 -0.0003 -0.00033 -0.00033 12 -0.00012 -0.00012 -0.00012 15 -5.9E-05 -5.9E-05 -5.9E-05 20 -1.8E-05 -1.8E-05 -1.8E-05      321  Table 308. The z-component of the collision-induced dipole moment at an internuclear distance of 2.457 bohr for geometry 50-0-95-55 Intermolecular Distance (bohr)   z (singlet)  z (triplet)  z (quintet) 5  -0.00536 0.001311 0.015168 6  -0.00293 -0.00181 0.000389 8 -0.00064 -0.00061 -0.00059 10 -6.2E-05 -7.2E-05 -7.3E-05 12 -3.9E-06 -3E-06 -4.4E-06 15 -1.4E-05 -1.4E-05 -1.4E-05 20 -3.4E-06 -3.4E-06 -3.4E-06  Table 309. The z-component of the collision-induced dipole moment at an internuclear distance of 2.646 bohr for geometry 50-0-95-55 Intermolecular Distance (bohr)   z (singlet)  z (triplet)  z (quintet) 5  -0.01338 -0.00415 0.016108 6  -0.00272 -0.00111 0.00208 8 0.000315 0.000349 0.000414 10 0.000378 0.000378 0.000378 12 0.000202 0.000201 0.000202 15 6.77E-05 6.75E-05 6.76E-05 20 2.32E-05 2.32E-05 2.31E-05  Table 310. The z-component of the collision-induced dipole moment at an internuclear distance of 2.929 bohr for geometry 50-0-95-55 Intermolecular Distance (bohr)   z (singlet)  z (triplet)  z (quintet) 5  -0.01802 -0.0074 0.017905 6  -0.00238 -0.00027 0.003984 8 0.001413 0.001474 0.001585 10 0.000903 0.000905 0.000906 12 0.000446 0.000446 0.000446 15 0.000166 0.000166 0.000166 20 5.56E-05 5.57E-05 5.56E-05      322  Table 311. The z-component of the collision-induced dipole moment at an internuclear distance of 3.213 bohr for geometry 50-0-95-55 Intermolecular Distance (bohr)   z (singlet)  z (triplet)  z (quintet) 5  0.009469 0.009469 0.030927 6  0.001672 0.001672 0.005971 8 0.002175 0.002175 0.002341 10 0.001245 0.001245 0.001249 12 0.000606 0.000606 0.000607 15 0.000232 0.000232 0.000232 20 7.78E-05 7.78E-05 7.78E-05  Table 312. The z-component of the collision-induced dipole moment at an internuclear distance of 3.3 bohr for geometry 50-0-95-55 Intermolecular Distance (bohr)   z  (singlet)   z (triplet)  z (quintet) 5  -0.01783 0.00287 0.039205 6  0.003059 0.004356 0.007796 8 0.002278 0.002357 0.002529 10 0.001319 0.001321 0.001326 12 0.000643 0.000642 0.000643 15 0.000243 0.000249 0.000247 20 8.28E-05 8.28E-05 8.29E-05             323             REFERENCES              324  REFERENCES  [1] L. Frommhold, Collision-Induced Absorption in Gases, (Cambridge University Press, New York, 2006).  [2] J. 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