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This is to certify that the
dissertation entitled

THE ENTHALPY OF FROMATION AND OUADRUOLE
MOMENTS OF ACETYLENE AND DOUBLY SUBSTITUTED
ACETYLENES AND ACETYLIDES, INTERMOLECULAR
INTERACTIONS, AND COLLISION-INDUCED MOLECULAR
FRAME DISTORTIONS

presented by

Dorothy J. Gearhart

has been accepted towards fulfillment
of the requirements for the

Doctoral degree in Chemistry

 

 

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Date

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THE ENTHALPY OF FORMATION AND QUADRUPOLE MOMENTS OF
ACETYLENE AND DOUBLY SUBSTITUTED ACETYLENES AND ACETYLIDES,
INTERMOLECULAR INTERACTIONS, AND COLLISION-INDUCED
MOLECULAR FRAME DISTORTIONS

By

Dorothy J. Gearhart

A DISSERTATION
Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of
DOCTOR OF PHILOSOPHY
Department of Chemistry

2006

ABSTRACT
TI-IE ENTHALPY OF FORMATION AND QUADRUPOLE MOMENTS OF
ACETYLENE AND DOUBLY SUBSTITUTED ACETYLENES AND ACETYLIDES,
INTERMOLECULAR INTERACTIONS, AND COLLISION -INDUCED
MOLECULAR FRAME DISTORTION S
By
Dorothy J. Gearhart
This thesis is broken into five chapters. The first chapter is a short summary of

the calculations for this report. The second chapter concerns the calculation of CCSD(T)
heats of formation at the complete basis set limit for HCCH, FCCF, ClCCCl, linear
LizCz, planar LizCz, linear NazCz, and planar Na2C2. CCSD(T) optimized geometries and
CCSD(T) vibrational frequencies are calculated to aid in the calculation of the heats of
formation. Core-valence, spin-orbit coupling, and relativistic corrections are made to the
calculated heats of formation. Calculated values are compared to experimental and
theoretical values with success. The third chapter concerns the calculation of SCF,
DFI‘(B3LYP), and CCSD(T) molecular quadrupole moments of HCCH, FCCF, ClCCCl,
linear Li2C2 planar Li2C2, linear Na2C2, and planar N aZCZ. The linear molecules have one
unique component of the quadrupole moment whereas the planar molecules have two.
The quadrupole moments of HCCH, FCCF, and ClCCCl are compared to experimental
and theoretical results of other researchers with success. Molecular quadrupoles are
broken into partition of charges to compare algebraic charges of terminal atoms. The
fourth chapter concerns the development of the third—order two-body interaction energies
of molecules. The second-order two-body and the third-order two-body interaction

energies are derived and developed using the Unsold Theorem. Numerical estimates are

made for the two-body second- and third-order interaction energies for two— and three-

body configurations as a function of internuclear separation. The fifth chapter concerns
the frame distortion of CH4 in CH4+X pairs where X is N2, H2, or He. The methane
molecule is approached by each molecule or atom, and the frame distortion is calculated.
Additional energy calculations are made to determine energy influences in the methane

pairs.

To my mother, Marjory

Acknowledgments

I would like to take this opportunity to thank both my advisers Dr. Katherine
Hunt and Dr. James Harrison for their academic support and friendly guidance during my
years as a student at Michigan State University. I would also like to thank my guidance
committee, Dr. Ned Jackson and Dr. Gary Blanchard for their advise and gentle
incentives along my path to graduation. I would like to acknowledge the faculty and staff
of Michigan State University’s Department of Chemistry. Special thanks to Lisa
Dillingham and Janet Haun for their constant encouragement. Special thanks also to Paul
Reed, Dr. Tom Carter, and Dr. T om Atkinson for maintaining the departmental
computers and for their technical advise. I would like to thank the many friends I have
gathered while attending Michigan State University, especially those of the
computational chemistry research groups. Without their support, I would not have
completed my studies.

Finally, I would like to thank all the members of my family for their inspiration

and support.

TABLE OF CONTENTS

List of Tables .................................................................................. x
List of Figures ................................................................................. xx
CHAPTER 1
INTRODUCTION ............................................................................ 1
CHAPTER 2
THE ENTHALPY OF FORMATION OF DOUBLY HALOGENATED
ACETYLENES AND ALKALI ACETYLIDES ......................................... 5
2. 1 Introduction .................................................................. 6
2.1.1 Method ...................................................... 7
2. l .2 Computational Details .................................... 9
2.1.3 Corrections ................................................. 9
2.1.4 Extrapolations to CBS Limit ............................ 13
2.2 HCCH ........................................................................ 16
2.3 FCCF ......................................................................... 23
2.4 ClCCCl ....................................................................... 31
2.5 Linear LizCz .................................................................. 37
2.6 Planar LizCz .................................................................. 43
2.7 Linear Na2C2 ................................................................. 49
2.8 Planar NazCz ................................................................. 55
2.9 Discussion .................................................................... 6O
2. 10 References .................................................................... 62
CHAPTER 3
QUADRUPOLE MOMENTS OF HCCH, FCCF, CICCCI, LizCz, and
NazCz ........................................................................................... 64
3. 1 Introduction ................................................................... 65

vi

3.1.1 Computational Details .................................... 67

3.1.2 Multipole Expansion ...................................... 68
3.1.3 Method ...................................................... 70
3.2 HCCH ........................................................................ 72
3.3 FCCF .......................................................................... 74
3.4 CICCCI ........................................................................ 76
3.5 Li2C2 ........................................................................... 78
3.5.1 Molecular Quadrupole Moment ......................... 78
3.5.2 Linear ....................................................... 79
3.5.3 Planar ....................................................... 80
3.6 Na2C2 ......................................................................... 82
3.6.1 Linear ....................................................... 82
3.6.2 Planar ....................................................... 83
3.7 Partition Quadrupole Moment into Point Charges ..................... 84
3.8 Discussion .................................................................... 85
3.9 References ................................................................... 95
CHAPTER 4
HIGHER-ORDER VAN DER WAALS INTERACTIONS AND
NONLINEAR RESPONSE TEN SORS .................................................... 97
4. 1 Introduction ................................................................... 98
4.1.1 Interaction Energy ......................................... 98
4.1.2 Perturbation Theory ....................................... 102
4.2 Second-Order Energy Correction ......................................... 104
4.3 Third-Order Correction to the Energy .................................... 109

vii

4.3.1 Third-Order Terms in the Static

Dipole oonrB .......................................... 111
4.3.2 First-Order Terms in the Dipole of Both
A and B ..................................................... 114
4.3.3 First-Order Terms in the Static Dipole
of System A or B, But Not Both ........................ 120
4.3.4 Zero-Order Term in the Static
Dipole Moments of Systems A and B .................. 134
4.4 Numerical Estimates ........................................................ 138
4.4.1 HCl Dimer .................................................. 143
4.4. 1.1 Second-Order Two-Body
Terms .............................................. 147
4.4.1.2 Third-Order Two-Body
Terms .............................................. 148
4.4.2 HCl Trimer ................................................. 150
4.4.2.1 Third-Order Three-Body
Terms .............................................. 153
4.4.2.2 Third-Order Two-Body
Terms ............................................. 155
4.5 References ................................................................... 156
CHAPTER 5
MOLECULAR FRAME DISTORTIONS IN CH4+X PAIRS, WITH
X=He, N2, OR H2 ............................................................................. 157
5.1 Introduction .................................................................. 158
5.1.1 Approaches ................................................ 159
5.1.2 Methods .................................................... 163
5.1.3 Computational Details .................................... 163
5.2 CH4-N2 ....................................................................... 163

viii

5.2.1 Linear Approach .......................................... 164

5.2.2 Perpendicular Approach ................................. 170

5.3 CH4-H2 ....................................................................... 176

5.3.1 Linear Approach .......................................... 176

5.3.2 Perpendicular Approach ................................. 178

5.4 CH4-He ....................................................................... 179

5.5 Discussion .................................................................... 180

5.6 References .................................................................... 1 82
APPENDIX A

RAW DATA FOR CHAPTER 2 ............................................................. 184
APPENDIX B

RAW DATA FOR CHAPTER 3 ............................................................ 209
APPENDIX C

RAW DATA FOR CHAPTER 5 ........................................................... 219

ix

LIST OF TABLES

Table Page
CHAPTER 2

2-1. Optimized Geometries of HCCH as a Function of Basis Set .................. 17
2-2. Vibrational Frequencies of HCCH ................................................. 18

2-3. Calculated Dissociation Energies and Heats of Formation
for HCCH as a Function of Basis Set ............................................. 18

2-4. Heats of Formation for HCCH Extrapolated to the Complete
Basis Set Limit ........................................................................ 19

2-5. Core-Valence Correction for HCCH as a Function of Basis
Set ...................................................................................... 20

2-6. Core-Valence Correction for HCCH at the Complete Basis
Set Limit ........................................................................................................ 20

2-7. AH; Values Corrected for Core-Valence Correlation, Spin-
Orbit Coupling, and Relativistic Effects for HCCH at the

Complete Basis Set Limit for each Method ...................................... 22
2-8. Comparison of Heats of Formation of HCCH ................................. 23
2-9. Optimized Geometries of FCCF as a Function of Basis Set ................... 24
2-10. Vibrational Frequencies of FCCF .................................................. 25

2-11. Calculated Dissociation Energies and Heats of Formation
for FCCF as a Function of Basis Set .............................................. 25

2-12. Heats of Formation for FCCF Extrapolated to the Complete
Basis Set Limit ....................................................................... 27

2-13. Core-Valence Correction for FCCF as a Function of Basis
Set ...................................................................................... 28

2-14. Core-Valence Correction for FCCF at the Complete Basis
Set Limit .............................................................................. 28

2-15.

2-16.

2-17.

2-18.

2-19.

2-20.

2-21.

2-22.

2-23.

2-24.

2-25.

2-26.

2-27.

2-28.

2-29.

2-30.

AHf Values Corrected for Core-Valence Correlation, Spin-
Orbit Coupling, and Relativistic Effects for FCCF at the

Complete Basis Set Limit for each Method ...................................... 29
Comparison of Heats of Formation of FCCF ..................................... 30
Optimized Geometries of ClCCCl as a Function of Basis Set ................. 31
Vibrational Frequencies of ClCCCl ............................................... 32
Calculated Dissociation Energies and Heats of Formation

for ClCCCl as a Function of Basis Set ............................................ 32
Heats of Formation for ClCCCl Extrapolated to the Complete

Basis Set Limit ........................................................................ 33

Core-Valence Correction for ClCCCl as a Function of Basis
Set ...................................................................................... 34

Core—Valence Correction for ClCCCl at the Complete Basis
Set Limit .............................................................................. 35

AHf Values Corrected for Core-Valence Correlation, Spin-
Orbit Coupling, and Relativistic Effects for ClCCCl at the

Complete Basis Set Limit for each Method ...................................... 36
Comparison of Heats of Formation of ClCCCl .................................. 37
Added Diffuse Function Exponents to Create Li AUG-CC

PVXZ Basis Sets ..................................................................... 38
Optimized Geometries of Linear Li2C2 as a Function of

Basis Set ............................................................................... 38
Vibrational Frequencies of Linear Li2C2 .......................................... 39

Calculated Dissociation Energies and Heats of Formation
for Linear Li2C2 as a Function of Basis Set ....................................... 40

Heats of Formation for Linear Li2C2 Extrapolated to the
Complete Basis Set Limit ........................................................... 41

Core-Valence Correction for Linear Li2C2 as a Function of
Basis Set ............................................................................... 41

xi

2-31.

2-32.

2-33.

2-34.

2-35.

2-36.

2—37.

2-38.

2-39.

240.

2-41.

2—42.

2-43.

2-44.

2-45.

2-46.

Core-Valence Correction for Linear Li2C2 at the Complete

Basis Set Limit ....................................................................... 42
AHf Values Corrected for Core-Valence Correlation, Spin-

Orbit Coupling, and Relativistic Effects for Linear Li2C2 at the

Complete Basis Set Limit for each Method ....................................... 42
Comparison of Heats of Formation of Linear Li2C2 ............................. 43
Optimized Geometries of Planar Li2C2 as a Function of

Basis Set ............................................................................... 44
Vibrational Frequencies of Planar Li2C2 .......................................... 44
Calculated Dissociation Energies and Heats of Formation

for Planar Li2C2 as a Function of Basis Set ....................................... 46
Heats of Formation for Planar Li2C2 Extrapolated to the

Complete Basis Set Limit ........................................................... 46
Core-Valence Correction for Planar Li2C2 as a Function of

Basis Set ............................................................................... 47
Core-Valence Correction for Planar Li2C2 at the Complete

Basis Set Limit ........................................................................ 47
AH; Values Corrected for Core-Valence Correlation, Spin-

Orbit Coupling, and Relativistic Effects for Planar Li2C2 at the

Complete Basis Set Limit for each Method ...................................... 48
Comparison of Heats of Formation of Planar Li2C2 ............................. 48
Energy Difference Between Linear and Planar Li2C2 ........................... 49

Added Diffuse Function Exponents to Create Na AUG-CC

PVXZ Basis Sets ..................................................................... 49
Optimized Geometries of Linear Na2C2 as a Function of

Basis Set .............................................................................. 50
Vibrational Frequencies of Linear Na 2C2 ........................................ 50

Calculated Dissociation Energies and Heats of Formation
for Linear Na 2C2 as a Function of Basis Set ..................................... 52

xii

2-47.

Heats of Formation for Linear Na 2C2 Extrapolated to the

Complete Basis Set Limit ........................................................... 52
2—48. Core-Valence Correction for Linear Na 2C2 as a Function of

Basis Set ............................................................................... 53
2-49. Core-Valence Correction for Linear Na 2C2 at the Complete

Basis Set Limit ........................................................................ 53
2-50. AHf Values Corrected for Core-Valence Correlation, Spin-

Orbit Coupling, and Relativistic Effects for Linear Na 2C2 at the

Complete Basis Set Limit for each Method ....................................... 54
2-51. Comparison of Heats of Formation of Linear Na 2C2 ........................... 54
2-52. Optimized Geometries of Planar Na 2C2 as a Function of

Basis Set ............................................................................... 55
2-53. Vibrational Frequencies of Planar Na 2C2 ......................................... 56
2-54. Calculated Dissociation Energies and Heats of Formation

for Planar Na2C2 as a Function of Basis Set ...................................... 57
2-55. Heats of Formation for Planar N a2C2 Extrapolated to the

Complete Basis Set Limit ........................................................... 57
2-56. Core—Valence Correction for Planar Na2C2 as a Function of

Basis Set ............... ' ................................................................ 58
2-57. Core-Valence Correction for Planar Na2C2 at the Complete

Basis Set Limit ........................................................................ 58
2-58. AHf Values Corrected for Core-Valence Correlation, Spin-

Orbit Coupling, and Relativistic Effects for Planar Na2C2 at the

Complete Basis Set Limit for each Method ....................................... 59
2-59. Comparison of Heats of Formation of Planar Na2C2 ............................ 59
2-60. Energy Difference Between Linear and Planar Na2C2 .......................... 60
2-61. Dissociation Energies and Heats of Atomization for All Molecules .......... 61
CHAPTER 3
3-1. Quadrupole Moment, 9 (eafi ),of HCCH ......................................... 73

xiii

3-2. HCCH Quadrupole Moments ...................................................... 74
3-3. Quadrupole Moment, 9 (e33 ),of FCCF .......................................... 75
3-4. FCCF Quadrupole Moments ........................................................ 76
3-5. Quadrupole Moment, (9 (eafi ),of CICCCI ........................................ 77
3-6. ClCCCl Quadrupole Moments ...................................................... 78
3-7. Quadrupole Moment, (9 (ea?) ),of Linear Li2C2 .................................. 79
3-8. Quadrupole Moment, (922 (eafi ),of Planar Li2C2 ................................. 80
3-9. Quadrupole Moment, (9,, (eafi ),of Planar Li2C2 ................................. 81
3-10. Quadrupole Moment, (9 (e33 ),cf Linear Na2C2 ................................. 82
3-11. Quadrupole Moment, 922 (ea?) ),of Planar N a2C2 ................................ 83
3-12. Quadrupole Moment, (9,, (ea?) ),of Planar Na2C2 ............................... 84
3-13. Linear Molecules (XCCX) Partitioned into Point Charges ..................... 86
3-14. Planar Molecules (X2C2) Partitioned into Point Charges ....................... 93
CHAPTER 4
4-1. Values for Use in Approximations ................................................ 149
CHAPTER 5
5-1. CH4-N2 Linear Approach ............................................................ 169
5-2. Dipole and Potential from Molecular Distortion in the

CH4-N2 Linear Approach ............................................................ 170
5-3. CH4-N2 Perpendicular Approach ................................................... 175
5-4. Dipole and Potential from Molecular Distortion in the

CPL-N2 Perpendicular Approach ................................................... 176

xiv

5-5. CH4-H2 Linear Approach ............................................................ 177

5-6. Dipole and Potential from Molecular Distortion in

CH4-H2 Linear Approach ............................................................ 177
5-7. CH4-H2 Perpendicular Approach ................................................... 178
5-8. Dipole and Potential from Molecular distortion in the

CH4-H2 Perpendicular Approach ................................................... 178
5-9. CH4-He Approach .................................................................... 179
5-10. Dipole and Potential from Molecular Distortion in

CI-I4-He Approach .................................................................... 180
APPENDIX A
A-l. Heats of Formation of Atoms and Molecules ....................................... 185
A-2. HCCH Energies, De, and Do Values ................................................. 186
A-3. Extrapolated HCCH Energies, De, Do Values ...................................... 187

A-4. Core-Valence and Extrapolated Core-Valence HCCH Energies, De,
Do, Values with Unfrozen Core ...................................................... 188

A-5. Core-Valence and Extrapolated Core-Valence HCCH Energies, De,

Do, Values with Frozen Core ......................................................... 188
A-6. Optimized Geometries for FCCF ............................................ , .......... 189
A-7. FCCF Energies, De, and Do Values .................................................. 189
A-8. Extrapolated FCCF Energies, De, and Do Values .................................. 190
A-9. Core-Valence and Extrapolated Core-Valence FCCF Energies, De,
Do, Values with Unfrozen Core ...................................................... 191
A-10. Core-Valence and Extrapolated Core-Valence FCCF Energies, De,
Do, Values with Frozen Core ....................................................... 192
A-1 1. Optimized Geometries for ClCCCl ................................................. 193
A-12. ClCCCl Energies, De, and Do Values .............................................. 193
A-l3. Extrapolated ClCCCl Energies, De, and Do Values .............................. 194

XV

A-l4. Core-Valence and Extrapolated Core-Valence ClCCCl Energies, De,

Do, Values with Unfrozen Core .................................................... 195
A-15. Core-Valence and Extrapolated Core-Valence ClCCCl Energies, De,

Do, Values with Frozen Core ...................................................... 195
A-l6. Optimized Geometries for Linear Li2C2 .......................................................... 196
A-l7. Extrapolated Geometries of Linear Li2C2 ........................................................ 196
A-18. Linear Li2C2 Energies, De, and Do Values ......................................... 197
A-19. Extrapolated Linear Li2C2 Energies, De, and Do Values ........................ 197
A-20. Core-Valence and Extrapolated Core-Valence Linear Li2C2 Energies,

De, Do, Values with Unfrozen Core ............................................... 198
A-21. Core-Valence and Extrapolated Core-Valence Linear Li2C2 Energies,

De, Do, Values with Frozen Core .................................................. 198
A-22. Optimized Geometries for Planar Li 2C2 ............................................ 199
A-23. Extrapolated Geometries of Planar Li2C2 .......................................... 199
A—24. Planar Li2C2 Energies, De, and Do Values ......................................... 200
A-25. Extrapolated Planar Li2C2 Energies, De, and Do Values ........................ 200
A-26. Core-Valence and Extrapolated Core-Valence Planar Li2C2 Energies,

De, Do, Values with Unfrozen Core ............................................... 201
A-27. Core-Valence and Extrapolated Core-Valence Planar Li2C2 Energies,

De, Do, Values with Frozen Core .................................................. 201
A-28. Optimized Geometries for Linear Na 2C2 .......................................... 202
A-29. Extrapolated Geometries of Linear Na 2C2 ......................................... 202
A-30. Linear Na 2C2 Energies, De, and Do Values ....................................... 203
A-31. Extrapolated Linear Na 2C2 Energies, De, and Do Values ....................... 203

A-32. Core-Valence and Extrapolated Core-Valence Linear Na 2C2 Energies,
De, Do, Values with Unfrozen Core ............................................... 204

xvi

A-33. Core-Valence and Extrapolated Core-Valence Linear Na 2C2 Energies,

De, Dc, Values with Frozen Core .................................................. 205
A-34. Optimized Geometries for Planar N a2C2 ........................................... 206
A-35. Extrapolated Geometries of Planar N a2C2 .......................................... 206
A-36. Planar N a2C2 Energies, De, and Do Values ........................................ 207
A-37. Extrapolated Planar Na2C2 Energies, De, and Do Values ........................ 207
A—38. Core—Valence and Extrapolated Core-Valence Planar Na2C2 Energies,

De, Do, Values with Unfrozen Core ............................................... 208
A-39. Core-Valence and Extrapolated Core-Valence Planar N a2C2 Energies,

De, Do, Values with Frozen Core .................................................. 208
APPENDIX B
B-l. SCF Quadrupoles of HCCH ........................................................... 210
8—2. CCSD(T) Quadrupoles of HCCH ..................................................... 210
B-3. DFT(B3LYP) Quadrupoles of HCCH ................................................ 210
8-4. Quadrupoles of FCCF .................................................................. 212
8-5. Quadrupoles of ClCCCl ................................................................ 212
B6 Quadrupoles of Linear Li2C2 .......................................................... 213
B—7. 6),; Component of Planar Li2C2 Quadrupole ........................................ 214
B-8. (9,, Component of Planar Li2C2 Quadrupole ........................................ 215
B—9. Quadrupoles of Linear Na2C2 ......................................................... 216
8-10. (92, Component of Planar N a2C2 Quadrupole ...................................... 217
B-1 1. (9,, Component of Planar N a2C2 Quadrupole ...................................... 218
APPENDIX C

C-l. CH4-N 2 Perpendicular Approach using aug-cc-pvtz basis set for
5.0 and 4.0 Angstrom Separations ................................................. 220

xvii

C-2. CH4-N2 Perpendicular Approach using aug-cc-pvtz basis set for
3.5 and 3.25 Angstrom Separations ................................................ 220

C-3. CPL-N2 Perpendicular Approach using aug—cc-pvtz basis set for
3.0 Angstrom Separations ........................................................... 221

C-4. CH4-N2 Linear Approach using aug-cc-pvtz basis set for 5.0 and
4.0 Angstrom Separations ........................................................... 221

C-5. CHu-N2 Linear Approach using aug-cc-pvtz basis set for 3.5 and
3.25 Angstrom Separations ......................................................... 222

C-6. CH4-N2 Linear Approach using aug-cc-pvtz basis set for 3.0
Angstrom Separations ............................................................... 223

C-7. CH4-H2 Perpendicular Approach using aug-cc-pvtz basis set for
4.0 and 3.5 Angstrom Separations ................................................. 223

C-8. CH4-H2 Perpendicular Approach using aug-cc-pvtz basis set for
3.25 and 3.0 Angstrom Separations ................................................ 224

C-9. CH4-H2 Perpendicular Approach using aug-cc-pvtz basis set for
2.75 and 2.5 Angstrom Separations ................................................ 224

010. CH4-H2 Perpendicular Approach using aug—cc-pvtz basis set for
2.25 Angstrom Separations ......................................................... 225

C-1 1. CH4-H2 Linear Approach using aug-cc—pvtz basis set for 4.0 and
3.5 Angstrom Separations ........................................................... 225

C-l2. CH4-H2 Linear Approach using aug-cc-pvtz basis set for 3.25 and
3.0 Angstrom Separations ........................................................... 226

C-13. CH4-H2 Linear Approach using aug-cc-pvtz basis set for 2.75 and
2.5 Angstrom Separations ........................................................... 226

014. CH4-H2 Linear Approach using aug-cc-pvtz basis set for 2.25
Angstrom Separations ............................................................... 227

015. CH4-He Approach using aug-cc-pvtz basis set for 4.0 and 3.5
Angstrom Separations ............................................................... 227

016. CH4- He Approach using aug-cc-pvtz basis set for 3.25 and 3.0
Angstrom Separations ............................................................... 228

xviii

C-17. CH4-He Approach using aug-cc-pvtz basis set for 2.75 and 2.5
Angstrom Separations ............................................................... 228

C-18. CPL-He Approach using aug-cc-pvtz basis set for 2.25 Angstrom
Separations ............................................................................ 229

xix

LIST OF FIGURES

Figure Page
CHAPTER 2

2-1. Vibrational Modes of Linear Li2C2 ................................................ 39
2-2. Vibrational Modes of Planar Li2C2 ................................................ 45
2-3. Vibrational Modes of Linear Na2C2 ................................................ 51
2-4. Vibrational Modes of Planar N a2C2 ................................................ 56
CHAPTER 3

3-1. Diagram of Point Charge not located at Origin .................................. 68
3-2. HCCH Density Difference Plot .................................................... 87
3-3. FCCF Density Difference Plot ...................................................... 88
3-4. CICCCI Density Difference Plot ................................................... 90
3-5. Linear LiCCLi Density Difference Plot ........................................... 92
3-6. Planar Li2C2 Density Difference Plot ............................................. 94
CHAPTER 4

4-1. Representation of the 3-0 term ...................................................... 114
4-2. Representation of the 1-1 terms .................................................... 120

4-3. Representation of the 1-0 term in the Third-Order Energy. . . . . . . . . . . . . . .........134

4-4. Representation of the Third—Order Pure Dispersion Energy .................... 137
4-5. First Configuration of the HCI Dimer ............................................. 144
4-6. Second Configuration of the HCl Dimer .......................................... 144
4-7. Third Configuration of the HCl Dimer ............................................ 146
4-8. First Trimer Configuration .......................................................... 150

XX

4-9. Second T rimer Confi guration ...................................................... 150
4-10. Third Trimer Configuration (the triangle formation) ........................... 151
CHAPTER 5
5-1. Reflection Planes in CH4 ............................................................ 160
5-2. Diagram of Linear Approach, Viewed Locking Down on Plane A ........... 161
5-3. Diagram of Linear Approach, Viewed Looking Down on Plane B ............ 161
5-4. Diagram of Perpendicular Approach, Viewed Looking Down on Plane A... 162
5-5. Diagram of Perpendicular Approach, Viewed Looking Down on Plane B... 162
5-6. Energy as a Function of Bond Angle at 5.0 A Separation

for Methane-Nitrogen: Linear Approach .......................................... 165
5-7. Energy as a Function of Bond Angle at 4.0 A Separation

for Methane-Nitrogen: Linear Approach .......................................... 166
5-8. Energy as a Function of Bond Angle at 3.5 A Separation

for Methane-Nitrogen: Linear Approach .......................................... 167
59 Energy as a Function of Bond Angle at 3.25 A Separation

for Methane-Nitrogen: Linear Approach .......................................... 168
5-10. Energy as a Function of Bond Angle at 3.0 A Separation

for Methane-Nitrogen: Linear Approach .......................................... 169
5-11. Energy as a Function of Bond Angle at 5.0 A Separation

for Methane-Nitrogen: Perpendicular Approach ................................. 171
5-12. Energy as a Function of Bond Angle at 4.0 A Separation

for Methane-Nitrogen: Perpendicular Approach ................................. 172
5-13. Energy as a Function of Bond Angle at 3.5 A Separation

for Methane-Nitrogen: Perpendicular Approach ................................. 173
5-14. Energy as a Function of Bond Angle at 3.25 A Separation

for Methane-Nitrogen: Perpendicular Approach ................................. 174
5-15. Energy as a Function of Bond Angle at 3.0 A Separation

for Methane-Nitrogen: Perpendicular Approach ................................. 175

xxi

CHAPTER 1

INTRODUCTION

1.1 Introduction

The purpose of Chapter 2 of this thesis is to report the most accurate heats of
formation data for dilithium and disodium acetylide. Accurate enthalpies of
formation values are important for thermodynamic study of molecules. In general,
chemical theory can calculate the heat of formation to within 2 kcal/mol of
experimentally determined heats of formation. In order to calculate the heat of
formation, accurate vibrational frequencies are needed. For the dilithium and
disodium acetylides, this information is not available. Optimized geometries,
vibrational frequencies and the heats of formation are calculated. The HCCH, FCCF,
CICCCI geometries, vibrational frequencies, and heats of formation are calculated as
reference points to check accuracy. The optimized geometries and heats of formation
for all molecules discussed are extrapolated to the complete basis set limit using four
extrapolation equations. A comparison of the results for each extrapolation method
was made. Core-valence, spin-orbit coupling, and relativistic corrections are added to
attempt to determine the most accurate heats of formation.

In Chapter 3, the quadrupole moments of HCCH, FCCF, CICCCI, Li2C2, and
Na2C2 are calculated. For these molecules, the quadrupole moment is the first non-
vanishing multipole moment, and as the dipole moment provides information about
the electron distribution in a dipolar molecule, the quadrupole moment provides
information about the electron distribution for these doubly substituted acetylenes and
acetylides. For HCCH, FCCF, CICCCI, and the linear acetylides, there is only one
unique component for the quadrupole moment, but for the planar acetylides, the

quadrupole moment has two unique components. After calculating the quadrupole

moment for each molecule, the quadrupole moment is broken down into its various
components. Discussions are made concerning inferences that can be made between
the quadrupole moment components and the electron distribution in the molecule.
The quadrupoles are then partitioned into algebraic charges and compared in an
attempt to draw conclusions about the nature of the bonding in the disubstituted
acetylides. The shortcomings of the theory are discussed.

Chapter 4 and Chapter 5 concern van der Waals forces and the influences of those
interactions. Van der Waals forces explain the boiling points of noble gases and
collision induced absorption of compressed gases. In Chapter 4, the two body second
order and two body third order interaction energies are derived. The interaction
energies are broken into categories and physical explanations for each type of
interaction are given. Using the Unsold’s Theorem, numerical estimates are made
including estimates for newly developed terms in the third-order interaction energy.
Using HCI as an example, the two body second- and third-order interaction energies
are compared for two configurations. The three and two body third-order interaction
energies are compared for three HCl configurations.

Finally, Chapter 5 deals with the effects of two body interactions. CH4+X
compressed gases, where X=N2, H2, or He, are studied due to their intriguing
behavior when examined using IR. Each pair exhibits unusual absorption in the far-
infrared region of their spectrum. There is belief that some of this absorption is due
to induced dipoles from distortion of the molecular frame of CH4. This distortion is
due to the collision of CH4 with N2, H2, or He. Two collision paths are defined and

calculations to determine the distortion of the methane molecule caused by these

approaching molecules are made. Electrostatic interactions within the CH4+X pairs
are discussed. In addition, the dipole moment and potential difference due to

distortion are calculated as are the interaction energies within the pairs.

Chapter 2

The Enthalpy of Formation of Doubly Halogenated Acetylenes and Alkali Acetylides

2.1 Introduction

Gas phase Li2C2 and Na2C2 are intriguing compounds for which there is no
experimental thermochemical or geometric data available and little to no theoretical
information available. During an investigation into the electronic properties of these
doubly alkalized acetylides, the two geometry configurations of these acetylides were
shown to be viable minima on the potential energy surface]. Although a linear
configuration is the form of acetylene and many substituted acetylenes, the lowest energy
arrangement for Li2C2 and N a2C2 is a doubly bridged molecule. In this report, the
geometries for Li2C2 and Na2C2 at both the linear and doubly bridged configurations are
tabulated, and the respective enthalpy of formation for each species is calculated.
Because there is a lack of experimental and theoretical data available for these two metal
acetylides, we calibrated our calculations by determining the geometry and the enthalpy
of forrnaticn for acetylenez‘3 and the two doubly substituted halogenated acetylenes,
ClCCCl3 and FCCF“, and then compared those results to those found by previous
researchers.

There have been several experimental studies for HCCH, CICCCI, and FCCF.
Acetylene, due to its commonality, is well documented theoretically and experimentally,
and its properties have been used in comparative studies to determine the accuracy of
theoretical methods. CICCCI and FCCF, on the other hand, are very reactive molecules
making such extensive study harder. Although the equilibrium geometry and the
enthalpy of formation for CICCCl are known, the experimental heat of formation for
dichloroethyne has error bars which are approximately a fifth of the experimental value.

Due to its wider range of experimental values, FCCF is even more intriguing. There are

two experimental heats of formation available, which describe the enthalpy of formation
of difluoroacetylene as both an endothermic or exothermic process.

2.1.1 Method

For the general reaction, Y2X2(g) —-> 2X(g) + 2Y(g) , the first step taken in

determining the enthalpy of formation for Y2X2 was to determine the energy change

after the reaction moves from reactant to products. The total energy of the reactants and
products are a sum of the electronic energies, as well as translational, vibrational, and
rotational contributions. Since the products are atoms, only the reactant has rotational
and vibrational contributions. Assuming that Y2X2 is linear, as in the case of HCCH,
FCCF, CICCCI, linear Li 2C2, and linear Na2C2, the reactant only has two rotational

degrees of freedom.

 

 

 

 

Eproducts=2Ex+ 333 +2Ey+ 313$ . (1)
translations translations
K \
3n-5
3 hv- hv'
Emma“, =EYXXY+ —RT + RT +NAZ —J-+ J .(2)
2 “'7‘ _ 2 hv-
HE rotations J ex J _
translations
. \ "bT 1.
vibragns

In the case of doubly bridged Li2C2 and Na2C2, the reactant has three rotational

degrees of freedom.

 

 

 

 

 

f \
3n-6
3 3 hv- hv-
Ereactant = EYZXZ + ERT + ERT +NA 2 21+ th .(3)
U ‘V‘.’ W" i exp[—J]—l

anslatrons rotations k T

. K b 2.
vibrations

The difference in the initial and final states gives the energy change, and Equation (5)

shows the generic relation between the energy change and the enthalpy of atomization.
AB = Eproducts ‘ Ereactants' (4)
AH am, = AE + AnRT . (5)

Shown in Equation (6) is the relation between the energy change and the enthalpy of

atomization for the reactions described in this manuscript.
AHaltm = AE + 3RT . (6)
In any reaction, the enthalpy of reaction is related to the heat of formation of the
products and reactants through Equation (7).

AHrxn = Z AHf (products)- 2 AHf (reactants). (7)
For the specific case of Y2X2 , the heat of reaction is described by Equation (8).

Ana", = 2m, (x(g))+ 2qu (Y(g))— AHf (Y2x2(g)). (8)

Equation (8) can be rearranged to express the enthalpy of formation of Y2X2 as a sum of

the heats of formation of the constituent atoms and the enthalpy of atomization.

AHf (Y2x2(g))= 2AHf (x(g))+ 2AHf (Y(g))— AHatm . (9)
The enthalpies of formation for the atoms are nonzero because the atoms are not the
standard state of the elements. A table of the standard enthalpy of formation for the
atoms is in the appendix. Using experimental heats of formation for the atoms and the
calculated heat of atomization, the enthalpy of formation for the reactant, Y2X2 , was

determined.

2.1.2 Computational Details

Using the MOLPRO 20005 and Gaussian 986 software packages, we calculated
geometry optimization, fundamental vibrational frequency, and single point energy.
Dunning ccrvxz. AUG-CC-PVXZ, and D-AUG-CC-PVXZ basis sets7'8‘9‘m were
used, as well as augmented basis sets which were constructed on site by augmenting the
Li and Na Dunning CC-PVXZ basis sets with even-tempered diffuse functions. X is the
cardinal index of the basis set. Therefore, the CC-PVXZ basis set where X equals 2
refers to the CC-PVDZ basis set. Since the core-valence correlation was not captured
using the suite of Dunning polarized valence basis sets, Dunning’s CC—PCVXZ and
AUG-CC-PVXZ basis sets were used to determine core-valence correlation corrections.

The standard enthalpy of formation and optimized geometries for each of the
aforementioned molecules were calculated as a function of basis set at the CCSD(T) level
and then extrapolated to the complete basis set limit. After extrapolation, the
thermochemical values were corrected for spin-orbit coupling, core-valence correlation,
and mass dependent relativistic effects. The atomic spin-orbit effects were computed
using atomic energy levels tabulated by Charlotte Moore”. Core-valence correlation was
determined as a function of basis set from CCSD(T) single point energy calculations and
extrapolated to the basis set limit. The mass dependent relativistic energy corrections
were determined from a Hartree-Fock calculation using the Cowan-Griffin approach from
expectation values of the mass-velocity and Darwin terms using a CC-PVDZ basis set.
2.1.3 Corrections

The non-relativistic Hamiltonian commonly used in quantum calculations only

accounts for the kinetic energy of the electrons and the electrostatic potential energy due

 

to nuclear-nuclear, nuclear-electron, and electron-electron interactions. Since there is no
consideration of spin or relativistic effects, any consideration of the mass-velocity
correction, Darwin term, or spin-orbit coupling corrections is considered relativistic. The
only non-relativistic correction made to the enthalpy of formation in this manuscript was

the addition of core-valence correlation.
The dissociation energy (De) for these acetylene and doubly substituted acetylene

molecules was calculated from CCSD(T) energy calculations in which the non-valence
shell electrons were frozen. In addition to this inability of the core electrons to be excited
to unoccupied molecular orbitals, the Dunning basis sets used were not designed to pick
up core-valence correlation. To include core-valence correlation absent from the
CCSD(T) single point energy calculations, a core-valence correction was determined
using the Dunning CC-PCVXZ and AUG-CC-PCVXZ basis sets. These polarized core-
valence basis sets are different from polarized valence basis sets because of additional
gaussian functions, which have been added to the polarized valence basis sets. These
added gaussian functions have exponents, which place them in the electron shell directly
below the valence shell. Because these new basis functions are located in the “core” of
the atom or molecule, additional information concerning the core-valence interactions
can be extracted. The additional functions were added to the carbon, lithium, and

fluorine ls orbital and the sodium and chlorine 2s and 2p orbitals.
We calculated the D2 for each molecule twice using the polarized core-valence

basis sets. The first calculation was done using a set of CCSD(T) single point energy
calculations where the atomic and molecular electronic cores were frozen, and the second

was made from a set of CCSD(T) calculations in which the core electrons were unfrozen.

10

The energy difference between the two dissociation energies is representative of the core-
valence correlation, and this energy difference is considered the core—valence correction.

Often with lighter atoms and molecules during quantum calculations, the
relativistic effects within the atom or molecule are ignored. In general, this
approximation is valid because for light species the energy differences for relativistic
effects between energy states or geometric orientations are negligible. Even though this
approximation tends to diminish the importance of the relativistic effects, the contribution
of relativistic effects to the total energy of an atom or molecule is significant. Therefore,
we made two corrections. The first correction is for the change in the mass of the
electrons and the resulting change in the velocity due to the mass change. The second
correction is for the “Zitterbewegung” or “trembling” of the electrons, which is the
relativistic smearing of an electron’s charge. In the Dirac theory, this correction is often
referred to as the Darwin term. The Darwin term correction can be thought of as a
correction for the “size” of the electrons.

As stated above, the mass-velocity and the Darwin term relativistic energy
contributions were determined using a Hartree-Fock wavefunction consisting of CC-
PVDZ basis functions. We added the mass—velocity and Darwin term operators to the
non-relativistic Hamiltonian as a perturbation, and the first-order energy correction was

computed. Having determined these corrections for the atoms and the molecule of
interest, a correction to the De was calculated producing the mass-dependent relativistic

energy correction.
The remaining relativistic energy correction is spin-orbit coupling. Resulting

from the interaction between the magnetic moments created by an electron’s spin and

11

orbital angular momentum, this splitting in the energy levels is generally referred to as

the fine structure, and it is the cause for the unique optical spectra of atoms. A carbon
atom in its ground electronic state is 3P. Due to the spin-orbit interaction, this 3P state

divides into sets of one 3 P0 state, three 3 PI states, and five 3P2 states. For halogens such

as chlorine and fluorine, the ground state is a 2 P state, which divides into four 2Py and
2

two 2 P}, states. For hydrogen, lithium, sodium, acetylene, and the doubly substituted
2

acetylene molecules, the ground state does not split. The multiple energy levels created
by this splitting are not degenerate, and since the possibility that a ground state atom can
be in one of these ground state energy levels, the distribution must be taken into account.
The single point energy calculations for the atoms and molecules used in this
study are run at zero degrees Kelvin (0 K), and only the lowest lying state is taken into
consideration. At this temperature, the probability of an atom being in any of the higher
energy levels is equal to a Boltzmann distribution. Therefore, the average energy of the

ground-state atoms is described by Equation (10).

(E) = _9;_ng where, q = 2.:gie-giB . (10)

After completing the derivative, the average energy takes the form of Equation (11).
Zgiei
i

=—___, (11)
Zgic—SIB
i

(E)

Since the temperature is 0 K, the average energy is equal to the mean energy as seen in

Equation (12).

12

Zgiei
(E) = lZgi °

 

(12)

Because the De values are calculated for the lowest lying energy states of the molecule

and its constituent atoms, the spin-orbit correction for HCCH, Li2C2, and Na2C2 is the
difference between the energy of the lowest lying state of carbon and the average energy

of the carbon atom. Equation (13) shows this relation.

AB.-. = 2(Ecarbon)-2Ewbo- (13)
The spin-orbit correction for ClCCCl and FCCF is equal to the spin-orbit correction for
acetylene and the doubly substituted acetylides plus the difference between the energy of

the lowest lying state of the halogen and the average energy of the halogen atom.

Equation (14) expresses this relation.

ABs-o = 2(Ecarbon) - 2Ecarbon + 2<E halogen > — 25 halogen . (14)
The tables of atomic energy levels that Charlotte Moorell published list the energy of
these spin-orbit states. Using these values, the correction for average energy of the atoms
(the spin-orbit correction) was subtracted from the dissociation energy.
2.1.4 Extrapolations to CBS limit

Extrapolated values are done using two extrapolation techniques: extrapolating
only the property to the basis set limit, P(oo) , and calculating the property from

extrapolated energies of the molecule and its constituent atoms at the basis set

limit, P(E(oo)). For example, in order to determine the DC of a molecule at the basis set

limit, first the molecular and atomic energies are calculated as a function of basis set.

From these single point calculations, the De is calculated as a function of basis set. To

13

extrapolate to the basis set limit, one could either extrapolate the property (De) to the
basis set limit, or one could individually extrapolate the single point molecular and
atcrnic energies to the basis set limit and calculate the De from those extrapolated

energies.

This concept of extrapolating a property to the basis set limit is dependent on a
property of the variational theorem. For the non-relativistic time-independent
Hamiltonian operator, there is a lowest energy eigenvalue. In quantum mechanics, the
wavefunction of an atom or molecule is approximated. As the wavefunction becomes
more accurate for the system in question, the resulting eigenvalue approaches this lowest
value. The point where an infinite basis set is used is referred to as the basis set limit,
and it is the limit of accuracy for the particular method used. Since the energy converges
in this manner, this same methodology is applied to the calculation of properties by
extrapolating to the basis set limit.

Many researchers have investigated the relation between basis set size and the
convergence of the energy of a system. The Feller equation, as Martin calls it, originates
from an empirical development. There is no quantitative development made to reach that
nonlinear equation except to note how well the equation models the behavior of the
system. The Schwartz equations originate from a quantitative derivation of how the
energy converges as higher order angular momentum functions are added to the basis set.
The Martin'3 paper compares the behavior of each extrapolation equation for small

molecules.

14

For each of the extrapolation techniques, the CCSD(T) values were fit to four
different nonlinear equations. Dunning and associates use an exponential equation

originated by Feller12 where X is again the cardinal index and Otis a variable power.
P(X)= P(oo)+Ae‘°‘X. (Feller) (15)
In a recent paper, Martinl3 offers three alternative polynomial equations for nonlinear

fitting referred to as Schwartz-type formulas derived from the works of Schwartz”,

Kutzelnigg's’w, Morgan”, and Hill”.

 

P(X)= P(oo)+—A—. (Schwartza) (16)
(met
P(X)=P(oo)+ A + B 6.(Schwartz6) (17)
(x+i)“ (mi)
A

 

(Schwartz4) (18)

Both P(oo) and P(E(oo)) methods should not necessarily produce the same value.
For the Schwartz4 and Schwartz6 methods, the P(oo) and P(E(oo)) results are more likely
to be equivalent because the extrapolation equations for the atoms and molecules are
consistent. If the error in the complete basis set (CBS) limit is consistent for all the
extrapolated properties and energies, then the results produced will be consistent. Feller
and Schwartzor equations allow more freedom for the extrapolation equation because of
the variable a. exponent. This freedom provides an Opportunity for the fit for the atoms to
be tighter than the fit for the molecule, and this inconsistency will cause differences

between the P(oo) and P(E(oo)) results.

15

Removing the data points from calculations in which smaller basis sets were used
is another option in extrapolating to CBS limit. Using the most accurate data points for
extrapolation can aid in producing a more accurate extrapolated value. For this report,
many smaller data sets were created from each complete data set collected. For example,
the energy of the carbon atom was calculated using a spectrum of basis sets ranging from
CC-PVDZ to CC-PV6Z. The original set contained all the data points calculated. Then,
subsequent sets were created by systematically removing the data calculated at smaller
basis sets such as the CC-PVDZ and CC-PVTZ basis sets. By removing carbon atom
energies calculated at smaller basis sets, the only values remaining are the most accurate
values. Extrapolations were done for each of these data sets and listed to show the
change in the extrapolated values as the less accurate values were removed. The
extrapolated core-valence correction was extrapolated using only the results from the
larger basis sets.

2.2 HCCH

Table 1 lists calculated CCSD(T) optimized geometries for HCCH as a function
of the Dunning unaugmented and augmented CC-PVXZ basis sets. Table 1 also tabulates
HCCH extrapolated bond distances and an experimental geometry provided by Strey and
Mills”. The notation for each extrapolation designates both the nonlinear fitting equation
and the cardinal indices of the basis sets included in the extrapolation. Therefore, a
Schwartz4(Q56) value is an extrapolated value using Equation (18) or the Schwartz4
equation from extrapolated properties calculated using basis sets where the cardinal index

X is equal to the set {Q,5,6}.

l6

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Table 1. Optimized Geometries of HCCH as a Function of Basis Set
. RCC RCH . RCC RCH
Basrs Basrs

(arts) (ang) (ang) (ans)
CC-PVDZ 1.22877 1.07899 AUG-CC-PVDZ 1.22998 1.07866
CC-PVTZ 1.20973 1.06372 AUG-CC-PVTZ 1.21016 1.06387
CC-PVQZ 1.20634 1.06337 AUG-CC-PVQZ 1.20686 1.06355
CC-PVSZ 1.20557 1.06303 AUG-CC-PVSZ 1.20587 1.06314
CC-PV6Z 1.20534 1.06300 AUG-CC-PV6Z 1.20556 1.06312
Feller(DTQ56) 1.2054 1.0631 Feller(DTQ56) 1.2057 1.0633
Feller(TQ56) 1.2053 1.0628 Feller(TQ56) 1.2054 1.0628
Feller(Q56) 1 .2052 1.0630 Feller(Q56) 1.2054 1.0631
Schwartza(DTQ56) 1.2051 1.0631 Schwartza(DTQ56) 1.2053 1.0632
SchwartzuCTQ56) 1.2052 1.0626 SchwartzaCT Q56) 1.2052 1.0623
Schwartza(Q56) 1.2052 1.0630 Schwartza(Q56) 1.2053 1.0631
Schwartz4(DTQ56) 1.2042 1.0617 Schwartz4(DTQ56) 1.2045 1.0618
Schwartz4(TQ56) 1.2048 1.0630 Schwartz4(TQ56) 1.2051 1.0631
Schwartz4(Q56) 1.2050 1.0629 Schwartz4(Q56) 1.2051 1.0629
Schwartz4(56) 1.2051 1.0630 Schwartz4(56) 1.2052 1.0631
Schwartz6(DTQ56) 1.2050 1.0634 Schwartz6(DTQ56) 1.2052 1.0635
Schwartz6(TQ56) 1.2052 1.0628 Schwartz6(TQ56) 1.2052 1.0629
Schwartz6(Q56) 1.2052 1.0631 Schwartz6(Q56) 1.2053 1.0632
Schwartz6(56) 1.2051 1.0630 Schwartz6(56) 1.2052 1.0631
‘gexperi mental 1 .2033 1.0605

 

In Table 1, the augmented and unaugmented basis sets carbon-carbon bond length

has converged to three decimal places by the 5-zeta basis set, and the carbon-hydrogen

bond length has converged to three decimal places by the Q-zeta basis set. The

extrapolated bond lengths confirm that at the 6-zeta basis set both the augmented and

unaugmented basis sets are very close to the basis set limit. There is no general trend in

the extrapolated values as optimized geometries calculated at smaller basis sets are

removed. Compared to the experimental geometry, the basis sets are not converging

towards the Strey and Mills experimental geometry. Although the geometry at the basis

set limit is not the true HCCH geometry, the CCSD(T) method limit is within three

thousandths of an angstrom, which is a small deviation and will have no effect on the

calculated heat of formation.

 

Using the Strey and Mills experimental geometry listed in Table 1, we completed
single point CCSD(T) calculations for the acetylene molecule. Taking advantage of
experimental vibrational frequencies provided by Schimanouchi19 listed in Table 2, and
using standard heats of formation for the constituent atoms provided by Chase2 listed in

the appendix, we calculated the enthalpy of formation of HCCH as a function basis set.

 

 

 

Table 2. Vibrational Frequencies of HCCH
Symmetry Frequency (cm'l)
Cis-bend (11“) 730
Trans-bend (Hg) 612
CC stretch(2g) 1974
s-CH stretch (2g) 3374
a-CH stretch (21,) 3289

 

 

 

 

Shown in Table 3 are dissociation energies of HCCH at 0 K and heats of

formation for HCCH at 298K for the unaugmented and augmented CC-PVXZ Dunning

 

 

 

 

basis sets.
Table 3. Calculated Dissociation Energies and Heats of Formation for HCCH
as a Function of Basis Set

. De AHf

Basrs Set (kcal/mol) (kcal/mol)
CC—PVDZ 370.20 89.23
CC-PVTZ 393.35 66.08
CC-PVQZ 399.59 59.84
CC-PVSZ 401.47 57.96
CC-PV6Z 402. 19 57.24
AUG-CC-PVDZ 371.09 88.34
AUG-CC-PVTZ 394.56 64.87
AUG-CC-PVQZ 400.09 59.35
AUG-CC-PVSZ 401.73 57.70
AUG-CC-PV6Z 402.33 57.1 1

 

 

 

 

 

Augmentation of the CC-PVXZ basis set does not alter the magnitude of either

the De or the heat of formation greatly, and the enthalpy of formation is near convergence

for both basis sets. Note that as the De increases, the enthalpy of formation decreases.

l8

This behavior is due to the mathematical nature of the relation between the two values.

Equation (9) shows the inverse dependence of the enthalpy of formation on the enthalpy
of atomization in which the enthalpy of atomization is merely the Dc with thermal
corrections. Table 4 lists the dissociation energies and enthalpies of formation at the

complete basis set limit for each nonlinear extrapolation equation for both P(oo) and

P(E(oo)) methods.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Table 4. Heats of Formation for HCCH Extrapolated to the Complete Basis
Set Limit

Basis sets AHf (°°) AHf (E(°°»

CC-PVXZ (kcal/mol) (kcal/mol)
Feller(DTQ56) 57.18 57.18
Feller(Q56) 56.79 56.82
Schwartza(DTQ56) 56.03 56.03
Schwartzor(Q56) 56.51 56.57
Schwartz4(DTQ56) 56.81 56.81
Schwartz4(56) 56.48 56.48
Schwartz6(DTQ56) 56.25 56.25
Schwartz6(56) 56.48 56.48

AUG-CC-PVXZ

Feller(DTQ56) 57.16 57.18
Feller(Q56) 56.77 56.78
Schwartza(DTQ56) 56.29 56.34
Schwartzor(Q56) 56.57 56.59
Schwartz4(DTQ56) 56.39 56.39
Schwartz4(56) 56.48 56.48
Schwartz6(DTQ56) 56.30 56.30
Schwartz6(56) 56.48 56.48

 

 

Table 4 has two columns of data. The rightmost column contains the enthalpy of
formation calculated using extrapolated energies for the constituent atoms and the
acetylene molecule. The left data column contains the heat of formation where values for
the enthalpy of formation were extrapolated to the complete basis set limit. As required,

the Schwartz4 and the Schwartz6 results are consistent for each method at both suites of

19

basis sets, and when only the two best points are used for extrapolation, the Schwartz4
and Schwartz6 results are equal to each other. Looking at the Feller and Schwartza
results, the differences between the two methods for each equation is minute. For all the
extrapolation methods and fitting equations, the extrapolated value was within 1 kcal/mol
of the enthalpy of formation computed at the AUG-C C -PV6Z basis set.

To correct for missing core-valence correlation, we used the Dunning
unaugmented CC-PCVXZ basis sets, and the results are shown as a function of basis set

and tabulated in Table 5.

 

 

 

Table 5. Core-Valence Correction for HCCH as a Function of Basis Set
Basis Sets Core-Valence Correction
(kcal/mol)
CC-PCVDZ _ 1.13
CC-PCVTZ _ 1.88
CC-PCVQZ -2.28
CC-PCVSZ 4.41

 

 

 

 

Using the data in Table 5, we extrapolated the core-valence correction to a complete basis

set for both P(oo) and P(E(oo)) extrapolation techniques. Table 6 lists the corrections.

 

 

 

 

 

Table 6. Core-Valence Correction for HCCH at the Complete Basis Set Limit
Basis sets ACV(°°) ACV(E(°°»
CC-PCVXZ (kcal/mol) (kcal/mol)
Feller(T Q5) -2.46 -2.47
Schwartza(TQ5) -2.50 -2.51
Schwartz4(Q5) -2.51 -2.51
Schwartz6(Q5) -2.51 -2.51

 

 

 

As mentioned before, not all data points were included in the extrapolations.
Only corrections calculated with the larger basis sets were kept for extrapolation, and
those extrapolated corrections were used as core-valence corrections. Table 6 illustrates

the core-valence corrections at the basis set limit. From the table we can see, the core-

20

valence correction extrapolations are consistent between methods and differ only slightly
between extrapolation equations.

In Table 7, we collected enthalpy of formation values corrected for relativistic
effects (0.25 kcal/mol), spin-orbit coupling (0.17 kcal/mol), and the core-valence
corrections from Table 6. As we can now see, the relativistic corrections to the energy
were not negligible. We added the core-valence corrections to the enthalpy of formation

values from Table 4. We calculated the Feller(TQS) core-valence correction using the
P(oo) method and combined it with the Feller(DTQ56) and Feller (Q56) heat of
formation values calculated using the P(oo) method. This methodology was used for all
extrapolation equations and methods. Table 7 includes also the difference between the

calculated enthalpy of formation and the excepted experimental value of 54.19 i019

kcal/mol provided by Chasez.

21

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Table 7. AH; Values Corrected for Core-Valence Correlation Spin-Orbit
Coupling and Relativistic Effects for HCCH at the Complete Basis Set Limit for
each Method
Difference from Difference
Basis sets Corrected experimental Corrected from
AHf (co) value AHf (E(oo)) experimental
value
CC-PVXZ (kcallmol) (kcallmol) (kcallmol) (kcallmol)
Feller(DTQ56) 55.14 0.95 55.14 0.95
Feller(Q56) 54.74 0.55 54.77 0.58
Schwartza(DTQ56) 53.95 -0.24 53.94 ~0.25
Schwartza(Q56) 54.43 0.24 54.48 0.29
Schwartz4(DTQ56) 54.72 0.53 54.72 0.53
Schwartz4(56) 54.39 0.20 54.39 0.20
Schwartz6(DTQ56) 54. 16 -0.03 54. 16 -0.03
Schwartz6(56) 54.39 0.20 54.39 0.20
AUG-CC-PVXZ
Feller(DTQ56) 55.12 0.93 55.14 0.95
Feller(Q56) 54.73 0.54 54.74 0.55
Schwartza(DTQ56) 54.21 0.02 54.25 0.06
Schwartzor(Q56) 54.49 0.30 54.50 0.3 1
Schwartz4(DTQ56) 54.30 0.1 1 54.30 0.1 1
Schwartz4(56) 54.39 0.20 54.39 0.20
Schwartz6(DTQ56) 54.22 0.03 54.22 0.03
Schwartz6(56) 54.39 0.20 54.39 0.20

 

 

 

 

 

 

The first thing to note is the loss of accuracy with the removal of points calculated
at smaller basis sets. When only the best data points were used, the error increased by at
least 0.1 kcallmol; however, regardless of the basis sets used, the resulting enthalpy of
formation is within 1 kcallmol of the experimental value. In addition, the difference
between the two extrapolation methods is very small, presumably due to the high
convergence of the values before extrapolation. The more accurately the fitted equation
matches the true convergence curve, the less discrepancy exists between the two

extrapolation methods, P(oo) and P(E(oo)). Of the four extrapolation equations, the best

performing equations are the Schwartz-type equations. These three equations yield

results, which fall within the experimental error bars. However, the Feller equation does

22

not provide such accuracy. These general trends are consistent with Martin’sl3 findings
in his study of the DC of HCCH. In Table 8, we collect the best-extrapolated heats of

formation from the AUG-CC-PVXZ basis sets from this work, as well as experimental

and select theoretical results from the literature.

 

Table 8. Comparison of Heats of Formation of HCCH

 

 

 

 

AHf EI‘I‘OI'
kcallmol kcallmol

”This work 54.73 0.54
brhis work 54.21 0.02
”This work 54.30 0.11
“This work 54.22 0.03
3Parthiban(W1) 54.41 0.22
31>arthiban(wz) 54.48 0.29
2experiment 54.19 $0.19

 

alFeller(Q56) extrapolation of augmented basis set values with P(oo) method

bSchwartza(DTQ56) extrapolation of augmented basis set values with P(oo) method
cSchwartz4(DTQ56) extrapolation of augmented basis set values with P(oo) method
dSchwartz6(DTQ56) extrapolation of augmented basis set values with P(oo) method

 

 

 

The acknowledged values for the enthalpies of formation are the experimental
values provided by Chasez. As mentioned earlier, most of the results from this work are
within the error bars associated with Chase's work. The Parthiban3 values are calculated
in a similar manner as this report using the Schwartz4, 6, and or equations, but the initial
geometries and vibrational frequencies for HCCH were calculated using ab initio and
DFT methods. In this study, the experimental geometries and fundamental vibrational
frequencies from the literature were used.

2.3 FCCF

Experimental values for FCCF are difficult to determine as stated by Parthiban3 et

al. ,

23

“. . .there remains no direct determination of the heat of formation of any
haloacetylene-these species are simply too unstable and/or reactive to allow for
such measurements.”
Because FCCF is so reactive, there are no reliable experimental data to use for
comparison. Therefore, we used the equilibrium geometry and vibrational frequencies
from theoretical work done by Breidung20 et al. Table 9 and Table 10 list these values.
Calculated CCSD(T) optimized geometries as a function of the CC-PVXZ and AUG-CC-
PVXZ basis sets are listed in Table 9 along with the extrapolated geometries and the

reference geometry provided by Breidung.20

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Table 9. Omimized Geometries of FCCF as a Function of Basis Set
Basis Sets RCC RCF Basis RCC RCF
(ang) (ang) (ang) (ans)
CC-PVDZ 1.20969 1.30039 AUG-CC—PVDZ 1.21 157 1.30594
CC-PVTZ 1.19277 1.28859 AUG-CC-PVTZ 1.19280 1.28948
CC-PVQZ 1.18918 1.28586 AUG-CC-PVQZ 1.18951 1.28613
CC-PVSZ 1.18857 1.28512 AUG-CC-PVSZ 1.18868 1.28524
Feller(DTQS) 1.1883 1.2850 Feller(DTQS) 1.1886 1.2851
Feller(TQS) 1.1884 1.2849 Feller(TQS) 1.1884 1.2849
Schwartza(DTQ5) 1.1878 1.2844 Schwartza(DTQ5) 1.1881 1.2845
Schwartza(TQ5) 1.1884 1.2847 Schwartza(TQ5) 1.1882 1.2847
Schwartz4(DTQ5) 1.1872 1.2844 Schwartz4(DT Q5) 1 . 1872 1.2841
Schwartz4(TQ5) 1.1875 1.2844 Schwartz4(TQ5) 1 . 1878 1.2845
Schwartz4(Q5) 1.1881 1.2845 Schwartz4(Q5) 1.1880 1.2845
Schwartz6(DTQ5) 1.1877 1.2844 Schwartz6(DTQ5) 1 . 1880 1.2844
Schwartz6(TQ5) 1 . 1884 1.2846 Schwartz6(TQ5) 1 . 1882 1.2846
Schwartz6(Q5) l . 1881 1.2845 Schwartz6(Q5) 1 . 1880 1.2845
20reference 1.186 1.2835

 

 

The optimized geometries in Table 9 show that the augmentation of CC-PVXZ
basis set does not affect the convergence of the basis set. Like the HCCH geometries, the
carbon-carbon and carbon-fluorine bonds, if rounded, converges to three decimal places
by the 5-zeta basis set. Note the extrapolated geometries differ by less than a thousandth

of an A from the 5-zeta result for both the unaugmented and augmented basis sets. Also,

24

note the CCSD(T) optimized geometries are converging to a value that differs only
0.002A from the reference value.

We used the unaugmented, augmented, and doubly augmented Dunning basis sets
to make single point CCSD(T) calculations at the reference geometry provided in Table
9. We used vibrational frequencies provided by Breidung20 et al listed in Table 10 and
the heats of formation for the fluorine and carbon atoms provided by Chase2 to determine

the enthalpy of formation for FCCF.

 

 

 

 

 

Table 10. Vibrational Frequencies of FCCF
Symmetry Frequencies (cm'l)
Cis-bend (1'1“) 276
Trans-bend (l'Ig) 284
s-CF stretch (22g) 792
a-CF stretch (2“) 1375
CC stretch (22) 2527

 

 

Table 11 lists the calculated standard enthalpy of formation values at 298 K and

dissociation energies at 0 K as a function of basis set.

 

 

 

 

 

 

Table 11. Calculated Dissociation Energies and Heats of Formation
for FCCF as a Function of Basis Set
- Dc AHf
Basrs sets (kcallmol) (kcallmol)

CC-PVDZ 343.01 43.15
CC-PVTZ 370.27 15.89
CC-PVQZ 378.47 7.70
CC-PVSZ 381.10 5.06
AUG-CC-PVDZ 344.37 41.80
AUG-CC-PVTZ 372.44 13.73
AUG-CC-PVQZ 379.67 6.50
AUG-CC-PVSZ 381.65 4.51
D-AUG-CC-PVDZ 346.83 39.33
D-AUG-CC-PVTZ 373.75 12.42
D-AUG-CC-PVQZ 380.36 5.80
D-AUG-CC-PVSZ 38 1 .87 4.29

 

 

 

 

25

Unlike HCCH, the dissociation energies of FCCF are not as closely converged at
these basis set levels. The heats of formation show that the difference between the Q-zeta
and 5-zeta results is a large percentage of the calculated value. The level of augmentation
seems to aid in the rate of convergence, but the differences between unaugmented,
augmented, and doubly augmented values are small but significant in comparison to the
magnitude of the reported 5-zeta value. Therefore, the addition of higher angular
momentum functions to the wave function influences the enthalpy of formation of FCCF

greatly. Table 12 illustrates heats of formation values extrapolated to the basis set limit.

26

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Table 12. Heats of Formation for FCCF Extrapolated to the Complete Basis
Set Limit
Basis sets AHf (co) AHf (E(°o))
CC-PVXZ (kcallmol) (kcallmol)
Feller(DTQ5) 4.00 4.08
Feller(TQ5) 3.81 3.87
Schwartza(DTQ5) 1.55 1.76
Schwartzon(TQ5) 2.88 3.01
Schwartz4(DTQ5) 4.24 4.24
Schwartz4(Q5) 2.92 2.92
Schwartz6(DTQ5) 2.52 2.52
Schwartz6(Q5) 2.92 2.92
AUG-CC-PVXZ
Feller(DTQ5) 3.88 4.08
Feller(TQ5) 3.76 3.93
Schwartza(DTQ5) 2.23 2.71
Schwartzoc(TQ5) 3.22 3.57
Schwartz4(DTQ5) 3.05 3.05
Schwartz4(Q5) 2.90 2.90
Schwartz6(DTQ5) 2.44 2.44
Schwartz6(Q5) 2.90 2.90
D-AUG-CC-PVXZ
Feller(DTQ5) 3.75 4.03
Feller(TQ5) 3.84 3.10
Schwartzo.(DTQ5) 2.42 3.08
Schwartzoc(TQ5) 3.52 4.04
Schwartz4(DTQ5) 2.61 2.61
Schwartz4(Q5) 3.06 3.06
Schwartz6(DTQ5) 2.44 2.44
Schwartz6(Q5) 3.07 3.07

 

 

 

Varying from around 1 kcallmol to over 3 kcallmol between extrapolation
equations and from 0.06 kcallmol to 0.8 kcallmol between methods, the results in Table
12 seem erratic. This behavior is true only because all the extrapolated results describe
the heat of formation of FCCF as only a few kcallmol above zero. Due to the relative
magnitude of the calculated enthalpy of formation, the difference in extrapolated values

is a large percentage of the heat of formation.

27

 

In all the single point calculations on FCCF and its component atoms, the Is core

on carbon and fluorine was frozen. Table 13 reports the calculated core-valence

corrections tabulated as a function of basis set for the unaugmented and augmented

polarized core-valence basis sets.

 

 

 

 

 

Table 13. Core-Valence Correction for FCCF as a Function of Basis Set
Core-Valence

Basis Sets Correction

(kcallmol)
CC-PCVDZ -1.40
CC-PCVTZ -2.29
CC-PCVQZ -2.7l
CC-PCVSZ -2.85
AUG-CC-PCVDZ -1.99
AUG-CC-PCV'IZ -2.48
AUG-CC-PCVQZ -2.75
AUG-CC-PCVSZ -2.86

 

 

 

As seen in Table 13, the level of augmentation has little effect on the core-valence

correction. In Table 14, the values of the core-valence correction are extrapolated to the

basis set limit using the same procedure as previously discussed.

 

 

 

 

 

 

 

 

 

Table 14. Core-Valence Correction for FCCF at the Complete Basis Set Limit

Basis sets ACV(oo) ACV(E(oo))

CC-PCVXZ (kcallmol) (kcallmol)
Feller(TQ5) -2.90 -2.90
Schwartzoc(TQ5) -2.95 -2.94
Schwartz4(Q5) -2.95 -2.95
Schwartz6(Q5) -2.95 -2.95

AUG-CC-PCVXZ

Feller(TQ5) -2.93 -2.93
Schwartzoc(TQ5) -2.98 -2.99
Schwartz4(Q5) -2.94 -2.94
Schwartz6(Q5) -2.94 -2.94

 

 

Between the two extrapolation methods, the difference between the core-valence

corrections is negligible; and amongst extrapolation equations, the difference in the

correction is less than 0.1 kcallmol. Table 15 lists heat of formation values corrected for

28

 

the core-valence correlation, spin-orbit coupling (0.94 kcanol), and the relativistic

effects (0.70 kcallmol) not originally included in the values in Table 11.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Table 15. AHf Values Corrected for Core-Valance Correlation, Spin-Orbit
Coupling, and Relativistic Effects for FCCF at the Complete Basis Set Limit for
each Method
Basis sets Corrected AHf (co) Corrected AHf (E(°°))
CC-PVXZ I (kcallmol) (kcallmol)
Feller(DTQ5) 2.7 l 2.79
Feller(TQ5) 2.52 2.58
Schwartzor(DTQ5) 0.21 0.41
Schwartza(TQ5) 1.54 1.67
Schwartz4(DTQ5) 2.94 2.94
Schwartz4(Q5) 1.62 1.62
Schwartz6(DTQ5) 1.21 1.21
Schwartz6(Q5) 1.62 1.62
AUG-CC-PVXZ
Feller(DTQ5) 2.59 2.79
Feller(TQ5) 2.47 2.64
Schwartzct(DTQ5) 0.88 1.37
Schwartzct(TQ5) 1.88 2.23
Schwartz4(DTQ5) 1.75 1.75
Schwartz4(Q5) 1.60 1.60
Schwartz6(DTQ5) 1.14 1.14
Schwartz6(Q5) 1.60 1.60
D-AUG-CC-PVXZ
Feller(DTQ5) 2.46 2.74
Feller(TQ5) 2.56 1.81
Schwartzor(DTQ5) 1.08 1.73
Schwartzct(TQ5) 2.18 2.69
Schwartz4(DTQ5) 1.31 1.31
Schwartz4(Q5) 1.76 1.76
Schwartz6(DTQ5) 1.14 1.14
Schwartz6(Q5) 1.76 1 .76

 

 

 

 

Even though the results in Table 15 show that the heat of formation values
between methods change by substantial percentages, the results describe the enthalpy of

formation for FCCF as an endothermic process. We used the D—AUG-CCPVXZ basis set

extrapolations performed using the P(oo) method to produce a list of calculated and

29

experimental heats of formation. The results are presented in Table 16. Theoretical
contributions by Bauschlicher4 and Parthiban3 and experimental results provided by

Chase2 and Ehlert2| are also listed in Table 16.

 

Table 16. Comparison of Heats of Formation of FCCF

 

 

 

AHr
kcallmol
3’This work 2.46
l”This work 1.08
cThis work 1.31
“This work 1.14
3r>arthiban(wr) 0.48
3l>arthiban(w2) 1.38
4Bauschlicher 1.433
2experiment 5.00 $5.00
21experiment -45 16.00

 

aFeller(DTQ5) extrapolation of doubly augmented basis set values with P(oo)
method

bSchwartzaCDTQS) extrapolation of doubly augmented basis set values with P(oo)
method

cSchwartz4(DTQ5) extrapolation of doubly augmented basis set values with P(oo)
method

dSchwartz6(DTQ5) extrapolation of doubly augmented basis set values with P(oo)
method

 

 

 

The computational work done by Baushlicher4 and Parthiban3 is a mixture of
CCSD(T), DF'T, and other ab initio calculations. The discrepancies between theoretical
values produced in this research and other computational methods are due mainly to a use
of different starting geometries and vibrational frequencies and a different core-valence
correction. Because we calculated the heat of formation values using many similar
techniques, it is not surprising that the values are within a kcallmol of each other. Since
the experimental values are inconsistent and the error bars broad, all of the calculated

heats of formation fall within experimental accuracy.

30

2.4 CICCCI

Table 17 is a collection of CCSD(T) optimized geometries as a function of

augmented and unaugmented basis sets, the experimental geometry of ClCCClzz, and

geometries extrapolated to the complete basis set limit.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Table 17. Optimized Geometries of CICCCI as a Function of Basis Set
. RCC RCC] . RCC RCCl
Basrs Basrs
(ang) (ang) (ang) (ang)
CC-PVDZ 1.22442 1.66458 AUG-CC-PVDZ 1.22560 1.66890
CC-PVTZ 1.20872 1.64891 AUG-CC-PVTZ 1.20889 1.64883
CC-PVQZ 1.20530 1.64339 AUG-CC-PVQZ 1.20551 1.64339
CC-PVSZ 1.20506 1.63965 AUG-CC-PVSZ 1.20533 1.63970
Feller(DTQ5) 1.2048 1.6379 Feller(DTQ5) 1.2050 1.6390
Feller(TQ5) 1.2050 1.6318 Feller(TQ5) 1.2053 1.6319
Schwartza(DTQ5) 1.2043 1.6339 Schwartza(DTQ5) 1.2046 1.6365
Schwartzot(TQ5) 1.2050 1.6162 SchwartzaCTQS) 1.2053 1.6162
Schwartz4(DTQ5) 1.2037 1.6406 Schwartz4(DTQ5) 1.2037 1.6400
Schwartz4(TQ5) 1.2040 1.6386 Schwartz4(TQ5) 1.2043 1.6387
Schwartz4(Q5) 1.2049 1.6366 Schwartz4(Q5) 1.2052 1.6367
Schwartz6(DTQ5) 1.2042 1.6379 Schwartz6(DTQ5) 1.2045 1.6382
Schwartz6(TQ5) 1.2054 1.6353 Schwartz6(TQ5) 1.2057 1.6354
Schwartz6(Q5) 1.2049 1.6366 Schwartz6(Q5) 1.2052 1.6367
22experimental 1.205 1.635

 

The augmentation of the CC-PVXZ basis set lends little to the accuracy and speed

of convergence of the optimized geometries listed in Table 17. The optimized geometries

are slower to converge than HCCH and FCCF, but the carbon-carbon bond distance has

converged to within three decimal places by the 5-zeta basis set, and the carbon-chlorine

bond has converged to within two decimal places by the Q-zeta. Interestingly, it seems

the optimized geometries are converging to within 0.001A of the experimental geometry

and the converged geometry is within 0.004A of the 5-zeta optimized geometry.

31

 

 

 

 

 

Table 18. Vibrational Frequencies of ClCCCl
Symmetry Frequency (cm’l)
Cis-bend (1'1“) 172
Trans-bend (Hg) 333
s-CCl stretch (2g) 477
a-CCI stretch (2,.) 988
CC stretch (2g) 2234

 

 

 

We used the experimental geometry provided in Table 17, the Schimanouchil9
vibrational frequencies provided in Table 18, and the standard heats of formation for the
carbon and chlorine atoms provided by Chase2 to compute the heat of formation at 298K
and dissociation energy at 0 K for ClCCCl. We compiled these results as a function of
the suite of Dunning unaugmented, augmented, and doubly augmented basis sets. Table

19 reports our results.

 

 

 

 

 

 

 

 

Table 19. Calculated Dissociation Energies and Heats of Formation for ClCCCl
as a Function of Basis Set

. De Mi

Basrs sets (kcallmol) (kcallmol)
CC-PVDZ 306.97 97.89
CC-PVTZ 332.28 72.58
CC-PVQZ 341.61 63.24
CC-PVSZ 345.74 59.12
AUG-CC-PVDZ 306.89 97.97
AUG-CC-PVTZ 334.55 70.31
AUG-CC-PVQZ 342.76 62.10
AUG-CC-PVSZ 346.32 58.54
D-AUG-CC-PVDZ 308.71 96.14
D-AUG-CC-PVTZ 335.61 69.25
D-AUG-CC-PVQZ 343.35 61.51
D-AUG-CC-PVSZ 346.62 58.24

 

 

If we compare the calculations in Table 19 to the corresponding tables for HCCH
and FCCF, we notice a general trend. Regardless of basis set, the heats of formation
calculations seem to be essentially independent of the level of augmentation. Similar to

FCCF, for ClCCCl neither the dissociation energy at 0 K nor the heat of formation at 298

32

K converges, but the change in values between the Q- and 5-zeta basis sets is only 10
percent of the computed value for ClCCCl compared to 40 percent for FCCF. We made

extrapolations of the heat of formation to the complete basis set limit. Table 20 reports

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

our results.
Table 20. Heats of Formation for CICCCI Extrapolated to the Complete Basis
Set Limit
Basis sets AH, (co) AH, (E(oo))
CC-PVXZ (kcallmol) (kcallmol)
Feller(DTQ5) 56.91 57.33
Feller(TQ5) 55.86 56.39
Schwartzor(DTQ5) 52.26 53.31
Schwartzoc(TQ5) 53.08 54.29
Schwartz4(DTQ5) 59.60 59.60
Schwartz4(Q5) 55.77 55.77
Schwartz6(DTQ5) 55.99 55.99
Schwartz6(Q5) 55.78 55.78
AUG-CC-PVXZ

Feller(DTQ5) 57.45 57.49
Feller(TQ5) 55.81 56.22
Schwartzor(DTQ5) 54.53 54.62
Schwartzoc(TQ5) 53.51 54.43
Schwartz4(DTQ5) 58.19 58.19
Schwartz4(Q5) 55.65 55.65
Schwartz6(DTQ5) 56.03 56.03
Schwartz6(Q5) 55.65 55.65

D-AUG-CC—PVXZ
Feller(DTQ5) 57.31 57.33
Feller(TQ5) 55.84 55.12
Schwartza(DTQ5) 54.72 54.75
Schwartzor(TQ5) 53.86 54.45
Schwartz4(DTQ5) 57.75 57.75
Schwartz4(Q5) 55.58 55.58
Schwartz6(DTQ5) 55.91 55.91
Schwartz6(Q5) 55.59 55.59

 

 

 

 

 

Table 20 shows that the Feller, Schwartz4, and Schwartz6 extrapolation equations
consistently extrapolate to a higher value than the Schwartza equation. Using only the

values from larger basis sets, the resultant enthalpy drops. There is little difference

33

within methods, but there are differences of up to several kcallmol between extrapolation
equations.

To be physically viable, the extrapolated values for the enthalpy of formation can
not be higher than the best unextrapolated value. In Table 20, the Schwartz4(DTQ5)
extrapolated value exceeds the unextrapolated CC-PVSZ heat of formation. This result is
in italics and is the first example of how these extrapolation equations are fallible.
Although that value is the only result higher than the unextrapolated value in Table 20,
other values do lie very close to the unextrapolated values. These results in bold italics,
and they occur when data points calculated at smaller basis sets are included in the
extrapolation. This problem does seem to 1i ght itself once the smaller data points are
removed from involvement in the extrapolations.

Table 21 contains the core-valence correction to the values in Table 14 due to the
frozen 1s, 28, and 2p core on chlorine and frozen ls core on carbon in the atomic and

molecular CCSD(T) energy calculations.

 

 

 

 

 

Table 21. Core-Valence Correction for ClCCCl as a Function of Basis Set
Core-Valence
Basis Sets Correction
(kcallmol)
CC-PCVDZ -2.09
CC-PCVTZ -2.61
CC-PCVQZ -2.81

 

 

The core-valence correction is close to convergence. Table 22 contains the core-valence

correction at the basis set limit.

34

 

 

 

 

Table 22. Core-Valence Correction for ClCCCl at the Complete Basis Set Limit
Basis sets ACV(°°) ACV(E(°°))
CC-PCVXZ (kcallmol) (kcallmol)
Feller(DTQ) -2. 93 -I.23
Schwartza(DTQ) -3.06 2.09
Schwartz4(TQ) -2.92 -2.92
Schwartz6(TQ) -2.92 -2.92

 

 

 

 

Notice in Table 22 that between methods for the Feller extrapolations, the energy
difference is 1.7 kcallmol, and the value for the core-valence correction changes sign
between methods for the Schwartza extrapolations. The cause for this sign change, an
unusual result, depends, most likely, on how closely the data had converged before fitting
of nonlinear equations to the data points. Notice that the ACV(oo) value for the Feller
and Schwartza fitting equations is approximately equal to the results for Schwartz4 and
Schwartz6. As mentioned earlier, since the Feller and Schwartza fitting equations have
variable exponents, there is a chance that the relative consistency in how tightly the
atomic and molecular data was fit could cause inconsistency in the extrapolated result.

This discrepancy between the two methods for the Schwartz and Feller equations

would have a tremendous impact on their respective corrected AH, (E(oo)) values in
Table 23. As stated earlier, instead of calculating the corrected AH, values by

combining the enthalpy of formation and core-valence corrections from like methods and

extrapolation equations, we added a constant -2.92 kcallmol core-valence correction to all
the extrapolated AH, values. Table 23 contains the enthalpy of formation values for

ClCCCl after we added the core-valence correction, spin-orbit coupling correction (1.85

kcallmol), and the correction due to relativistic effects (0.99 kcallmol).

35

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Table 23. AH, Values Corrected for Core-Valance Correlation, Spin-Orbit
Coupling, and Relativistic Effects for ClCCCl at the Complete Basis Set Limit
for each Method
Basis sets Corrected AH, (co) Corrected AH, (E(oo))
CC-PVXZ (kcallmol) (kcallmol)
Feller(DTQ5) 56.83 57.25
Feller(TQ5) 55.7 8 56.31
Schwartzor(DTQ5) 52.18 53.23
Schwartzor(TQ5) 53.00 54.21
Schwartz4(DTQ5) 59.52 59.52
Schwartz4(Q5) 55.70 55.70
Schwartz6(DTQ5) 55.91 55.91
Schwartz6(Q5) 55.70 55.70
AUG-CC-PVXZ
Feller(DTQ5) 57.37 57.41
Feller(TQ5) 55.73 56.14
Schwartzor(DTQ5) 54.45 54.54
Schwartza(TQ5) 53.43 54.35
Schwartz4(DTQ5) 58.1 1 58.1 1
Schwartz4(Q5) 55.57 55.57
Schwartz6(DTQ5) 55.95 55.95
Schwartz6(Q5) 55.57 55.57
D-AUG-CC-PVXZ
Feller(DTQ5) 57.23 57.25
Feller(TQ5) 55.76 55.04
Schwartzoc(DTQ5) 54.64 54.67
Schwartzor(TQ5) 53.78 54.37
Schwartz4(DTQ5) 57.67 57.67
Schwartz4(Q5) 55.50 55.50
Schwartz6(DTQ5) 55.83 55.83
Schwartz6(Q5) 55.51 55.51

 

 

Table 24 presents a few values from Table 23 along with results from the works

of Parthiban3 and Chase2 for comparison.

36

 

 

 

Table 24. Com arison of Heats of Formation of CICCCI
AHf EITOI‘
kcallmol kcallmol

“This work 57.23 7.13
l*rhis work 54.64 4.54
cThis work 57.67 7.57
this work 55.83 5.73
3Parthiban(w1) 56.46 6.36
3Parthiban(W2) 56.21 6.11
2experiment 50.1 :10

 

 

 

alFeller(DTQ5) extrapolation of doubly augmented basis set values with P(°°)
method

bSchwartzot(DTQ5) extrapolation of doubly augmented basis set values with P(°o)
method

°Schwartz4(DTQ5) extrapolation of doubly augmented basis set values with P(oo)
method

dSchwartz6(DTQ5) extrapolation of doubly augmented basis set values with P(oo)
method

 

 

 

All the heat of formation values in Table 24 are within the experimental error
bars. The calculated values from this manuscript differ from Parthiban’s work due to
Parthiban’s use of calculated DFT geometries and frequencies. Having completed the
three molecules for which there is experimental data available, it is evident that the
procedure used for evaluation is adequate for determining the enthalpy of formation for
HCCH and the halogenated acetylenes. Almost all values fall very near the experimental
result, and most fall within experimental error bars.

2.5 Linear L12C2

We generated augmented lithium CC-PVXZ basis sets for CCSD(T) calculations
single point, geometry optimization, and vibrational frequency calculations by adding
even—tempered basis functions to the Dunning lithium CC-PVXZ basis sets. Table 25
lists the added gaussian function exponents for the constructed AUG-CC-PVDZ, AUG-

CC-PVTZ, and AUG-CC-PVQZ basis sets.

37

 

 

 

 

 

 

 

 

Table 25. Added Diffuse Function Exponents to Create Li AUG-CC-
PVXZ Basis Sets
AUG-CC-PVDZ AUG-CC-PVTZ AUG-CC-PVQZ
S 0.01122 0.01104 0.01067
P 0.01047 0.007649 0.007978
D 0.04765 0.03560 0.03207
F --- 0.08627 0.07383
G --- --- 0.1253

 

 

 

 

To calculate the heat of formation, we used the equilibrium geometry and

 

fundamental vibrational frequencies. Table 26 lists CCSD(T) optimized geometries,
extrapolated geometries, other theoretical geometeries, and the geometry used for the

heat of formation calculations.

 

 

 

 

 

 

 

 

 

 

 

 

 

Table 26. Optimized Geometries of Linear Li2C2 as a Function of Basis Set
Basis RCC (ang) RCLi (ang)
CC-PVDZ 1.2765 1.9255
CC-PVTZ 1.2577 1.8939
CC-PVQZ 1.2540 1.8939
Feller(DTQ) 1.2531 1.8939
Schwartza(DTQ) 1.2522 1.8939
Schwartz4(DTQ) 1.2515 1.8876
Schwartz4(TQ) 1.2519 1.8939
Schwartz6(DTQ) 1.2520 1.8963
Schwartz6(TQ) 1.2519 1.8939
23MP2 1.267 1.883
74RHF 1.2314 1.8845
USED 1.2531 1.8943

 

 

Table 26 demonstrates that the optimized geometries are close to convergence.
The extrapolated values are within 0.002A of the optimized geometry calculated at the
highest basis set and are within a few thousandths of an angstrom of the value used in the
heat of formation calculations. The results of other researchers are within 0.01A for the
carbon lithium bond and 0.03A of the carbon-carbon bond. We calculated the vibrational

frequencies using the CC-PVTZ and AUG-CC-PVTZ basis sets wherein the basis sets

38

had all the higher angular momentum functions intact. These values are shown in Table

 

 

 

 

27.
Table 27. Vibrational Frequencies for Linear L12C2
Basis Sets

Symmetry CC-PVTZ AUG-CC-PVTZ

(D) Cis-bend (11“) 86 86

(E) Trans-bend (fig) 147 157

(A) s-CLi stretch (2g) 582 577

(C) a-CLi stretch (21,) 737 733

(B) CC stretch (2g) 1943 1937

 

 

 

 

 

Figure 1 provides a visual description of the vibrational modes. Each of these
vibrations is the same type of vibration for linear four atom molecules. The A vibration
is normally referred to as the symmetric carbon-lithium stretches, and the vibration C is
the asymmetric carbon-lithium stretch. The B stretch is the carbon-carbon stretch, and
the D and E vibrations are doubly degenerate cis and trans-bends, respectively. Because
there are no imaginary frequencies, the linear configuration is a viable geometric

minimum on the Li2C2 potential energy surface.

 

<—> <——b <——§ _,4_‘__.

Li—C—C—Li Li—C—C—Li Li—C—C—Li

A B C

LTi_T:i—l i Li—A—q—Iti

 

 

 

Figure l: Vibrational Modes of Linear Li2C2

Using the CCSD(T) vibrational frequencies calculated at the AUG-CC-PVTZ

basis set described in Table 27, the geometry RCC = 1.2531 A, RCLi = 1.8943 A from

39

Table 26, and the heats of formation of atomic carbon and lithium provided by Chase2
from the appendix, we calculated dissociation energy at 0 K and heat of formation at 298

K for linear Li2C2. Table 28 shows our results.

 

Table 28. Calculated Dissociation Energies and Heats of Formation for
Linear Li2C2 as a Function of Basis Set

 

 

 

 

 

Basis sets De AHr
(kcallmol) (kcallmol)

CC-PVDZ 291.44 130.69
CC-PVTZ 315.98 106.16
CC-PVQZ 323.21 98.92
aCC-PV(QZ-SZ) 325.12 97.01
bCC-PV(QZ-6Z) 326.04 96.10
AUG-CC-PVDZ 298.01 124.12
AUG-CC-PVTZ 318.99 103.14
AUG-CC-PVQZ 324.20 97.93
°AUG~CC-PV(QZ-SZ) 325.80 96.33
dAUG-CC-PV(QZ-6Z) 326.43 95.71

 

atLi CC-PVQZ basis set with C CC-PVSZ basis set

bLi CC-PVQZ basis set with c CC-PV6Z basis set
cLi AUG-CC-PVQZ basis set with C AUG-CC-PVSZ basis set
dLi AUG-CC-PVQZ basis set with C AUG-CC-PV6Z basis set

 

 

 

The enthalpy of formation is not converged by the Q-zeta basis set for the
unaugmented and augmented basis sets. Table 28 shows also results from mixed basis
set calculations. In these calculations, we allowed the carbon basis set to progress
beyond the Q-zeta level, but the lithium basis set did not progress pass the Q-zeta level.
These basis sets are unusual, so the extrapolations in Table 29 do not include those
values. We did not attempt extrapolation of the augmented basis sets. Since the
constructed basis sets were not optimized, the extrapolation behavior would be unlike
that of the Dunning optimized basis sets. We used the data from the unaugmented basis
sets, excluding the mixed basis set calculations, to extrapolate to the complete basis set

limit. Table 29 reports the extrapolations.

40

 

 

 

 

 

 

 

 

Table 29. Heats of Formation for Linear Li2C2 Extrapolated to the
Complete Basis Set Limit

Basis sets AHf (°°) AHf (E(°°))

CC-PVXZ (kcallmol) (kcallmol)
Feller(DTQ) 95.90 95.94
Schwartza(DTQ) 92.78 92.87
Schwartz4(DTQ) 96.33 96.33
Schwartz4(TQ) 94.75 94.75
Schwartz6(DTQ) 94.16 94.16
Schwartz6(TQ) 94.77 94.77

 

 

 

The extrapolated enthalpies of formation in Table 29 range from around 93 to

around 97 kcallmol. Again, any extrapolated heat of formation that is higher than the

best-unextrapolated value is not physically viable and highlighted in red. Keep in mind

that we determined the highlighted values in Table 29 from extrapolations of the pure

basis sets. The extrapolated value is higher than the value from unextrapolated mixed

basis set values. Previously for HCCH, FCCF, and ClCCCl by the 5-zeta basis set, the

enthalpy of formation was near convergence. It seems that the same may be true for Li.

Without the 5-zeta basis set, the extrapolations are not as precise as previous examples.

To correct for the core-valence correlation of the 1s shell on carbon and lithium,

we made core-valence corrections. Table 30 presents those results as a function of basis

set.

 

Table 30. Core-Valence Correction for Linear L12C2 as a Function of Basis Set

 

 

 

Basis Sets Core-Valence Correction
(kcallmol)
CC-PCVDZ -1 . 15
CC-PCVTZ -2.09
CC-PCVQZ -2.62
aCC-PCV(QZ-SZ) -2.74

 

 

aLi with CC-PCVQZ basis set C with cc-pCVSZ basis set

 

The core-valence correction did not converge, but the magnitude of the correction

is approximately the same as for the halogenated molecules. Table 31 reports complete

41

 

basis set extrapolation values that we calculated using the results calculated using the

three unmixed basis sets.

 

 

 

 

 

Table 31. Core-Valence Correction for Linear Li2C2 at the Complete Basis Set
Limit
Basis sets ACV(oo) ACV(E(oo))
CC-PCVXZ (kcallmol) (kcallmol)
Schwartzor(DTQ) -4.65 1.07
Schwartz4(TQ) -2.93 -2.93
Schwartz6(TQ) -2.93 -2.93

 

 

 

Similar to CICCCI, the differences between the methods and extrapolation
equations are large. These discrepancies may be due to differences among the nonlinear
fit equations and the fitting methods created by inconsistencies in how accurate the fit for
each equation and the method is to the data. In previous core-valence calculations, such
as, HCCH and FCCF, we used a 5-zeta basis set. Similar to the ClCCCl molecule, we set
the ACV correction for all the extrapolated linear Li2C2 heat of formation values to —2.93
kcallmol. Table 32 shows the values of the enthalpy of formation that we added after the
missing core-valence correlation (-2.93 kcallmol), spin-orbit coupling (0.17 kcallmol)

and relativistic effect (0.19 kcallmol) corrections.

 

 

 

 

 

 

 

Table 32. AH, Values Corrected for Core-Valance Correlation, Spin-Orbit
Coupling, and Relativistic Effects for Linear Li2C2 at the Complete Basis Set
Limit for each Method

Basis sets Corrected AH, (co) Corrected AH, (E(°°))
CC-PVXZ (kcallmol) (kcallmol)
Schwartzor(DTQ) 90.21 90.28
Schwartz4(TQ) 92.18 92.18
Schwartz6(DTQ) 91.59 91.59
Schwartz6(TQ) 92.20 92.20

 

 

 

The enthalpy of formation of linear Li2C2 seems to be substantially larger than the
enthalpy of formation of the previous acetylenes. The largest inconsistency between

extrapolation equations is the range of 2 kcallmol between the Schwartza and Schwartz6

42

extrapolation equations. Table 33 presents selected values from Table 32, the value for
the enthalpy of formation calculated from the unextrapolated enthalpy of formation in

Table 28, and core-valence correction at the largest basis set.

 

Table 33. Comparison of Heats of Formation of Linear L12C2

 

 

 

Basis Set AH, (co)
kcallmol
2”This work 90.21
l*rhis work 92.18
cThis work 91.59
dUnextraEflated value 93.30

 

aSchwartzcr(DTQ) extrapolation of unaugmented basis set values with P(°°) method
bSchwartz4(TQ) extrapolation of unaugmented basis set values with P(oo) method
cSchwartz6(DTQ) extrapolation of unaugmented basis set values with P(°o) method
dAH, from AUG-CC-PV(QZ-6Z) using ACV from CC-PCV(QZ-SZ) basis set

 

 

 

Table 33 shows that the unextrapolated result from the AUG-CC-PV(QZ-6Z) heat
of formation and the CC-PCV(QZ-6Z) core-valence correction is the largest heat of
formation. The enthalpy values from Table 33 are generally around 92 kcallmol, but the
Schwartzor result is closer to 90 kcallmol.

2.6 Planar L12C2

The doubly bridged or planar Li2C2 complex is the lowest energy configuration
for the Li2C2 species with a 7.3 kcallmol (11.7 millihartree) energy difference between
the linear and doubly bridged geometries. To find the equilibrium geometry, we
calculated CCSD(T) optimized geometries for the planar Li2C2 species. These optimized
geometries are extrapolated to the complete basis set and compared to previously
determined theoretical geometries. Table 34 lists the geometry used in the heat of

formation calculations.

43

 

 

 

 

 

 

 

 

 

 

 

Table 34. Optimized Geometries of Planar Li2C2 as a Function of Basis Set

Basis RCC (ang) RCLi (ang)
CC-PVDZ 1.2932 2.0406
CC-PVTZ 1.2730 2.0072
CC-PVQZ 1.2686 2.0049
Feller(DTQ) 1.2673 2.0047
Schwartza(DTQ) 1.2661 2.0045
Schwartz4(DTQ) 1.2660 1.9989
Schwartz4(TQ) 1.2660 2.0036
Schwartz6(DTQ) 1.2661 2.0053
Schwartz6(TQ) 1.2660 2.0053
24MBPT(2) 1.2808 1.9954
23MP2 1.282 2.017
USED 1.2673 2.0086

 

 

 

 

The optimized geometries in Table 34 are close to converging, and the
extrapolated geometries are within 0.003A of the best-unextrapolated geometry and
within 0.001A of the geometry used to calculate the heat of formation. The literature
values from previous studies differ by 0.02A for the carbon-carbon bond and 0.002A for
the carbon-lithium bond. Table 35 lists the vibrational frequencies of planar Li2C2 at the
CC-PVTZ and AUG-CC-PVTZ basis sets. To conserve computation time, we removed
the d and f gaussian functions from the lithium AUG-CC-PVTZ basis set, but we retained

them for the carbon AUG-CC-PVTZ basis set.

 

 

 

 

Table 35. Vibrational Frequencies for Planar Li2C2
Basis Sets
Symmetry CCePVTZ AUG-CC-PVTZ
vibration A (B3,,) 66 64
vibration B (B2,) 199 193
vibration C (83g) 306 302
vibration D (Ag) 564 557
vibration E (Blu) 685 681
vibration F (Ag) 1799 1793

 

 

 

 

 

Again, there are no imaginary frequencies to indicate that the planar complex is

not a minimum. Figure 2 is a visual representation of the vibrational frequencies of

planar Li2C2. Vibration A is a side view of the doubly bridged molecule. The second
carbon atom is located directly behind the labeled carbon. Note that the vibration C
resembles the type of motion which might move the planar species towards the formation

of the linear species.

 

Ll-> Li-> Li-p
A
<-C <—C-<—C C—C
L1-> Li» «L1
X --------- -> Z --------- -> Z --------- ->
A B C
A
L1 Li L1
C C C—C COC
i
Li Li L1
Z --------- 9 Z --------- -> Z --------- ->
D E F

 

 

 

Figure 2: Vibrational Modes for Planar Li2C2

Using the geometry RCC = 1.2673 A, RLic = 2.0086 A, ACCLi = 71.6 from Table

34 and vibrational frequencies calculated at the CCSD(T) level using the mixed AUG-
CC-PVTZ basis set, we calculated dissociation energies at 0 K and heat of formation

values at 298 K for the doubly bridged species. Table 36 shows our results.

45

 

Table 36. Calculated Dissociation Energies and Heats of Formation for Planar
Li2C2 as a Function of Basis Set

 

 

 

 

 

Basis Set De AH,

kcallmol kcallmol

cc-onz 300.86 119.87
CC-PVTZ 324.15 96.57
CC-PVQZ 330.84 89.89
aCC-PV(SZ-QZ) 332.62 88.11
bCC-PV(6Z-QZ) 333.47 87.26

AUG-CC-PVDZ 304.54 116.19
AUG-CC-PVTZ 326.28 94.45
AUG-CC-PVQZ 331.52 89.21
°AUG-CC-PV(SZ-QZ) 333.12 87.61
dAUG-CC-PV(6Z-QZ) 333.77 86.96

 

 

aLi CC-PVQZ basis set with C CC-PVSZ basis set

bLi CC-PVQZ basis set with C CC-PV6Z basis set
cLi AUG-CC-PVQZ basis set with C AUG-CC-PVSZ basis set
dLi AUG-CC-PVQZ basis set with C AUG—CC-PV6Z basis set

 

The unaugmented and augmented basis sets are not converged by the Q-zeta basis

set. We extrapolated the data from the unaugmented basis sets, excluding the mixed

basis set calculations, to the complete basis set limit. Table 37 shows the results of this

 

 

 

 

 

 

investigation.
Table 37. Heats of Formation for Planar L12C2 Extrapolated to the Complete
Basis Set Limit

Basis sets AH, (co) A501...»

CC-PVXZ (kcallmol) (kcallmol)
Feller(DTQ) 87.19 87.23
Schwartzor(DTQ) 84.43 84.50
Schwartz4(DTQ) 87.36 87.36
Schwartz4(TQ) 86.03 86.03
Schwartz6(DTQ) 85.52 85.52
Schwartz6(TQ) 86.05 86.05

 

 

 

 

Again, the Schwartz4(DTQ) and Feller extrapolation results in an unphysical

value higher than the unextrapolated values.

46

 

 

To correct for the frozen ls core on lithium and carbon, we calculated core-
valence corrections using the Dunning CC-PCVXZ basis sets. Table 38 reports the

results of these calculations.

 

 

 

 

 

 

Table 38. Core-Valence Correction for Planar Li2C2 as a Function of Basis Set
Basis Sets Core-Valence Correction
(kcal/mol)
CC-PCVDZ -1.31
CC-PCVTZ -2.19
CC-PCVQZ -2.68
aCC--PCV(QZ-SZ) -2.79
aLi with CC-PCVQZ basis set C with CC-PCVSZ basis set

 

 

Similar to the linear species, the core-valence correction for planar Li2C2 has not
converged. Yet, the core-valence correction for the planar molecule differs by less than
0.1 kcallmol from the linear core-valence correction. Table 39 reports the core-valence

correction extrapolated to an infinite basis set.

 

 

 

 

Table 39. Core-Valence Correction for Planar L12C2 at the Complete Basis Set
Limit

Basis sets ACV(°°) ACV(E(°°))

CC-PCVXZ (kcallmol) (kcallmol)
Feller(DTQ) -3.30 ~2.95
Schwartza(DTQ) —4.43 1.14
Schwartz4(TQ) -2.96 -2.96
Schwartz6(TQ) -2.96 -2.96

 

 

 

 

Similar to the linear species, the extrapolated values for the variable exponent
equations are erratic, so the core-valence correction used for the planar molecule is -2.96
kcallmol. Table 40 reports the enthalpy of formation for the planar species after we
added the core-valence corrections (-2.96 kcallmol), spin-orbit coupling (0.17 kcallmol),

and relativistic corrections (0.15 kcallmol).

47

 

Table 40. AH, Values Corrected for Core-Valance Correlation, Spin-Orbit
Coupling, and Relativistic Effects for Planar Li2C2 at the Complete Basis Set Limit

 

 

 

 

 

for each Method
Basis sets Corrected AH, (co) Corrected AH, (E(°°))
CC-PVXZ (kcallmol) (kcallmol)
Schwartzor(DTQ) 81.79 81.87
Schwartz4(TQ) 83.39 83.39
Schwartz6(DTQ) 82.89 82.89
Schwartz6(TQ) 83.41 83.41

 

 

 

The heat of formation results reported in Table 40 span a range of 2 kcallmol.

Table 41 reports the better values from Table 40 and the combination of the heat of

formation calculated with the AUG-CC-PV(QZ-6Z) basis set and the core-valence

correction calculated using the CC-PCV(QZ-5Z) basis set.

 

Table 41. Comparison of Heats of Formation of Planar Li2C2

 

 

 

Basis Set AH,
kcallmol

aThis work 81.79

brhis work 82.89

cThis work 83.41

dUnextrapolated value 84.49

 

 

aSchwartza(DTQ) extrapolation of unaugmented basis set values with P(°°) method
bSchwartz4(T Q) extrapolation of unaugmented basis set values with P(oo) method
cSchwartz6(DTQ) extrapolation of unaugmented basis set values with P(oo) method
dAH, from AUG-CC-PV(QZ-6Z) using ACV from CC-PCV(QZ-SZ) basis set

 

 

Table 41 shows that the extrapolated heats of formation are a kcallmol smaller

than the unextrapolated heat of formation using larger basis sets. Table 42 shows a

comparison of the enthalpy of formation for Li2C2 at its two ground-state geometries.

48

 

 

Table 42. Energy Difference between Linear and Planar Li2C2

 

 

 

 

 

Planar Li2C2 Linear Li2C2 Difference

Basis Set AH, AH, kcallmol
“This work 81.79 90.21 8.42
”This work 83.39 92.18 8.79
cThis work 83.41 92.20 8.79
‘Unextrapolated value 84.49 93.30 8.81

 

aSchwartzr1(DTQ) extrapolation of unaugmented basis set values with P(oo) method
bSchwartz4(TQ) extrapolation of unaugmented basis set values with P(oo) method
cSchwartz6(TQ) extrapolation of unaugmented basis set values with P(°o) method
“AH, from AUG-CC-PV(QZ-6Z) using ACV from CC-PCV(QZ-SZ) basis set

 

 

 

We compared the enthalpy of formation for both the linear and planar species and
found the energy difference between the two configurations to be 8 kcallmol. We found
the energy difference between the two configurations using De values with no thermal
corrections to be 7 .3 kcallmol.

2.7 Linear Na2C2

Using the same process followed for calculating the properties of the Li2C2
molecules, we calculated the molecular geometries, vibrational frequencies, and enthalpy
of formation of both Na2C2 species. We generated augmented CC-PVXZ basis sets by
adding even-tempered basis functions to the sodium Dunning CC-PVXZ basis sets.
Table 43 lists the gaussian function exponents added for augmentation of the CC-PVDZ,

CC-PVTZ, and CC-PVQZ basis sets.

 

 

 

 

 

 

 

Table 43. Added Diffuse Function Exponents to Create Na AUG-CC-PVXZ
Basis Sets
AUG-CC-PVDZ AUG-CC-PVTZ AUG-CC-PVQZ

8 0.00995 0.00942 0.00790

P 0.00648 0.00602 0.00323

D 0.0228 0.0296 0.02740

F --— 0.05588 0.0560

G --- --- 0.0861

 

 

 

 

 

 

49

We calculated CCSD(T) single point energy, geometry optimization, and

vibrational frequency using these generated basis sets. Table 44 reports the calculations.

 

 

 

 

 

 

 

 

 

 

Table 44. Optimized Geometries of Linear Na2C2 as a Function of Basis Set

Basis RCC (ang) RCN a (ang)
CC-PVDZ 1.2777 2.2427
CC-PVTZ 1.2592 2.2373
CC-PVQZ 1.2560 2.2378
Feller(DTQ) 1.2553 2.2376
Schwartz0t(DTQ) 1.2546 2.2376
Schwartz4(DTQ) 1.2533 2.2366
Schwartz4(TQ) 1.2541 2.2382
Schwartz6(DTQ) 1.2544 2.2387
Schwartz6(T Q) 1.2541 2.2382
USED 1.2549 2.2341

 

 

Table 44 shows that both the carbon-carbon and carbon-sodium bond length are

converged to within 0.004A by the Q-zeta basis set. The extrapolated values differ less

than 0.003A from the unextrapolated geometries, but the carbon-sodium bond length

differs by 0.004A from the geometry used to calculate the enthalpy of formation. We

calculated CCSD(T) vibrational frequencies at the CC-PVTZ and AUG-CC-PVTZ basis

sets where the sodium basis set is void of any higher order angular momentum functions

above a p orbital. Table 45 lists these frequencies.

 

 

 

 

 

 

Table 45. Vibrational Frequencies of Linear Na 2C2
Basis Sets
Symmetry CC-PVTZ AUG-CC-PVTZ
(D) Cis-bend (nu) 45cml 51 cm"
m) Trans-bend (rig) 85i cm" 69 cm'1
(A) s-CNa stretch (23g) 278 cm" 275 cm"
(C) a-CNa stretch (2,) 475 cm'1 473 cm"
(B) CC stretch (2,) 1922 cm'I 1907 cm"

 

 

 

Figure 3 illustrates the motion of atoms during each of the vibrations. Table 45 shows

that the trans-bend or vibration E is imaginary at the CC-PVTZ basis set, but it is real

50

 

when the basis set is augmented. Therefore, there is a question as to the whether the
linear Na2C2 is a viable minimum. The vibration E diagram, Figure 3 shows that the
trans-bend vibration is similar to the motion the linear molecule might take to form the
planar configuration. Maybe the potential energy surface calculated using the CC-PVTZ

basis set does not accurately describe the shallow well that is the linear Na2C2 minimum.

 

Na—C—C—Na Na—C—C—Na Na—C—C—Na

A B C

t
Na—T:i—l$a Nf—$:i—Ila

 

 

 

Figure 3: Vibrational Modes of Linear Na2C2
Using the optimized geometry obtained in previous work', RCC = 1.255 A, RCNa =

2.234 A, and CCSD(T) vibrational frequencies from Table 45, we calculated the enthalpy
of formation. Table 46 presents the heats of formation at 273 K and dissociation energies

at 0 K for linear Na2C2 as a function of basis set.

51

 

 

 

 

 

 

 

 

Table 46. Calculated Dissociation Energies and Heats of Formation
for Linear Na 2C2 as a Function of Basis Set
Basis sets De AHf
(kcallmol) (kcallmol)
CC-PVDZ 258.94 137.70
CC-PVTZ 280.18 1 16.46
CC-PVQZ 287.59 109.05
aCC-PV(QZ-5Z) 289.62 107.02
bCC-PV(QZ-6Z) 290.57 106.07
AUG-CC-PVDZ 265.19 131.45
AUG-CC-PVTZ 283.78 1 12.86
AUG-CC-PVQZ 288.92 107.73
cAUG-CC-PV(QZ—5Z) 290.53 106.1 1
dAUG-CC-PV(QZ-6Z) 291.23 105.41
aNa CC-PVQZ basis set with C CC-PVSZ basis set
bNa CC-PVQZ basis set with c CC-PV6Z basis set
cNa AUG-CC-PVQZ basis set with C AUG-CC-PVSZ basis set
dNa AUG-CC-PVQZ basis set with c AUG-CC-PV6Z basis set

 

The enthalpy of formation is not converged by the Q-zeta basis set for either the

unaugmented or augmented basis sets. The extrapolations shown in Table 47 do not

include the larger basis sets above the unaugmented Q-zeta basis set.

 

 

 

 

 

 

 

 

 

Table 47. Heats of Formation for Linear Na 2C2 Extrapolated to the
Complete Basis Set Limit

Basis sets AHf (°°) AHf (E(°°))

CC-PVXZ (kcallmol) (kcallmol)
Feller(DTQ) 105.09 105.65
Schwartzct(DTQ) 100.79 102.13
Schwartz4(DTQ) 107.1 7 107.1 7
Schwartz4(TQ) 104.78 104.78
Schwartz6(DTQ) 103.88 103.88
Schwartz6(TQ) 104.80 104.80

 

 

 

The extrapolated enthalpies of formation in Table 47 range from about 100 to
about 107 kcallmol. Most of the extrapolated enthalpies of formation are very close to

the 105.41 kcallmol value that we found using the AUG-CC-PV(QZ-6Z) basis set.

52

To correct for the core-valence correlation of the 2s and 2p shells on sodium and

the frozen 1s orbital on carbon, we made core-valence corrections. Table 48 presents

those results as a function of basis set.

 

 

 

 

 

 

Table 48. Core-Valence Correction for Linear Na2C2 as a Function of Basis Set
Basis Sets Core-Valence Correction
(kcallmol)
CC-PCVDZ -1.14
CC-PCVTZ -0.74
CC-PCVQZ -0.74
aCC-PCV(QZ-SZ) -0.97
“Na with CC-PCVQZ basis set C with CC-PCVSZ basis set

 

 

The core—valence correction converges, and the magnitude of the correction is
much smaller than that of the halogenated molecules and the two lithium complexes. The
mixed basis set calculation provides a correction that is larger than the unmixed basis set.
This result is unusual, but the mixing of basis sets does not insure a proper convergence
to the basis set limit. Table 49 reports the complete basis set extrapolation values using

the results that we calculated using the three unmixed basis sets.

 

 

 

 

 

Table 49. Core-Valence Correction for Linear Na2C2 at the Complete Basis
Set Limit
Basis sets ACV(oo) ACV(E(oo))
CC-PCVXZ (kcallmol) (kcallmol)
Feller(DTQ) -0.74 _---
Schwartzor(DTQ) -0.74 ----
Schwartz4(TQ) -0.73 -0.73
Schwartz6(TQ) -0.73 -073

 

 

 

The core-valence correction for the Feller(DTQ) and Schwartza equations using
the P(E(oo)) method are not reported because the resulting values are not realistic.

Similar to the lithium species, the differences between the methods for Schwartza and
Feller extrapolation equations are large. As done previously in the ClCCCl and the

lithium species, the ACV correction for all the extrapolated AH, we set values to -0.73

53

kcallmol. Table 50 shows the values of the enthalpy of formation after we added the

missing core-valence correlation (-0.73 kcallmol), spin-orbit coupling (0.17 kcallmol),

and relativistic effect (0.21 kcallmol) corrections.

 

 

 

 

 

 

 

 

 

Table 50. AH, Values Corrected for Core-Valance Correlation, Spin-Orbit
Coupling, and Relativistic Effects for Linear Na 2C2 at the Complete Basis
Set Limit for each Method

Basis sets Corrected AH, (co) Corrected AH, (E(oo))
CC-PVXZ (kcallmol) (kcallmol)
Feller(DTQ) 104.74 ----
Schwartzor(DTQ) 100.44 101.78
Schwartz4(TQ) 104.43 104.43
Schwartz6(DTQ) 103.53 103.53
Schwartz6(T Q) 104.45 104.45

 

The enthalpy of formation of linear Na2C2 seems to be about 10 kcallmol larger

than the enthalpy of formation of linear Li2C2. The largest inconsistency between

extrapolation techniques is a 7 kcallmol difference between Schwartza and Schwartz4

extrapolation results. Table 51 presents selected values from Table 50 and the value for

the enthalpy of formation calculated from the enthalpy of formation and core-valence

correction at the largest basis set.

 

Table 51. Comparison of Heats of Formation of Linear Na2C2

 

 

 

Basis Set AH, (00)
kcallmol
21This work 104.74
bThis work 100.44
°This work 104.43
“This work 104.45
°Unextrapolated value 104.82

 

method

 

alFeller(DTQ) extrapolation of unaugmented basis set values with P(oo) method
bSchwartzrr(DTQ) extrapolation of unaugmented basis set values with P(oo)

°Schwartz4(T Q) extrapolation of unaugmented basis set values with P(oo) method
dSchwartz6(T Q) extrapolation of unaugmented basis set values with P(oo) method
°AH, from AUG-CC-PV(QZ-6Z) using ACV from CC-PCV(QZ-5Z) basis set

 

54

 

 

Table 51 shows the unextrapolated result from the AUG-CC-PV(QZ-6Z) heat of
formation and the CC-PCV(QZ-5Z) core-valence correction to be closer to the fixed
exponent extrapolations.

2.8 Planar Na2C2

The doubly bridged or planar N a2C2 complex is the lowest energy configuration
for the Na2C2 species with a 11.2 kcallmol (17.8 millihartree) energy difference between
the linear and doubly bridged geometries. Table 52 lists CCSD(T) optimized geometries
as a function of basis set. Table 52 lists also the optimized geometries, extrapolated
geometries to the basis set limit, and the geometry that we used to calculate the enthalpy

of formation for planar Na2C2.

 

 

 

 

 

 

 

 

 

 

 

Table 52. Optimized Geometries of Planar Na 2C2 as a Function of Basis Set

Basis RCC (ang) RCNa (ang)
CC-PVDZ 1.2958 2.3612
CC-PVTZ 1.2761 2.3543
CC-PVQZ 1 .27 16 2.3497
Feller(DTQ) 1.2703 2.3394
Schwartza(DTQ) 1.2689 2.2883
Schwartz4(DTQ) 1.2691 2.3498
Schwartz4(TQ) 1 .2690 2.3470
Schwartz6(DTQ) 1.2690 2.3459
Schwartz6(TQ) 1 .2690 2.3470
USED 1 .2704 2.3427

 

 

Both the carbon-carbon bond length and the carbon-sodium bond length have
converged to two decimal places by the Q-zeta basis set. The extrapolated geometries
vary only 0.001A from the carbon-carbon bond length we used to calculate the heat of
formation, but the sodium-carbon bond length differs by as much as 0.008A from the
bond length that we used to calculate the heat of formation. Even a geometry difference
of this size will not affect the enthalpy of formation to any substantial amount. Table 53

lists the CCSD(T) vibrational frequencies for the planar sodium species. Similar to our

55

investigation of linear Na2C2, we removed the d and f gaussian functions from the sodium

CC-PVTZ and AUG-CC-PVTZ basis set for the frequency calculations.

 

 

 

 

Table 53. Vibrational Frequencies of Planar Na 2C2
Basis sets
Symmetry CC-PVTZ AUG-CC-PVTZ

vibration A (B3,) 42 cm" 48 cm'1
vibration B (B2u) 122 cm'l 119 cm '
vibration C (ng) 297 cm'l 296 cm '
vibration D (Ag) 276 cm'1 275 cm”1
vibration E (Blu) 462 cm'1 460 cm '
vibration F (Ag) 1790 cm'1 1782 cm'1

 

 

 

 

 

The vibrational frequencies listed in Table 53 are not imaginary. This data leads
to the conclusion that both the linear and planar configurations are rrrinima on the
potential energy surface. Figure 4 is a visual representation for the planar Na2C2,

vibrational frequencies.

 

 

Na» Na» Na»
A
4-C +C+C E—C
Na» Na-> «Na
X --------- -> Z --------- + Z --------- ->
A B C
Na Na Na
C~I~C C—C C0 C
i
Na I‘Ia Na
2 --------- -> Z --------- -> Z --------- ->
D E F

 

 

Figure 4: Vibrational Frequencies of Planar Na2C2

Using the geometry, RCC=1.2704 A, RNaC=2.3427 A, ACCNa=74.31 from Table

52 and vibrational frequencies calculated at the CCSD(T) level using the AUG-CC-

56

PVTZ basis set from Table 53, we calculated dissociation energies at 0 K and heat of

formation values at 298 K for the doubly bridged species. Table 54 reports the results.

 

Table 54. Calculated Dissociation Energies and Heats of Formation for
Planar Na2C2 as a Function of Basis Set

 

 

 

 

 

Basis Set De AH,
kcallmol kcallmol
CC-PVDZ 271.95 123.412
CC-PVTZ 292.44 102.921
CC-PVQZ 299.14 96.226
aCC-PV(SZ-QZ) 301.03 94.33
bCC-PV(6Z-QZ) 301.97 93.39
AUG-CC-PVDZ 275.87 119.50
AUG-CC-PVTZ 295.01 100.35
AUG-CC-PVQZ 300.09 95.28
°AUG-CC-PV(SZ-QZ) 301.71 93.65
dAUG-CC-PV(6Z-QZ) 302.40 92.96

 

 

a‘Na CC-PVQZ basis set with C CC-PVSZ basis set

bNa CC-PVQZ basis set with C CC-PV6Z basis set
cNa AUG-CC-PVQZ basis set with C AUG-CC-PVSZ basis set
dNa AUG-CC-PVQZ basis set with C AUG-CC-PV6Z basis set

 

Table 55 presents extrapolations to the complete basis set limit for the enthalpy of

formation using the unaugmented unmixed basis set results.

 

 

 

 

 

 

 

 

 

Table 55. Heats of Formation for Planar Na2C2 Extrapolated to the Complete
Basis Set Limit

Basis sets AHf (°°) AHf (E(°°))

CC-PVXZ (kcallmol) (kcallmol)
Feller(DTQ) 92.98 93.39
Schwartztx(DTQ) 89.54 90.49
Schwartz4(DTQ) 94.27 94.27
Schwartz4(TQ) 92.36 92.36
Schwartz6(DTQ) 91.65 91.65
Schwartz6(TQ) 92.38 92.38

 

The difference between the heat of formation between the linear and planar Na2C2

species is approximately 11 kcal/mol.

57

 

 

To correct for the frozen 2s and 2p core on sodium and ls core on carbon, we
calculated core-valence corrections using the Dunning CC-PCVXZ basis sets. Table 56

reports the results of these calculations.

 

 

 

 

 

 

Table 56. Core-Valence Correction for Planar Na2C2 as a Function of Basis Set
Basis Sets Core-Valence Correction
(kcallmol)
CC-PCVDZ -l.09
CC-PCVTZ -0.57
CC-PCVQZ -0.7 1
aCC-PCV(QZ-SZ) -0.97
aNa with CC-PCVQZ basis set C with CC-PCVSZ basis set

 

 

Like the linear species, the core-valence correction for planar Na2C2 has not
converged, but more importantly, the core-valence correction for the planar molecule
differs from the linear molecule at the T-zeta basis set. Otherwise, the core-valence
correction for both the linear and planar species is approximately the same. Table 57

reports the core—valence correction extrapolated to an infinite basis set.

 

 

 

 

Table 57. Core-Valence Correction for Planar Na2C2 at the Complete Basis
Set Limit
Basis sets ACV(°°) ACV(E(°°))
CC-PCVXZ (kcallmol) (kcallmol)
Feller(DTQ) -0.64 ----
Schwartzor(DTQ) -0.64 ----
Schwartz4(TQ) -0.80 -0.80
Schwartz6(TQ) -0.80 -0.80

 

 

 

 

Similar to the linear species, the extrapolated values for the variable exponent
equations are erratic or non-physical. For the planar molecule, we used the core-valence
correction of -0.73 kcallmol. Table 58 reports the enthalpy of formation for the planar
species after we added the core-valence correction (-0.73 kcal/mol), spin-orbit coupling

(0.17 kcallmol), and relativistic corrections (0.21 kcallmol).

58

 

 

 

 

 

 

 

Table 58. AH, Values Corrected for Core-Valance Correlation, Spin-Orbit
Coupling, and Relativistic Effects for Planar Na2C2 at the Complete Basis Set
Limit for each Method

Basis sets Corrected AH, (0°) Corrected AH, (E(oo))
CC-PVXZ (kcallmol) (kcallmol)
Feller(DTQ) 92.63 93.04
Schwartzor(DTQ) 89.19 90.14
Schwartz4(TQ) 92.01 92.01
Schwartz6(DTQ) 91.30 91.30
Schwartz6(TQ) 92.03 92.03

 

 

 

 

The heat of formation results reported in Table 58 span a range of 5 kcallmol.
Table 59 reports the better values from Table 58 and the combination of the heat of
formation calculated with the AUG-CC-PV(QZ—6Z) basis set and the core-valence

correction calculated using the CC-PCV(QZ-6Z) basis set.

 

Table 59. Comparison of Heats of Formation of Planar Nang

 

 

 

Basis Set AH,
kcallmol

aThis work 92.63

I’rhis work 89.19

°This work 92.01

“This work 92.03

°Unextmolated value 92.37

 

aFeller(DTQ) extrapolation of unaugmented basis set values with P(°o) method
bSchwartzo.(DTQ) extrapolation of unaugmented basis set values with P(oo) method
cSchwartz4(T Q) extrapolation of unaugmented basis set values with P(oo) method
dSchwartz6(TQ) extrapolation of unaugmented basis set values with P(oo) method
eAH, from AUG-CC-PV(QZ-6Z) using ACV from CC-PCV(QZ-SZ) basis set

 

 

 

Table 60 lists the difference in the enthalpy of formation between the linear and planar

Na2C2 species.

59

 

Table 60. EnergLDifference Between Linear and Planar Na2C2

 

 

 

 

 

Planar Li 2C2 Linear Li2C2 Difference

Basis Set AH, AH, kcallmol
alFeller(DTQ) 92.63 104.74 12.1 1
aSchwartzt3t(DTQ) 89.19 100.44 1 1.25
aSchwartz4(TQ) 92.01 104.43 12.42
aSchwartz6(TQ) 92.03 104.45 12.42
bUnextrapolated value 92.37 104.82 12.45

 

aExtrapolations of CC-PVXZ basis set values using P(oo) method
I’AH, from AUG-CC-PV(QZ-6Z) using ACV from CC-PCV(QZ-SZ) basis set

 

 

 

 

Similar to the lithium molecules, the energy difference between the heats of formation for

the linear and planar sodium species is a kcallmol larger than the differences between the
D, values where there is no correction for thermal effects.

2.9 Discussion

The heats of atomization at zero Kelvin (0 K) for ClCCCl, FCCF, and HCCH are
very similar values. The dissociation energy difference between HCCH and ClCCCl is
approximately 60 kcallmol, whereas the difference between the enthalpies of formation is
closer to 4 kcallmol. The difference between the FCCF and ClCCCl dissociation
energies is approximately 35 kcallmol, but the difference between the enthalpies of
formation are around 40 kcallmol. This behavior is primarily due to differences in the

enthalpy of formation of the constituent atoms and the size of the zero point energy.

60

 

Table 61. Dissociation Enegies and Heats of Atomization for All Molecules

 

 

 

 

 

 

 

 

Linear Planar Linear Planar

HCCH FCCF ClCCCl Li2C2 Li2C2 Na2C2 Na2C2

Dc (kcallmol) 408° 388b 355° 335d 342d 300° 310°

Do (kcallmol) 392° 379b 348° 331d 337d 296° 305°
AHam

(kcallmol) 393° 380b 348° 330d 338d 296° 306°

 

 

method

method

method

 

method

a'Schwartzot(Q56) extrapolation of augmented basis set values with P(oo) method
bSchwartzot(T Q5) extrapolation of doubly augmented basis set values with P(oo)

cSchwartzchI‘ Q5) extrapolation of doubly augmented basis set values with P(°°)
dSchwartzoi(DTQ) extrapolation of unaugmented basis set values with P(oo)

°Schwartza(DTQ) extrapolation of unaugmented basis set values with P(oo)

 

All the substituted acetylenes and acetylene itself have dissociation energies of

the same magnitude and very different heats of formation. This behavior is primarily due

to the contributions from the vibrational frequencies of the molecule discussed and the

heats of formation of its constituent atoms. The core-valence corrections for all the

molecules, except the sodium molecule, are comparable.

61

 

2.10 References

1. D. Gearhart, J. F. Harrison, K. L. C. Hunt ( In preparation)

2. Chase, M.W., Jr., NIST-JANAF Therrnochemical Tables, Fourth Edition, J. Phys.
Chem. Ref. Data, Monograph 9, 1998, 1-1951.

3. S. Parthiban, J. M. Martin, J. F. Liebman, Mol. Phys, 100, 453 (2002).

4. C. W. Jr. Baushlicher,A. Ricca, J. Phys. Chem. A, 104, 4581 (2000).

5. Werner, H. J.; Knowles, P. J .; Almof, J.; Amos, R. D.; Deegan, M. J. 0.; Elbert, S. T.;
Hampel, C.; Meyer, W.; Peterson, K.; Pitzer, R.; Stone, A. J .; Taylor, P. R.; Lindh, R.;
Mura, M. E.; Thorsteinsson, T. Molpro, a package of ab initio programs.

6. Frisch, M. J .; Trucks, G. W.; Schlegel, H. B.; Scuseria, G. E.; Robb,M. A.;
Cheeseman, J. R.; Zakrzewski, V. G.; Montgomery, J. A., Jr.; Stratmann, R. E.; Burant, J.
C.; Dapprich, S.; Millam, J. M.; Daniels, A. D.; Kudin, K. N.; Strain, M. C.; Farkas, 0.;
Tomasi, J.; Barone, V.; Cossi, M.; Cammi, R.; Mennucci, B.; Pomelli, C.; Adamo, C.;
Clifford, S.; Ochterski, J.; Petersson, G. A.; Ayala, P. Y.; Cui, Q.; Morokuma, K.;
Malick, D. K.; Rabuck, A. D.; Raghavachari, K.; Foresman, J. B.; Cioslowski, J.; Ortiz, J.
V.; Stefanov, B. B.; Liu, G.; Liashenko, A.; Piskorz, P.; Komaromi, 1.; Gomperts, R.;
Martin, R. L.; Fox, D. J .; Keith, T.; Al-Laham, M. A.; Peng, C. Y.; Nanayakkara, A.;
Gonzalez, C.; Challacombe, M.; Gill, P. M. W.; Johnson, B.; Chen, W.; Wong, M. W.;
Andres, J. L.; Gonzalez, C.; Head-Gordon, M.; Replogle, E. S.; Pople, J. A. Gaussian 98,
Revision A.6; Gaussian, Inc., Pittsburgh, PA, 1998.

7. TH. Dunning, Jr., J. Chem. Phys, 90, 1007 (1989).

8. DE. Woon and TH. Dunning, Jr., J. Chem. Phys, 98, 1358 (1993).

9. R. A. Kendall, T. H. Dunning, R. J. Harrison, J. Chem. Phys, 96, 6796 (1992).

10. D. E. Woon, K. A. Peterson, T. H. Dunning, Unpublished.

11. Moore, C. B. Atomic energy levels; Natl. Bur. Stand. (US); 1949,circ. 467.

12. D. Feller, D., J Chem Phys, 96, 6104 (1992).

13. J. M. L. Martin, Chem. Phys. Lett. 259, 669 (1996).

14. C. Schwartz, in: Methods in computational physics, ed. 3.]. Alder (Academic Press,
New York, 1963).

15. w. Kutzclnigg. Theor. Chim. Acta 68, 445 (1985).

62

16. W. Kutzelnigg and J .D. Morgan 111, J. Chem. Phys. 96, 4484 (1992); 97, 8821 (1992)
(E)-

17. RN. Hill, J. Chem. Phys. 83, 1173 (1985).
18. G. Strey, I. M. Mills, J Mol Spec, 59, 103 (1976).

19. Shimanouchi, T., Tables of Molecular Vibrational Frequencies Consolidated Volume
II, J. Phys. Chem. Ref. Data, 1972, 6, 3, 993-1102.

20. J. Briedung, T. Hansen; W. Thiel, J Mol Spec, 179, 73 (1996).
21. T. C. Ehlert, J. Phys. Chem., 73, 949 (1969).

22. Tables of interatomic distances and configuration in molecules and ions; The
Chemical Society: London, 1958.

23. P. Schleyer, J. Phys. Chem, 94, 5560 (1990).

24. Jaworski, A., Person, W., Adamowicz, L., Bartlett, R., International Journal of
Quantum Chemistry Symposium, 1987, 21, 613.

63

Chapter 3

Quadrupole Moments of HCCH, FCCF, CICCCI, Li2C2, and Na2C2

3.1 Introduction

An important facet of intermolecular interactions and key in understanding
intermolecular interactions is the permanent electronic moment of a molecule. For the
HCCH, FCCF, CICCCI, Li2C2, and Na2C2 molecules, the first permanent non—vanishing
electric moment is the quadrupole moment, and since quadrupole moments have been

shown to play an important role in chemistryLz'z‘IA's'“:7

, the quadrupole moments of these
molecules influences the interactions of these molecules with others. Although
quadrupole moments of molecules are difficult to determine experimentally, they are very
simple to calculate. Unfortunately, the composition of and chemical implications of

quadrupole moments and the basis set and electron correlation dependence of quadrupole

moment calculations are still under investigation.
From the work of Lawson and Harrisons, it was shown for the 13202;) molecule,

the quadrupole moment was heavily influenced by the level of correlation in the
calculation. Using a T-AUG-CC-PVSZ basis set, the quadrupole moment at the MRCI
level was 0.4759 ea?) , whereas the RHF result reported a value of 1.055 eag . It was

later shown that this large difference was due to the insufficient description of it bonding

in the P2 molecule within the SCF calculation. The Lawson prediction of the quadrupole

moment of the C12 (12; ) molecule showed the differences between the SCF (2.2846

eag ) and MRCI (2.2998 eag ) calculated values were minute which is due to the lack of

7: bonding in C12.
From the same work, the size of the basis set used in calculations greatly

influenced the magnitude and sometimes the sign of the calculated quadrupole moments.

65

For all the addressed molecules, the convergence of the quadrupole moments with

unaugmented basis sets was considerably slower than that of the augmented basis sets.

For P2(12; ) the difference between the MRCI quadrupole moment calculated at the CC-

PVDZ and CC-PVSZ basis set was 0.6822 ea?) , whereas the difference between the

AUG-CC-PVDZ and AUG-CC-PVSZ quadrupole moments was only 0.1888 atomic
units. The difference between the AUG-CC-PVSZ and CC-PVSZ quadrupole moments
was 0.0978 atomic units. From this data, it was determined that the addition of diffuse s,
p, and d functions upon augmentation not only allowed the quadrupole to converge faster,
but the augmentation of the basis sets was necessary for a proper description of the
quadrupole moment of diatomics.

In this manuscript we studied seven molecules. Two of those molecules have two
different geometry types, a linear and a doubly bridged! geometry. For the linear I
molecules—acetylene, the doubly halogenated acetylenes, and the linear doubly alkalized
acetylides—there is only one unique quadrupole making the description of the
quadrupole simple, but during our investigation of the quadrupole moments of the doubly
alkalized acetylides, the existence of a second geometric minimum on the potential
energy surface was reaffirmed. The second geometry, a doubly bridged geometry, has
two unique quadrupole moments, making the interpretation slightly more complicated.

Therefore, we hope that further investigation of the quadrupole moments of these
alkalized acetylides, as well as the doubly halogenated acetylenes, will provide additional
enlightenment about the relationship between the magnitude and the sign of the
quadrupole moment, as well as the molecule’s electronic structure. The intention of this

discussion is to investigate the basis set dependence of the quadrupole moments for the

66

aforementioned molecules and the influence of different correlation methods on the
calculated results.

Although there has been at least one theoretical study on the FCCF and ClCCCl
quadrupole moments, experimental quadrupole moments of the doubly halogenated
acetylenes have yet to be determined due to the high reactivity of these molecules. The
metal acetylides have neither experimental nor calculated quadrupole moment results
with which to compare our results. Therefore, similar to Chapter 2, because of the
limited information for the Li2C2, Na2C2, FCCF, and ClCCCl quadrupole moments, we
calculated the quadrupole moment of a heavily studied molecule, HCCH, as a gauge for
the accuracy of the calculated quadrupole moments in this manuscript.

3.1.1 Computational Details

Using Molpro2000, we determined all CCSD(T), DFT(B3LYP) and SCF single
point and geometry optimization calculations. We used the Dunning CC-PVXZ, AUG-
CC-PVXZ, and D-AUG-CC-PVXZ basis sets, as well as AUG-CC—PVXZ basis sets
constructed on site from Dunning CC-PVXZ basis sets in our calculations. We used a
field gradient of 0.0001 atomic unit to make field gradient calculations. We determined
quadrupole densities using locally written codes.

We used the Z-axis as the internuclear axis for the linear molecules. For the

linear molecules, the quadrupole moment defined is the traceless quadrupole moment so

1 . .
that (9 xx = (9 y, = -2- (922 , and by convention the (9,; component is taken as the

quadrupole moment.
For the nonlinear molecules, the metal atoms lie on the Z-axis while the carbon

atoms lie on the Y-axis. The quadrupole moment of the planar molecules is traceless so

67

that®xx + (9yy + (9n = 0 , but since (9,, 7E (9,0,, both of the two quadrupole moment

components, (922 and (9,,, are needed for a full description of the molecular quadrupole

moment.
3.1.2 Multipole Expansion

The definition of the quadrupole moment stems from an expansion of the
electrostatic potential. Figure 1 illustrates a charged particle Q located at a distance
5 from the origin with a potential at a point P equal to the magnitude of Q divided by the
distance between Q and the point P. Equation (1) gives the mathematical representation

of the relationship.

 

C2 r

Origin

 

 

 

Figure 1: Diagram of Point Charge not located at Origin
) Q Q (1)

<1)R=—=—:—.
( |'r‘| 'R-El

Expanding the potential around 8'" = 0 results in the expression in equation (2),

<D(R)=%—Qsa(Va %]+%Q(% (32,2, —8a,382 D6737, 3+ . (2)

68

1 . . .
where V a, = 51—?— and the Greek indices are summed over the three Cartesran vectors.
a

Equations 3 and 4 define the 013 component of the quadrupole moment, Dag, and the or

component of the dipole moment, pa.
1
(9,15 = Qt; (38(185 - 5,1582 B . (3)

pa = Qea (4)

Even though these purely mathematical definitions of the electronic moments do
not provide much chemical insight into the concept of multipoles, a discussion of the
most commonly recognized electronic moment, the dipole moment, customarily involves
an argument about differences in electronegativity between the two atoms of a diatomic
molecule. Therefore, the dipole is perceived as the difference between the charge
distribution of a molecule and the charge distribution of the atoms. This interpretation is
very different from the process used to calculate the molecular dipole moment, which
involves the summing of the electronic and nuclear contributions to the overall dipole

moment as shown in equation (5) where n(?) is the electron density and Z k is the atomic

number of the Kth nucleus.
NUC
u, = — n(‘r‘)r,,dv + szRKa. (5)
K=l
If the atoms of the molecule are separated by a large distance, so that the atoms
are not interacting, the dipole moment is zero. Rearrangement of the expression for the
dipole results in equation (6) .
NUC

In atoms (i: )rqu = Z ZkR Ka - (6)
K=l

69

Since this expression holds at all distances, equation (5) can be rearranged into equation

(7).
u, = — n('r‘)radV + n,,oms(i=)radv,or ug’ = — ISn(i")radV . (7)

Now the dipole is represented as an integral involving the charge redistribution occurring

at molecular formation.

ua = 113' (8)

Using this same rationale, an expression for the quadrupole moment as a function
of the charge redistribution upon molecular formation results in equation (9),

6,, = ogtgms + 63,, . (9)
where the components of the quadrupole moment of a molecule are represented by
equation (10).

atoms

Gatfioms= 2:199;
(10)

(93,3 = -5 ll}8n(f)(3rar,3 — fiafirz 11V.
In the molecular dipole moment the dipole due to the atoms is zero, atoms can have
quadrupole moments, so it is important to consider this influence.
3.1.3 Method

There are two independent theoretical methods to determine the quadrupole
moment. The first method is to determine the expectation value of the quadrupole
moment or the average value of the property by evaluating equation (11) where the

quadrupole moment operator is defined by equation (12).

(w|é>,,,,|\il)=ea,,. (11)

7O

A l N A A A2 INK A A A2
(9,1,3 =§Zl3rairl3i —r, +§KZBraKrI3K —r,(. (12)
i= =1

Although the expectation value of the quadrupole moment is the most accurate
method, at the time of these calculations, there was no software to allow this type of
calculation to be done at the CCSD(T) level. Instead we calculated the CCSD(T)
quadrupole moments using a finite field calculation. If the energy of a system is

expanded as a function of the field gradient, the expression for the energy takes the form

of equation (13).
dB 1 82E
E F' E F ——F' F'
( ): “ii—FEB “B 2311,3811,” 78 (13)
1 03B
PEBFifiFE9+

 

+58 wary, 81:;

The derivatives of the energy with respect to the field gradient are assigned physical

interpretations. Equation (14) reflects the quadrupole moment as the leading term in the
expansion of the energy as a function of the field gradient. Equation (14) includes a few
of the trailing terms. We interpret these trailing terms as the change in the quadrupole as

a function of the field gradient.

 

 

 

BE 1
_-—e
817,, 3 “‘3
32E __13®mi_ 1
——C . . 14
arises”, 3 or“, 3 “5'75 ( )
83B _ _1 a__®__a[i _1 G
argflasaargg 3 85,811,, 3 “3:75;”

71

If we substitute the results of equation (14) into equation (13) (which is the energy
of a molecule as a function of the field gradient), the quadrupole moment and other terms
take the form of equation (15).

E(F')= 130 -—1-

l l
39aBFEB-gcaawsf‘3fi6‘fiGaaweSxJEBPisFEcp-m - (15)

If we calculate the energy of a molecule with a positive and negative field gradient, the

difference between the two energies takes the form of equation (16).

l l l
E(- FI) : EO + 3 aaflFdfi - g CaB;78FdBF;6 + EGanlethdBFyéFEo

l l l
— 15(17): 130 + 3 9011311313 + g (3115;751:3355 + 15 Ga3;y5;e¢pF&sP§aFE«p (16)

 

2 l
E(F')- 13(- 17) = 3911135113 + gGaa;ya;e¢F&aP§5Fé¢-

Examination of equation (16): By choosing a field gradient small enough to
cause the contributions of the trailing terms to be minute, the quadrupole moment is
isolated. A comparison of the results of both the expectation value of the quadrupole
moment and the finite field quadrupole moment shows that not only is there an ability to
select a gradient small enough to minimize the contribution of the higher order terms, but
there is also a quick check of both methods.

3.2 HCCH

We calculated the CCSD(T), DFT(B3LYP), and SCF quadrupole moments for the

nonvibrating HCCH molecule at the Strey and Mills9 experimental geometry, RCC =

1.2033 A and RCH = 1.0605 A. Table 1 lists each result as a function of basis set.

72

 

 

 

 

 

 

 

 

 

 

 

 

Table 1. Quadrupole Moment, 9 (eafi ),of HCCH
SCF CCSD(T) DFT(B3LYP)
Basis Set Field Field Field
<9 3 ) Gradient Gradient <9 “ ) Gradient

CC-PVDZ 5.2112 5.2112 4.5078 4.6171 4.6172
CC-PVTZ 5.4186 5.4187 4.8010 4.9014 4.9014
CC-PVQZ 5.4345 5.4345 4.8428 4.8981 4.8981
CC-PVSZ 5.4444 5.4444 4.8608 4.9090 4.9092
CC-PV6Z 5.4442 5.4442 4.8652 4.9079 4.91 15
AUG-CC-PVDZ 5.4689 5.4689 4.7721 4.8739 4.8739
AUG-CC-PVTZ 5.4577 5.4577 4.8451 4.9173 4.9171
AUG-CC-PVQZ 5.4456 5.4455 4.8570 4.8722 4.8723
AUG-CC-PVSZ 5.4445 5 .4445 4.8634 4.9078 4.9068
AUG-CC-PV6Z 5.4435 5.4435 4.8661 4.8903 4.8901
D-AUG-CC-PVDZ 5.4557 5 .4557 4.7584 4.8650 4.8643
D—AUG-CC-PVTZ 5.4456 5.4456 4.8333 4.8934 4.8940
D-AUG-CC-PVQZ 5.4441 5.4441 4.8359 4.8907 4.8896

 

 

Table 1 compares the two computational methods that we used to calculate the
SCF and DFT quadrupole moments. The results of the two methods should be equal, but
the difference between the results for the SCF quadrupole moments at each basis set is
slight and only affects the quadrupole moment at the fourth decimal place. We found that
the difference between some SCF quadrupole moments was zero. The DFT results differ
between the two methods at the third decimal place.

The SCF quadrupole moment converges to two decimal places by the 52 basis set
and is unaffected by augmentation by the QZ basis set. By the 52 basis set the CCSD(T)
quadrupole moment has converged to two decimal places. The two correlated methods
converge and agree to one decimal place, while the DFT(B3LYP) method converges to a

larger value than that of the CCSD(T) method. The difference between the correlated
and uncorrelated quadrupole moment is less than 1 eag. Table 2 is a tabulation of

experimental and calculated values for the quadrupole of HCCH.

73

 

 

 

 

 

 

 

 

Table 2. HCCH madrupole Moments
9(eag ) Method Reference
5.366 SCF 10
5.443 SCF l 1
5.460 SCF 12
5.426 SCF 13
5.05, 5.24 SCF 14
5.444 SCF This work
4.703 BLYP l 1
4.890 DFT(B3LYP) This work
4.925 CC 13
4.88 MP4 15
4.787, 4.795,4.806 MP4, QCISD, BD(T) 11
4.717 MRCI 16
4.866 CCSD(T) This work
4.71 10.14 Field gradient-induced birefringence 17
5.65 $0.07 Cotton-Mouton effect 18
4.03 10.30 Collision-induced far-infrared absorption 19

 

 

The range in experimental values for the quadrupole moment of HCCH is as wide
as the range in theoretical values. Although the SCF calculated results of other
researchers are comparable to the SCF quadrupole moment calculated in this work, the
correlated quadrupole moments vary within methods and even between methods. The
correlated quadrupole moments differ from the experimental values, but these values are
not corrected for molecular vibrations. By correcting the CCSD(T) quadrupole moment
for vibrational contributions”, the CCSD(T) quadrupole moment falls within
experimental error of the field gradient-induced birefringence quadrupole”.

3.3 FCCF

We examine FCCF as one of two doubly halogenated acetylenes in this chapter.

The doubly halogenated acetylenes are very reactive, making the calculation of the

quadrupole moment possibly the only route available for determining the quadrupole.

We used the equilibrium geometry provided by Briedung et alz', RCC = 1.186 A and

74

ch=1.2835 A, to calculate the SCF, CCSD(T), and DFT(B3LYP) quadrupole moments

as a function of basis set. Table 3 reports the results.

 

 

 

 

 

 

 

 

 

 

 

 

Table 3. Quadrupole Moment, (9 (eafi ),of FCCF
SCF CCSD(T) DFT(B3LYP)
Basis Set Field Field Field
<9 ‘2 > Gradient Gradient (9 a) Gradient
CC-PVDZ 0.9726 0.9726 0.4090 0.0534 0.0534
CC-PV'IZ 0.8457 -0.8457 0.4569 0.2783 0.2783
CC-PVQZ 0.8986 0.8986 0.5825 0.4419 0.4419
cc—vaz 0.9225 0.9225 0.6343 0.5104 0.5108
AUG-CC-PVDZ -1.0335 4.0335 0.7860 0.6003 0.6004
AUG-CC-PVTZ 0.9430 0.9429 -0.6485 0.5404 0.5391
AUG-CC-PVQZ 0.9130 0.9130 0.6357 0.5163 -0.5180
AUG-CC-PVSZ 0.9168 -0.9168 0.6400 0.5218 0.5245
D-AUG-CC-PVDZ 0.8547 0.8547 0.6367 -0.5128 0.5126
D-AUG-CC-PVTZ 0.9067 0.9067 0.6156 0.5060 0.5069
D—AUG-CC-PVQZ 0.9119 0.9119 -0.6331 0.5141 0.5152
D-AUG-CC-PVSZ 0.9157 0.9157 -0.6389 0.5182 0.5193

 

 

Interestingly, the substitution of fluoride for hydrogen in acetylene causes a
change in the sign of the quadrupole moment and lowers the magnitude greatly. We
discuss this result later. The change in sign is attributed to the size of the fluorine atom
quadrupole moment and the electronegativity of the fluoride atom.

The quadrupole expectation value for the SCF reproduces an equivalent to the
quadrupole that the field gradient method produces, as expected. Again, there is some
discrepancy in the DFT values, but in general they are consistent to the second decimal
place. Table 3 shows that the convergence is slow for all three methods. All the
calculated quadrupoles are dependent on the level of augmentation of the basis set,
especially the DFT(B3LYP) quadrupoles. The SCF quadrupoles have only converged to
one decimal place by the augmented 52 basis set, whereas the correlated methods have

converged to two decimal places by the augmented QZ basis set. The two correlated

75

quadrupoles are within 0.1 atomic unit of each other, and the uncorrelated quadrupole is
within 0.4 atomic units of the two correlated methods. Table 4 lists the results of this

work with other calculated results for the quadrupole of FCCF.

 

 

 

 

 

 

Table 4. FCCF Quadrupole Moments
6(ea3 ) Method Reference
-0.9085 SCF 22
-0.9157 SCF This work
-0.6524 MP2 22
-O.5193 DFT(B3LYP) This work
-0.6389 CCSD(T) This work

 

 

 

The results of this work and the results of Maroulis are equivalent to one decimal
place for the SCF quadrupole and are within 0.1 atomic unit of each other for the
correlated calculations. The FCCF calculated quadrupole moments were not corrected
for vibration.

3.4 CICCCI

We calculated SCF, CCSD(T), and DFT(B3LYP) quadrupoles for the second

doubly halogenated acetylene, CICCCI, at the experimental geometry, Rcc=l .205 A and

RCIC=1.635 A, as a function of the Dunning basis sets. Table 5 lists these results.

76

 

 

 

 

 

 

 

 

 

 

 

 

Table 5. Quadrupole Moment, 6 (ea?) ),of CICCCI
CCSD(T) DFT(B3LYP)
Basis Set Field Field Field
<9 a ) Gradient Gradient (8a) Gradient

CC-PVDZ 3.9317 3.9316 3.7337 4.1042 4.1037
CC-PVTZ 3.9343 3.9342 3.8524 3.9956 3.9960
CC-PVQZ 3.9253 3.9252 3.7892 3.9157 3.9167
CC-PVSZ 3.9353 3.9352 3.8054 3.8977 3.8949
AUG-CC-PVDZ 4.0586 4.0586 3.7755 3.8246 3.8236
AUG-CC-PVTZ 3.8568 3.8568 3.7970 3.7688 3.7674
AUG-CC-PVQZ 3.8783 3.8782 3.7595 3.7787 3.7787
AUG-CC-PVSZ 3.8988 3.8987 3.7672 3.7909 3.7905
D-AUG-CC—PVDZ 3.9381 3.9379 3.6686 3.7565 3.7563
D-AUG-CC-PVTZ 3.8219 3.8217 3.7492 3.7387 3.7384
D-AUG-CC-PVQZ 3.8685 3.8684 3.7478 3.77 12 3.7674
D-AUG-CC-PVSZ 3.9006 3.9005 3.7683 3.7918 3.7952

 

 

Remarkably, the quadrupole moment of the chlorinated acetylene has the same
sign and close to the same magnitude as that of HCCH. This similarity comes as a
surprise when compared to the quadrupole of FCCF. Therefore, the negative quadrupole
of FCCF is not solely due to electronegativity. The differences in quadrupoles calculated
by expectation value and field gradient are slight. We detected most differences in the
fourth decimal place within the SCF calculations. Again, the DFT results are not
equivalent, but still comparable.

The results for all calculations are slow to converge beyond the second decimal
place, but they are comparably close in magnitude. All the quadrupoles have converged
to one decimal place by the Q2 basis set. The SCF is converging to a higher value than
the correlated methods, and the difference between the correlated methods is less than
0.04 atomic units. The relative difference between the DFT and CCSD(T) results may be
an indication of the level of covalent bonding in the CICCCI molecule. Table 6 lists

other calculated results for the quadrupole moment of CICCCI.

77

 

 

 

 

 

 

Table 6. ClCCCl Quadrglole Moments
9(ea3) Method Reference
3.9140 SCF 22
3.9006 SCF This work
3.6612 MP2 22
3.7952 DFT(B3LYP) This work
3.7683 CCSD(T) This work

 

 

 

The SCF results of this work are equivalent to one decimal place with the SCF
result provided by Maroulis, and the correlated results are within 0.2 ea?) of the result

provided by Maroulis. Again, the quadrupole moment of ClCCCl was not corrected for
vibration.
3.5 LizCz

Since there is no experimental geometric information provided for dilithium
acetylide in the literature, we calculated an equilibrium geometry. In Chapter 2, we
explain the process used to calculate the equilibrium geometries and reaffirm that the

metal acetylides have two equilibrium geometries. First, the linear molecule geometry of

RCC = 12531134 . RCLi = 1.8943A , and the doubly bridged molecule geometry of RCC =

1.2673A , RCLi = 2.0086A , and 4 CCLi = 71.6° , seem to locate the lithium atoms very

far from the carbon atoms, so that the atoms seem to be hanging off a C2 species.
3.5.] Molecular Quadrupole Moment

Each geometry has its own quadrupole moment. The linear molecule, like the
other linear species discussed in this paper, has one unique component to the quadrupole
moment, which lies along the internuclear line, while the doubly bridged or planar
molecule has two unique quadrupole moment components, one parallel to the carbon

atoms and one parallel to the Li atoms.

78

3.5.2 Linear
We used the geometry, RCC = 1.531A , RCLi = 1.8943A, to calculate the

quadrupole moment of linear dilithium acetylide. Table 7 lists the results.

 

 

 

 

 

 

 

 

 

 

 

 

Table 7. Quadrupole Moment, (9 (ea?) ),of Linear LizCz
SCF CCSD(T) DFT(B3LYP)
Basis Set Field Field Field
(9“) Gradient Gradient <9“) Gradient
CC-PVDZ 35.921 1 35.9209 34.7707 34.2305 34.2295
CC-PV'IZ 36.1191 36.1189 35.1960 34.5417 34.5412
CC-PVQZ 36.3687 36.3685 35.4927 34.7938 34.7934
aCC-(SZ-QZ) 36.4222 36.4221 35.5717 34.8403 34.8388
bCC-(6Z-QZ) 36.4878 36.4877 35.6443 34.8973 34.8978
AUG-CC-PVDZ 36.6568 36.6566 35.7495 35.0863 35.0855
AUG-CC-PVTZ 36.5193 36.5191 35.6585 34.9204 34.9184
AUG-CC-PVQZ 36.5094 36.5093 35.6662 34.9233 34.9228
°AUG-CC-(SZ-QZ) 36.5083 36.5082 35.6749 34.9248 34.9241
dAUG-CC-(6Z-QZ) 36.5078 36.5077 35.6787 -- --
alLi CC-PVQZ basis set with C CC-PVSZ basis set
bLi cc-onz basis set with C CC-PV6Z basis set
cLi AUG-CC-PVQZ basis set with C AUG-CC-PVSZ basis set
dLi AUG-CC-PVQZ basis set with c AUG-CC-PV6Z basis set

 

One of the most interesting aspects of the Li2C2 quadrupole moment is the

 

magnitude. The magnitude of linear LizCz is nearly 10 times the magnitude of the
ClCCCl quadrupole moment. The quadrupole is so large that the differences in the
quadrupole moment between the unaugmented and augmented basis sets is minute. The
differences are only 2% of the total value at the DZ basis set and 0.05% at the largest
basis set. Again, the difference between the expectation value and field gradient methods
is minute for the SCF quadrupoles; however, the DFT results show the field gradient has
some strong influence beyond the first decimal place. Also note that the CCSD(T) results

are sandwiched between the DFT and SCF results as the SCF results are converging to

79

the higher value. The difference between the DFT and SCF results is less than 2 atomic

units which is around 6% of the total quadrupole moment.
3.5.3 Planar

Since there are two unique quadrupole moment components for the planar
quadrupole, a unique property of doubly bridged molecules with D21I symmetry, we

calculate the Z2 and YY components. By calculating two components, the third is
known since the quadrupole moment is traceless. For these calculations, keep in mind

that the carbon atoms lie on the Y axis and that the Li atoms lie on the Z axis. We used

the geometry provided in Chapter 2, RCC = 1.2673A, RCLi = 2.0086A, and A CCLi =

716°, to calculate the SCF, CCSD(T), and DFT @n for planar dilithium acetylide. Table

8 reports the results as a function of basis set.

 

 

 

 

 

 

 

 

 

 

 

 

Table 8. Quadrupole Moment, 9,, (ea?) ),of Planar Li2C2
SCF CCSD(T) DFT(BBLYP)
Basis Set Field Field Field
<9 3’: > Gradient Gradient (9 u ) Gradient
CC-PVDZ 26.4236 26.4235 25.8323 25.3724 25.3720
CC-PVTZ 26.2621 26.2619 25.8399 25.2043 25.2023
CC-PVQZ 26.2875 26.2874 25.9046 25.2641 25.2632
aCC-(5Z-QZ) 26.3001 26.2999 25.9366 25.2872 25.287 1
bCC-(6Z-QZ) 26.3076 26.3074 25.9538 25.3017 25.2999
AUG-CC-PVDZ 26.4269 26.4267 26.0478 25.4179 25.4198
AUG-CC-PVTZ 26.3124 26.3123 25.9502 25.2977 25.2967
AUG-CC-PVQZ 26.31 15 26.31 14 25.9538 25.3068 25.3067
cAUG-CC-(SZ-QZ) 26.3094 26.3092 25.9564 25.3070 25.3054
dAUG-CC-(6Z-QZ) 26.3089 26.3088 25.9578 25.3064 25.3051
aLi CC-PVQZ basis set with C CC-PVSZ basis set
bLi CC-PVQZ basis set with C CC-PV6Z basis set
cLi AUG-CC-PVQZ basis set with C AUG-CC-PVSZ basis set
dLi AUG-CC-PVQZ basis set with C AUG-CC-PV6Z basis set

 

 

Note the magnitude and sign of the quadrupole component. Like the linear

species it is large, but it is not as positive. We compared the unaugmented and

80

augmented quadrupoles and found no need for augmentation in order to calculate the
quadrupole component to three significant figures. The unaugmented and augmented
suites of basis sets are equivalent to the units place when rounded. The differences
between the two calculation methods are consistent with previous results in this study.
Also note, the SCF and CCSD(T) quadrupole moments are close in magnitude while the
DFT result is a slight outlier.

We used the same geometry to calculate the SCF, CCSD(T), and DFT results for

the YY component of the planar molecule’s quadrupole moment. Table 9 lists the

 

 

 

 

 

 

 

 

 

 

 

 

results.
Table 9. Quadrupole Moment, 3,, (ea% ),of Planar Li2C2
SCF CCSD(T) DFT(B3LYP)
Basis Set Field Field Field
<9 W > Gradient Gradient (9” > Gradient
CC-PVDZ 46.0693 46.0693 -15.8702 45.4046 45.4047
CC-PVTZ 46.1829 -16.1829 46.1661 45.6996 45.6995
CC-PVQZ 46.2317 -16.2317 46.2694 45.8119 -15.8110
‘CC-(SZ-QZ) 46.2663 46.2663 46.3300 45.8685 45.8674
bCC-(6Z-QZ) -l6.283l 46.2831 46.3687 45.9055 45.9046
AUG-CC-PVDZ 46.1994 46.1995 46.3309 45.8374 45.8384
AUG-CC-PVTZ 46.2843 46.2843 46.3798 45.9184 45.9165
AUG-CC-PVQZ 46.2761 46.2761 46.3706 45.9114 45.9088
“AUG-CC-(SZ-QZ) 46.2739 46.2739 46.3697 45.9099 45.9107
dAUG-CC-(6Z-QZ) 46.2736 46.2737 46.3699 45.9095 45.9112
a‘Li CC-PVQZ basis set with C CC-PVSZ basis set
bLi cc-onz basis set with c CC-PV6Z basis set
cLi AUG-CC-PVQZ basis set with C AUG-CC-PVSZ basis set
dLi AUG-CC-PVQZ basis set with c AUG-CC-PV6Z basis set

 

 

Again, due to the magnitude of the quadrupole, we determined little need for the
augmentation of the basis set. The two methods are consistent with the remainder of the
study. The SCF behaves well, while the DFT results differ in the second decimal place.
The YY component quadrupoles are nearly equivalent. The CCSD(T) and SCF YY

component are closer to each other than they are to the DFT result.

81

3.6 Na2C2

Similar to Li2C2, there are two viable minima on the potential surface. The linear

geometry is RCC = 1.2549A, RCNa = 2.2341A, and the planar geometry is RCC =

1.2704A, RCNa = 2.3427A, A CCNa = 74.3°. Again, the large separations between the
Na atoms and carbon atoms seem to describe two sodium ions attached to a C2 molecule.
3.6.1 Linear

Table 10 lists SCF, DFT, CCSD(T) quadrupole calculations at the aforementioned

linear geometry as a function of the Dunning basis sets.

 

 

 

 

 

 

 

 

 

 

 

 

Table 10. Quadrupole Moment, (9 (ea?) ),of Linear Na2C2
SCF CCSD(T) DFT(B3LYP)
Basis Set Field Field Field
<9 ZZ > Gradient Gradient <9 a) Gradient
CC-PVDZ 47.5156 47.5143 44.4711 42.4321 42.4258
CC-PVTZ 48.4753 48.4741 46.0046 43.2959 43.2893
CC-PVQZ 48.6964 48.6953 46.4453 43.5136 43.5060
aCC-(SZ-QZ) 48.6933 48.6922 46.5153 43.5297 43.5241
bCC-(6Z-QZ) 48.7012 48.7001 46.5590 43.5392 43.5260
AUG-CC-PVDZ 49.0474 49.0462 46.4530 43.8956 43.8890
AUG-CC-PVTZ 48.8505 48.8494 46.5903 43.6743 43.6638
AUG-CC-PVQZ 48.7065 48.7054 46.5634 43.5591 43.5563
cAUG-CC-(SZ-QZ) 48.6725 48.6714 46.5663 -- --
dAUG-CC-(6Z-QZ) 48.6368 48.6357 46.5490 -- --
“Na CC-PVQZ basis set with C CC-PVSZ basis set
bNa CC-PVQZ basis set with c CC-PV6Z basis set
cNa AUG-CC-PVQZ basis set with C AUG-CC-PVSZ basis set
dNa AUG-CC-PVQZ basis set with c AUG-CC-PV6Z basis set

 

 

Again, due to the magnitude of the quadrupole moment, the level of augmentation has a

small influence. The magnitude of the calculated quadrupole is larger than the magnitude
in the Li case. The SCF and CCSD(T) results are not nearly as equivalent as they were in
the Li case. The SCF converges to a higher value than the CCSD(T) results, and the DFT

quadrupole converges to the lowest value which is the same as in the Li case. The two

82

computation methods behave differently than previously in this study with the SCF
values deviating as much as the DFT values.
3.6.2 Planar

We used the calculated planar geometry for N a2C2, RCC = 1.2704A, RCNa =
2.342721, 2 CCNa = 743°, to calculate SCF, CCSD(T), and DFT quadrupoles at various

basis sets for the ZZ component of the quadrupole moment. Table 11 lists the results.

 

 

 

 

 

 

 

 

 

 

 

 

 

Table 11. Quadrupole Moment, on (eaf, ),of Planar Na2C2
SCF CCSD(T) DFT(B3LYP)
Basis Set Field Field Field
<6 32) Gradient Gradient (6 7“) Gradient
CC-PVDZ 36.3662 36.3655 35.3337 33.4682 33.4645
CC-PVTZ 36.4226 36.4218 35.7632 33.5028 33.4990
CC-PVQZ 36.4039 36.4032 35.8558 33.5168 33.5159
aCC-(SZ-QZ) 36.3964 36.3957 35.8854 33.5276 33.5265
bCC-(6Z-QZ) 36.3738 36.3732 35.8846 33.5310 33.5298
AUG-CC-PVDZ 36.7047 36.7040 36.0321 33.8974 33.8914
AUG-CC-PVTZ 36.4725 36.4719 35.9317 33.6177 33.6135
AUG-CC-PVQZ 36.3394 36.3745 35.8768 33.4561 33.4506
cAUG-CC-(SZ-QZ) 36.3619 36.3612 35.8777 33.5289 33.5323
dAUG—CC-(6Z—QZ) 36.3448 36.3441 35.8669 33.5225 33.5247
aNa CC-PVQZ basis set with C CC-PVSZ basis set
l’Na CC-PVQZ basis set with c CC-PV6Z basis set
cNa AUG-CC-PVQZ basis set with C AUG-CC-PVSZ basis set
dNa AUG-CC-PVQZ basis set with C AUG-CC-PV6Z basis Set

 

The magnitude of the Z2 component is, again, smaller than that of the linear
species, and again the level of augmentation seems unimportant with such a large
property. The two computational methods are behaving well while DFT still differs in
the second decimal place. The SCF and CCSD(T) results for the ZZ component are
close. DFT reports results with the lowest magnitude.

We used the same geometry to calculate the YY component at the SCF,

CCSD(T), and DFT levels. Table 12 shows the results.

83

 

 

 

 

 

 

 

 

 

 

 

 

Table 12. Quadrupole Moment, 9,, (ea?) ),of Planar Na2C2
SCF CCSD(T) DFT(B3LYP)
Basis Set Field Field Field
(9” > Gradient Gradient <9”) Gradient
CC-PVDZ -21.3517 -21.3515 -20.8689 -l9.6638 -l9.6634
CC-PVTZ -21.5251 -21.5249 -21.4l47 -20.0101 -20.0093
CC-PVQZ -21.5899 -21.5898 -21.6085 -20. 1510 -20. 1498
aCC-(SZ-QZ) -21.6039 -21.6039 -21.6500 -20. 1775 -20.1760
bCC-(6Z—QZ) -21.5957 -21.5957 -21.6660 -20. 1939 -20. 1977
AUG-CC-PVDZ -21.6850 ~21.6851 -21.7050 -20.31 14 -20.3091
AUG-CC-PVTZ -21.6563 -21.6564 -21.7 172 -20.2745 -20.2722
AUG-CC-PVQZ -21.57 14 -21.5898 -21.6713 -20. 1717 -20.2101
cAUG-CC-(SZ-QZ) -21.5844 -21.5844 -21.6729 -20.2068 -20.2082
dAUG-CC-(6Z-QZ) -21.5757 -21.5758 -21.6668 -20.2048 -20.2050
aNa CC-PVQZ basis set with C CC-PVSZ basis set
bNa CC-PVQZ basis set with C CC-PV6Z basis set
cNa AUG-CC-PVQZ basis set with C AUG-CC-PVSZ basis set
dNa AUG-CC-PVQZ basis set with C AUG-CC-PV6Z basis set

 

 

Again, due to the magnitude of the quadrupole moment, there is little need for the
augmentation of the basis set. The behavior of the two methods is consistent with the
remainder of the study. The SCF behaves well, while the DFT results differ in the second
decimal place. Note the quadrupoles are nearly equivalent with the magnitude of the
CCSD(T) and SCF YY component closer to each other than to the DFT result. Overall,
the YY component to the planar Na molecule quadrupole moment is larger in magnitude
than the YY components of the planar Li molecule.

3.7 Partition Quadrupole Moment into Point Charges

We used the algebraic definition of the quadrupole moment to partition the
quadrupole moments of the aforementioned molecules into a collection of charges.

When conceptualizing the dipole moment, the general argument made is that the dipole
consists of two charges that are opposite in sign, but equal in magnitude, and separated
by a distance R. The LiF dipole, for example, is discussed as having a positive charge

located at Li and a negative charge of equal magnitude located at F. The value of the

84

dipole is expressed as the magnitude of the charge times the internuclear separation.
Based on the principle that a dipole moment of a molecule is a collection of two charges
separated by a distance such that the total system has no charge, the quadrupole moment
of a molecule is broken up in much the same manner as four charges arranged such that
the system has no dipole. We partitioned the quadrupole moment of each molecule into a
collection of four charges, and we used equation (3) to calculate the magnitude of those
charges. Equation (3) shows that the quadrupole moment equals the contribution of each

charge in the system in the manner described in equation (17).

)J. (17)

cr 4 l 2 2 2
965 =Zqi(5(3faifpi—5up(xi +yi +Zi)
i=1

So, for linear molecules like HCCH, equation (18) describes the 9‘; component of the

quadrupole moment,

(92; =2<1H(Rcc +RCH)RCH- (18)
where, RCC is the carbon-carbon bond length and RCH is the carbon-hydrogen bond
length. For the nonlinear molecules like Li2C2, equation (19) describes the 6);;

component of the quadrupole moment, and equation (20) describes the 6),, component,
9;; = 9Li(2R12..iC “iRécj (19)

where, RCC is the carbon-carbon bond length and Rue is the carbon-lithium bond length.
®§ry = QLi(R12.iC +fiRéc] (20)

3.8 Discussion
Recall that the molecular quadrupole moment consists of contributions from the

atomic quadrupoles and the contribution from charge redistribution upon molecular

85

formation. To determine the atomic contributions, we calculated the atomic quadrupoles
of C(3Po), F(ZPO), and Cl(2Po), and we used these calculated atomic quadrupoles to
determine the molecular quadrupole due to charge redistribution. Table 13 lists the
results from determining the partitioning of each of the charge redistribution quadrupole

moments of the titled linear molecules into a collection of point charges.

 

 

 

 

 

 

 

 

 

Table 13. Linear Molecules (XCCX) Partitioned into Point Charges

®zz( 30% ) ®(zz)atomsa( €03 ) Ger( 308 ) QX(®zz)

HCCH 4.8558 2.8406 2.0152 +0104

FCCF 0.6389 4.1976 -3.5587 0.214

CICCCI 3.7685 5.8730 2.1045 0.063

Linear Li2C2 35.6787 2.8406 32.8381 +0771

““6” 46.5490 2.8406 43.7084 +0.785
N32C2

“Atomic quadrupoles C(l.4203 gag ), F(0.6785 gag ), Cl(l.5162 eag )

 

 

Beginning with HCCH, Table 13 lists the 22 component to the molecular
quadrupole moment that we calculated at the CCSD(T) level with an AUG-PV6Z basis
set. By subtracting out the contribution due to the atomic quadrupoles, the remaining
portion of the quadrupole moment is due to the redistribution of charge on the formation
of the molecule. Partitioning that quantity into algebraic charges shows the terminal
hydrogen atoms to be slightly positive. Positive hydrogen atoms are consistent with the
basic understanding of molecular bond formation. Electron density moves from
occupying, in this case, four individual atoms to surrounding one molecule in which
some of the electron density occupies the newly created bonds. It is evident that upon
molecular formation the electron density around the hydrogen atoms shifts toward the

carbon atoms. Figure 2 is a graphical representation of this redistribution of charge.

86

 

HCCH density difference

 

 

 

 

 

—8 . . . . . . . , . . . . .
-8 0
Molecular axis

 

 

 

Figure 2: HCCH Density Difference Plot.
Images in this dissertation are presented in color.

Figure 2 shows a plot with blue and red regions. The blue regions are descriptive

of areas that are electron deficient in comparison with the atomic electron distribution,

and the red regions are descriptive of electron rich areas. The electronegativity argument

holds in this example. Carbon is more electronegative which pulls electron density away

from the hydrogen atoms. This distribution of charge removes some of the shielding

87

from the hydrogen nucleus which results in a positive charge on hydrogen and an overall
positive quadrupole moment.

It is important to note the magnitude of the atomic quadrupoles of FCCF. After
removal of the atomic quadrupoles, the quadrupole moment due to redistribution is more
negative. When partitioned into charges, the fluorine atom has a negative charge twice
that of hydrogen. The density difference between the atomic and molecular densities is

plotted in Figure 3.

 

FCCF

 

 

 

 

 

Molecular axis

 

 

 

Figure 3: FCCF Density Difference Plot
Images in this dissertation are presented in color.

88

Figure 3 makes the electronegativity of fluorine obvious. In HCCH the electron
density is pulled from the hydrogen atom leaving the nucleus slightly unshielded;
however, in FCCF the electron density moves toward the fluorine atoms creating two
islands of negative charge at the terminus of the molecule. Islands of negative charge at
the terminus of the molecule are not present in the plot of HCCH. The charge
distribution shows clearly the negative quadrupole moment of the molecule and the
negative charge of the fluorine atoms.

ClCCCl is the second case of a doubly halogenated acetylene. Table 13 shows
the magnitude of the atomic quadrupoles to be larger than in the case of FCCF. This
discrepancy is not unusual if one considers that the atomic radii of chlorine are larger
than the radii of fluorine and that the quadrupole moment has a square dependence on
distance. The atomic quadrupoles are large enough that when removed from the
molecular quadrupole the quadrupole moment due to charge redistribution is negative.
Therefore, when the redistribution quadrupole is partitioned into point charges, the
charge on the chlorine atom is negative, which gives credence to the following
electronegativity arguments: Fluorine is the most electronegative element on the periodic
table followed by chlorine, and hydrogen has a small electronegativity in comparison to
chlorine and fluorine. The charge redistribution of the CICCCI molecule on molecular

formation is plotted in Figure 4.

89

 

 

ClCCCl

10 ....,fi...,-....

 

04

 

 

 

 

0 10

Molecular axis

Figure 4: CICCCI Density Difference Plot
Images in this dissertation are presented in color.

 

 

Chlorine is very electronegative, and it is pulling electron density toward it. The
molecular quadrupole of ClCCCl is positive, but this is true only because the quadrupole
moments of the constituent atoms are positive, and they compensate for the negative

quadrupole created during charge redistribution.

90

The last of the linear molecules that we discuss are the metal acetylides. Note
the enormous magnitude of the partitioned charges on the metal atoms. This type of
magnitude lends itself to the interpretation that the metal complexes are ionic in nature.
Also note that the partitioned charge on Na is larger than that on Li. This phenomenon is
explained by electronegativity arguments which state that sodium is easier to ionize into
positively charged particles than lithium. Figure 5 is a plot of the density difference for

the linear Li2C2 molecule.

91

 

Linear Li2C2

 

0-

 

 

 

 

-8 . . . . . - . . . . . . . . .
_8 0 8
Molecular axis

 

 

 

Figure 5: Linear LiCCLi Density Difference Plot
Images in this dissertation are presented in color.

The most noticeable feature of Figure 5 is the highly positive Li atoms. Most of
the electron density has moved to the carbon atoms leaving the Li nucleus unshielded.
Most of the electron density is surrounding the carbon atoms making the center of the

molecule very negatively charged. The density difference for the Na complex behaves

92

the same way; however, the positive regions will be more electron deficient and farther

reaching.

Table 14 contains the partitioned charges of the titled planar molecules.

 

 

 

 

 

 

 

 

 

 

 

 

 

Table 14. Planar Molecules (X2C2) Partioned into Point Chagg
®zz ®(zz)atomsa ®(zz)cr Clx(®zz)
Li2C2 25.958 2.841 23.117 0.844
Na2C2 35.867 2.841 33.026 0.870
@1717 ®(yy)at_or_n_sb ®(zz)cr (11491717)
Li2C2 - 16.370 -1.420 -l4.950 0.944
Na2C2 -21.667 -1 .420 -20.247 0.958

 

 

Again, the partitioned charges are large, but the partitioned charges are not equal between
the two components. This discrepancy, which has been shown by other research in the
laboratory, is due to induced dipoles in the molecule. Therefore, the model which
illustrates that the molecules can be partitioned into point charges does not adequately

describe the planar molecules. Figure 6 shows the density difference for the LiZC2

planar complex.

93

 

Planar L12C2

 

   

 

 

 

8 I v v I l 1
1 999 -
.l ’ . .2 - >7 \‘y l
O— I i ’ > I J.-
; a
. 2x -- ‘ .7." .
—3 . . . . . . . , . . . . . . .
—s 0 8

Li-Li axis

 

 

 

Figure 6: Planar L12C2 Density Difference Plot
Images in this dissertation are presented in color.

Figure 6 shows the planar molecule as what seems to be a collection of four

charged species with positively charged Li atoms and negatively charged carbon atoms.

94

3.9 References

1. A. D. Buckingham, Adv Chem Phys, 12, 107 (1967).
2. D. E. Stogryn; A. P. Stogryn, Mal Phys, 11, 371 (1966).
3. A. Krishnaji; B. Prakash; C. Vinod, Rev Mod Phys, 38, 690 (1966).

4. Bogaard, M. P.; Orr, B. J. MTP International Review of Science; Physical Chemistry
Series 2; edited by Buckingham, A. D.; Butterworths: London, 1975; Page 144.

5. Buckingham, A. D. Intermolecular Interactions: From Diatomics to Biopolymers;
edited by Pullman, B.; Wiley: New York, 1978; Page 1.

6. Maitland, G. C.; Rigby, M.; Smith, E. B.; Wakeham, W. A. Intermolecular Forces;
Oxford University Press: Oxford, 1981.

7. Gray, C. G.; Gubbins, K. E. Theory of Molecular Fluids, Volume 1: Fundamentals;
Claredon Press: Oxford, 1984.

8. D. B. Lawson; J. F. Harrison, Mol Phys, 93, 519 (1998).

9. G. Strey, M. J. Mills, J Mol Spec, 59, 103 (1976).

10. P. Biindgen; F. Grein; A. J. Thakhar, J Mol Structure, 334, 7 (1995).

11. A. J. Russell; M. A. Spackman, Mol Phys, 88, 1109 (1996).

12. G. Maroulis; A. J. Thakhar, J Chem Phys, 93, 652 (1990).

13. C. E. Dykstra, S. Y. Liu; D. J. Malik, Adv Chem Phys, 75, 37 (1989).

14. G. deLuca, N. Russo; E. Sicilia; M. Toscano, J Chem Phys, 105, 3206 (1996).
15. G. Maroulis, J Phys B, 26, 775 (1993).

16. J. Briedung; T. Hansen; W. Thiel, J Mol Spec, 179, 73 (1996).

17. Watson, J. N. Ph.D Thesis, University of New England, Arrnidale NSW, Australia,
1995.

18. H. King; H. Gerschka; W. Huttner, Chem Phys Lett, 96, 631 (1983).

19. I. R. Dagg, A. Anderson; W. Smith; M. Missio; C. G. Joslin; L. A. A. Read, Can J
Phys, 66, 453 (1988).

95

20. D. J. Gearhart; J. F. Harrison; K. L. C. Hunt, Int J Quan Chem, 95, 697 (2003).
21. Briedung, J.; Hansen, T.; Thiel, W. J Mol Spec 1996, 179, 73.

22. G. Maroulis; D. Xenides, Electric Moments and (hyper) polarizability of
X - C E C — X, X = F, Cl, Br, and I (preprint).

96

Chapter 4

Hi gher-order van der Waals Interactions and Nonlinear Response Tensors

97

4.1 Introduction

Van der Waals forces are the intermolecular forces that arise from electrostatic
interactions, induction, and fluctuations in the charge distribution within each molecule'.
Van der Waals interactions also affect the dipoles and polarizabilities of the interacting
moleculesz. In this chapter we derive the van der Waals interaction energy between two
molecules to third order. Our analytical results at second order are identical to known
results. Our results at third order are consistent with those of Piecuch, but we provide
new equations for the third-order two-body van der Waals interaction energy: We express
the fluctuation effects in terms of hyperpolarizabilities at imaginary frequencies. We also
make numerical estimates for the second- and third-order interaction energies due to
charge-density fluctuations and compare those results with various other second- and
third-order interaction energies.
4.1.1 Interaction Energy

Consider two molecules A and B which are well-separated, so that the electronic
overlap between the two molecules is essentially negligible. The energy of these two
weakly interacting systems is equal to the sum of the energy of molecule A (E A ), the
energy of molecule B (EB ), and the energy due to the interaction of A with B (EA+B ),
as shown in equation (1).

EAB=EA+EB+EA+B. (1)
We obtain the operator for the interaction between A and B by considering the interaction

between two classical charge distributions. If the two molecules are represented as

collections of positive and negative point charges, the classical electrostatic interaction

98

energy is E A+B = qud)? where in is the charge of the ith particle in system A and
i

(DiB is the potential at the ith particle due to molecule B. If the potential due to B is
expanded around an origin in A, the electrostatic interaction energy takes the form of

equation (2), where QA , 11A , and @A are the overall charge, dipole, and quadrupole

moment of system A, and FE and F33 are the field and field gradient generated by

system B. Each Greek subscript refers to the x, y, or 2 component of the electric moment

or field quantity.
1
ENE = QAoB 413 F0? 703;, F53 -... . (2)

In equation (2), we follow the Einstein Convention of Summations over Greek indices,

such that equation (2) equals equation (3).

EA+B =QA “’8 'H? F)? ‘11? F? Til? F28

1 B 1 A B 1 A B 1 1

-399, F1, 70,, F1, 70,, Fx, 709,, F’yE 769, F}; (3)
lABlABlAF-IBIAFB

—§®yszz _ngx Féx ‘3'sz zy -§®zz zz —'" °

The dipole and field are tensors of rank one (or vectors) described by equations (4) and
(5).
u=ltXX+uyy+uzL (4)
F=Fxx+Fyy+Fzz. (5)
The quadrupole and field gradient are tensors of rank two and can be represented in

matrix format as seen in equations (6) and (7).

99

®xx ®xy 6on

o: (9,, oyy 0y, . (6)
92x 92y 922
Fiat Pity 1‘12

F’= 1:}, FW By, . (7)
Fix Féy F22

R is a vector, between the origin in A and the origin in B, which points from B to
A. From the multipole expansion, equations (8), (9), and (10) give the potential, field,

and field gradient due to B, respectively.

 

B QB B 1 B
<0 (R)=—R—- up, +§Tafl 0043+... . (8)
a 1
1135811, ¢B(R)=‘Ta QB+Tap ug-Srw, 037+... . (9)
FB. 32 B _ B B 1 9B 10
tits-"513312739 (R)-' aBQ +1111” 117 "5 days yo+~~ - ( )

Equations (11) and (12) are explicit definitions of the first and second rank T tensors.

 

 

a 1 -R
T ____ a 11
a 812,12 R3 ( )
32 1 3RaRB 0,3112 lif a = B
“‘3 akaaRBR R5 were “5 0ifa¢B ( )

Assuming that systems A and B have no net charge, the leading terms of the interaction

energy depend on the dipole and quadrupole moments of both systems as shown in

equation ( 13).

100

A B 1 A B
EA+B =‘Pa Tali PB +3111: Tom 907
(13)
1 A B 1 A B
—39013 Tam [.17 +§®al3T01375 975 +... .
Using the dipole propagators or T tensors, we can determine the distance dependence of

the interaction energy on a particular interaction. For example, - 115‘ Tab 11? is the energy

of interaction between the dipoles on systems A and B. As the distance, R, between A

. . . . 1 .
and B is increased, the interactron energy decreases as —3— . For the second and thrrd
R

terms, the dipole-quadrupole and quadrupole-dipole interactions, the energy decreases as

1
——4-. Therefore, the long-range interaction energy between the two systems depends
R

more on the leading dipole-dipole term than on the dipole—quadrupole or quadrupole-

quadrupole terms.

Next, we construct a Hamiltonian operator for the two interacting systems, H AB ,
for use in the time-independent Schrbdinger equation, HAB‘P = E‘I’. In equation (14),
the unperturbed Hamiltonians for each of the two molecules, HA and H3 , are summed
into the unperturbed Hamiltonian H0 , and the portion of the Hamiltonian from which the
interaction energy originates (\A/AB) is treated as a perturbation. The form of the operator

VAB is determined from the classical electrostatic interaction energy, given in equation

(13).

101

fiABzfiA+fiB
M—7—J
H0
A B 1 A B
“Pa 1118 118 +31% TaB'y 907 (14)
+ .
1 A B 1 A B
"3311011100 117 +6909 Tums (975 +...

 

Y
A

VAB
In this work, we focus on the dipole-dipole interactions, which give the leading
contribution to the interaction energy at each order. Corrections to the first- and second-
order interaction energies due to higher multipolar effects have been given by

Buckingham3, Stogryn“, Piecuch6‘7’8'9 etc. Thus we take

HARM: rig—a: 11,, pg [9) =(r10+9,) |‘P)=EAB|‘P). (15)
fi——J
V1

4.1.2 Perturbation Theory

Rayleigh-Schrbdinger perturbation theory is based on the premise that, for a
particular system for which the eigenfunctions and eigenvalues of the time-independent
Schr6dinger equation are unknown, there exists a second system, similar to the first, for
which the eigenfunctions and eigenvalues are known. Using the eigenvalues and
eigenfunctions of the second system we may approximate the eigenfunctions and
eigenvalues of the first system. For example, we need to solve equation (15), but the
eigenfunctions and eigenvalues are unknown; however, we do know the eigenvalues and

eigenfunctions of equation (16),

I“101‘1’01 = Eol‘l’o)~ (16)

102

where ‘110 = |0) = ‘1’8 ‘1’}; = |0AOB) ,which is the product of the unperturbed ground state

wavefunctions of system A and system B, and where E0 = BOA + E8 is the sum of the

unperturbed ground state energies of systems A and B. Electron exchange between A
and B is neglected in the zeroth-order wavefunction.

Introducing an ordering parameter is, which we will eventually set equal to one,
we can expand the exact eigenfunctions and eigenvalues from equation (15) as series in A,

yielding equations (17) and (18).

EAB 43941331) +1.2 E32) +23 133” +... . (17)
1P=|irg°1>+x|rgll>+xz|w321>+x3|wg31>+... . (18)

Substituting back into equation (15), we derive equation (19).
(1‘10 +1.01) 011130)) + 1|r311>+ 1.411139%”):

(19)
(1339+ 1. E81) + 1.2 E82) +...) OPS») + AIMS”) + 1.2 I‘ll?) + ...)

If we expand the expressions and equate terms with equal powers of X“, we obtain

equations (20), (21), and (22):

from) = Eg°)|0) , for n = 0. (20)
frolir91>+vl|0)=Eg)|0)+Bg°)|~Pg)>,torn: 1. (21)
11481821) + 0411181) = 131,2) |0) + E91|wg)>+ 5304413)) , for n = 2. (22)

Using the condition that the mth correction to the unperturbed wavefunction is a sum of

the excited states of the unperturbed system, Tam) = Zefimhl’fio) where k is the kth
k¢0

103

excited state (so k #0), and using the result that the excited states are orthogonal to the

unperturbed ground-state wavefunction, (0| ‘11:?) = (0| m) = 0 , for m > 0, we can take the

scalar product of equations (20), (21), and (22) with W60) 2 | 0). From these scalar

product equations, we obtain equations (23), (24), (25) and (26).

 

 

lag» = E0 . (23)
Bf,” =(0|\‘71|0) . (24)
2 0 v k k v 0
k¢0 E0 “13k
E83): 2 <018|k><k181m><mlalo>,wm,=,1_Ep (,6,

k,m¢0 (E30) -E§?’)(EE,°’ —E$2’)
Equation (23) is an equality involving the unperturbed energy (in this case, the energies
of the noninteracting systems A and B). Equation (24) gives the first-order energy
correction, which yields the permanent dipole-dipole interaction, while equation (25) and
equation (26) give the second- and third-order energy corrections, respectively.

4.2 Second-Order Energy Correction

By defining an operator G known as the reduced resolvent,

é =(1—P0Xfio-EOT10-PO): XML. (27)
k¢O(E1< E0)

where P0 = IOXOI , the second-order energy correction takes form in equation (28).

B32) =_(o|9,<‘39,|o). (28)
For a pair of molecules A and B, the G operator is equal to a sum of three operators.

G = (GA +GB +GA$B) such that

104

GA = ZIJAOBXJAOBI. (29)
j¢0 EA E0

éB_ _ Z IOAmBXOgmBI. (30)
m¢0 EB m-EO

GAoB = IJAmBXJAmBl . (31)
..Eoer—Esuei—Ea

GA is the reduced resolvent for excitations of system A only, GB is the reduced
resolvent for excitations of system B only, and GAEBB is the reduced resolvent for
excitations in both system A and B. Substituting v, = 419 Tag it? into equation (28)
results in equation (32):
502) = (OAOBI 1‘ 113 Tats 11113) (GA +GB "LG/ma) 1 15? T7811 “63) 1013013) (32)
Expanding equation (32) results in equation (33).
Em (0 0,. (4:: a, 1131G1A-11A7T ti) IOAOB)

+<°A°Bl ('fiaT "11.511 “)9 B1— 5‘7 T76 51?) lOAOB> (33)

+(0A03| (413 T “11‘;
For simplicity, equation (33) is broken into portions of the second-order energy which

depend on excitations only in A (ESE) ), excitations only in B (E?) ), and excitations in
both A and B (1313298 ). The second-order energy correction now has the form in

equation (34), where Bf) is given in equation (35), egg) is given in equation (36) and

1353291313 given in equation (37).

105

B32) = Bf) + BS) + 13ng . (34)

Egg-(0410131 (“113‘ Tali 111331 9A (‘11? T76 8811014013} (35)
El.” =-<0AOB| (...: To a?) 93 (4145401003). (36>
E2298 ="<0A0B1 (”113 T913 £11133) GAeB 1" 1111A T75 11153) 10A0B>- (37)

Equation (38) for E?) which simplifies to equation (39), where 113013 the static ground-

state dipole moment of system B and 13$ is the energy of the mth-excited state of

molecule A.

 

 

 

E(2) _ _ (OAOB 1118 Tori 111133 ImAOBXmAOB 111$ Tye 918310.408) (38)
A — 2 EA _EA '
m¢0 m 0
0 “A m m ”A 0
5(3):- a5 T75 113011830 2 ( AIPal [AX 211111 A). (39)
m¢0 Em ‘50
If we expand equation (39) into a sum of two terms, it becomes equation (40).

l 0 11" m m 11" 0

E13)”; apTysuiiouis’OZ ( Al al :11 2111A)
m¢0 Em ’E0 (40)

 

1 B0 B0 (GA 1113 1mA><mA 11“? 10A)
-- T 8 11 #8 Z
2 “B Y B m¢0 158 "EOA

Since the Greek indices are dummy variables, by interchanging the labels [3 and 6 and

simultaneously interchanging a and y, we convert equation (40) into equation (41).

 

Egg.) = 'lTaB T78 1111330 11830 X (0A 1113‘ lmAXmAlfii 10A)
2

EA—E
m¢0 m 0 (41)

 

1 0 ftAm m 0A0
__ YsTaflugougoz < Al 71 2X A101 A).
m¢0 Em"EO

106

Now the sum over states in equation (41) has the form of the static polarizability.
If we use the definition of the static polarizability of system A given in equation (42),

equation (41) becomes equation (43).

 

A _ 2 (DA lfiélmAXmAlpy10A)+<OAIH¢1mA><mAIfia10A) . (42)

“W‘... Bi-EO Bi—Es

Eff) = __;_ (,5 T75 pf,” 11?” erg. (43)
Following the same procedure, we can expand and rearrange Eg) to obtain equation
(44).

1‘33) =-%T60Tv 81180 1190 “(I338 (44)
The final term in the second-order interaction energy is given in equation (45).

(2)
EA€BB — ‘TaBTrfi

 

X (0A Ilia ImAXOB ll‘lli IkBXmA luv IOAXkB 11‘510‘3) . (45)

m,k¢0 (Em—E0)+(Ek 'EO)

Using the integral relation”), i: If du 2 a * b and

3+1? a +u2 b2+u2

 

a=%,%

, _ , we can rearrange the denominator of equation (45) to obtain
32 +u2 a+lu a-lu

equation (46), where 133,0 = 133, 4:3 and B20 = BB 438.

 

 

if d0) + 1 A 1 + A l . (46)
Ek_—0 +1Emo= 0° Ekol ‘1’ 10h EkO -1(Dh Emo +1(.0h Emo - 1(1)”!
Substituting the right Side of equation (46) back into equation (45) and expanding, we

obtain equation (47).

107

(2) _
EAEBB ” GBTY

Lg (1902 Z<OA alfiél mAXOA 1111131 kBXmA 11111071st 111153103) (47)

k,m¢0

 

 

x A l + A l x B 1 + B l .
(Emo + imh) (EmO “1(1),” (BRO +1012) (EkO —1(Dh)
Equation (47) can be rearranged into equation (48).

(2)
EAEBB ‘ ’ aBTrfi _

1 (<oiltilmixmilailon ‘
(139,0 +ituh)
""0 + (GA 11131 mAXmA 1111104.)
A .
oo - h
xLodw K (13:10 10) ) ) B
(01311113 IkBXkB lfia IOB)
B .
X Z EEBkO +1(!)h) AB
”0 +(08111011‘3Xk811161081
\ K (13130—10011) U

 

 

(48)

 

 

\ .

 

 

 

 

 

 

From Orr and Ward”, the frequency-dependent polarizability of A is given by

equation (49).

.A «A «A eA
(1976”): X (0A 111a 1m: )(m 1117 10A) + (0111117 1’“: )(mA 111a 10A)
m¢0 Em0 -h(1) EmO +h0.)

 

(49)

The dipole operators are Herrnitian. If IOA) ,|OB) ,lmA) , and IkA) are real eigenstates,

then (0A Il’iélmA) = (mA IfiQIOA) and (081111331113) = (k3 |fig|03> . Using these
equalities and the definition of the frequency-dependent polarizability from equation

(49), equation (48) becomes equation (50).

h co
E93098 = _ 013111757“ [00 any A(1w)aB mambo). (50)

108

Summing equations (39), (40), and (46) gives the second-order energy correction as a
function of the static polarizabilities of A and B and the imaginary-frequency dependent

polarizability of A and B, as seen in equation (51).
2 1
131)- )- __ 20113 T780400 “$0 (1:335 “([1530 “1530 (127+ 27: _J::amy (im)a35(ico)dto). (51)

This result has been obtained previously by Buckingham3, Dalgarno'z, Longuet-
Higgins'3‘14'15, Stogryn4‘5, Piecuchm‘g‘9 , etc.
4.3 Third-Order Correction to the Energy

From equation (26), after substituting in the reduced resolvent, we see that the

third-order correction to the energy takes the form of equation (52).

B331 = 2(0|\‘q(oA +03 +GA$B)V (GA +03 +GA$B)v,|0). (52)
k,m¢0

The V operator is defined by V = \71 — E81). If we substitute the perturbation into the V

operator we have equation (53).

ill-BE,” =41:as05+(0A0slainsngloios). (53)
We can rearrange equation (53) into equation (54), where if = (if: — p.90.

Vi -E( ) =-FuT (18 111133 413° 11061—11133 -F: Top 1111330- (54)
If equation (54) is substituted into equation (52) and the result is expanded, a 27-term

expression is generated. Fortunately, not all 27 terms are non-zero. For example, the left

hand side of equation (55) equals zero, as shown below.
(0h?l GA 11:,“ Tag 11,? GA mo) = 0. (55)

Expanding equation (55), we obtain equation (56).

109

(0|)?l GA a: Tag a}? GA v1|0) =

.. "1 OBXmAOBl—A —B lnAOBXnAOBI ~
(voalvlzl A u T p Z VIIOAOB)-
m¢0 (EQO'ESD) a QB Bil-to (EQO’Egob

(56)

 

 

Equation (56) then simplifies to equation (57).
(6|v, GA a: Tag EEG/1‘ (7116):

T 13 Z (OAOB lvllmAOBXmA IEQIDAXOB IHIIBBIOBXDAOB IVIIOAOB) (57)
a I
m.n¢0 (E:1 —E8 )(E: _E8)

 

Now, because the (OB 1E1? [013) = 0as shown in equation (58), the term given on the left in
equation (55) does not contribute to the third-order interaction energy in equation (52).
(OB 11111331013) =(03 IfiglOB>-<OB 111113301013) = “(To ““1130 = 0- (58)
Similarly the term on the left in equation (59) also equals zero, as shown next.
(0|\‘7l GA :1: Tag pg” GAEBB v1|0)= 0. (59)
Expanding equation (59) and simplifying we obtain equation (60).

(0|‘71 GA :13 Tag 1130 GAEBB 9,16) =

T X (0,103 IvtlonBXmA lit'mnAxoB ngolkBXnAkB |V1|0 A08) (60)
“B k,m,n¢0 (E31 “E8 ”(BE ‘EgH-(E: —E6‘ )]

 

The matrix element (0A 111113301 k3) equals zero, as Shown in equation (61); therefore, the

term on the left in equation (59) does not contribute to the third-order energy.
(0131111530013)="EO(OA|RB)=0- (61)

Of the 27 terms, 12 equal zero and those terms are listed in equations (62) through (73).

(0|)?1 (3A a: T043 a"? GA 9,16) = 0 (0|\71GB it: Tag if} GB v1|0) = 0 (62-63)

110

(0|vl G Aug?" T ,3 ”133GB v1|0) =0 (0|v1GBp3Tg p5 °GA 9,16) = 0 (64-65)

(0|\71GB 113° T043 a}; GA i740) = 0 (0|\"/l GA a: Tag ago GB v1|0) = 0 (66—67)
(0|v1GApA0 T B a}; GA v1|0) =0 (0|vl GB Ti: Tag 1130 GB v1|0) = 0 (68-69)

(0Wl GA it: Tag pf,” GA€BB v1|0) = 0 (0|\"/1GB p.90 Tag a}? GA6913 v1|0)= 0 (70-71)
(0|i"/1GA<19B 1:: Tag 1130 GA v1|0) = 0 (0|v1GAGBBpA0TaB-pBBGB v1|0) = 00273)
The remaining 15 nonzero terms break into one of four categories based on their

dependence on the static dipole moment of system A and system B. Terms of order p in

pm and order q in ,uBO are classified as p-q terms in the discussion below.

4.3.1 Third-Order Terms in the Static Dipole of A or B
Listed in equations (74) and (75) are the 3-0 and 0-3 terms respectively. If we

expand equation (74), we obtain equation (76).

T3_ 0- (=0|\‘/1GB 11010111018118 BGB v1|0). (74)
T0_ 3—(--O|\71GA u: Tag uEOGA v1|0). (75)

(GIT/168 11.30 T1113 it? GB VHO) =
2 (voB 11"? T76 1L81°41‘13on kB liloO TaB up BIOAJBXOAJB lusT so noIOAOB) (76)
k,j¢0 (1313-13003;3 “150)

Equation (76) simplifies to equation (77).

(001103 1190188 it? GB Vilo)=
Oflkkujjuo (77)
TagT75T8¢p30p¢0ué10<Bl 8| BX Bl El BXBI 3| 3).
(Bk —EO)(EjB —E0)

111

We can now see that the term represented by equation (74) depends on the permanent
dipole of system A to order 3 and depends on the permanent dipole of system B to order
0.

We know that the static hyperpolarizability of system B has the form given in

equation (78)”.

(03 11111331 kB><kB Ifigl 113 >08 1113103)
(B? $1003}3 ~13?)
+ (OB 1111331 kBXkB 11131 JBWB 1111331013)
(Bi? 43803}3 -Er‘i)
+ (OB 1111153 I kBXkB 1111133 IJB><JB 1113 108)
(Bi? -Et‘i)(i~:}3 -E8)
(03 11151 kBXkB 1331 JB><JB 1111133103)
(Bi? —E8)(E? 458)
+ (OB 11131 kBXkB 1511331113108 1111331013)
(E? 4380233 438)
+ (OB 11131 kBXkB IF? 113108 1111331013) .
(E? 4:38 XE}3 438 ) (78)

 

B
111159 =

 

 

+

 

 

 

Reassigning the Greek indices in a similar manner to the reassignment done with the

second-order energy correction, equation (77) can be rearranged into equation (79).

112

(0191 GB 1130 Tag n53 GB vl|0) =

((03 111133 IkBXkB lit? lieXl's 113 lots)
(E? 438x13}3 -E8)

+ (“E 111131 kBXkB 11731 131013 1112133103)

(Bi? 458x13}3 —E8)

(03 lfigl kgkaBlfili 131310: 1115103)

1 A0 A0 A0 (Bk "BO )(E1 ”130)

31‘“ T“ T” ““ ”1 1“ kzo +(orlailkrsxksluilaxislailos)
(BE -Et’i)(13}3 433)

+ (OB lfigl kBXkB 11111331 jers lfiEIOB)
(El? $8105? 458')

+ (OB 11131 kBXkB 131113ij 11111103)

( (BE-E8)(E?-E3) )'

\

 

 

+

 

 

 

 

(79)

 

 

Now the matrix element on the left in equation (74) has the form given in equation (80).
.. B A0 —B B . 1 A0 A0 A0 B
<0|V1 G Pa Tali "[3 G V110) = ‘6'1111 1411 118 T08 T78 Tarp 1105(1) ' (80)

The second term in this classification is given in equation (75). By following the same
procedure, we can Show that the matrix elements on the left in equation (75) depend on
the permanent dipole of system B to order 3 and depend on the permanent dipole of
system A to order 0. If we expand equation (75), we get equation (81).

" A -A BO A "
T0—3 = (OlvlG 1111 T613 110 G V110)

(0A IfiyAlmAXmA 1391“».an 11191041) (81)

 

_ BO BO BO
4“" T15 T81 "1 "5 "‘1 21:0 (EA-E8)(EA—E8)
m," m n

Then as in equation (77), we can rearrange indices and represent equation (75) as

equation (82).

113

. A—A B0 A“ 1 B0 B0 B0A
(1)1le 1111 TaB PB G V1|O>=ETGB T78 Tarp 118 118 1111) Bayer (82)

The net contribution to the energy from this term is —%Ta5 T75 T8“, “[1330 pgo 113013373 .

From equations (80) and (82), the 3-0 and 0-3 terms refer to a classical interaction
between the two systems, where system A generates a static field, which polarizes system
B through the B-hyperpolarizability of B, and vice versa. Figure 1 shows the 3—0 terms in

a graphical manner.

 

Figure 1: Representation of the 3-0 term.
Images in this dissertation are presented in color.

In the figure, the three solid black lines represent the field generated by A. The dotted
green lines represent the polarization of system B through its B-hyperpolarizability.
There has been previous study of this contribution to the third-order energy by Stogryn‘1‘5
and Piecuch6'7’8'9.
4.3.2 First-Order Terms in the Dipoles of Both A and B

The next category of nonzero terms in the third-order energy correction are the l-
1 terms, two terms that are first-order in the ground-state dipoles of both A and B. Listed

in equations (83) to (88) are those terms.

114

TH, = (0M GA a: Tag p133 GB v1|0). (83)

THJ, = (0071 GB 11;,“ Tag it"? GA vl|0). (84)
TH", =(0|\7l GA p30 Tag a}? GA6198 9.16). (85)
T1_1,d=—(0|V1GA€BB uAO T g pg’ GA v1|0). (86)
T1- 1, (=0|v1GBu§T,,g pg” GATE v1|0). (87)
T1_ 1, =(0|v1GA$Bn:,ATag pg 0GB v1|0). (88)

We begin with equation (83). If we expand the term on the left and simplify, we obtain
equation (89).
(0|)?l GA a: Tag 11,? GB v1|0)=

GummuooukkpBo (89)
T08 TyfiTgrpllerpBo Z (A171 AX AI (11 Ax BI 01 BX 3' I B).
m,k¢0 (Em’EO )(Ek "Eo)

If we expand equation (84) and simplify, we obtain equation (90).
(0|vl GB a: Tag a}? GA v1|0) =

0 m m 0 0 k k 0 (90)
TGBTYSTwllyAOPgo 2 (AIM? I AX Al"? I AX B11151 31113111131 13).
m,k¢0 (Em -E0 )(Ek 1‘30)

In equation (90), if we interchange y and e and Simultaneously interchange 5 and (p ,
then equation (90) becomes equation (91).
(0|vl GB a: Tag fig’ GA v1|0) =

T A0 B0 2 (OAIFQ ImAXmAh‘tA |0A)(OB|p3 IkBXkBIPBIOB) (91)

Tali Tao 78118 118
m,k¢0 (Em-E0 )(Ek BO)

115

Then the Tm and T1“, terms both depend on the 8 component of the ground-state dipole
of system A and the 8 component of the ground-state dipole of system B. Taking
equation (85), after expanding and simplifying, we obtain equation (92).

(0|?!1 GA “:0 Tat; LII? GA€913 Odo) =

(OAlaé‘ImAxmAIu9I0A><oBIaEIkakBIamoB) (92>

 

T T TsfiAougo
“‘3 8'" 7 °‘ “2:0 (E8—E8)<E$-E8+EE-E8>

In equation (92), interchanging e and a while simultaneously interchanging Band (p
yields equation (93).
(0M GA (130 T04; LIE GAEBB \‘Il|o) =

T8 Tan T 5 {12° M21530 2 (0A lfiyAlmAXmA IHQIOAXOB IfiSIkBXkB lfiglon). (93)
(p v m,k¢0 (E3, —E3)(E§; -53 +55 _Eg)

 

Similarly, if we expand and simplify equation (87), we obtain equation (94).
(0h?l GB n: T043 (130 GA613 Odo) =

Ta TQBngAopgo Z (OA|fi9|mA><mAlufil0A)(OBlfiEIRBXkBmmoB) (94)
(P 7 Y m.k¢0 (BE-E8)(E$-E8+EE—E8)

 

In equation (94), we interchange e and 7 while simultaneously interchanging B and 6;
we also assume that the dipole matrix elements are real. This yields equation (95).
(OK/1 GB it: Tell; pg“ GA63B \‘Il|o) =

(OA Ifié‘l mAXmA IPQIOAXOB Ifim 193ka IfigIOB) (95)

 

T T T “A0 30 2:
78 043 sous P5
m,k¢0 (131k3 -E8 )(EQ 453‘ +13? —E8 )

If we add equations (93) and (95), collect like terms, and simplify, we obtain equation

(96).

116

(o|\"/1GB a: Tag p30 GAEBB \‘/l|o)+(o|\7l GA “3‘0 TaB if? GA€913 Vl|o) =

. Ao BO (0A Ill? I mAXmA I“: IOAXOB Ifig I kaxka lfig I013) (96)
T18 Tali Tao us #5 2 B B A A ,
m,k¢0 (Bk 'Eo )(Em -EO )

 

Expanding and simplifying equation (86) results in equation (97).
(0W1 GA$13 “3° T0,, a}? GA \‘11|0)=

(GA lfii‘l mA XmA IHQIOAXOB Iii? l 193st hit? IOB> (97)

 

BO A0
T TaT u u
A” A *° °‘ 33 (E3—E8)<E$—E8+EE—E8>

To rearrange equation (97) so that it depends on the 8-component of the permanent dipole
of system A and the 8-component of the permanent dipole of system B, we interchange (p
with 8 and a with 8; we also assume that the dipole matrix elements are real. This gives
equation (98).

(oh?1 GA61913 “:0 TaB a}? GA \71|0) =

(0A I99 l mA XmA luyAIOAXOB liq};3 I kBXkB lamog) (98)

 

T T51" #:1530qu
“B Y 8" 8 ago (E9.—E8><E$—E8+EE-E8>

Expanding equation (88) and simplifying, we obtain equation (99).

(o|\"/1GA‘BB a: Tap (150 GB \71|0) =

B0 A0 (OA In? I mAXmA I“: IOAXOB Iii? I kexkn Ifig ‘03) (99)
TysTapTwup ”5 X B B A A B B
k,m¢0 (Bk -Eo )(Em -E0 +Ek —E0)

 

O

Rearranging Greek indices so that equation (99) depends on the s-component of the
permanent dipole of system A and the 8-component of the permanent dipole of system B
and assuming that the dipole matrix elements are real, we obtain equation (100).

(0|\7l GA6913 ii: Tap “30 GB \71|0) =

T a T a T .80 “A0 z (”A “‘9'me ITAIOAXOB lfifi lka><ke |fi$l03> W»
a 8 a
Y ‘P k,m¢0 (BE —E8>(E:. 423 +13}: 4355

 

117

We combine equation (100) with equation (98), collect like terms, and obtain equation
(101).
(0|{11GA63B E8 Tag (130 GB \71|0)+(0|\71 GAEBB ”A“ Tag 11'}? GA (Mo) =

B0 A0 (OA '1“: ImAXmA I”? IOAXOB 1111133 IkBXkB 1831013) (101)
T15 Tali Tao P5 118 2 B B A A
k.m¢0 (Ek "Eo )(Em -EO )

 

If we add equations (101), (96), (89), (91) we obtain equation (102).

Tl—l,a +T1-1,b +T1—1,c +T1-1,d +T1—1,e +T1-1,f

BO A0
= T78 TaB Tap P5 P8

((0,, |fi¢lmA)(mA lflc'iIOAXOB 1811331 kBXRB 1831013) 1
(El? -BR)(E$ 438)
+ (GA 11191 mAXmA lfli‘lOAXOB 1831 kBka 18153108)
X Z (EE-ERXEQ-Eé‘)
k.m¢0 + (0A 1117A I mAXmA I“: M X013 1113 lkBXkB 113133 I013)
(Elk3 -E(1i)(E$ 433‘) ' (102)
+ (0A lfiél mAXmA IHflOAXOB W? l kBXkB “131013)
\ (BE-ngEQ—Eé) 2

 

 

 

 

 

 

The product of two polarizabilities, one on A and one on B, is shown in equation (103).

.. . . )
(<0A InilmAxmA Infill/8X03 luEIkBXkB Ingloal
(BE -E8)(E$ 438)

+ (0A lfifi‘lmAXmA IMO/axon lfigl kBXkB lfiEIOB)
A Gap— 2 (BE-E8)(E$—EOA)
Ya - A x x

k.m¢0 + (0A I“? ImAXmA I“: lOAXOB IPEIRBXRB but}:3 1013)

(BE -E8)(E3 -E6‘)

+ (0A 189 ImAXmA W IOAXOB 1% lkBXkB 1831013)
( (EE-EEXEQ-Eé‘) )

 

 

a . (103)

 

 

 

 

Thus the sum of all of the 1-1 terms is given by equation (104).

118

Tl-l.a +T1-1,b +T1-1,c +T1-1,d +T1—1,e +T1-1.f

B0 A0 A B
= 78 TaB Tarp 115 148 “ya “Bo

(104)

We can further simplify equation (104). As shown in equation (105), the dipole on A

produces a field that acts on B.

A0 B A B0 A B A BO
148 Tao “Bo TaB aya T75 118 = F(p “Bo TaB aya T76 118 °
Ev—d
FA
‘1’

That field then induces a dipole in B as seen from equation (106).
A B A B0 B A BO
F(p “Bo TaB “ya T78 P5 = 113 TaB “ya T78 1‘8 °
H—J
B
”B
The induced dipole in B then generates a field that acts on A.
B A BO B A BO
PB Tail “ya T75 148 = Fa “ya T76 145 -
k—V—J
F3?
The field from the induced dipole in B induces a dipole in A.

B
Fa ayaTy8 H80 =uy Tyaltao -
EH

w?"

The dipole induced in A then interacts with the permanent dipole of B.

B0 A B0
11? T78 “6:135“
BF’

A
PS

This explanation of the 1-1 terms is represented in Figure 2.

119

(105)

(106)

(107)

(108)

(109)

 

 

Figure 2: Representation of the 1-1 terms.
Images in this dissertation are presented in color.

The solid black line from A to B represents the field produced by the permanent
dipole of A. That field induces a dipole in B, via the a-polarizability of B, as shown in
dotted red lines. This induced dipole of B generates a field represented by the solid red
line from B to A. This generated field influences A through the polarizability of A, as
shown by dotted red lines. In turn, the induced dipole of A generates a reaction field
which then acts back on B as shown by a solid red line from A to B. The same induction

mechanism operates beginning with the field from the permanent dipole of B. The net

contribution to the energy is ‘TaB Tyfi Tw “£0 011133,p 01¢}, (11,30. This contribution to the

third-order energy has been analyzed by Stogryn“'5 and Piecuch6‘7‘8‘9.

4.3.3 First-Order Terms in the Static Dipole of System A or B, But Not Both
The next category of nonzero terms includes the 1—0 and 0-1 terms. Listed in

equations (110)-(112) are the 1-0 terms, which depend on the dipole of A to order 1 and

the dipole of B to order 0.
T1—0,a =<0|V1GB if: TaB if); GAeB VIIO). (110)
Ti—o,b =(0|\71GAeB if? Tap if? GB Vllo). (111)

120

TH),C = (01v, GA61913 1190 TaB BE GAGBB 9,10). (112)
If we expand and simplify equation (110), we obtain equation (113).

" B —A —B A®B " AO
Tl—O,a =<OIV1 G 14a TaB “B G V1I0> = TaB T78 Tag) 147

x<°A|fi9|mAXmAlfi9|0A><OBlfi§|13>03IHEIkBXkBmmoB) (113)
<13}3 —E8><E$ -E6‘ +13}? —E6‘)

 

We interchange e and 7 while simultaneously interchanging q) and 8. Then equation
(113) becomes equation (114), which depends on the e-component of the permanent
dipole of system A (as above).
“ B -A —B A®B " A0
(OIVl G “a TaB “B G VlIO) = TaB T76 Teq) 148

(GAInélmAXmAIatloxxos1113;11me88888311113103) (114)
(E? -E3><E$ -E6‘ +El? —E6‘>

 

We know from equation (46) that we can rewrite equation (114) as equation (115), given

that F is defined by equation (116).

2 F

B B A A B B =
m,j,k¢0(Ej -Eo)(Em"Eo +Ek ‘50)

 

 

 

 

 

 

 

 

(
FX B1 B I
(Ej ‘50)
f 1 1 (115)
h
_ dw X + .
41: I: mgw ((EQ-Bg‘nhw) (EQ-BA—ith
I 1 1
X B B - + B B -
K \(Ek -E0 +1710) (Ek -E0 -lh(0) )

F = Tab T76 Tam “30 (DA I89 I mAXmA I11? IOA)

(116)
><(03|1‘13|1'3X1'B I31? I kBXkB I11? I°B>~

121

Expanding the integrand in equation (116) and then taking the first term T1,. 15 of the

result we obtain equation (117).

A0
T1,115 = (18 T75 Tarp P8

2 (0A |fi9lmAXmA IfiYAI0A><OB IT‘TEIJBXJ'B IiIEflkakB |fi5 |oB) (117)
m.j.k¢0 (E? 433 ‘1' 173(0ng 438‘ + 171(1))(Eg3 — E8)

 

Since we have assumed that the eigenstates are real, we can rearrange the numerator to

convert equation (117) into equation (118) below. First we interchange the summation
indices j and k. Then we replace (OB IfigIkakB |fi€| jBXjB [(1503) by
831181180388888311131038 and we replace <oAIT91mA><mA 1191011) by
(0A IfiyAImAXmA |a£|oA). This gives equation (118).

TU 15 = T013 T75 Tarp ”SAG
m.j.k¢o (Efi3 -E8 +ihm)(E,A, 433 +ihm)(EE _Eg)

 

We can transform each term in the integrand of equation (115), so that the integrand in
equation (115) is a sum of the four terms T1115, T2,; 15, T1115, and T4,)”, given in

equations (1 18)-( 121).

T211 15 : T013 T75 Tetp 11:0
x 2 (0A I113 ImAXmA IfiyA IOAXOB I11? IJ'BXJB lfi§|k3)(k3 IT‘SIOB) (119)
m,j,k¢0 (E? 438 + ”160ng -E8 -ihco)(EE — fig)

 

A0
T3,115 = TaB T75 Tao 148

2 (GA IfivAImAXmA IBQIOAXOB |fi3hn><jg IEEIkBXkB IfigIOB) (120)
m.j.k¢0 (E? -E8 -ihm)(E$ —E0A +lhw)(EjB _ 138)

 

122

T4,115 = TaB T78 Tarp 1190
2 (GA I113 lmAXmA I9? IOAXOB I133 IjB><jB I81? IkBXkB lfiis3 I03) (121)
m,j,k¢0 (Bi.3 —E8 —ihw)(E$ 433‘ —ihm)(13}3 438)

 

The next term comes from equation (111). Expanding and simplifying, we obtain
equation (122)
A A B _ _ A
(DIV! G ® “(1A TaB 111133 GB V1IO> =TaB T75 Tap 14er

x Z (OAIfiYAImAXmAIBQIOAXOBIfiglkBXkBIfigljBXjBlfig‘OB) (122)
"‘J'kio (E? -E8><E3-E8+E£—E3)

 

Using equation (122) and following the same procedure as was used for equation (115),
we can re-express the denominator in equation (122) as an integral over frequencies,
where the integrand is a sum of four terms with different denominators. With appropriate
relabelin g of the matrix elements in the numerator, we obtain the following four

equations for the terms generated from equation (122).

A0
T1,122 = Tap Tye; T38 its

2 (0A I11? ImAXmA I113 IOAXOB I513 IJBXJB IE? I kBXkB Ifip I013) (123)
j,k,m,.o (Bi? 438 +ihm)(E$ —1~:f,"+iiim)(E§3 —E0A )

 

A0
T2,122 = (18 T75 T88 11;,

x z (”A lfié‘lmAXmA Ifii‘IOAXoB 111311003 ITEITBXTB 1a, 10,). 024)
j,k,m¢O (E? 438 +ihm)(E$ -E6‘ -ih0))(E? -E6‘)

 

A0
T3.122 = TaB T78 Tarp us

2 (OAlfli‘lmAXmAlfié‘loixoalfiglBXBIfifIkBXkBlfiroB) (125)
13me (Elf9 - E3 )(Ef; - 56‘ armada}3 — 56‘ —iha))

 

123

A0
T4,122 = Tat} T15 Tao 148

Z (“A If}; lmA )(mA W‘ IOA X08 Iflgl jB )(J'B IE? Ike )(ks I105 IO 3). (126)
j,,,,,,,,.0 (E;3 — 55x15}; — 55‘ —irzw)(Ef - 156‘ —iha))

 

The final l-O term comes from equation (112). Expanding and simplifying equation
(112), we obtain equation (127).
(0'01 GAGBB “:0 1013 El? GA$B VlIO> = T06 T78 T811) ”90

2 (0A IfivA I mA><mA I11? IOAXOB IfigljB ><1B IE1]; IkBXkB I113 I013) (127)
j,k,m¢0 (E3. 438 +Ei? —E8)<E$ -E8 +E}3 -E8)

 

We know from the work of Craig and Thirunamachandranl6 that we can represent the
denominator of equation (127) by use of equation (128).

1
(5:3. -E6‘ +15]? —E€)(E:2 —Eé‘ +Ef—Eé‘)

 

 

 

I 1 + 1 I
(5",: - 156‘ +17%») (Eff, — 155‘ —ihw)
I d‘" B B B B
0° (Ek -E0 +ih0))(Ej —E0 +1710)
1
(El? — E8 4118ng 4:59 —ihw) ,

(128)

 

+

 

 

 

 

 

\ K /
By reassigning Greek indices so that the result will depend on the e-component of the
permanent dipole moment of A, we obtain equation (129)

" A$B A0 —B A$B " A0
(OIVl G “a 103 1113 G V1I0> = T08 T75 Tao 118

2 (GA Ifi¢ImA><mA IfiQIOAXOB IfingBXjB IEgIkBXkB IfigIOB> (129)
1.k.m¢0 (5;: -E6‘ +EE -Eg)(E$ _ng +13? _Eg)

 

124

Substituting equation (128) into (129) gives an integral with four frequency-dependent
terms in the integrand. Rearranging the matrix elements in the numerator, we obtain the
following equations for these four terms in the integrand.

A0
T1,129 = Tali T78 ch) 143

x 2 (0A Iii? ImAXmA I113 IOAXOB 11183 IJ'BXJ'B 1831 kBXkB |0f33|03) (130)
j.k.m¢0 (Ei’i‘. -‘EoA +ihm)(E}3 438 + ihmeE 438 +ih00)

 

A0
T2,129 = 1013 T76 Tap Hg

2 (0A IfiélmAXmA IfiyAIOAXOB IfigljBXJB IfigIkBXkB I113 '08) (131)
13km" (EB "53‘ --ihco)(E}3 438 +ih00)(EE 428 + ihco)

 

A0
13,129 = TaB Ty8 qu, “a

Z (0AIfii‘lmAXmAlfiéloAXOBIfililiaxiaInglkgxkglaglog) (132)
j.k,m¢0 (E8 436‘ +ihco)(13}3 +38 -ihm)(EE 438 —iha))

 

A0
T4,129 = TaB T76 Tarp 118

2 (CA In: I mAXmA IfiyA IOA XOB I111];3 11B )(13 IE]: I kB XkB I1183 IOB) (133)
j,k,m¢0 (13$ 450 -i’i00)(13}3 —E3 —ihco)(Ef(3 —E8 -ih(0)

 

If we use equations (118)-(121), (123)-(126), and (130)-(133), then for the sum of the 1-0
terms we obtain equation (134). By factoring equation (134), we obtain equation (135).
Then from equations (136) and (137), we can see that equation (135) can be re-written as
equation (138), which gives the contribution from the 1-0 term in terms of the imaginary

frequency-dependent polarizability of A and B-hyperpolarizability of B.

125

A0
TaB T15 Tap 148

(IOA I11? I mAXmA I83 IOAIIOB Il‘lis3 I J'BXJ'B I11? I kBXRB I113 IOBI

 

(13}3 438 +ihco)(E31 -E8‘ +ihm)(BE —Eg)

(”A I83 I mAXmA Ifii‘ IOA X03 Ilia? IJB><JB Ifiii I kBXkB I83 I08)

 

(12}3 458 + ithEé. 456‘ -ihco)(EE -E8)

IOA I8? I mAXmA I113 IOAXOB I83 IJ'BXJ'B I51? I kBXkB IBEIOB)

 

(Bf,3 —B8 —ihco)(E,’i‘, -E8 mamas? 438)

(CA I83 I mAXmA If“? IOAXOB IP13 IJ'BXJ'B I51? I kBXkB Iii? I013)

 

(E? 458 —ihm)(E$ —E(,A —112co)(1~:}3 —E8)

(0A I11? I mAXmA I113 IOAXOB I113 IJB>(jB Iii? I kBXkB IBIS3 I013)

 

(BB 458 +ihm)(B,A, 456A +ihco)(E? —E0A)

(0A I83 I mAXmA I11? IOA X03 IBEIJ'BXJ'B Wis; I kB><kB IlfiiiB I013)

 

(Bi? 43%? +1»...ng —E8 —ihw)(E? -53)

(0A lfiyAlmAXmAlfiQIOAXOBIfiEIJBXJB WilkBXkalfiglon)

 

(BE -B8)(E,A, 433‘ 4.010005}3 —E0A -—ihc0)

(0A I119 I mAXmA Ifi¢I°A>(OB Ilfiii3 IJBIIJB I5? I kBXkB I113 I013)

 

(13},3 438mg 453‘ —i1ico)(B}3 - BOA —ihc0)

(0A Ifii‘ I mAXmA I113 IOAXOB Ifiis3 “Exits I53 I kBXI‘B I515 I013)

 

(EA—153‘ “more? 438 +1120in3 -E8 + ihm)

(0A Ifiél mA><mA lfii‘IOAXOB Iii? IJ'BXJ'B in}? | '93an IBEIOB)

 

(EA, 433‘ —i7100)(E‘1i3—E(I)3 +ihm)(Ei3 - E8 +1121»)

(OA I11? I mAXmA I119 IOAXOB Ifiii IJB><JB W: I kBXI‘B Ifigloai

 

(133 453‘ +ihao)(1~:}3 438 -irmo)(13‘k3 458 —iht0)

(OA I83 I mAXmA I51? IOAXOB I111}:3 IJ'BXJ'B I53 I kBXkB IBEIOB)

 

 

( (EA, -E6‘ -ihai)(B}3 438 —ih00)(EE 438 —ihc0)

126

+

+

+

+

+

+

+

+

+

+

‘1

 

(134)

A0
T08 T76 Temp 148

(oiluilmixmiw 10.0 (oilu 1mA><m11uiloA>J

(Em ”Eo —Jh0)) (Em -Eo +ihco)
(OB IBJ‘B IJBXJB I110 I kBXI‘B I116 I03) I
(EB —E0 —lh(D)(Ek -E0 -lh0))
+ (03 Illa IJBXJB I“? IkBXkB I? IOBI
(EJB _EO —lh(1))(Ek “E0)
xi 2 I” + (OBIJJSIIBJBXIJBB IEBZIkBIékB IFJJIOB) do,
4“ j,k,m¢0 °° (E,- —E0 )(E1. -Eo +ihm)
+ (OH Illa IJBXJB I144) IkBXkB I116 I013)
(E? —E0 +lh(0)(Ek —E0 +1710)
+ (03 IFTIJBXJB I113: IkBXkB I115 I03)
(13}3 E0 )(Ek E0 ”71(0)

+ (OB I”? IJBXJB I813 IkBXkB I113 I013)
\ k (E? -E0 +lhO))(Ek -E(B)) ) }

 

 

 

 

 

 

 

.(135)

 

 

 

 

 

The dipole polarizability for A at the imaginary frequency ico is given by equation (136),

 

 

A )_ X (OAlualmAXmAluyA |0A> MAI» ImAXmAlual 0A).

61,117 (1m _ (136)
m¢o (Em -Eo -lh(t)) (Em _EO 4'th)

and the B-hyperpolarizability with 0);: i0), (02- — 0, and (00: 1'00 is given by equation

(137), according to Orr and Ward' '.

127

((03 IFIJIJBXJB IfiglkBXkB IPiJIOB) I

(13}3 453 -ih(t))(EE 438 -ih(0)
+ (“B IPIJ IJBXJB |fi§|k3Xk3 I11}? I03)
(E? 458 —ihm)(EE— 4130)
+ (OB II'l‘PI-JBXJB I“? IkB)B(kB IHB I08)
—E —EB + 'h
8386(-iw:iw,0)= Z (1:? 0)(Ek" o z (0)
308111511003 11(06ka 16103)
(13}3 438 +ih03)(Ek -E8 +ih(0)
+ (Osln8|13)<13 lib: IkBXkB Ins IOB)
(E? —E0 )(Ek —E0 "17103)
+ (OB I“? IJBXJB III-(J3 I kBXkB I113 I013)
( (Bi-53 ”rimming-E03) ). (137)

 

 

 

 

 

 

 

 

The net contribution to the energy from the 1-0 terms is
3 A0 h °° A . B . ,.
EE _-)0 — — a3 TY5 Tm, 118 ‘47; Lm0m(zm)Bfia¢(—zm,zm,0)dm , (138)

to third order in the interaction.
This same type of manipulation can be done in the case of the 0-1 terms to
generate a term that depends on the imaginary-frequency polarizability of B and the [3-

hyperpolarizability of A. Listed as equations (139) - (141) are the 0-1 terms.

T04,a = (0|\"/1 GA B: Tag 51133 GA$13 (7)6). (139)
T0_l,b=(0|V1GA®B'1I§TaB it}? GA v1|0). (140)
T04“, = (0|\‘I1 GAG9B 6;} Tag 113° GA€913 v1|0). (141)

If we expand equation (139), we obtain equation (142).

128

TO-l,a = (13 T76 Tap 111830
0 aA —A AAO 0 AB ' ' ABo (142)
Z < A I“? ImAXmA Iva lnA><nA Incl 4X 3 111, IJBXJB lwpl B)
j,m,n¢0 (E; “E8 XE: “E8 +13? -E(I)3)
By reassigning Greek indices so that the expression depends on the (p-component of the

permanent dipole of system B in equation (142), we obtain equation (143).

B
TO-l,a = T05 T75 T80) 140)

2 (DA lit? ImAXmA In: InAan I11? IOAXOB ll’ilii3 lJBXJB I11? IOB) (143)
j,m,n¢0 (E91 “E8 )(EnA "EDA 'J' E? _E8)

Using the same relation as in equation (115), the denominator can be rewritten as in

 

 

equation (144).
l —
(153. -E6‘)(E:.‘ 438 +13? 456’)
f N
_L_
(BA —EA)
3“ 0 (144)

h 1 1
— d00 x +
4n I: “E? -E0A +ih00) (E9 433‘ -ih0)):|

 

 

 

 

 

1 l
x +
I _(E? 458 +ih00) (E? 438 —ih0)):l)
Beginning with equations (143) and (144), we can rearrange the numerator to generate

equations (145)-(148), with the assumption that the dipole matrix elements are real.

B
TO—l,al = Tali T78 Tap 11¢

L fa... Z (GAlfiilmAXmAlfia‘lnAanIaé‘loixoslagligxi'gIngloB)(145>
4" °° j,m,n¢0 (Bf? -E6‘)(E$ —133‘+ihm)(1=.}3 =58 +ih00)

X

129

B
TO-Laz = 1913 T75 T89 “9

xi L“ d0, 2 (OAIIAEImAXmAII"aInA><nAIfiyIOAXOBIBSIJBXJBIBBIOB) (146)
4" °° j,m,n¢0 (EQ— BA)(En —BO —ih(0)(Ej —E0 +ihm)

 

B
TO—l,a3 = T68 T78 T80 11.,

XI’. L” d0) 2 (OAIIAYImAXmAII’laInAXnAIFaA IOAXOBlufljBXJBIfiSIOB) (147)
47! j,m,n¢0 (En “E0 )(Em -E8 +ih00)(E? -E0 -ih(1))

 

B
T04,1114 = (1B T78 T88 11.,

x_.h_ [>0 do) 2 (OAIH‘: ImAXmAI-I’ia InA><nA|llA IOAXOBlugI-JBXJBIBB I03) (148)
411: 0° j,m.n¢0 (Em —E8 )(En —E8 -1h(D)(EjB -E0 —ih(o)

 

Tm), in equation (140) becomes the sum of terms in equations (149)-(152), after
expanding, simplifying, and substituting for the denominator and rearranging the

numerator.

B
T0-1.b1 = TaB T78 T 5896

xi f do, 2 (OAI“8ImAI(mAIJ*TIAAXUAIHaIOAXOBIMIJBXJBIBBIOB) (149)
4“ 0° j,m,n¢0 (Em ‘EO )(En -E0 +ih00)(Ej -Eo +ih0))

 

B
T0-1.b2 = Tali T76 Tap “(p

X—h- fo dw z (OAIBSA ImAXmAII‘“? InA><nAIfiaIOA)<OB IHBIJBXJB I115IOB>(150)
41: j,m,n¢0 (EA- E8)(EnA -E8 +ihw)(Ej —E0 —ihm)

 

B
TO-l,b3 = 1013 Ty8 T88 liq,

Xifodco 2 (GAIN: ImAXJJlAIJJf,A InAXnAIBQ IOAXOBIFJ: |JB)(JB|“BIOB) (151)
41!: 0° j,m.n¢0 (En ’Eo )(Em—E8 -1h(0)(Ej -E8 +1710»

B
T0-1,b4 = TaB Ty8 Tetp 11¢

X1 If d0) 2 (OAIfia ImAXmAIp‘Y InAXnAIPeA IOA)<OBI“§IJB><JBI118I08). (152)
4n 0° j,m,n¢0 (En —E0 AXEA “E0 4710003? -E0 -ihc0)

 

130

After expansion, the last 0-1 term, T111,c from equation (141), is given by equation (153).

To—1.c =(OI‘710A®B E1? Tap #11130 GAAB VI|°)=T as T16 T211 "3°
X Z (OA I11? lmA><mA life InAXnA lfia |0A><OB I116 IJ'BXJ'B I115 log) “53)
m,n,j¢0 (Em "'EO A+E?" “E0 XE: _EO A+E?‘ "'EO)
We can use equation (154) to replace the energy denominator in equation (153).

1
(E1’i1 433‘ +13}3 4353x139 433‘ +13}3 —E8)

 

 

 

 

{( 1 N \
(Ea-138 +1hm)(E.’.‘ -E8‘ +ihco)
1 (154)
h 0° +
2; Led“) 1 (15:, —58 41111115: -E6‘ 412m);

1 1
X +
\ (13}3 438 +1111») (13}3 -E8 —ihco) )

 

 

 

 

Then we can rearrange each of the terms and obtain T114,c as a sum of four terms (To-1,c1 to

T01“) given in the following four equations:

BO
TO-l,cl = T111} T118 Tap “(p

xi 1:” (11,) Z (OAluYA lmAXmAl“€ InAanlF‘aIOAXOBIHBIJBXJBIfiBIOB) (155)
411 ”NO (59-511 +ihm)(Em —Eo +1hm)(E}3 —Eo +1711»)

BO
T0—l.c2 = Tali T75 T811) 14¢

x._h_ [x (1(1) 2 <0A|fi7|mA><mAll1éA lnAXnA Illa IOAXOB BIAB ljBXjB his I03) (156)
4“ 0° 11m1n¢0 (En A_EA +ih60)(Em -E0 +ih(1))(El-3 ~Eo -—ih(o)

BO
TO—l .c3 : TaB T18 Tap Ftp

_n°°d(o4J‘_ (OA lilo lmAXmAluéA lnAanlfiy IOAXOB I115 ljBXjB IfiB '08) (157)
°° j”m "‘0 (En A—EOA "hmem "Eo "ithEj -Eo Him)

131

TO-l,c4 : TaB T78 Tecp P130
1de z <°Alfi9ImA><mAIuélnA><nAIa¢IoA><oBIa€IjB><jBInflow “58>
4n °° '

X
j,m,n¢o (139-133 -ihm)(E§; — £3 -ihm)(E}3 438 —ih(o)

Summing the terms in equations (145)-(148), (149)-( 152), and (155)-(158), we obtain the

net T04 term given in equation (159).

132

TO—l = Tali T78 Tap F50
‘1
(“A le ImAXmA lFa JFAXM lFe IOAXOB |F6 lJB J<JB IFB JOBJ
(E11 -—EO )(Em —Eo +1ha11)(13§3 -—E0 +1711»)

(OAIFeA lmAXmAlFa lnAJ<nA|F§A IOAXOBJFEIJBXJBIFBIOBJ
(Em -E0 A)(EJJA 425‘ —111111)(E}3 —E0 +1111»)

 

 

(0:1le ImAXmA lFa IFAXFA lFe IOA X013 JFBJJ JJBXJB |F6 IOBJ +
(E, -E0 A)(EA —E0 +1hm)(E}3—Eo —ih(o)

 

(0A JFEA I mAXmA JFa lnAXnA JFy IOA X013 IFB IJBXJB |F6IOBJ
(Em —EO )(En —E0 —1hc11)(E}3 -1~:0 —iha))

(0A lFe ImAXmA le IFAXFA lFa |0A X013 |F6 lJBJ<JB lFB IOBJ
(Em —EA)(13n —E0 +ihw)(EJ~ —E0 +1111»)

(0A lfis ImAJ<mA |F1 |nAJ(nA IFa IOAXOB IFB IJBXJB |F6 IOBJ
(Em —E0 )(1—3n —E0 +ihm)(EJ- -E0 —ihm)

Jummto (OAlfialmA)(mAlB$ IFAXFAJFe IOAXOB|F6 JJBXJBIFBIOBJ
(EJJA —EO )(Em —E01hw)(EJ —Eo +1111»)

(0A IFa lmAXmA|F1 IDAXBA IFe IOAXOB IFB IJBXJB |F6 IOBJ
(En A-EAxEm 433‘ —ihco)(EJ- —Eo -ihm)

(0A IF lmAXmA IF lnAJ(nA|Fa IOAJ(OB IF IJBXJB IFB|OBJ
(En —E0 +1h(D)(Em —Eo 'i'lh(1.))(E‘lj3 —EO +ihm)

 

 

 

fiL:dmZ

 

 

 

(0A JFAI mAJ<mA IFe l“AJ<"A IFa IOAXOB BJFB IJB XJB |F6 IOBJ
(En -E0 “710005". "E0 +ii'zco)(E‘li3 -Eo —ih(o)

(0A lFa ImAXmA lFe IFAXFA IFyA IOAXOB |F6 lJB XJB IFB IOBJ
(E0 —E0 —ihw)(Em —E0 —ih(n)(E§3 —E0 +ihm)

(0A lFa lmAXmA lFe IFAXFA le IOAXOB BJFB IJBXJB |F6 IOBJ
K (En -E0 —lh(0)(Em -E8 -ih0))(Ej -E0 -lh(0) )

 

 

 

 

 

(159)

133

Factoring and substituting the definition of the imaginary-frequency dependent
polarizability of B and B-hyperpolarizability of A, we obtain equation (160) for the 0-1

term in the third-order energy.
3) BO 71 °° A . ,. 13 .
EEH = — (13 T75 qu, uq, :1; .[mBaYJ—1w,zw,0)a33(1w)dm. (160)

Figure 3 represents the 1-0 term. System A sets up a fluctuating field represented by the
two red dashed lines and a static field due to its permanent dipole moment. B responds
nonlinearly to the fields from A through its frequency-dependent B-hyperpolarizability, as

represented by the green dotted lines.

“

 

   

~‘

Figure 3: Representation of the 1-0 term in the Third-Order Energy.
Images in this dissertation are presented in color.

4.3.4 Zero-Order Term in the Static Dipole Moments of Systems A and B.

The final third-order term is the nonzero 0—0 term, listed in equation (161).
TH = (0M GAeB EA TaB HE GA698 W0). (161)

Expanding equation (161), we get equation (162).

134

T0_o = (0W1 GA63B a: Tap 11,? GAABB \“11|o)= Tag T75 TSJ,

 

l l
x 2 (162)
1,1,m,n¢»[<E$ -E1’1‘ +15? —E8>J[<E1A 436‘ + BE -E1’1’)J
X<°A JFJA I mAXmA IE1? InAJ(nA IF;A IOA X013 IFJS3 IJBJ<JB IKE IkBXkB JFJJ IOBJ-
If we use the analog of equation (46), then the energy denominator can be represented as

the form given in equation (163).

 

l 1
[(Eg 421'? +13? -E8)J[(E:;‘ 458 +13}? —E€)J_
ff 1 J ] J
+ X

103;: —E6‘ +1111») (1391 -E(,A-1h1»)
(

 

 

 

1 1
+ x (163)
\(EJA -E{,‘+1r11»’) (15';A -E(,A -1h1»’)J

(fir 1:111» J: 111» 1 1
Joe? J

+
438 +1121») (1353—138 —iha))

 

 

 

 

1 l
+
I (BE - 138 +1111») (13L3 - EB -1121»’)J J
Using equation (164), we can rearrange equation (163) into equation (165).

j:[1(x)+f(-x)]dx = 2fmf(x)dx. (164)

135

 

 

‘ ‘ {if
mama-6111 (B9-E8+E13—E8> 2“

(r

1 J J

 

(E3, 438 +1111»)(E}3 —E8 +1111»)

1

 

+

 

fmdw fwdm' L (EJ’J’:I 433* +1111»)(E}3 -E8 -1h1»)J

{

 

1 J J

 

+

(E: -E(} -1-ih(1)’)(Elk3—E(1)3 +ihm’)

1

 

 

 

\ ( (E9 —E8‘ +1111»’)(E,£3 438 -1h1»’))

 

 

\

By use of equation (165), we obtain equation (166).

h 2
TO—O = T1113 Tyfi Tao (‘2?)

x 1:: do» [:0 dd 2

j,k,m,n¢0

(f

 

K

 

 

 

 

1 \
(E9, -E8 +1711»)(E}3 —E8 +1711»)
+ 1
3 —E3 +1111»)(E}3 —E8 —1h1»)J
( 1 \

 

(E3 —E3 +1r11»’)(E{(3 —E8 +1111»)
1

 

 

 

+
( (13:,A —E6A +1h1»’)(EE —E8 -ihm’) )

X<°A va ImAXmA lFa l“AJ("A IFe IOAJ
NOB |F6 IJBXJB IFBB “‘8st IF» IOBJ

 

J

(165)

(166)

Equation (166) can be rewritten as equation (167), where the b tensor elements are given

by equation (168).

T0_ 0=(O|V1GA$BE11§‘TGB 115 BGA61313 V1|O)= Tap T75 Tap

tzt‘: «01:66

136

X[b59¢( (10); —im— in)’ ,iw')+b53¢(— 1(1) 1(1) iw', in) ')]

9,861»;- -’11»—11» ,’,11»)+b‘JAa£(11»--11»+11»’,—11»’)] (167)

0 ftAm m EAn n fiAO
b1?y[3(‘(0)1+(02J3w21w1J= 2 (Al 0| AX Al ‘1' AX Al 5' A)

.(168)
A A
111,11160 (E31 ‘EQ ‘FFDI +032JJ(E11 ‘50 4‘01)

 

The result in equation (167) is new. It gives the third-order pure dispersion energy as a

double integral over imaginary frequencies, since E6330 = —T0_0. Figure 4 shows a

graphical representation of the 0-0 term. Both A and B are mutually polarized by each
other through specific terms in the frequency-dependent hyperpolarizability.

This interaction energy is a nonlinear analog of the second-order van der Waals
dispersion energy. A fluctuation in the charge density of molecule A polarizes B
nonlinearly, via [3, inducing a fluctuating charge in B. The field from the fluctuating
charge in B acts on A, shifting the energy of A in a way that depends on the B-
hyperpolarizability of A, because it depends on the correlation of the original fluctuation
in the charge density of A and the nonlinear response to the field from B. There is a
corresponding contribution to the energy with the roles of A and B reversed. Equation
(167) provides an efficient means of computing the third-order dispersion energy for

various A-B pairs.

 

Figure 4: Representation of the Third-Order Pure Dispersion Energy.
Images in this dissertation are presented in color.

137

The effects of hi gher—multipole interactions can be derived by replacing the dipole
operators with polarization density operators, replacing the dipole propagators Tap with

point-to-point dipole propagators Tag (r, r') , and then Taylor-expanding about the

vectors RA and R3 as origins of the spatial variables.
4.4 Numerical Estimates

Although the derivation of the third—order interaction energy provides insight into
the types of interactions that occur at third order, estimates of the magnitude of these
interactions will aid in their interpretation. To determine rough, approximate values of
the third-order terms, we have assumed that all excitation energies are approximately
equal to the ionization potential. This approximation allows us to simplify the
perturbation expressions and find numerical values for the frequency-dependent
polarizability and hyperpolarizability. The definition of the static polarizability is shown

in equation (169).

 

 

O k k 0 0 k k 0 .
aa=2< 1161 >< WI >+< 11111 x 1161 1. 111»
1,10 (511 '50) (511 '50)
Since the eigenstates are real, (OlpaIkalyfilO) = (OlyfllkalpaIO) . We are assuming

all excitation energies are approximately equal to the ionization potential,
(Ek - E0) = Ionization Potential (IP). Then, we solve for the sum of the matrix elements

as shown in equation (170).

a
Z<olflalk><klnfllo>=%IP- (170)
11:60

Equation (171) gives the frequency-dependent polarizability. Now that we have solved
for the sum of the matrix elements in the static polarizability, we can use those results to

approximate the numerical value of the imaginary-frequency dependent polarizability.

138

 

. _ 1 1
aaB(JwJ-[JP+W+JP_JaJl§)(0lflalkJ<k|#B|0J- 111»

Substituting in the expression for the sum of the matrix elements, we have equation

(172).

2
a»B(in= W “J l + 1 J: an» I? ' (172)
2 IP+im IP-ico 11>2 +032

The same method can be used to find a numerical value for the frequency-

 

 

dependent hyperpolarizability. Starting with the static hyperpolarizability in equation

(173), we can again solve for the sum of the matrix elements.

’(0lflalJXJIIIAkaIfion)J(Oluylixilfifllkxklfian) ‘
(Ej-EoXEk ‘50) (Ej-EoXEk ‘50)
J(OIFBIJ'XJIfialkJFIFyIOJJ(OIFJIJ'XJlfialkaIFBIOJ .117.»
(Ej—EOXEk -Eo) (EJ—EOXEk —Eo)
+ (OI/Jal J'J<J'|il'y|kJ(k|FB|0J + (OIFBI J'XJ'IF'ylkalfialOJ
K (E; ‘50)(511 ‘50) (E; ‘50)(Ek ‘50) )

 

 

 

flaflr = Z

j,k¢0

 

 

 

 

Because the eigenstates are assumed to be real, we can equate pairs of matrix elements.

2 (oluanXJIHsIkafiyIO)
6.», =g Z +<0|J1Y|JXJJFaJkaIFBJOJ' (174)
"”‘° +<0|FB|JXJJFYJJ‘inFaJOJ

Then if 60497 denotes the operator that takes the average of cyclic permutations of a, [3,

and y in the expression that follows, we obtain equation (175).

— . ._ ,. 13
€an 1.2 §0|u6|1J(J|Fp|kJ(kluy|0J = "TAP? (175)
J, 9‘5

Equation (176) gives the frequency-dependent hyperpolarizabilityl '.

139

( (OI/(6| J‘XJ'IfifllkalfileJ ‘
(E) -E» -h(w1 +0)2J)(Ek -E» 4160»)
(Olflrl iXJ‘IfifllkalfialO)
(E,- -E» +hwszk -E» +h(w1 +wz))
(DWI J‘XilfialkalfileJ
(E,- -E» +hw1)(Ek -Eo “7102)
fig (-(w1+wzJ.w1,wzJ= Z . ._ .
a y j,k¢0 + (Oll‘rllxl 11%|ka J/‘fliOJ
(Ej ‘Eo +hw2XEk '50 ‘hle
(OI/(6| JJ<J|firlkJ<k|fi6|0J
15,. —E» -h(w1 +wz))151 -E» -hw1)
J (OIFBIJ‘XJIF’rIkalfiaIOJ
\ (EJ- —E0 +111»,)(Ek —E0 +h(1»l +102» J

 

 

.( 176)

 

 

 

 

We use equation (175) and also approximate each sum of the form

Z<O|ua| 1X jIIJIBI k)(k|fiy|0) by the average over the cyclic permutations of Greek
1.11:0

subscripts. Approximating the excitation energies by the ionization potential (as above)
gives equation (177), in units with h =1.

BaBy(—(<D1+<02J1(011(°2J=

PEEL [Pg 3IP4 — IP2 (11112 + 111% +031c02) , (177)
3 (IP2 - (61 + 1112)2)11p2 «6121a?2 — 6%)

Equation (178) gives the approximation for the specific B-hyperpolarizability needed in

this study.

 

4 2 2
BQBJ(-11»;11»,0)=B“FY 31p ”P 1» . (178)
3 (IP2+c02)2

As for the b tensors, a generic approximation is shown in equation (179).

BafiyIPZ

. 179
6(IP+11»)(IP+11»’) ( )

 

b(— ico;(i(o’ — im),-io)’) =

140

The specific frequency-dependent b tensors needed in this work are approximated in

equations (l80)-(183).

 

 

 

 

b(11»,-(11»’—11»),—11»’) = B?” 1P2 . . (180)
6(IP +11»)(IP +1co’)

b(11»,-(—11»’—11»),11»’)= 6(IP 3:311:41») . (181)

b(—11»,-(i1»'+11»),—11»’) = 6 (IF Egyxlrf+ 11»’) . (182)

b(-11»,-(—1»1’+11»),11»’) = 6 B “‘37 1P2 (183)

(IP-i1»)(IP—11»’)'
Equation (51) gives the known result for the second-order energy correction.
Equation (184) approximates the second-order energy correction in terms of quantities

that are more readily available.

 

 

 

( \
190110611.
2 1 BO B0 A
Eg)=-E GB T75 +113 115 (lay _ . (184)
A B 2 2 °° l
+—a a IP IP ‘ “ ‘dw
07 BS A BL. 2 2 2 2
( 2n (IPAHo Inga») )

Now we will approximate the third-order energy in terms of readily available
quantities. The classical contribution to the third-order energy is the sum of the 3-0, 0-3,

and 1-1 terms, given in equation (185).

(3) _. 1 BO BO BO A
Eclass-—ETQBTYFT5‘P “13 "5 11¢ B1179
1
--F90 FAQ 11.30 T1118 T78 T81» 13113381» (185)

6

BO A0 A B
"116 “a T65 T78 Tao “ya “Bip-

141

The non-classical contribution to the third-order energy is the sum of the 1-0, 0-1, and O-

0, terms, given in equation (186).

3 A0 ’1 °° A . B . .
Eircinclassical=_T0~13 T75 TS‘P “3 a L» dm a“? (1(1)) Bfia‘l’ (-1a51w’0)

h °° . . .
——TaB T75 Teq) “3027.: L1» dco 13:73 (—za);za),0) 011335 (1(1))
"TaB T75 T8!»
T1 2 m m [bJAJJ‘JJJs (in); — in) — ico', iw’)+ b?” (ico; — 110 + 110', — ico’)]
x[—) L do)": da)
21: oo 00

x [b33113 (i0); — in) - ito', ico')+ bFBB‘P (- ionic) - ico', JfD'JJ
(186)

This includes both induction-dispersion effects (the first two terms) in equation (186) and
pure dispersion effects (the last term). Substituting the approximations converts equation
(186) into equation (187).

13‘” T T A0 a__11Ay B1338» h 3113;; + ngm2 IPX
nonclassicalz T013 Y5 8‘P “8 T£:dm 2 2 2 2

(131132032 2
_ “fl T75 Tap 1130—36 Saw? ’1 TL:d(D 31p A '1']? 2[PB
(IP2 + (1112)2 IPB + (1)2

BA B IP13 (187)
‘ afi T78 T8(p[ “78611) A1853: 2]

 

 

 

( J J 1
+
[(IPA +iw)(IPA —iw’) (IPA + iw)(IPA +1103]

)1 2 .. ..
x(§;] Lmdmimdm'

 

 

 

 

J 1
\xJ:(IPB +iw)(IP3 —1112’)+ (1P3 -in(”’ B “MAJ /

Therefore, the total third-order energy is approximated by equation (188).

142

3 1 130 BO BO
E( )=—-6' (1B T75 Tarp PB “8 “(p Bays

1 A0 A0 A0 B
‘gfla “y “a TaB T75 Tetp Bfiérp

BO A0 A B
‘98 “a TM 76 Tarp “ya “Bo

 

—T T T A0 “017 B___B_6¢ h fl!- (311’1‘31 + “’1ng 1P:
016 76 so “a co 2 2
( (IPfi + (1)2)2 11)A + (’3

B0 _B___a§ Says 71 TM]: 3IPX+IPim2 [P182

 

 

 

 

— T T
\(IPK + m2)2 [PB + 03
A 2 B 2
Boys IPA [3580) IPB
"T08 T75 T80) 6 6
- 188
r 1 + l \ ( )
(IPA +ia))(IPA —iw’) (IPA +iw)(IPA +iw’)_

h 2 ... ...
X(2_1t) Lodmimdco’

 

 

 

 

 

1 1 '
\XLIPB + iw)(IPB —iw’) + (1P3 —ia))(IPB -—iw’)d )
Equation (188) gives a general approximation for the third-order interaction energy.
4.4.1 HCI Dimer
Using the HCl dimer as an example, we compare the two-body third-order
interaction energy to the two-body second-order interaction energy for two specific
geometrical configurations. The first is a configuration in which the centers of mass of

both HCl molecules lie on the Z-axis and the internuclear axes of both molecules run

parallel to the Z-axis. This configuration is shown in Figure (5).

143

Figure 5: First Configuration of the HCl Dimer
Images in this dissertation are presented in color.

The second configuration has the centers of mass of the two molecules remaining on the
Z-axis, with both of the HCl molecules rotated 90 degrees, so that the dipole moments of
the molecules are parallel to each other and perpendicular to the Z—axis, as shown in

Figure (6).

X

Figure 6: Second Configuration of the HCl Dimer
Images in this dissertation are presented in color.

There are two sets of coordinate systems involved in these configurations. The
first is the laboratory coordinate system. In Figures 5 and 6, the Z-axis vector of the lab
frame points from the center of mass of the first HCl molecule to the center of mass of
the second HC] molecule. The dipole propagator Tap and hi gher-order T tensors are

written in the laboratory coordinate system. In addition to the laboratory coordinate

system, there is a molecular coordinate system for each of the two HCl molecules. The Z
-axis of the molecular coordinate system lies along the bond.

If an HCl molecule is oriented parallel to the pair-fixed Z-axis, the only nonzero
component of the HCl dipole is the Z component, but if the molecule is rotated in the
plane defined by the pair-fixed X- and Z-axes, the dipole has both Z and X components.
It is not until the molecule has rotated 90 degrees that the dipole of HC] has only an X
component. A rotation matrix is needed to calculate all of the components of the dipole
if the molecule is not oriented parallel to an axis in the pair-fixed frame. This rotation

matrix, R, is shown in equation (189) where 9 is the angle of rotation.

Rn ny sz cos(0) 0 sin(6)
R = Ryx Ryy Ryz = O l O . (189)
R2; R0 Rzz — sin(6) 0 cos(6)

Using this rotation matrix, one can find the dipole from equation (190), where ,u' is the

dipole in the new coordinate system and p. is the dipole in the old coordinate system.

H, = (P'xRxx +P'yRyx +P'szx )x’
+erny 441,12yy + “212,, )y (190)
+(I'lxsz +“yRyz +szzz)z"

(The primes are the new coordinate system.)

such that

,u’ = - sin(6)yzx' + Oy’ + cos(6l),uzz’.

Note: In this situation the old coordinate system is the molecular frame’s coordinate
system, the new system is the lab frame, and (11) is negative.

Again, if the HCl molecule lies parallel to the Z-axis, then the nonzero components of the

polarizability are a“, 01”, and 0122. Due to the symmetry of the HCl molecule, an = 01”.

145

In addition, the nonzero components of the hyperpolarizability are [3222, Bm, Bm, Bzxx,
Byyz, Byzy, and Bzyy. Also due to the symmetry of the HCl molecule, Bm = [3m = [3m =
Byyz = [3,2, = [32”. As the molecule rotates, the components of the polarizability and
hyperpolarizability change as well. The components of the polarizability in the new
coordinate system are given by equation (191), where the Greek indices refer to Cartesian

coordinates.

(1:11;! = aaBRaa’R my. (191)
The hyperpolarizability transforms under rotation as described by equation (192).

Binary = BafiyRaa’RBB’R w' . (192)

Referring back to the laboratory coordinate system, if the arrangement of the
centers of mass changes, the dipole propagators change as well. For example, in Figure
(7), one of the HCl molecules has been moved off the laboratory Z-axis. The dipole
propagators in this case are shown in equations (193)-(198), where (p is the angle that the
vector between HCl molecules makes with the Z-axis and R is the distance between the

centers of mass of the two molecules.

A

 

HCI ‘9 Z

V

Figure 7: Third Configuration of the HCl Dimer.
Images in this dissertation are presented in color.

146

 

 

T..<R.¢>=——,—— (193)
R

TX), = Tyx = 0. (194)

sz(R.¢)=sz(R.¢>= 3°°S‘:)§“““”’. (195)

l

Tyy(R)=—E—3. (196)

Tyz = sz = 0. (197)
2 _

Tzz (R, (P) = 3COSR(3¢) l - (198)

4.4.1.1 Second-Order Two-Body Terms
Recall from the previous section that the second-order interaction energy for two

bodies is approximated by equation (199).

 

 

 

( \
B
113011900135
2 1 BO BO A
E8)=-E GBTY5 +1.”; [.15 (lay . (199)
h A B 2 2 °° l
+_“GY“B51PAIPBL.‘ 2 2” 2 2d“)
( 21: (IPAm XIPBm) )

The second-order two-body interaction energy is a sum of a classical induction effect

given by equation (200) and a dispersion effect approximated by equation (201). The

notation E (m’n) is used for the mth-order interaction energy for 11 bodies.

 

(2.2) _ 1 (FAO A0 B B0 B0 A )

Eclassical "' —§TGB T75 a “y 0136 +143 “6 any (200)
(2.2) __ h A B 2 2 ”L .1. .

EdisPe'Sio“ — “‘3 T75 21? any “Ba IPA IPBJ:°"(IP,112, + 012 1ng +012)“. (201)

147

We have used Mathematica 4.0 to evaluate the expressions in equations (200) and
(201). The second-order two-body interaction energy for the collinear configuration is

given in equation (202), and the result for the stacked configuration is given in equation

 

 

(203).
2,2 _ -90.7O
AE£olli)near - R6 ' (202)
2,2 _ -91.58
AEEmcr)... - R6 - (203)

4.4.1.2 Third-Order Two-Body Terms

The third-order interaction energy for two bodies is approximated by equation
(188). The third-order two-body energy is a sum of classical induction, induction-
dispersion, and pure dispersion interactions. Equation (204) shows the classical
induction interactions, equation (205) describes the induction-dispersion energy, and

equation (206) describes the pure dispersion interactions.

3.2 BO A0 A B
Egnd ) = ‘116 lie TaB T78 Tarp “ya “[310
BO BO BO A
l T05 T78 Tarp “B ”8 Prp Bays (204)

6 +.../2° 119° 1.901.. n. T... as...

(3 2) A0 “:1 [3113380
Eind-disp = ‘TaB T115 T80 “8 _3

x_r._ °°d 3mg.rpg..2 1p},

oo 2 2
41: (IPg+0)2Y IPA'H”

BO “[1335 [3de
" TaB T75 T80 “<11 “—3 _

 

(205)

xi fa dw 31Pfi+n>§m2 [pg
00 2 2'
411 (11%“to IPBH”

148

A 2
(3,2) _ BayeIPA
Edisp " “TaB T75 T89[ 6 ][

[383811111313
6

1 1 (206)

 

27:

K \
[ h )2 oo oo , [(IPA +iw)(IPA —ia)’) +(IPA +ia))(IPA +1203]
x Lwdmiwdm .

 

 

l l
\X[(IPB +ia))(IPB -—ia)’) +(IP13 -iw)(IPB -i(0’):l}

 

Again, using Mathematica 4.0 to evaluate the expressions, we have obtained the

third-order two-body interaction energy for the collinear configuration as shown in

equation (207) and the energy for the stacked configuration as shown in equation (208).

The experimental values for the dipole, a-polarizability, B-hyperpolarizability, and

ionization potential are listed in Table l.

 

 

 

 

Table 1. Values for Use in Approximations
Value in Value in
Property atomic units electron volts References
112 0.4643 Maroulisrr
on 17.96 Maroulis"
a,“ = 01,, 16.17 Maroulis17
B... -lO.6 Maroulis"
Bl“ = Bl” = 3"“ = 2.2 Maroulis”
Byzy = Bxxz = Byyz
[P 0.46852 12.749 CRC‘8

 

 

 

 

 

 

 

 

 

 

 

(3,2) _ 492.35
collinear _- R9 ‘
(3.2) _68.90
stacked _ R9 '

(207)

(208)

There is a much larger difference at the third order for the HCl dimers. Within the

approximation used here, there is a stronger interaction for the collinear dimer than for

the stacked dimer.

149

4.4.2 HCl Trimer

Using the HCl trimer as an example, we have estimated the third-order three-body
interaction energy and the third-order two-body interaction energy, and we have
compared their values for three geometrical configurations. The first configuration is
shown in Figure 8, where the centers of mass of three HCl molecules lie on the laboratory

Z—axis and the internuclear axis for each molecule runs parallel to the Z-axis.

00¢

Figure 8: First Trimer Configuration.
Images in this dissertation are presented in color.

X

The second configuration has the three centers of mass on the Z-axis, but each of the HCl
molecules is rotated 90 degrees, so that the internuclear axis runs parallel to the

laboratory X-axis as shown in Figure 9.

 

Figure 9: Second Trimer Configuration.
Images in this dissertation are presented in color.

The final arrangement has the HCl molecules arranged such that the centers of mass

generate an equilateral triangle with the internuclear axis for each molecule still parallel

150

to the laboratory Z-axis. This configuration, known as the triangle formation, is shown in

Figure 10.

Y

 

 

Figure 10: Third Trimer Configuration (the triangle formation).
Images in this dissertation are presented in color.

For three-body interactions, we need three dipole propagators. The first dipole

propagator connects systems A with B. These tensors are described by equations (209)—

(213).
- 2
35m -1
T33 = ((30.11)) _ (209)
Rab
TQB=TQB=0 (210)
3cos s‘n
TQB =TZI§B = ((0:12 1 (¢ab)_ (211)
ab
1
TAB = _ _ (212)
W 3
Rab
2 _
T33 =——3°°s (fab) 1. (213)
Rab

151

A second dipole propagator connects system A with system C. These propagators are

described in equations (214)-(218).

 

 

 

38in2 —l
TS‘A 2 (fear) . (214)
RC8
T3“ =T§A =0. (215)
3cos sin
TSZA = TzCiA = ((aca; (wca). (216)
RC3
1
TCA = ——. (217)
W 3
RC8
2
Boos —1
T5». = (gca) _ (218)
RC3

The third and final dipole propagator is needed to connect system B with system C.

These propagators are described by equations (219)-(223).

 

 

 

3s'n2 —1
T2? = 1 ((é’bc) . (219)
Rbc
133C = T5,? = 0. (220)
3 'n
TXBZC =TZBXC = COS(%RC;SI (wa) . (221)
be
BC _ l
Ty), —-—R—3—. (222)
be
3 2 -1
TszC = cos ((3%) . (223)
Rbc

152

4.4.2.1 Third-Order Three-Body Terms

The third-order three-body terms were derived in terms of nonlocal
susceptibilities by Li and Hunt”. Li and Hunt separated the third-order three-body
energy into three categories: energy due to induction, energy due to a mix of induction

and dispersion forces, and energy due solely to dispersion interactions.

(3.3)_ (3,3) (3,3) (3.3)
E —Eind +15i+d +Edisp. (224)

The induction energy consists of classical forces, which fall into one of three sub-

categories.

(3.3)_ (3,3) (3,3) (3,3)
Eind .13srf +Ehyp +13tbf . (225)

The first component is the static reaction field energy, which is a sum of three

components represented in equations (226) and (227).
ES?” =E§3€1+E§3€r§+5§3€é- (226)
133,3; = - Idr---drVP6°‘ (r)-T(r, r’)-aB(r', r')-T(r', r')-aC(r",r‘")-T(ri",r").135‘ (rV) .
(227)
The second is the third-body field terms described by equations (228) and (229).
(3.3) _ (3.3) (3.3) (3.3) (3.3) (3.3) (3.3)
Etbf - Etbf,ABC + Etbf,ACB + Etbf,BAC + Etbf,BCA + Etbf,CAB + Etbf,CBA .(228)
13:32ch = - J’dr---dr"P(l)3(r)-T(r,r’)-01C(r',r')-T(r",r").1113(r"',ri")-T(ri",rv 136‘ (H).
(229)

The third contribution to the classical induction energy comes from the three-body

hyperpolarization effects, described by equations (230) and (231).

(3,3) _ (3,3) (3,3) (3,3)
Ehyp _ Em}, + Ehyp’B + Eth. (230)

153

130,3) =-i J.dr--~dr"BA(r,r’,r')[T(r,r")-P(§3(r')]
"”‘A 2 xmririb-P§(r‘V)11T(r'.rV)P€(rV)1
1 ---dr"BA(r.r’.r')[T(r.r')-PoC(r')]

2 x [T(r', ri" ) . p53 (r‘V )] [T(r', r" m? (rV )1.

(231)

The induction-dispersion interaction energy is described by equations (232) and (233).

3,3
Ei(+d) = E(A---B)—)C + E(C---A)-—)B + E(B---C)—)A . (232)

{BBAW (r', r', r : i010),OI‘Y5 (r', r')\

(3,3 __ h B p ’ . '
E(A--)-B—)C) ——--2; [)mdcoj'drmdrv x05£(r ,r'v,10)(0Tsp(r‘v,r’)

Kx'r,,,q,(r,rv)POSPu")

 

 

( B \ (233)
B3,,“ (rt, r" r : i0)00,OTY5 (1,, r')

__ 17“” Japan xag,(r~,riv,immrefl(ritr3

\XT‘I‘P (r,rV)1>(§3p (rV)

 

 

The dispersion energy is described by equation (234).

T rV,riV -aC riv,r";' wT ”,r'
13535) = l fdm jar-um" Tr ( ) ( “D (r ) (234)
P 7‘ B I I, - I A V, .
xa (r ,r ,100)-T(r ,r)-a (r,r ,rw)
Combining equations (224)-(234), we calculated the third-order three-body
interaction energy at each of the three geometrical configurations. From the integrals
over imaginary frequencies, we used the approximations to the frequency-dependent

values of 01 and B described above. The results are listed in equations (235)-(237).

AE(3’3) _ _ 1369.34

collinear - R9 (235)
(3,3) _ 223.68
AEstacked - ——R9 - (236)

154

AE(3.’3) = ___-_, (237)

4.4.2.2 Third-Order Two-Body Terms

Finally, we compared the third-order three-body energies from equations (235)-
(237) with the third-order two-body energies to generate the full third-order two-body
results, we added three third-order two-body equations. as shown in equation (238).

(3,2) _ (3,2) (3,2) (3,2)
E —EAB +133C +EAC . (238)

Listed in equations (239)-(241) are the results for the calculation of the third-order two-

 

 

 

body terms.
£313.... =-981:'965- (239)
AEiifr’... = ”1:593 (240)
1359.53,... - 6313331 (241)

155

4.5 References

1. F. London, Z. Phys. 63, 245 (1930); Z. Phys. Chem. Abt. B 11, 222 (1930).

2. AD. Buckingham, Propriétés Optiques et Acoustiques des Fluids Comprimés et
5A.;:tions Intermoléculaires (Centre National de la Recherche Scientifique, Paris, 1959), p.
3 . A. D. Buckingham, Adv. Chem. Phys. 12, 107 (1967).

4 . D. E. Stogryn, Phys. Rev. Lett. 24, 971 (1970); J. Chem. Phys. 52, 3671 (1970).

5. D. E. Stogryn, MOLPhys. 22, 81 (1971).

6. P. Piecuch, Chem. Phys. Lett. 110, 496 (1984).

7. P. Piecuch, Mol. Phys. 59, 1067 (1986).

8. P. Piecuch, Mol. Phys. 59, 1085 (1986).

9. P. Piecuch, Moi. Phys. 59, 1097 (1986).

10. Lafuente, R. Dissertation requirement for PhD.

11. B. J. Orr, J. F. Ward, Mol Phys, 20, 513 (1971).

12. A. Dalgamo, Adv. Chem. Phys. 12, 143 (1967).

13. H. C. Longuet-Higgins, Proc. R. Soc. London Ser. A 235, 537 (1956)

14. H. C. Longuet-Higgins and L. Salem, Proc. R. Soc. London Ser. A 259, 433 (1961).
15. H. C. Longuet-Higgins, Discuss. Faraday Soc. 40, 7 (1965).

16. Craig, D. P.; Thirunamachandran, T. Chem. Phys. Lett., 80, 14 (1981).

17. G. Maroulis, J Chem Phys. 108, 5432 (1998).

18. David R. Lide, ed., CRC Handbook of Chemistry and Physics, Internet Version 2006,
<http://www.hbcpnetbase.com>, Taylor and Francis, Boca Raton, FL, 2006.

19. X. Li, K. L. C. Hunt, J. Chem. Phys.105, 4076 (1996).

156

Chapter 5

Molecular Frame Distortions in CH4+X Pairs, with X=He, N2, or H2

157

5.1 Introduction

Compressed gas mixtures containing CH4-H2, CH4-N2, and CH4-He, although
pertinent to astrophysics mainly due to the gaseous composition of the atmosphere of
Titanl'2’3’4, Satum‘u, and Neptune“, are of interest in chemical theory because of their
collision-induced absorption in the infrared and far-infrared region. Induced dipoles
produced during collisions within the compressed gas cause nearly all compressed gases
to absorb infrared radiation. This behavior is true even if the individual constituents of
the gas are infrared inactive6’7’8. The same is also true of CH4-N2, CH4-H2, and CH4-He
gas mixtures. Although the symmetric vibrational modes of CH4 and the bond stretch of
the homonuclear diatomics are infrared inactive, part of the collision-induced absorption
in the infrared region is due to interacting CH4-N2, CH4-Hz, and CH4-He pairsg'm'l 1.
Absorption in the far-infrared is due to translational and rotational modulation of the
collision-induced dipoles. These collision-induced dipoles originate from four sources:
multipolar induction, dispersion forces, intermolecular exchange forces, and frame
distortions“ "2.

Although theorists have been able to predict the collision-induced infrared (IR)
spectra and IR absorption intensities of other simple gas mixtures, they have been unable
to predict absorption intensities correctly in the range8 150 cm"1 — 450 cm'1 of the IR
spectra of CH4-N2, CH4-H2, and CH4-He mixtures“ "'3. The discrepancy between
experiment and theory is due to an incomplete treatment of the forces between molecules
in these gas mixtures. Previous theoretical examinations of other gas mixtures have taken

into consideration the more commonly studied causes of collision-induced absorption:

multipolar induction, dispersion forces, and intermolecular exchange forces, but the

158

influence of frame distortion has been investigated in only one case thus far: Weiss12
studied frame distortion effects in He collisions with CO2. He concluded that in CO2-He
gas mixtures frame distortions due to collisions between the He atom and CO2 molecule
generated induced dipoles of comparable size to exchange force-induced dipoles.

Collision-induced frame distortion in CH4 mixtures has the potential to produce
dipoles similar to those found in CO2-He gas mixtures”. In the CH4 molecule, the
bending constants are smaller than those in CO2. Frame distortion in CH4 will generate a
dipole, due in large part to the bond dipoles in CH4. In the equilibrium geometry, the
bond dipoles cancel due to symmetry, but the movement of any hydrogen from its
equilibrium position will generate a dipole". Not only will the frame distortion generate
a dipole in the CH4 molecule, but the CH4 dipole generated by distortion may in turn
influence the collision partner, adding to the electron cloud distortion in the collision
partner. It is expected that in CH4-He gas mixtures the frame-distortion dipole will be on
the order of 0.003-0.03 Debyes. In this chapter, we study frame distortion in CH4-N2,
CH4-H2, and CH4-He pairs due to interactions within the mixed pairs.
5.1.1 Approaches

To examine the frame distortion of CH4, we need to see the effects of approaching
molecules on the methane molecular frame. Figure 1 shows that methane has two
vertical planes of symmetry, A and B, such that plane A bisects the H1, H2, and C atoms,
and plane B bisects H3, H4, and C atoms. Using this diagram, one can describe the

collision paths used in this study.

159

 

 

 

 

 

 

 

 

 

Figure 1: Reflection Planes in CH.

In our work on collisions of helium and methane, we have positioned the helium
atom on the C2 axis, which bisects both HCH bond angles in methane. This axis is the
line labeled L, where the two planes intersect, in Figure 1. In the work on collisions of a
diatomic molecule with methane, we considered two approaches termed linear and
perpendicular. In the linear approach, the nuclei (they are both terminal) of the
homonuclear diatomic lie in both planes A and B along line L. This configuration is

shown in Figure 2 and Figure 3.

160

 

 

 

 

 

Figure 2: Diagram of Linear Approach, Viewed Looking Down on Plane A
Rotating the geometry in Figure 2 90 degrees about the axis bisecting the HCH bond

angles, we reach the view in Figure 3.

 

 

 

 

 

Figure 3: Diagram of Linear Approach, Viewed Looking Down on Plane B
In the perpendicular approach, plane B contains the nuclei of the diatomic, and
plane A bisects the homonuclear diatomic bond. This configuration is shown in Figure 4

and Figure 5.

161

 

 

 

 

 

Figure 4: Diagram of Perpendicular Approach, Viewed Looking Down on Plane A
Rotating the geometry in Figure 4 90 degrees about the axis bisecting the HCH bond

angles, we get Figure 5.

 

 

 

 

 

Figure 5: Diagram of Perpendicular Approach, Viewed Looking Down on Plane B
Using both paths of approach, the center of mass of the diatomic molecule, or the

helium atom, was moved closer to the carbon atom along the line L.

162

5.1.2 Methods

At specific separations, single-point calculations were made at a range of Hl-C-
H2 bond angles. By fitting the energy values versus the H1-C-H2 bond angle to a third-
order polynomial, the bond angle at the minimum of the potential well was determined;
this is the optimized bond angle at that particular separation of helium, or the diatomic,
from methane. Using this optimized bond angle, we calculated the dipole due to the
collision-induced frame distortion, as well as the energy change on the potential surface
due to the frame distortion. The dipole was calculated using equation (1), where
131,13, F3, and {'4 are unit vectors along the bonds in the CH4 molecule, and the potential
change was calculated from a potential energy surface generated by Marquardt”.

Bind =Bbond(fr+?2+f3+f4)- (1)
The bond dipole was determined as described in a paper by Hollenstein” et al.
5.1.3 Computational Details

Single point CCSD(T) calculations were performed with the Gaussian9816
software package using the Dunning aug-cc-pvtz basis sets for carbon'7"8, hydrogen”,

”"8, and helium"). Mathematica 4.020 was used to fit the raw data to the third-

nitrogen
order polynomial and to determine the optimized bond angle, as well as to determine the
potential and interaction energies.
5.2 CI-I4-N2

For this study, the experimental geometries for methane and nitrogen were used,
as provided by the 86th edition of the CRC Handbook of Chemistry and Physics“. The
methane bond lengths are RCH = 1.087 A, and all the HCH bond angles are equal to

109.47120 except for the Hl-C-H2 bond angle, which is varied as described below. For

163

nitrogen, the bond length is RNN = 1.9077 A. Using these parameters, we collected two
sets of single-point energy calculations: single-point energy values from the linear
approach and single point energy values from the perpendicular approach. The linear
approach is discussed first.
5.2.1 Linear Approach

The separation between the N2 center of mass and the methane center of mass was
set to distances of 5.0 A, 4.0 A, 3.5 A, 3.25 A, and 3.0 A. At each distance, the H1-C-H2
bond angle was varied from 1040 to 1140 in increments of 1°, also including the angle
109.47120. The minimum of the curve in each slice is the optimized Hl-C-H2 bond
angle for the respective nitrogen-methane separation. Using the aforementioned data
points, the minimum of the potential curve was determined. Plots of the total energy of

the system as a function of the H1-C-H2 bond angle can be seen in Figures 6-10.

164

 

CH4-N2 Linear Approach

104105106107108109110111112113114
-149.8212 1 ‘r 1 1* P P 1 i 1

-149.8213 1
-149.8214 7
-149.8215 ~ K
-149.8216 - __1
449.8217 — *5 an
449.8218 —
-149.8219 .
449.8220 .
-149.8221 —— -- ~-

 

Energy (H)

 

 

 

HCH Bond Angle

 

 

 

Figure 6: Energy as a Function of Bond Angle at 5.0 A Separation for Methane-
Nitrogen: Linear Approach

A first glance at Figures 6-8 may give the impression that the H1-C-H2 bond
angle is unaffected by the approach of the nitrogen molecule, but a closer examination
reveals the bond angle has changed from its equilibrium value. At a 5 A separation, the

optimized bond angle is 109.340 compared to the equilibrium bond angle of 109.47120.

165

 

CH4-N2 Linear Approach

104105106107108109110111112113114
449.8212 ‘ 4 i i 4 4 + i i 1
-149.8213
-149.8214 —
-149.8215 —
-149.8216 4
-149.8217 -
-149.8218 2
-149.8219 4
-149.8220 ~ ~---——— ~4—

 

Energy (H)

 

 

 

 

 

HCH Bond Angle

 

Figure 7: Energy as a Function of Bond Angle at 4.0 A Separation for Methane-
Nitrogen: Linear Approach

At a 4 A separation, the optimized bond angle is 109.190. This data suggests that

the bond angle is becoming more acute as the nitrogen approaches.

166

 

 

CH4-N2 Linear Approach

104105106107108109110111112113114
-149.8172 L i i i 4 i i i i 1
-149.8173
-149.8174 —
-149.8175 ~
-149.8176 ~
-149.8177 -1
-149.8178 -
-149.8179 ~
-149.8180 —
-149.8181 -
-149.8182 4

 

 

@521

Energy (H)

 

 

 

 

HCH Bond Angle

 

 

 

Figure 8: Energy as a Function of Bond Angle at 3.5 A Separation for Methane-
Nitrogen: Linear Approach

When the separation is decreased to 3.5 A, the bond angle Opens to an angle of

109.870.

167

 

CH4-N2 Linear Approach

104105106107108109110111112113114
4498096111: 41111:
-149.8098
-149.8100 ~
-149.8102 ~
-149.8104 .
-149.8106 2
-149.8108 —
-149.8110 —
-149.8112 -
-149.8114 ~ -———— ______

HCH Bond Angle

 

  
   
 

 

[+ 3.25 angj

 

Energy (H)

 

 

 

 

 

 

 

Figure 9: Energy as a Function of Bond Angle at 3.25 A Separation for Methane-
Nitrogen: Linear Approach

At 3.25 A the bond angle opens to 111.340, and at 3 A the bond angle opens to
114.830, as shown in Figure 10. (In this case, we expanded the set of the H1-C-H2.bond
angles to include angles that range from 104 to 119 degrees in one degree increments.)

The range shown in Figure 1018 1090 to 119°.

168

 

 

 

 

 

 

CH4-N2 Linear Approach
109110111112113114115116117118119
449.7941 1 I ' . I ' I '
-149.7943
A -149.7945 —
33,
5 449.7947 ~
2 449.7949 —
LIJ
-149.7951 2
449.7953 1
-149.7955 A——----— —- _-
HCH Bond Angle

 

 

 

Figure 10: Energy as a Function of Bond Angle at 3.0 A Separation for Methane-
Nitrogen: Linear Approach

It is evident that as the nitrogen molecule approaches methane, in the
configurations shown in Figures 2 and 3, the H1-C-H2 angle decreases at first, and then
increases as N2 approaches more closely. Table 1 lists the optimized bond angle at each

N2-CH4 separation.

 

 

 

 

Table 1. CH4-N2 Linear Approach
Distance (A) Distorted bond angle
5.00 109.34
4.00 109.19
3.50 109.87
3.25 l l 1.34
3.00 l 14.83

 

 

 

We have calculated the potential energy difference between the equilibrium CH4
structure and the distorted CH4 structure twice, once with a potential surface generated by

Marquardt and Quackl4 and again using Gaussian98. Also calculated twice was the

169

dipole moment, once using Gaussian98 and a second time using equation (1). These
results are listed in Table 2 where 11 is the dipole moment and V(YXY) is the potential

energy difference between the unperturbed and distorted methane molecule.

 

 

 

 

 

Table 2. Dipole and Potential from Molecular Distortion in the CH4-N2 Linear
Approach

Gaussian 98 Quack et a1” Gaussian 98 Quack et a1”
distance (A) u (a.u.) u (a.u.) V(YXY) (H) V(YXY) (H)
5.00 2.9356E-04 2.E-04 5.E-07 4.07070E-07
4.00 6.2890E-04 4.7E-04 2.0E-06 1.86641E-06
3.50 -8.9378E—04 -6.7E-04 4.0E-06 3.96732E-06
3.25 -4.2071E-03 -3.lE-03 8.79E-05 9.78689E-05
3.00 -1.2189E-02 —9.01E—03 7.014E-04 7.03584E-04

 

 

 

 

 

Table 2 shows that the potential energy differences due to the distortion of the methane
molecule are of the same magnitude. For the Marquardt potential energy surface and
Gaussian98, the difference between the values is less than 20 percent. These results seem
reasonable considering that the Marquardt potential is based on a MRD-CI potential
energy surface calculated using CC-PVDZ basis sets for carbon and hydrogen and that
the Gaussian98 values are only converged to 1x10‘7 Hartrees. The difference between the
dipole moment values is more noticeable; however, the simple bond-dipole model
provides a good approximation to the actual dipole.
5.2.2 Perpendicular Approach

Using the same procedure as used for the linear approach, we performed the same
collection of calculations for the perpendicular approach of the N2 molecule to the CH4

molecule.

170

 

 

 

11.11 t

 

CH4-N2 Perpendicular Approach

104105106107108109110111112113114
449.8210 1 111%4411
449.8211 3
449.8212 ~
449.8213 ~
449.8214 — K
449.8215 ~
449.8216 ~
449.8217 ~
449.8218 4
449.8219 1
449.8220 J- —— —-—- 4

HCH Bond Angle

 

Energy (H)

 

 

 

 

 

 

Figure 11: Energy as a Function of Bond Angle at 5.01 Separation for Methane-
Nitrogen: Perpendicular Approach

At 5A separation, N2 has very little effect on the geometry of the CH4 molecule. The
optimized angle is 109.480, which is only a hundredth of a degree different from the
magnitude of the equilibrium bond angle. In contrast, the magnitude of the equilibrium

bond angle of CH. changed by 0.130 for the linear approach of N2 at 5A.

171

 

CH4-N2 Perpendicular Approach

104105106107108109110111 112113114
449.8215'111111111
449.8216
449.8217 ~
449.8218 —
449.8219 —
449.8220 ~
449.8221 4
449.8222 ~
449.8223 ~
449.8224 4 - - —— - - __,_

HCH Bond Angle

Figure 12: Energy as a Function of Bond Angle at 4.0 A Separation for Methane-
Nitrogen: Perpendicular Approach

 

Energy (H)

 

 

 

 

 

 

In fact, when the distance is decreased to 4.0 A, the magnitude of the minimized bond
angle is almost equivalent to the magnitude of the tetrahedral angle, at 109.470, which

suggests that the bond angle is essentially unaffected by the N2 molecule at this distance.

172

 

 

CH4-N2 Perpendicular Approach

104105106107108109110111112113114
449.8214'111111111
449.82151
449.8216—
449.8217~
449.8218~
44982191
449.8220~
449.8221 1
44932221
-149.8223~——~~-—<~-——

 

 

[-O- 3.5 agg]

 

Energy (H)

 

 

 

 

HCH Bond Angle

 

 

 

Figure 13: Energy as a Function of Bond Angle at 3.5 A Separation for Methane-
Nitrogen: Perpendicular Approach

It is not until the distance is decreased to 3.5 A that the H-C-H angle expands to 109.680.

173

 

CH4-N2 Perpendicular Approach

104105106107108109110111112113114
-149.8203 ' l + [L l % 1L % w‘ 1‘
-149.8204
-149.8205 4
-149.8206 ~
-149.8207 ~
-149.8208 4
-149.8209 *

-149.821 -
-149.8211 -
-149.8212 -
-149.8213 1
-149.8214 ~ ——* * m~ ——

 

 

+ 3.25 angj

 

Energy (H)

 

 

 

 

HCH Bond Angle

 

 

 

Figure 14: Energy as a Function of Bond Angle at 3.25 A Separation for Methane-
Nitrogen: Perpendicular Approach

At 3.25 A the H-C—H angle is disturbed to 109.960, which becomes noticeable as shown

in Figure 14.

174

 

CH4-N2 Perpendicular Approach

104105106107108109110111 112113114
-149.8172L + l ‘fil l i l ll

 

-149.8174 ~
-149.8176 a
-149.8178 ~

 

 

-149.818 ~

Energy (H)

-149.8182 -
449.8184 -

 

 

 

 

-149.8186 ~ ——~~~ — -~-—- - —-
HCH Bond Angle

 

 

 

Figure 15: Energy as a Function of Bond Angle at 3.0 A Separation for Methane-
Nitrogen: Perpendicular Approach

Finally, at 3.0 A, the distortion angle is 110.810. In contrast to the distortion on
the linear approach, the bond angle distortion for the perpendicular approach is less
extreme. Also, squeezing of the bond angle as the N2 first approaches is seen in the
linear approach but not in the perpendicular approach. Listed in Table 3 are the Nz-CH4

distances and the distortion angles at these distances.

 

 

 

Table 3. CH4-N; Perpendicular Approach
distance (A) distorted bond angle
5.00 109.48"
4.00 109.47°
3.50 109.630
3.25 109.960
3.00 110.810

 

 

 

 

Listed in Table 4 are the dipoles and potential differences calculated using Gaussian98

and the Marquardt potential surface.

175

 

 

 

 

 

 

 

 

Table 4. Dipole and Potential from Molecular Distortion in the CH4-Nz
Perpendicular Approach

Gaussian 98 Quack et a1" Gaussian 98 Quack et a1"
distance (A) p (a.u.) p (a.u.) V(YXY) (H) V(YXY) (H)
5.00 0.E+00 -l.9699E-05 0.0000000 3.07966E-09
4.00 0.E+00 2.6861E—06 0.0000000 1.28879E-09
3.50 -3.E-04 -3.5564E-04 7 .E-07 5.95173E-07
3.25 -8.3E-04 -l.0958E-03 5.9E-06 5.84582E-06
3.00 -2.2E-03 -3.0091E-03 4.42E-05 4.43188E-05

 

 

Table 5 shows that the potential values agree more closely between methods than the
dipole moments. It is also noticeable that for the 5.0 A and 4.0 A values that the
magnitudes of the dipole and potential difference for Gaussian98 are so small that they
are reported as zero.
5.3 CH4-Hz

For this portion of the study, again, the experimental geometry of methane was
used, RCH = 1.087 A with all the HCH bond angles equal to 109.47120 except for the H1-
C-H2 bond angle, which is described below. For molecular hydrogen, the geometry used
was RH“ = 0.7414 A. These geometries are provided by the CRC Handbook of Chemistry
and Physics” . Using these parameters, two sets of single-point energy calculations were
again performed: single-point energy values from the linear approach and single-point
energy values from the perpendicular approach. The linear approach is discussed first.
5.3.1 Linear Approach

The separation between the H2 center of mass and the CH4 center of mass was set
to distances of 4.0 A, 3.5 A, 3.25 A, and 3.0 A, 2.75 A, 2.50 A, and 2.25 A. At each
distance, the Hl-C-HZ bond angle was varied from 1040 to 1140 in increments of 1°, also

including the angle 109.4712". The minimum of the curve in each slice is the optimized

176

Hl-C-HZ bond angle for the respective hydrogen-methane separations. Our results are

 

 

 

listed in Table 5.
Table 5. CH4-H2 Linear Approach
distance (A) distorted bond angle
4.00 109.53“
3.50 109.620
3.25 109.770
3.00 110.120
2.75 110.880
2.50 112.480
2.25 115.530

 

 

 

 

Table 5 shows that as the H2 molecule approaches the CH4 molecule, the bond angle

under investigation widens. Table 6 lists the potential differences and dipoles of the

 

 

 

 

 

 

 

distorted CHa molecule.
Table 6. Dipole and Potential from Molecular Distortion in CH4-H2 Linear
Approach

Gaussian98 Quack et a1I4 Gaussian98 Quack et a]14

distance (A) p. (a.u.) p (a.u.) V(YXY) (H) V(YXY) (H)
4.00 -8.E—05 -l.3164E-04 1.E-07 8.27122E—08
3.50 -2.E-04 -3.3323E-04 6.E-07 5.22746E-07
3.25 -5.lE-04 -6.6946E-04 2.2E-06 2.10298E-06
3.00 -1.lE-03 -l.4552E—03 1.04E-05 1.03475E-05
2.75 -2.4E-03 -3. 167 1E-03 4.90E-05 4.90498E-05
2.50 -5.04E-03 -6.7967E-03 2.224E-04 2.23087E-04
2.25 -1.02E-02 -l.3809E-02 8.952E—04 8.97715E-04

 

 

 

Agreement of the distortion potential energy differences is very good. The bond-dipole

model is useful as a first approximation, but it is not very accurate.

177

5.3.2 Perpendicular Approach
Using the same procedure as for the linear approach, similar sets of data were
calculated and collected for the perpendicular approach of the H2 molecule toward the

CH4 molecule. Table 7 lists the bond angle at each of the H2-CH4 separations.

 

 

 

Table 7. C114 -H2 Perpendicular Approach
distance distorted bond angle

(A) (degrees)

4.00 109.40

3.50 109.360

3.25 109.380

3.00 109.510

2.75 109.920

2.50 111°

2.25 113.390

 

 

 

 

Again, we see a shrinking of the bond angle as the diatomic first approaches CI-Ia,

but at 3.00 A, the angle begins to expand.

 

 

 

 

Table 8. Dipole and Potential from Molecular Distortion in the CH4-H2
Perpendicular Approach
Gaussian98 Quack et a1” Gaussian98 Quack et a1"
distance (A) p (a.u.) u (a.u.) V(YXY) (H) V(YXY) (H)
4.00 l.E-04 1.5934lE-04 2.E-07 1.21E-07
3.50 l.E-04 2.48828E-04 2.E-07 2.24E-05
3.25 2.E-04 2.04088E-04 2.E-07 1.97E-07
3.00 -8.E-05 -8.68616E—05 l.E-07 3.67E-08
2.75 -7.5E-04 -l.00600E-03 5.0E-06 4.96E-06
2.50 -2.6E-03 -3.438l6E-03 5.76E-05 5.78E-05
2.25 -6.57E-03 -8.8763lE-03 3.764E-04 3.78E-04

 

 

 

 

 

 

In Table 8, there is a stray value at 3.50 A separation which might be due to some
mathematical error in the potential energy surface. Otherwise, the potentials are close in

value and the dipoles are of the same magnitude.

178

5.4 CH4-He

For this portion of the study, again, the experimental geometry of methane was
used, RCH = 1.087 A with all the H-C-H bond angles equal to 109.47120 except for the
Hl-C-HZ bond angle, which is varied as described below. This geometry is provided by
the CRC Handbook of Chemistry and Physics” . Using these parameters we calculated
the energies for separations between the He atom and the methane center of mass of 4.0
A, 3.5 A, 3.25 A, and 3.0 A, 2.75 A, 2.50 A, and 2.25 A. At each distance, the mom
bond angle was varied from 1040 to 1140 in increments of 10 also including the angle
10947120. The minimum of the curve in each slice is the optimized Hl-C-H2 bond

angle for the respective helium-methane separation. The results are listed in Table 9.

 

 

 

Table 9. CH4-He Approach
distance distorted bond angle
(A) (degrees)
4.00 109.46"
3.50 109.460
3.25 109.470
3.00 109.550
2.75 109.810
2.50 110.550
2.25 112.350

 

 

 

 

The potential differences and dipole moments are shown in Table 10.

179

 

 

 

Table 10. Dipole and Potential from Molecular Distortion in CH4-He
Apm'oach

Gaussian98 Quack et a1” Gaussian98 Quack ct a1]4

distance (A) u (a.u.) p. (a.u.) V(YXY) (H) V(YXY) (H)
4.00 0.E+00 2.50696E-05 0.E+00 4.21E—09
3.50 0.E+00 2.50696E—05 0.E+00 4.21E-09
3.25 0.E+00 2.6861 lE-06 0.E+00 1.29E-09
3.00 -l.E-04 -l.76431E-04 2.E-07 1.48E-07
2.75 -5.5E-04 -7.59173E-04 3.E-06 2.70E-06
2.50 -1.8E-03 -2.42282E-03 3.E-05 2.87E—05
2.25 -4.80E-03 -6.50056E-03 2.E-04 2.04E-04

 

 

 

 

 

 

 

Again, Gaussian98 is not sensitive enough to measure the smaller potential
differences and dipoles. The values are of the same magnitude and the potential energies
agree more closely than the dipole values.

5.5 Discussion

We have proven that frame distortion occurs as N 2, H2, or He approaches a CH4
molecule. The change in the equilibrium bond angle produces a dipole moment on
methane, thus proving an additional mechanism for absorption in the far-IR or IR by
compressed gas mixtures containing methane. This is the principal conclusion from our
work.

In addition, we have found that the methane Hl-C-H2 bond angle first narrows
and then widens as N2 approaches in the “linear” configuration shown in Figure 2 and
Figure 3. At the separations of 4 A and 5 A, the bond angle decreases in magnitude from
the tetrahedral angle. This behavior may be due to the interaction of the negative
quadrupole of nitrogen and the slightly positive charge of the hydrogen atoms in the
methane molecule. In the presence of N2 at 3.5 A, 3.25 A, and 3.0 A, the methane bond

angle widens, probably due to steric effects associated with electron repulsion. On the

180

perpendicular approach of N2, we have found that the bond angle simply increases as the
intermolecular distance decreases.

However, this behavior does not mirror the methane-hydrogen approaches. The
reverse is seen. On the perpendicular approach, we find first a slight closing and then the
reopening of the bond angle, while the linear approach shows only a widening of the
bond angle. This behavior would follow since in the linear conformation the positive
portion of the hydrogen quadrupole is facing the C114 molecule; while in the
perpendicular conformation, the negative portion of the H2 quadrupole moment can
interact with the partially positively charged H nuclei in CH4. At shorter range, steric

effects predominate.

181

5.6 References

l. J. L.Hunt, J. D. Poll, D. Goorvitch, and R. H. Tipping, Icarus , 55, 63 (1983)
2. C. P. McKay, J. B. Pollack, and R. Courtin, Icarus 80, 23 (1989).
3. A. Borysow and C. Tang, Icarus 105, 175 (1993).

4. L. M. Trafton, Planetary Atomospheres: The Role of Collision-Induced Absorption
(World Scientific, Singapore, 1998), pp. 177—194, chap. 6.

5. A. Borysow, L. Frommhold, Astrophysical Journal, 304, 849, (1986).

6. H. L. Welsh, in MTP Int. Reveiew of Science-Physical Chemistry, Spectroscopy, Vol.
3, edited by A. D. Buckingham and D. A. Ramsey (Butterworths, London, 1972), Series,
1, Chap. 3, pp. 33—71.

7. L. Frommhold, Collision-induced Abosrbtion in Gases (Cambridge University Press,
Cambridge, NY, 1993).

8. M. Buser, L. J. Frommhold, J Chem. Phys. 122, 24301 (2005).

9. Phenomena Induced by Intermolecular Interactions, edited by G. Birnbaum (Plenum,
New York, 1985).

10. Collision- and Interaction-Induced Spectroscopy, NATO ASI series C, Vol. 452,
edited by G. C. Tabisz and M. N. Neuman (Kluwer, Dordrecht, 1995).

11. M. Buser, L. Frommhold, M. Gustafsson, M. Moraldi, M. H. Champagne, and K. L.
C. Hunt, J. Chem. Phys. 121, 2617 (2004).

12. S. J. Weiss, Phys. Chem. 86, 429 (1982).
13. X. Li, M. H. Champagne, and K. L. C. Hunt, J. Chem. Phys. 109, 8416 (1998).
14. Marquardt, M. and Quack, M. J Chem Phys, 109,10628 (1998).

15. H. Hollenstein, R. R. Marquardt, M. Quack, and M. A. Suhm, J Chem Phys 101,
3588, (1994).

16. Frisch, M. J .; Trucks, G. W.; Schlegel, H. B.; Scuseria, G. B.; Robb,M. A.;
Cheeseman, J. R.; Zakrzewski, V. G.; Montgomery, J. A., Jr.; Stratmann, R. B.; Burant, J.
C.; Dapprich, S.; Millam, J. M.; Daniels, A. D.; Kudin, K. N.; Strain, M. C.; Farkas, O.;
Tomasi, J.; Barone, V.; Cossi, M.; Cammi, R.; Mennucci, B.; Pomelli, C.; Adamo, C.;
Clifford, S.; Ochterski, J .; Petersson, G. A.; Ayala, P. Y.; Cui, Q.; Morokuma, K.;

182

Malick, D. K.; Rabuck, A. D.; Raghavachari, K.; Foresman, J. B.; Cioslowski, J .; Ortiz, J.
V.; Stefanov, B. B.; Lin, G.; Liashenko, A.; Piskorz, P.; Komaromi, I.; Gomperts, R.;
Martin, R. L.; Fox, D. J .; Keith, T.; Al-Laham, M. A.; Peng, C. Y.; Nanayakkara, A.;
Gonzalez, C.; Challacombe, M.; Gill, P. M. W.; Johnson, B.; Chen, W.; Wong, M. W.;
Andres, J. L.; Gonzalez, C.; Head-Gordon, M.; Replogle, E. S.; Pople, J. A. Gaussian 98,
Revision A.6; Gaussian, Inc., Pittsburgh, PA, 1998.

17. TH. Dunning, Jr. J. Chem. Phys. 90, 1007 (1989).

18. R.A. Kendall, T.H. Dunning, Jr. and RJ. Harrison, J. Chem. Phys. 96, 6769 (1992).
19. DE. Woon and TH. Dunning, Jr. J. Chem. Phys. 100, 2975 (1994).

20. Mathematica 4.0, Wolfram Research Inc., Champaign IL, 1999.

21. David R. Lide, ed., CRC Handbook of Chemistry and Physics, Internet Version 2006,
<http://www.hbcpnetbase.com>, Taylor and Francis, Boca Raton, FL, 2006.

183

APPENDIX A

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0000.000 0000.0 0002.000 00000.0 0800000- 008000.00- 0000.020- 6.0005200
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0...00 00:00 00.000 000.0.0 0.000000- 0000.00.00- 00.00008- E00003
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00000000- 00000.0 2.00.. 00000000- 000000 00000.. 005009000
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00.000 0.0000 00.000 00000.0 .0.0000.00.- 000000000- 08003? 00300-8000
00.000 00000.0 00.00 00000.0 0000000.00.- 000.000.00- 0000000000: 0050-8
00.000 0.0000 00.000 00.000 0000000.00.- 0000000.00- 0000000000- 0050-8
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00.000 00000.0 00.000 .0800 00000..0.- 00000.00- 00000000- .00-00.33000
00.000 00000.0 .0.000 00000.0 00000..0.- 00000.00- 00000000- 60.005350
00.000 00000.0 00.000 00000.0 00000.0. - 0000000- .0000000- .00-0.05.0200
00.000 00000.0 00.00 00000.0 00000..0.- 00000.00- .0000000- .00-0.02.0".
00.000 00000.0 00.00 00000.0 00000.07 00000.00- 00000000- .00-00.50-8000
0.000 00000.0 00.000 00000.0 00000..0.- 0.00000- 0.000.000- 0050-09000
0...00 00000.0 00.000 00000.0 00000.0.- 00000.00- 00000.000- 0.500-803
00.000 000.0.0 00.000 00000.0 00000.0.- 00000.00- 00000000- 0900-09000
00.000 00000.0 00.000 00000.0 00000. .0.- 0000000- 00000000- .00-00.598
00.000 00000.0 00.000 00000.0 00000..0.- 00000.00- 00. 00.000- 00060-8
00.000 00000.0 ...000 00000.0 00000..0.- 00.00.00- 00000000- 00500-8
00.000 00000.0 0. .000 000.0.0 00000..0.- 00.00.00- 00000000- 00500-8
39508— moobudm 3:508— moobua: moo-Pa: mop—Ham moo-Pa: 30m mmmam
00 00 00 00 .000. .50. 02 .000. .52 0 00.2.0. .02 ...-.0000

 

 

 

 

 

 

 

 

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00000.000- 0.0000 00000.0 00.00.. 00000000- 00000.0 00000.0 00000.. 002.590....
.0000000- 00000.0 .000.0 00000.. 00000000- 00.000 00000.0 00000.. 0.50.0903
00000000- 00000.0 00000.0 00000.. 00.00000- 00000.0 00.000 00000.. 0050-090...
00000000- 00000.0 00.000 0.000.. .0000.000. 00000.0 00000.0 0.000.. 00.00
00000000- .0000.0 00.000 00000.. 00000000- 00000.0 0.0000 .0000. 0000
00000000- 00000.0 00000.0 .0.00.. 0.000.000- 00000.0 00000.0 00000.. 00>..-8
0.000.000- 00000.0 00000.0 0.000.. 00000000- .8000 00.000 00000.. 0.50-8
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00.000 00.000 00.000 00000.0 0000000..0.- .00000000- 0000000000- 00208-0...
00.000 00000.0 .0.000 0.0000 00000000.- 0000.00.00- 00.0000000- 0.50-8.0...
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00.000 00000.0 00.00 00000.0 00000000.- 000000000- 0000000000- .00-00.-..
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208

 

APPENDIX B

CHAPTER 3 RAW DATA

209

 

Table l. SCF Quadrupoles of HCCH

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Property Calculations Field Gradient calculations
Basis sets Property 6) Field Grad ('9 0.0001 -0.0001 0
cc-pVDZ 5.21 121 5.21 120 46.82520 46.82625 46.82572
cc-pV'I‘Z 5.41863 5.41865 46.84884 46.84993 46.84938
cc-pVQZ 5.43455 5.43455 46.85385 46.85494 46.85439
cc-pVSZ 5.44436 5.44435 46.85497 46.85606 46.85551
cc-pV6Z 5.44423 5.44420 46.8551 1 46.85620 46.85565
aug-cc-pVDZ 5.46892 5.46890 46.82790 46.82899 46.82844
aug-cc-pV’I‘Z 5.45771 5.45770 46.84961 46.85070 46.85016
aug-cc-pVQZ 5.44559 5.44555 46.85402 46.8551 1 46.85456
aug-cc-pVSZ 5.44453 5.44450 46.85499 46.85608 46.85553
aug-cc-pV6Z 5.44352 5.44350 46.8551 1 46.85620 46.85566
d-aug-cc-pVDZ 5.45567 5.45565 46.82825 46.82934 46.82879
d-aug-cc-pVTZ 5.44564 5.44565 46.84971 46.85080 46.85025
Table 2. CCSD(T) Quadrupoles of HCCH
Field Gradient Calculations
Basis sets Field Grad 9 0.0001 -0.0001 0
cc-pVDZ 4.50785 47.10868 47.10958 47.10913
cc-pVTZ 4.80095 47.18708 47.18804 47.18756
cc-pVQZ 4.84280 47.20880 47.20977 47.20929
cc-pVSZ 4.86080 47.21527 47.21624 47.21576
cc-pV6Z 4.86520 47.21747 47.21844 47.21795
aug-cc-pVDZ 4.77215 47.1 1908 47.12004 47.1 1956
aug-cc-pV’I‘Z 4.84505 47.19161 47.19258 47.19161
aug-cc-pVQZ 4.85700 47.21048 47.21 145 47.21096
aug-cc-pVSZ 4.86345 47.21597 47.21695 47.21646
aug-cc-pV6Z 4.86615 47.21791 47.21888 47.21839
d-aug-cc-pVDZ 4.75840 47.12003 47. 12098 47. 12050
d-aug-cc-pVTZ 4.83335 47.19201 47.19297 47.19249
Table 3. DFT(B3LYP) Quadrupoles of HCCH
Property calculations Field Gradient Calculations.
Basis sets Property ('9 Field Grad 9 0.0001 -0.0001 0
cc-pVDZ 4.61713 4.61715 47.281 17 47.28209 47.28163
cc-pVTZ 4.90139 4.90140 47.31166 47.31264 47.31215
cc-pVQZ 4.89813 4.89810 47.31741 47.31839 47.31790
cc-pVSZ 4.90903 4.90915 47.31933 47.32031 47.31982
cc-pV6Z 4.90791 4.91 155 47.31968 47.32066 47.32017
aug-cc-pVDZ 4.87387 4.87390 47.28640 47.28737 47.28688
aug-cc-pV’I‘Z 4.91726 4.91715 47.31266 47.31365 47.31315
aug-cc-pVQZ 4.87216 4.87225 47.31767 47.31864 47.31815
aug-cc-pVSZ 4.90775 4.90680 47.31939 47.32037 47.31988
aug-cc-pV6Z 4.89025 4.89015 47.31988 47.32086 47.32037
d-aug-cc-pVDZ 4.86497 4.86435 47.28709 47.28806 47.28757
d-aug-cc-pVTZ 4.89342 4.89400 47.31290 47.31388 47.31339
d-aiggc-pVQZ 4.89071 4.88965 47.31768 47.31865 47.31816

 

 

 

 

 

 

210

 

 

 

Table 4. Quadrupoles of FCCF

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

SCF Quadrpoles of FCCF
Property calculations Field Gradient Calculations.
Basis sets Property (9 Field Grad 6) 0.0001 -0.0001 0
cc-pVDZ -0.97258 -0.97255 -274.47696 -274.47676 -274.47686
cc-pVTZ -0.84572 -0.84570 -274.56734 -274.56717 -274.56726
cc-pVQZ -0.89863 -0.89860 -274.58903 -274.58885 -274.58894
cc-pVSZ -0.92249 -0.92250 -274.5947l -274.59453 -274.59462
aug-cc-pVDZ - 1 .03349 -1 .03 345 -274.49362 -274.49341 -274.4935 1
aug-cc-pVTZ -0.94298 -0.94295 -274.57137 -274.571 19 -274.57128
aug-cc-pVQZ -0.91302 -0.91305 -274.59007 -274.58989 -274.58998
aug-cc-pVSZ -0.91678 -0.91680 -274.59489 -274.59471 -274.59480
d-aug-cc-pVDZ -0.85468 -0.85465 -274.49569 -274.49552 -274.49560
d-aug-cc-pVTZ -0.9067 1 -0.90670 -274.57214 -274.57 196 —274.57205
d-aug-cc-pVQZ -0.91185 -0.91185 -274.59034 -274.59016 -274.59025
d-aug-cc-pVSZ -0.91572 -0.91570 —274.59491 -274.59473 -274.59491
Basis sets CCSD(T) Quadrupoles of FCCF
cc-pVDZ -0.40895 ~275. 12056 -275. 12048 -275. 12052
cc-pVTZ -0.45690 -275.39150 -275.3914l -275.39l45
cc-pVQZ -0.5 8250 -275.47592 -275.47580 -275.47585
cc-pVSZ -0.63430 -275.50423 -275.50410 -275.50416
aug-cc-pVDZ -0.78600 -275. 17600 -275. 17585 -275. 17592
aug-cc-pVTZ -0.64850 -275.41 182 -275.41 169 -275.41 176
aug-cc-pVQZ 063565 -275.48398 -275.48386 -275.48392
aug-cc-pVSZ -0.64005 -275.50746 -275.50733 -275.50739
d-aug-cc-pVDZ -0.63670 -275. 18089 ~275.18076 -275.18082
d-aug—cc-pVTZ -0.61565 -275.4l468 -275.4l455 -275.41461
d-aug-cc-pVQZ -0.63315 -275.48517 -275.48504 -275.48510
d-am-pVSZ -0.63885 -275.50798 ~275.50785 -275.50791
Basis sets DFT(B3LYP) Quadrupoles of FCCF
cc-pVDZ -0.05337 -0.05335 ~275.64631 -275.64630 -275.646298
cc-pVTZ -0.27828 -0.27830 -275.74266 -275.74261 —275.742634
cc-pVQZ -0.44190 -0.44190 -275.76648 -275.76639 -275.766429
cc-pVSZ -0.5 1039 05 1080 -275.77397 -275.77386 -275.773912
aug-cc-pVDZ -0.60025 -0.60045 -275.66684 -275.66672 -275.666775
aug-cc-pVTZ -0.54044 -0.53910 -275.74753 -275.74742 -275.747474
aug—cc-pVQZ -0.5 1631 -0.5 1800 -275.76799 -275.76789 -275.767939
aug-cc-pVSZ -O.52176 —0.52450 -275.77428 -275.774 1 7 -275.774224

 

 

 

 

 

 

211

 

 

Table 5. Quad rupoles of ClCCCl

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

SCF Quadrupoles of CICCCI
Property calculations Field Gradient Calculations.
Basis sets Property (E) Field Grad 6) 0.0001 -0.0001 0
cc-pVDZ 3.93167 3.93165 -994.64186 —994.64265 -994.64225
cc-pV'I‘Z 3.93428 3.93425 -994.69678 -994.69756 -994.69716
cc-pVQZ 3.92530 3.92525 -994.71019 -994.71097 -994.71057
cc-pVSZ 3.93531 3.93520 -994.71417 -994.7 1496 99471455
aug-cc-pVDZ 4.05863 4.05860 -994.64677 -994.64758 -994.64716
aug-cc-pVTZ 3.85683 3.85675 -994.69794 -994.69871 -994.69832
aug-cc-pVQZ 3.87828 3.87815 -994.71048 -994.71 126 -994.71086
aug-cc-pVSZ 3.89883 3.89875 -994.71425 -994.71503 ~994.71463
d-aug-cc-pVDZ 3.93806 3.93790 -994.64829 -994.64908 -994.64868
d-aug-cc-pVTZ 3.82185 3.82175 -994.69831 -994.69907 -994.69868
d-aug—cc-pVQZ 3.86852 3.86840 -994.71062 -994.71 139 -994.71099
d-aug-cc-pVSZ 3.90064 3.90050 -994.71427 -994.71505 -994.71465
Basis sets CCSD(T) Quadrupoles of CICCCI
cc-pVDZ 3.73365 -995.20710 -995 .20785 -995.20747
cc-pVTZ 3.85235 -995.43348 -995.43425 -995.43386
cc-pVQZ 3.78920 -995.50274 -995.50350 -995.5031 1
cc-pVSZ 3.80535 -995.52592 -995.52668 ~995.52629
aug-cc-pVDZ 3.77545 -995.24246 -995.24322 —995.24283
aug-cc-pVTZ 3.79695 -995.44800 -995.44876 —995.44837
aug~cc-pVQZ 3.75945 -995.50830 -995.50905 -995.50867
aug-cc-pVSZ 3.76720 -995.52878 -995.52954 —995.52915
d-aug-cc-pVDZ 3.66855 -995.24726 -995.24799 —99S.24762
d-aug-cc-pVTZ 3.74915 -995.45044 -995 .45 1 l9 -995.45080
d-aug-cc—pVQZ 3.74780 -995 .50935 -995.51010 -995 .50972
d-aug-ccopVSZ 3.76825 -995.52949 -995.53025 -995.52986
Basis sets DFT(B3LYP) Quadrupoles of CICCCI
cc-pVDZ 4.10418 4.10370 -996.37300 -996.37382 -996.37341
cc-pVTZ 3.99562 3.99600 -996.43262 -996.43341 -996.43301
cc-pVQZ 3.91567 3.91670 -996.44733 -996.4481 1 -996.44771
cc-pVSZ 3.89775 3.89495 -996.45723 -996.45801 -996.45761
aug-cc-pVDZ 3.82461 3.82360 -996.38013 —996.38089 -996.38050
aug-cc—pVTZ 3.76884 3.76740 -996.43419 -996.43494 -996.43455
aug-cc-pVQZ 3.77873 3.77875 -996.44784 -996.44859 996.4482]
aug-cc-pVSZ 3.79085 3.79055 -996.45746 -996.45822 -996.45783

 

 

 

 

 

212

 

 

Table 6. Quadrupoles of Linear QC;

 

SCF Quadrupoles of Linear Li2C2

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Property calculations Field Gradient Calculations.
Basis sets Property ('9 Field Grad (9 0.0001 -0.0001 0
cc-pVDZ 35.921 13 35.92090 -90.55783 -90.56501 -90.56142
cc-pV'I‘Z 36.1 1907 36.1 1890 -90.58353 -90.59075 -90.58714
cc-pVQZ 36.36871 36.36855 -90.5 8902 -90.59630 —90.59266
cc-(SZ-QZ) 36.42224 36.42210 -90.58997 -90.59725 -90.59361
cc-(6Z-QZ) 36.48779 36.48765 -90.59016 -90.59745 -90.59380
aug-cc-pVDZ 36.65677 36.65660 -90.56656 -90.57389 -90.57022
aug-cc-pVTZ 36.51928 36.51915 -90.58514 -90.59245 -90.58879
aug—cc-pVQZ 36.50940 36.50925 -90.58933 -90.59663 -90.59297
aug(SZ—QZ) 36.50828 36.50820 -90.59010 -90.59740 -90.59375
M6Z-QZ) 36.50784 36.50770 -90.59021 -90.59751 90.59385
Basis sets CCSD(T) Quadrupoles of Linear Li2C2
cc-pVDZ 34.77075 -90.84527 -90.85222 -90.84874
cc-pVTZ 35. 19595 -90.92645 -90.93349 -90.92997
cc-pVQZ 35.49270 -90.94952 -90.95662 -90.95306
cc-(SZ-QZ) 35.57165 -90.95594 -90.96305 -90.95949
cc-(6Z-QZ) 35.64430 -90.95843 -90.96555 -90.96199
aug-cc-pVDZ 35.74955 -90.86463 -90.87178 -90.86820
aug-cc-pV'I‘Z 35.65855 -90.93329 -90.94042 -90.93685
aug-cc-pVQZ 35.66620 -90.95 196 -90.95910 -90.95552
aug(SZ-QZ) 35.67495 -90.95730 -90.96443 -90.96086
jung-QZ) 35.67865 -90.95927 -90.96641 -90.96283
Basis sets DFT(B3LYP) Quadrupoles of Linear Li2C2
cc—pVDZ 34.23046 34.22950 -91. 1 1920 -91. 12604 -91. 12261
cc-pVTZ 34.54174 34.54115 -91.14847 -9l.15537 -91.15191
cc-pVQZ 34.79379 34.79340 -91. 15494 -91. 16190 -91. 15841
cc-(SZ-QZ) 34.84033 34.83875 -91 . 15668 -91 . 16365 -91. 16016
cc-(6Z-QZ) 34.89727 34.89780 —9 1 . 15708 -91. 16406 -91. 16057
aug-cc-pVDZ 35.08634 35.08545 -91. 12795 -91. 13497 -91. 13146
aug-cc-pVTZ 34.92038 34.91845 -91 . 15007 -91. 15705 -91. 15356
aug-cc-pVQZ 34.92334 34.92280 -9 1 . 15532 -91. 16230 -91. 15880
aug(SZ-QZ) 34.92483 34.92415 -91 . 15690 -91 . 16388 -9 1. 16039
aug(6Z-QZ) 34.63372 34.91735 -91. 15727 -91. 16425 -9 l . 16075

 

213

 

 

Table 7. 9.2.2. Component of Planar Li2C2 Quadrupole

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

SCF SCF 9,, Component of Planar Li2C2 Quadrupole
Property calculations Field Gradient Calculations.
Basis sets Property (9 Field Grad (9 0.0001 -0.0001 0
cc-pVDZ 26.42364 26.42345 -90.57130 -90.57658 -90.57394
cc-pVTZ 26.26211 26.26195 -90.59421 -90.59946 -90.59683
cc-pVQZ 26.28749 26.28735 -90.59907 -90.60433 -90.60169
cc-(SZ-QZ) 26.30007 26.29995 -90.59991 -90.60517 -90.60254
cc-(6Z-QZ) 26.30757 26.30740 -90.60005 -90.60531 -90.60267
aug-cc-pVDZ 26.42695 26.42670 -90.57578 -90.58107 -90.57842
aug-cc-pVTZ 26.31244 26.31235 -90.59501 -90.60028 -90.59764
aug-cc-pVQZ 26.31155 26.31140 90.59920 -90.60446 -90.60183
aug(SZ-QZ) 26.30941 26.30925 -90.59997 -90.60524 -90.60260
flwZ-QZ) 26.30894 26.30880 -90.60009 -90.60535 -90.60272
Basis sets CCSD(T) 6,, Component of Planar Li2C2 Quadrupole
cc-pVDZ 25.83225 -90.861 1 1 -90.86628 -90.86369
cc-pVTZ 25.83985 -90.94040 -90.94557 -90.94298
cc-pVQZ 25.90455 -90.96264 -90.96782 -90.96522
cc-(SZ-QZ) 25.93655 90.96884 -90.97403 -90.97143
cc-(6Z-QZ) 25.95380 -90.97124 -90.97643 -90.97383
aug-cc-pVDZ 26.04780 -90.87582 -90.88103 -90.87842
aug-cc-pVTZ 25.95015 -90.94588 -90.95107 -90.94847
aug-cc-pVQZ 25.95380 -90.96459 -90.96978 -90.96718
aug(SZ-QZ) 25.95645 -90.96993 -90.97512 -90.97252
me-QZ) 25.95780 -90.97194 -90.97713 -90.97453
Basis sets DFT(B3LYP) 92.1 Component of Planar L12C2 Quadrupole
cc-pVDZ 25.37237 25.37200 -9l.13161 -91.13668 -91.13414
cc-pV'I‘Z 25.20428 25.20235 -9l.15839 -91.16343 -91.16091
cc-pVQZ 25.26415 25.26320 -9l.16484 -91.16989 -91.16736
cc-(SZ-QZ) 25.28720 25.28715 —91.16660 -91.17 166 -91. 16913
cc-(6Z-QZ) 25.30170 25.29990 -91.16696 -91.17202 -91.16948
aug-cc-pVDZ 25.41788 25.41980 -91. 13640 -91. 14148 -91.13893
aug-cc-pVTZ 25.29770 25.29675 91.15965 -91.l6471 -91.16217
aug-cc-pVQZ 25.30683 25.30675 -91.16516 -91. 17022 -9l.16768
aug(SZ-QZ) 25.30699 25.30540 -91.16674 -91.17180 -91.16926
Lung-QZ) 25.30645 25.30510 -91.16710 -91.17216 -91.16963

 

214

 

 

Table 8. 9,1 Component of Planar Li2C2 Quadrupole

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

SCF SCF 9,, Commnent of Planar Li2C2 Quadrupole
Property calculations Field Gradient Calculations.
Basis sets Property 9 Field Grad (9 0.0001 -0.0001 0
cc-pVDZ - 16.06928 - 16.06925 -90.57555 -90.5723 -90.57394
cc-pVTZ —16.18292 -16. 18290 -90.59845 -90.59522 -90.59683
cc-pVQZ -16.23168 -16.23165 -90.60332 -90.60007 -90.60169
cc-(SZ-QZ) -16.26629 -16.26630 -90.60417 -90.60092 -90.60254
cc-(6Z-QZ) - 16.28308 -16.28310 -90.60431 -90.60105 -90.60267
aug-cc-pVDZ -16. 19942 -16. 19945 -90.5 8004 -90.57680 -90.57842
aug—cc-pV'I‘Z - 16.28427 - 16.28430 -90.59927 -90.59602 -90.59764
aug-cc-pVQZ - 16.27610 - 16.27615 -90.60346 -90.60020 -90.60183
aug(SZ-QZ) - 16.27387 - 16.27395 -90.60423 -90.60098 -90.60260
fl(6Z-QZ) - 16.27356 - 16.27365 -90.60435 -90.60109 -90.60272
Basis sets CCSD(T) 9,, Component of Planar Li2C2 Quadrupole
cc-pVDZ - 15.87020 -90.86528 -90.8621 1 -90.86369
cc-pVTZ - 16. 16610 -90.94460 -90.94137 -90.94298
cc-pVQZ - 16.26935 -90.96686 -90.96360 -90.96522
cc-(SZ-QZ) - 16.33000 -90.97307 -90.96980 -90.97 143
cc-(6Z-QZ) -16.36865 -90.97547 -90.97220 -90.97383
aug-cc-pVDZ -16.33085 —90.88006 -90.87680 -90.87842
aug-cc-pVTZ - 16.37980 -90.9501 1 -90.94683 -90.94847
aug-cc-pVQZ - 16.37065 -90.96882 -90.96555 -90.96718
aug(SZ-QZ) - 16.36970 -90.97417 -90.97089 -90.97252
flwZ-QZ) -16.36990 -90.97617 -90.97290 -90.97453
Basis sets DFT(B3LYP) (9,, Component of Planar Li2C2 Quadrupole
cc-pVDZ - 15.40464 - 15.40470 -9 l . 13569 ~91 . 13260 -91. 13414
cc-pVTZ -15.69964 -15.69950 -91.16248 -91.15934 -9l.16091
cc-pVQZ -15.81193 -15.81105 -91.16895 -91.16578 -9l.16736
cc-(SZ-QZ) -15.86851 - 15.86735 -91 . 17072 -91. 16754 -91.16913
cc-(6Z-QZ) - 15.90546 - 15.90465 -91 . 17108 —9 1 . 16790 -91 . 16948
aug-cc-pVDZ -15.83735 -15.83840 -91.14052 -91.13735 -91.13893
aug-cc-pVTZ -15.91843 -15.91650 -91.16377 -91.l6059 -91.16217
aug-cc-pVQZ -15.91138 -15.90880 -91.16928 -91.16610 —91.16768
aug(SZ-QZ) - 15.90995 - 15.91070 -91. 17086 -91 . 16767 -91 . 16926
_a_ug(62-QZ) 45.90946 -15.91120 -91.17122 -91.l6804 -91.16963

 

215

 

 

Table 9. Quadrupoles of Linear Na2C2

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

SCF Quadrupole of Linear Na2C2
Property calculations Field Gradient Calculations.
Basis sets Property (9 Field Grad (9 0.0001 0.0001 0
cc-pVDZ 47.51559 47.51435 -399.34140 -399.35091 -399.34615
cc-pVTZ 48.47527 48.47410 -399.37163 -399.38133 -399.37647
cc-pVQZ 48.69636 48.69525 -399.37900 -399.38874 -399.38386
cc-(SZ-QZ) 48.69327 48.69215 -399.37999 -399.38973 -399.38485
cc-(6Z-QZ) 48.70123 48.70010 -399.38019 -399.38993 -399.38505
aug-cc-pVDZ 49.04743 49.04625 -399.34785 -399.35766 ~399.35275
aug-cc-pVTZ 48.85053 48.84945 -399.37366 -399.38343 ~399.37853
aug-cc-pVQZ 48.70649 48.70540 -399.37953 -399.38927 -399.38439
Basis sets CCSD(T) Quadru le of Linear NazCz
cc-pVDZ 44.471 10 -399.63410 -399.64299 -399.63852
cc-pVTZ 46.00460 -399.71897 -399.72817 -399.72356
cc-pVQZ 46.44525 ~399.74362 -399.75291 -399.74825
cc-(SZ-QZ) 46.5 1530 69945024 -399.75954 -399.75487
cc-(6Z-QZ) 46.55900 ~399.75278 -399.76209 -399.75742
aug-cc-pVDZ 46.45305 -399.65245 -399.66174 -399.65708
aug-cc-pVTZ 46.59030 -399.72672 -399.73603 -399.73136
aug-cc-pVQZ 46.56340 -399.74660 -399.75591 -399.75 124
aug(SZ-QZ) 46.56630 -399.75197 -399.76128 -399.75661
flng-QZ) 46.54900 -399.75407 -399.76338 -399.75871
Basis sets DFT(B3LYP) Quadrupole of Linear Na2C2
cc-pVDZ 42.43210 42.42575 40059468 40060316 40059890
cc—pVTZ 43.29595 43.28935 400.62965 400.6383 1 400.6396
cc-pVQZ 43.5 1357 43.50595 40064037 40064907 400.64470
cc-(SZ-QZ) 43.52969 43.52410 40064239 40065109 40064672
cc-(6Z-QZ) 43.53921 43.52600 40064320 40065191 40064753
aug-cc-pVDZ 43.89563 43.88895 400.60179 400.6105? 400.60615
aug-cc-pVTZ 43.67435 43.66385 40063181 40064054 40063615
fl-cc-pVQZ 43.55906 43.55635 40064153 40065025 400.64586

 

216

 

 

Table 10. 92.21 Component of Planar Na2C2 Quadrupole

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

SCF SCF 92 Component of Planar Na2C2 Quadrupole
Property calculations Field Gradient Calculations.
Basis sets Property 6) Field Grad (9 0.0001 -0.0001 0
cc-pVDZ 36.36619 36.36545 -399.36385 —399.371 12 -399.36748
cc-pVTZ 36.42261 36.42180 -399.39214 -399.39942 -399.39577
cc-pVQZ 36.40388 36.40320 -399.39861 -399.40589 -399.40224
cc-(SZ-QZ) 36.39636 36.39570 -399.39946 -399.40674 -399.40309
cc-(6Z-QZ) 36.37383 36.37315 -399.39971 -399.40699 -399.40335
aug-cc-pVDZ 36.70472 36.70400 -399.36737 -399.37471 -399.37103
aug-cc-pVTZ 36.47252 36.47185 -399.393 19 -399.40048 -399.39683
aug-cc-pVQZ 36.33939 36.37450 -399.39891 ~399.40619 -399.40255
aug(SZ-QZ) 36.36187 36.36120 -399.39971 -399.40698 -399.40334
fl6Z-QZ) 36.34475 36.34410 -399.39994 -399.40721 -399.40357
Basis sets CCSD(T) (9z_z Component of Planar Na 2C2 Quadrupole
cc-pVDZ 35.33375 -399.65565 -399.66272 -399.65917
cc-pVTZ 35.76320 -399.73951 -399.74666 -399.74308
cc-pVQZ 35.855 85 -399.76308 -399.77026 -399.76666
cc-(SZ-QZ) 35.88545 -399.76948 -399.77666 -399.77306
cc-(6Z-QZ) 35.88460 -399.77202 -399.77919 -399.77560
aug-cc-pVDZ 36.03205 -399.67052 -399.67772 -399.6741 1
aug-cc-pVTZ 35.93175 -399.74568 -399.75287 -399.74927
aug-cc-pVQZ 35.87685 -399.76548 -399.77265 -399.76906
aug(SZ-QZ) 35.87775 —399.77085 -399.77803 -399.77443
flwZ-QZ) 35.86695 -399.77293 ~399.7801 1 39947651
Basis sets DFT(B3LYP) 6);; Component of Planar Na 2C2 Quadrupole
cc-pVDZ 33.46820 33.46450 400.61235 400.61904 400.61568
cc-pVTZ 33.50279 33.49895 400.64601 400.65271 400.64935
cc-pVQZ 33.5 1679 33.51595 400.65617 400.66287 400.65950
cc-(5Z-QZ) 33.52760 33.52655 400.65843 400.66513 400.66176
cc-(6Z-QZ) 33.53103 33.52985 400.65922 400.66593 400.66256
aug-cc-pVDZ 33.89736 33.89145 400.61673 400.62351 400.6201 1
aug-cc-pVTZ 32.97706 33.61345 400.64765 400.65437 400.65099
aug-cc-pVQZ 33.45609 33.45060 400.66397 400.67066 400.66730
aug(SZ-QZ) 33.52892 33.53235 400.65924 400.66594 400.66257
__aung-QZ) 33.52248 33.52465 400.65974 400.66645 400.66308

 

 

 

 

217

 

 

 

 

Table 11. But Component of Planar Na 2C; Quadrupole

 

SCF (9,. Component of Planar Na 2C2 Quadrupole

 

Property calculations

Field Gradient Calculations.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Basis sets Property (9 Field Grad (9 0.0001 -0.0001 0
cc-pVDZ -21.35167 41.35155 -399.36962 -399.36535 -399.36748
cc-pVTZ -21.52509 -21.52495 -399.39793 -399.39362 -399.39577
cc-pVQZ -21.58987 -2 1 .5 8980 -399.40440 -399.40009 -399.40224
cc-(SZ-QZ) -21.60388 -21.60385 -399.40526 -399.40094 -399.40309
cc-(6Z-QZ) -21.59570 -21.59570 -399.40551 -399.401 19 -399.40335
aug-cc-pVDZ -21.68500 41.68505 -399.37320 -399.36887 -399.37103
aug-cc-pVTZ ~21.65632 —21.65635 ~399.39900 -399.39467 ~399.39683
aug-cc-pVQZ -2 1.57 139 -2 1 .5 8985 -399.4047 1 -399.40039 -399.40255
aug(SZ-QZ) -2 1 .5 8440 -21.58440 -399.40550 -399.401 19 -399.40334
fing-QZ) -21.57575 -2 1 .575 80 -399.40573 -399.40142 -399.40357
Basis sets CCSD(T) 91.2 Component of Planar Na 2C2 Quadrupole
cc-pVDZ -20.86890 -399.66126 -399.65709 —399.65917
cc-pVTZ -21.41470 —399.74522 -399.74094 -399.74308
cc-pVQZ -21.60855 -399.76883 -399.76451 -399.76666
cc-(SZ-QZ) -21.65005 -399.77523 -399.77090 -399.77306
cc-(6Z-QZ) -21.66595 ~399.77777 -399.77344 -399.77560
aug-cc-pVDZ 41.70495 -399.67629 -399.67 195 -399.6741 1
aug-cc-pV’I‘Z -21.71720 -399.75145 -399.74710 -399.74927
aug-cc-pVQZ -21.67125 -399.77 123 -399.76690 -399.76906
aug(SZ-QZ) -21.67290 -399.77661 -399.77227 -399.77443
au (6Z-QZ) -21.66680 -399.77869 -399.77435 -399.77651
Basis sets DFT(B3LYP) (9,_z Comment of Planar Na 2C2 Quadrupole
cc-pVDZ -19.66378 -19.66335 400.61765 400.61372 400.61568
cc-pVTZ 40.01005 40.00925 400.65136 400.64736 400.64935
cc-pVQZ -20. 15 104 -20. 14980 400.66153 400.65750 400.65950
cc-(SZ-QZ) -20. 17747 -20. 17605 400.66379 400.65975 400.66176
cc-(6Z-QZ) -20. 19390 -20. 19765 400.66459 400.66055 400.66256
aug-cc-pVDZ «20.31 136 -20.30910 400.62215 400.61808 400.6201 1
aug-cc-pVTZ 49.92045 -20.27225 400.65303 400.64898 400.65099
aug-cc-pVQZ -20. 17 168 -20.21010 400.66286 400.65882 400.66083
aug(SZ-QZ) -20.20684 40.20820 400.66460 400.66056 400.66257
au (6Z-QZ) -20.20483 -20.20495 400.66511 400.66107 400.66308

 

218

 

APPENDIX C

CHAPTER 5 RAW DATA

219

 

Table 1. CPL-N2 Perpendicular Approach using aug-cc-pvtz basis set for 5.0 and 4.0A Separations

 

 

 

 

 

5.0 A separation 4.01 separation

Angle Energy Angle Energy

Hartrees Hartrees
104 -149.821128 104 -149.821584
105 -149.8213782 105 -149.821834
106 -149.8215775 106 -l49.822033
107 -149.8217278 107 -149.822183
108 -149.8218247 108 -149.82228
109 -149.8218733 109 -149.822328
109.4712 -149.821879 109.4712 -149.822334
109.48 -149.821879 109.47 -149.822334
110 -149.8218723 109.5 -149.822334
111 -149.821822 110 -149.822327
112 -149.8217228 111 -l49.822276
113 -149.8215751 112 -149.822177
114 -149.8213791 113 -149.822029
114 —149.821833

 

 

 

 

 

Table 2. CH4-N2 Perpendicular Approach using aug-cc-pvtz basis set for 3.5 and 3.25 Angstrom

 

 

 

 

 

 

Separations
3.5 A separation 3.25 A separation .

Angle Energy Angle Energy

Hartrees Hartrees
104 -149.8214405 104 -149.820398
105 -149.821699 105 -149.8206748
106 -149.8219064 106 -149.8209002
107 -149.8220646 107 -149.821076
108 -149.8221694 108 -149.8211983
109 -149.8222256 109 —149.8212715
109.4712 -149.8222348 109.4712 -149.8212886
109.6 -149.8222354 109.96 —149.8212946
109.63 -149.8222354 l 10 -149.8212946
110 -149.822232 111 -149.8212679
111 -149.822189 112 -149.8211917
112 -149.8220969 113 -149.8210665
113 -149.8219561 114 -149.8208925

114 -149.8217668

 

 

 

220

 

 

 

 

Table 3. CPL-N2 Perpendicular Approach using aug-cc-pvtz basis set for 3.0 A Separations

 

 

3.0 A separation

 

 

 

 

 

Angle Energy

Hartrees
104 -149.817287
105 -149.817612
106 ~149.817884
107 ~149.818105
108 ~149.818273
109 -149.81839
109.4712 -149.818428
110 -149.818457
110.8 -149.818473
110.81 -149.818473
111 -149.818472
112 -149.818437
113 -l49.818352
114 -149.818218

 

 

Table 4. CPL-N2 Linear Approach using aug-cc-pvtz basis set for 5.0 and 4.0 A Separations

 

 

 

 

 

 

 

5.0 A separation 4.0 A separation
Angle Energy Angle Energy
Hartrees Hartrees
104 -149.8213045 104 -149.8212801
105 -149.8215469 105 -149.8215138
106 -149.8217386 106 -149.8216971
107 -149.8218813 107 -149.8218317
108 -149.821971 108 -149.82l9l41
109 -149.8220125 109 -149.8219483
109.34 -149.8220153 109.19 -149.8219493
109.4712 -149.8220148 109.2 -149.8219493
110 -149.8220045 109.4712 -149.8219474
111 -149.8219474 110 -149.8219335
112 -149.8218415 111 -149.8218698
113 -149.8216872 112 -149.8217576
114 -149.8214848 113 -149.8215973
114 4498213892

 

 

 

221

 

 

 

Table 5. CPL-N2 Linear Approach usinfig aug-cc-pvtz basis set for 3.5 and 3.25 A Separations

 

 

 

 

 

3.5 A separation 3.25 A separation

Angle Energy Angle Energy

Hartrees Hartrees
104 -149.8172923 104 -149.8097708
105 -149.8175600 105 -149.8101183
106 -149.8177772 106 -149.8104143
107 -149.8179450 107 -149.8106596
108 -149.8180613 108 -149.8108530
109 -149.8181288 109 -149.8109963
109.4712 -149.8181435 109.4712 -149.8110463
109.87 -149.8181473 110 -149.8110891
109.9 -149.8181473 111 -149.8111318
110 -149.8181469 111.34 -149.8111350
111 -149.8181159 111.4 -149.8111350
112 -149.8180362 112 -149.8111248
113 -149.8179080 113 ~149.8110682
114 -149.8177316 114 -149.8109623
115 -149.8108076
116 -149.8106042
117 -149.8103525
118 -l49.8100529
1 19 -149.8097056

 

 

 

 

222

 

 

 

 

 

 

 

 

Table 6. CPL-N2 Linear Approach using aug-cc-pvtz basis set for 3.0 A Separations
3.0 A separation
Angle Energy
Hartrees

99 -149.7885192
100 -149.7893414
101 -149.7901081
102 -149.7908196
103 -149.7914759
104 - 149.7920775
105 - 149.7926245
106 -149.7931173
107 -149.7935562
108 -l49.7939410
109 -149.7942725
109.4712 -149.7944103
1 10 -149.7945508
1 1 1 -149.7947763
1 12 -149.7949491
1 13 -149.7950697
114 -149.7951382
114.8 -149.7951558
114.83 -149.7951558
115 -149.7951551
116 -149.7951206
1 17 -149.7950350
1 18 -149.7948988
119 -149.7947122

 

 

Table 7. CPL-H2 Perpendicular Approach using aug-cc-pvtz basis set for 4.0 and 3.5 A Separations

 

4.0 A separation

3.5 A separation

 

 

 

 

 

 

 

Angle Energy Angle Energy

Hartrees Hartrees
104 41.6130504 104 41.6130603
105 41.6132963 105 41.613304
106 41.6134914 106 41.613497
107 41.6136375 107 41.613641
108 41.6137305 108 41.6137321
109 41.6137751 109 41.6137749
109.4 41.613779 109.36 41.6137781
109.4712 41.6137789 109.4 41.6137781
110 41.6137701 109.4712 41.6137779
111 41.613716 110 41.6137682
112 41.613613 111 41.6137123
113 41.6134615 112 41.6136077
114 41.6132618 113 41.6134546
114 41.6132534

 

223

 

 

 

Table 8. CH4-1-12 Perpendicular Approach using aug-cc-pvtz basis set for 3.25 and 3.0 A Separations

 

 

 

 

 

 

 

3.25 A separation 3.0 A separation
Angle Energy Angle Energy
Hartrees Hartrees

104 41.6127939 104 41.6119134
105 41.6130384 105 41.6121644
106 41.6132322 106 41.6123647
107 41.613377 107 41.6125158
108 41.6134691 108 41.6126143
109 41.6135127 109 41.6126642
109.38 41.6135163 109.4712 41.6126705
109.4 41.6135163 109.5 41.6126706
109.4712 41.6135162 109.51 41.6126706
110 41.613507 110 41.6126646
111 41.6134521 111 41.6126159
112 41.6133484 112 41.6125184
113 41.6131964 113 41.6123723
114 41.6129962 114 41.612178

 

 

Table 9. CH4-H2 Perpendicular Approach using aug-cc-pvtz basis set for 2.75 and 2.5 A Separations

 

 

 

 

 

 

2.75 A separation 2.5 A separation
Angle Energy Agle Energy
Hartrees Hartrees
104 41.6095553 104 41.6038628
105 41.6098284 105 41.6041942
106 41.6100504 106 41.6044737
107 41.6102228 107 41.6047025
108 41 .6103426 108 41 .6048783
109 41.6104132 109 41.6050039
109.4712 41.6104289 109.4712 41.6050445
109.9 41.6104342 110 41.6050789
109.92 41 .6104342 1 1 1 41.6051035
110 41.6104341 112 41.6050781
111 41.6104054 113 41.605003
112 41.6103277 114 41.6048785
113 41.610202
114 41.610027

 

 

 

224

 

 

 

 

Table 10. CPL-H2 Perpendicular Approach usingaug-cc-pvtz basis set for 2.25 A Separations

 

 

 

 

 

2.25 A separation
Angle Energy
Hartrees

104 41.5910364
105 41 .59 15061
106 41 .591921 8
107 41 .5922841
108 41.5925918
109 41 .5928469
109.4712 41 .5929459
1 10 41 .5930493
111 41.5931991
1 12 41.5932968
1 13 41 .5933428
1 13.39 41.5933468
113.4 41.5933468
1 14 41 .5933373
1 15 41 .5932808

 

 

 

Table 11. C1-14-H2 Linear Approach using aug-cc~pvtz basis set for 4.0 and 3.5 A Separations

 

 

 

 

 

 

 

4.0 A separation 3.5 A separation

Angle Energy Lngle Energy

Hartrees Hartrees
104 41.613106 104 41.6131301
105 41.6133592 105 41.6133881
106 41.6135615 106 41.6135952
107 41.613714? 107 41.613753
108 41.6138145 108 41.6138573
109 41.6138659 109 41.6139131
109.4712 41.6138728 109.4712 41.613922
109.5 41.6138729 109.6 41.6139227
109.53 41.6138729 109.62 41.6139227
110 41.6138676 110 41.6139191
111 41.6138199 111 41.613875?
112 41.6137236 112 41.6137832
113 41.613578 113 41.6136419
114 41.6133844 114 41.6134523

 

 

 

 

225

 

 

Table 12. CPL—H; Linear Approach using aug-cc-pvtz basis set for 3.25 and 3.0 A Separations

 

 

 

 

 

3.25 A separation 3.0 A separation
Angle Energy Angle Energy:
Hartrees Hartrees
104 41.6128024 104 41.6117048
105 41.613068? 105 41.6119908
106 41.6132838 106 41.6122251
107 41.6134495 107 41.6124095
108 41.6135615 108 41.6125398
109 41.613624? 109 41.6126209
109.4712 41.613637 109.4712 41.6126411
109.7? 41.6136394 110 41.6126516
109.8 41.6136393 110.1 41.6126519
110 41.613638 110.11 41.6126519
111 41.613601? 110.12 41.6126519
112 41.6135161 111 41.6126322
113 41.6133816 112 41.6125632
114 41.6131985 113 41.612444?
114 41.6122773

 

 

 

 

 

Table 13. CH4-H2 Linear Approach using aug-cc-pvtz basis set for 2.75 and 2.5 A Separations

 

 

2.75 A separation

2.5 A separation

 

 

 

 

 

 

 

 

AngLe Energy Angle Energy

Hartrees Hartrees
104 41.6087869 104 41.6018304
105 41.6091178 105 41.6022598
106 41 .6093959 106 41.6026339
107 41.6096231 107 41.6029548
108 41 .6097951 108 41.603218
109 41.6099169 109 41.6034289
109.4712 41.6099554 109.4712 41.6035074
1 10 41.6099872 1 10 41.6035862
110.88 41.6100069 111 41.6036904
110.89 41.6100069 112 41.6037419
110.9 41 .6100069 112.48 41.603748
111 41.6100066 112.49 41.603748
112 41.6099753 112.5 41.603748
113 41.609893? 113 41.6037411
1 14 41 .6097622 1 14 41 .6036886

 

226

 

 

Table 14. CPL-H2 Linear Approach using aug-cc-pvtz basis set for 2.25 A Separations

 

2.25 A separation

 

 

 

 

 

 

 

 

 

 

 

Angle Enggy
Hartrees
104 41.586458?
105 41.5870911
106 41.5876632
107 41.5881776
108 41.5886288
109 41.5890235
109.4712 41.589185
110 41.5893603
1 1 1 41 .5896398
112 41.5898626
113 41.5900292
114 41.5901403
115 41 .5901964
115.5 41.590204
1 15.53 41.5902041
116 41.5901981
117 41.590146
Table 15. CH4-He Approach using aug-cc- pvtz basis set for 4.0 and 3.5 A Separations
4.0 A separation 3.5 A separation
Agle Energy Angle Energy
Hartrees Hartrees
104 43.3408519 104 43.340872?
105 43.3411011 105 43.341121?
106 43.3412995 106 43.3413198
107 43.3414488 107 43.3414688
108 43.3415449 108 43.3415646
109 43.3415925 109 43.3416121
109.46 43.3415978 109.46 43.3416172
109.4712 43.3415978 109.4712 43.3416172
109.5 43.341597? 109.5 43.3416171
110 43.3415906 110 43.3416099
111 43.3415394 111 43.3415584
112 43.3414392 112 43.3414581
113 43.3412905 113 43.3413092
114 43.3410935 114 43.341112

 

 

 

 

227

 

 

 

Table 16. CH4-He Approach using aug-cc-pvtz basis set for 3.25 and 3.0 A Separations

 

 

 

 

 

 

 

 

3.25 A separation 3.0 A separation
ALgle Energy flgle EnergL
Hartrees Hartrees

104 43.340??? 104 43.3403936
105 43.341026? 105 43.3406476
106 43.3412256 106 43.340850?
10? 43.3413754 107 43.3410045
108 43.3414721 108 43.3411051
109 43.3415203 109 43.3411574
109.47 43.3415258 109.4712 43.341164?
109.4712 43.3415258 109.5 43.3411648
109.5 43.3415258 109.55 43.3411648
110 43.3415189 110 43.3411599
111 43.3414682 111 43.3411132
112 43.3413686 112 43.3410169
1 13 43.3412204 1 13 43.3408722
l 14 433410239 114 43.3406792

 

 

Table 17. CPL—He Approach using aug-i-pvtz basis set for 2.75 and 2.5 A Separations

 

 

 

 

 

2.75 A separation 2.5 A separation
Angle Energy Angle Energy
Hartrees Hartrees
104 43.3392281 104 43.3360642
105 43.3394966 105 43.3363742
106 43.3397138 106 43.3366321
107 43.3398813 107 43.3368391
108 43.3399956 108 43.3369924
109 43.3400609 109 43.3370956
109.4712 43.340074] 109.4712 43.337125?
109.8 43.3400772 110 43.3371478
109.81 43.3400772 110.5 433371549
1 10 43.3400762 1 10.54 43.337155
111 43.3400419 110.55 43.337155
112 43.3399582 111 43.3371495
113 43.3398255 112 43.3371009
114 43.3396441 113 43.3370025
1 14 43.3368546

 

 

 

 

228

 

 

 

Table 18. CPL-He Approach using aug-cc-pvtz basis set for 2.25 A Separations

 

2.25 A separation

 

 

 

 

Angle Energy
Hartrees
104 43.3280358
105 43.3284556
106 43.3288205
107 43.3291319
108 43.3293874
109 43.32959
109.4712 43.3296649
1 10 43.3297394
1 1 1 43.329835?
1 12 43.329879
1 12.3 43.3298825
1 12.34 43.3298825
l 12.35 43.3298825
113 43.3298712
1 14 43.32981 1

 

 

229

 

 

 

   

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