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Opportunity lnetltulon W ”3-9.1 ENERGIES, POLARIZATION, AND POLARIZABILITIES OF MOLECULES INTERACTING AT LONG OR INTERMEDIATE RANGE By Xiaoping Li A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Chemistry 1 994 EXER( Th POlanzal: range Centres}- order R fOT COllis momenu Slmbols ABSTRACT EN ERGIES, POLARIZATION, AND POLARIZABILITIES OF MOLECULES INTERACTING AT LONG OR INTERMEDIATE RANGE by Xiaoping Li This thesis presents results for the energies, interaction-induced polarization and polarizabilities of a set of molecules (two or three) interacting at long or intermediate range. Collision-induced dipoles and polarizabilities have been determined for pairs of centrosymmetric linear molecules interacting at long range. The analysis is complete to order R‘7 in the intermolecular separation for collision-induced dipoles and to order R”6 for collision-induced polarizabilities. For each of the polarization mechanisms, angular momentum algebra has been used to obtain compact results in terms of 6-j and 9-j symbols. Numerical results have been obtained for the polarizabilities of the pairs H2...H2, qu-Nz, and N2---N2. The nonlocal polarizability density 0t(r, r';(o) and hyperpolarizability densities such as B(r, r', r";co 1,0) 2) play an important role in this research. The linear response tensor a(r,r';co) gives the polarization P(r,oa) induced at point r in a molecule by the electric field F(r',co) acting at another point r'. The hyperpolarizability density B(r, r’, r";(o 1 ,co 2) describes the distribution of the hyperpolarizability in molecules. A method of computing the B hyperpolarizability density has been developed based on its connection to a set of auxiliary functions (DLM(k,m) that determine van der Waals interaction eni expressions ft the damped d asechnd. Three-b interacting at intermolecula the interactie and polarizat tablished l effects, The '1 static reactio and penurha POlallZaIiOn‘ densities are from its inter lowest-order mOleCular Cl The th ground‘Stan An analhllca llle rESUlIS {:1 lS ShOWn lli. Zer0~ While threfibodE interaction energies. For the hydrogen atom in the Is state, the method yields analytical expressions for the B hyperpolarizability density. The results have been used to compute the damped dispersion-induced dipole in one hydrogen atom, due to its interactions with a second. Three-body energies and the interaction-induced polarization for molecules interacting at long or intermediate range have been analyzed, assuming that intermolecular exchange effects are negligible. The analysis is complete to third order in the interactions. Distinct polarization mechanisms that contribute to three-body energies and polarization have been identified and clear physical interpretations have been established. These include dispersion, induction, and combined dispersion-induction effects. The induction effect fiirther contains three different polarization mechanisms: the static reaction field, third-body field, and hyperpolarization. Both reaction-field theory and perturbation analysis are used to derive the equations for three-body energies and polarization, giving equivalent results. Polarizability density and hyperpolarizability densities are employed to characterize the nonlocal response of a molecule to the fields from its interacting partners. Thus the results include the direct modifications of the lowest-order electrostatic, induction, and dispersion effects, due to overlap of the molecular charge distributions. The three-body dispersion energy is calculated for a model system, interacting ‘ ground-state hydrogen atoms, to illustrate how overlap modifies three-body interactions. An analytical expression for the damped triple-dipole dispersion energy is obtained and the results are compared to those fiom the long-range Axilrod-Teller-Muto expression. It is shown that the damped dispersion energy converges as interatomic distances approach zero, while the Axilrod-Teller-Muto equation diverges. The angular dependence of the three-body dispersion energy is also changed appreciably, due to overlap of the charge distributions among interacting hydrogen atoms. I vs 0 direction a My 1 Harrison 1' and Dr .\‘. My ' friendship Sper these stud ACKNOWLEDGMENTS I would like to thank my research advisor, Dr. Katharine L. C. Hunt, for her direction and encouragement in this research and in the preparation of this thesis. My thanks are extended to the members of my guidance committee, Dr. James Harrison (who served as second reader), Dr. Richard Schwendeman, Dr. Jeffrey Ledford, and Dr. Milton Smith for their time and kindly help. My thanks also go to members of Hunt's group for their stimulating discussions and fiiendship. Special thanks are given to my wife, Yang Hong, for her constant support during these studies. iv LIST OF 1 LIST OF 1 l. I..\TRO ll COLL LINE.1 ”l D) b) b) TABLE OF CONTENTS LIST OF TABLES viii LIST OF FIGURES ix I. INTRODUCTION 1 II. COLLISION-INDUCED DIPOLES FOR PAIRS OF CENTROSYMMETRIC LINEAR MOLECULES AT LONG RANGE 7 2.1 Introduction 7 2.2 A Spherical Tensor Analysis 8 2.3 Summary and Discussion 12 III. COLLISION-INDUCED POLARIZABILITIES FOR PAIRS OF CENTROSYMMETRIC LINEAR MOLECULES AT LONG RANGE: THEORY ANDNUMERICALRESULTS FOR H2---H2, H2---N2,ANDN2---N2 16 3.1 Introduction , 16 3.2 Changes in Polarizability Induced By Long-Range Interactions Between Two Centrosymmetric Linear Molecules 22 3.3 Collision-Induced Changes in Scalar Polarizabilities 33 3.4 Collision-Induced Changes in Anisotropic Polarizabilities 35 3.5 Approximations for Dispersion Coefficients 39 3.6 Numerical Results for qu-Hz, qu-Nz, and N2---N2 45 3.7 Summary and Discussion 48 IV. CALCULATIONS OF CONTRACTED HYPERPOLARIZABILITY DENSITIES 57 4.] Introduction 57 4.2 A Method of Calculating Contracted Susceptibility Densities 60 4.3 Application to Hydrogen Atoms 63 4.4 Summary and Discussion 69 V DISPERf EFFECI~ 5.1 Int 52D: 5.3 A; \I .\'O.\'AI AN AP] MM 62 \1 6681 VII. NON: DIPO, 7.1\ 7.2.\' 73T 74T 75T 76] VIII. EFH. EXL 8.11 8.2. V. DISPERSION DIPOLES AND QUADRUPOLES FOR PAIRS OF ATOMS: EFFECTS OF OVERLAP DAMPING 5.1 Introduction 5.2 Damped Dispersion Dipoles and Quadrupoles 5.3 Application to Hydrogen Atoms VI. NONADDITIVE THREE-BODY ENERGIES, DIPOLES, AND FORCES: AN APPROACH BASED ON NONLOCAL RESPONSE THEORY 6.1 Introduction 6.2 Nonadditive Three-body Dispersion Energy 6.3 Nonadditive Induction and Induction-Dispersion Energies 6.4 Nonadditve Dispersion Dipoles, Classical Induction, and Induction- Dispersion Dipoles 6.5 The Electrostatic Interpretation of Nonadditive Three-body Forces on Nuclei 6.6 Summary and Discussion VH. NONADDITIVE THREE-BODY INTERACTION ENERGIES AND VIII. DIPOLES: PERTURBATION ANALYSIS 7.1 Nonadditve Three-body Interaction Energy at Second Order 7.2 Nonadditve Three-body Energy at Third order: "Circuit" Tenns 7.3 Third-body Perturbation of Two-body Interactions 7.4 Three-body Polarization at Second Order 7.5 Three-body Polarization: "Circuit" Terms 7.6 Three-body Polarization: "Noncircuit" Terms EFFECTS OF OVERLAP DAMPING ON THREE-BODY DISPERSION ENERGIES 8.1 Introduction 8.2. Dispersion Energy for Three Interacting S-state Atoms 8.3. Application to Interacting Hydrogen Atoms 73 73 76 81 93 93 95 101 105 110 113 120 120 122 125 130 134 138 151 151 152 158 Appendix A Appendix 8 Appendix C Appendix D Appendix E Appendix A Appendix B Appendix C Appendix D Appendix E vii 166 168 174 178 183 3.1. 3.2. 3.3. 3.4. 3.5. 3.6. 3.7. 3.8. Expansion coefficients for y-tensor components, Eq. (18). Expansion coefficients for B-tensor components, Eq. (19). LIST OF TABLES Coefficients for A2202L in Eq. (46). Coefficients for AzzzatL in Eq. (49). Coefficients for A242xL in Eq. (50). Molecular properties (in a.u.) used to calculate collision-induced polarizabilities. Dispersion energy coefficients (in a.u.) used to calculate collision-induced polarizabilities. 27 27 38 38 39 43 44 Test of the constant ratio approximation for dispersion polarizability coefficients by comparison with accurate ab initio results (Ref. 39) for H2---H2. Results in a.u. for the coeflicients of R—6. . Long-range contributions to collision-induced polarizabilities A018, Aug, and Act? for Hz-nHz, H2---N2, and N2---N2. . Local dispersion dipole coefficients and quadrupole coefficients (in a.u.) for H(ls)-~H(1s). viii 54' 81' 83' LIST OF FIGURES 5.1. The dipole moment induced in one hydrogen atom by dispersion interactions with a second, displaced by a distance R along the z axis. (1) u5(R) defined by Eq. (49), and (2) 117 (R) defined by Eq. (50). 87 5.2. The damping functions x6(R) and x7 (R) defined by Eq. (51) and Eq. (50), respectively. (1) x6 (R), and (2) x7 (R). 88 5.3. The leading term in the local quadrupole moment 96 induced in one hydrogen atom by dispersion interactions with a second, displaced by a distance R along the z axis. 89 5.4. The leading term in the pair quadrupole moment of a H- - - H given by Eq. (56). 90 8.1. The triple-dipole dispersion energy of interacting ground-state hydrogen atoms in the geometry of an equilateral triangle with R the length of a side. (1) the damped dispersion energy from Eq. (36), and (2) the undamped form from Eq. (39). 162 8.2. The damping function X111 for the triple-dipole dispersion energy of interacting ground-state hydrogen atoms in the geometry of an equilateral triangle. 163 8.3. The triple-dipole dispersion energy for interacting ground-state hydrogen atoms in the geometry of an isosceles triangle, as a function of the angle 9 between the two equal sides R. (1) AE(3)(1,1, 1) with R = 3, (2) AE(3)(1,1, 1) with R = 4, and (3) 11139130”) with R = 4. 164 ix This 11: induced char electrons be Polarizabilit In C h; Symmetric I kg in the ‘ determined nonuniform of these po “”115 of 6- indUced Iin Chap 30 for pa and Seton induction and dISPe lite).- are E 11113 Work CHAPTER I INTRODUCTION This thesis is concerned with the theory of intermolecular forces and the interaction- induced changes in molecular properties such as dipoles and polarizabilities. The work focuses on two or three molecules interacting at long or intermediate range. The intermolecular separation is assumed to be sufficiently large that overlap between molecular charge distributions is weak and the effects due to exchange of electrons between molecules are negligible. Changes in the energy, polarization, and polarizability of the interacting molecules are analyzed. In Chapter II, the collision-induced dipoles are determined for pairs of centro- symmetric linear molecules interacting at long range. The analysis is complete to order R'7 in the intermolecular separation. Through this order, the collision-induced dipoles are determined by quadrupolar [1] and hexadecapolar induction [2-5] , effects of nonuniformity in the local fields [3-5], back-induction [4], and dispersion [4, 6-8]. For all of these polarization mechanisms, spherical tensor analysis yields the dipole coefficients in terms of 6-j and 9-j symbols. The results are expected to be usefirl in simplifying collision- induced line shape analyses. Chapter 111 gives the long-range contributions to the collision-induced polarizability A01 for pairs of centrosymmetric linear molecules through order R'6, including the first- and second-order dipole-induced-dipole (DID) interactions [9], higher-multipole induction, effects of the nonuniformity of the local fields [10, 11], hyperpolarization [12], and dispersion [12-16]. The results have been obtained using spherical tensor analysis and they are given in terms of 6-j and 9-j symbols. The polarization mechanisms included in this work give rise to isotropic rototranslational Raman scattering and to simultaneous rotational transitions on two interacting molecules; both are collision-induced phenomena. Transitions \ treated in th' multipole m1 contributior estimate the polarization der Waals n C hap: B hweTpOI; and quadm; 10 a set of a quantum m the work )1 In CI for Pfilrs o and quadn reduce to the few“ the local QUadmp< the lead] Transitions with A] up to i4 are produced by the R"5 and R"6 polarization mechanisms treated in this work. For the pairs H2---H2, H2---N2, and Nz-HNZ, ab initio results for multipole moments and susceptibilities have been used to evaluate the classical induction contributions to A01, and a constant ratio approximation [4, 12, 14] has been used to estimate the dispersion contributions. The relative contributions to Act from different polarization mechanisms are discussed for R values ~0.5 - 1 a.u. outside the isotropic van der Waals minimum of the pair potential. Chapter IV presents a method of calculating the B hyperpolarizability density and the B hyperpolarizability density, which determine the damped dispersion-induced pair dipole and quadrupole of interacting molecules, respectively [17]. These densities are connected to a set of auxiliary functions denoted by LM(k,co) that have been determined via a quantum mechanical variational method [18-21]. For the hydrogen atom in the Is state, the work yields analytical results for the B and B hyperpolarizability densities. In Chapter V, the damped dispersion-induced dipoles and quadrupoles are computed for pairs of S-state atoms. It is shown that the equations for damped dispersion dipoles and quadrupoles are convergent as the interatomic separation R goes to zero, while they reduce to the corresponding equations from the multipole expansion at long range. Using the results given in Chapter IV, analytical expressions are obtained for the leading term in the local dispersion dipole x7D7R'7 and the leading term in the local dispersion quadrupole x6 M6R'6 for a pair of ground-state hydrogen atoms; here D-, and M6 are the leading long-range dipole and quadrupole coefficients [22-26], respectively, and x7 and x6 are the damping firnctions. The functions 31-, and x6 are distinct, but both of them drop to ~0.85 at the van der Waals minimum for H2 in the triplet state (R = 7.85 a.u.). The leading three dispersion dipole coefficients and the leading three dispersion quadrupole coefficients are also estimated and they compare well with the results fi'om ab initio calculations [24-26]. C harm: the interactii interacrions that contribi indua‘ion. a htpemolan; nonuniform accounts fo dispersion e body forces 190(1) forces derivatises [29, 30] F1 dispersion 1 Oflhat nucl ihfll is, they Charge dist conjecture Chapter VI contains an analysis of nonadditive three-body interaction energies and the interaction-induced polarization. The analysis is complete through third order in the interactions. A reaction-field method is used to identify various polarization mechanisms that contribute to three-body energies and polarization. These include dispersion [27, 28], induction, and combined dispersion-induction effects. The polarizability density and hyperpolarizability densities are used to describe the nonlocal response of a molecule to a nonuniform external field or a local field due to neighboring molecules. Thus this approach accounts for the direct modifications of the lowest-order electrostatic, induction, and dispersion effects due to overlap of the molecular charge distributions. Nonadditive three- body forces are also analyzed in this chapter. An electrostatic interpretation of the three- body forces acting on nuclei is given based on a chain of relations between property derivatives with respect to nuclear coordinates and linear and nonlinear response tensors [29, 30]. For a group of three molecules A, B, and C, it is shown that the three-body dispersion force acting on a nucleus in molecule A results from the electrostatic attraction of that nucleus to the dispersion-induced polarization of the electrons in molecule A itself; that is, the three-body dispersion force on a nucleus in A depends only on the perturbed charge distribution of molecule A. This generalizes Hunt's proof [31] of F eynman's conjecture [32] on the origin of two-body dispersion forces to three-body forces. In contrast to the dispersion forces, the three-body induction and induction-dispersion forces on a nucleus in A depend not only on the perturbed charge density of A, but also on that of B and C. In Chapter VII, the time-independent perturbation theory is used to derive equations for three-body energies and polarization. The results are shown to be equivalent to those obtained in Chapter VI from the reaction-field method. Finally, in Chapter VIII the three-body dispersion energy [27, 28] is calculated for interacting ground-state hydrogen atoms. The calculation includes the direct effects of short-range C3" [33] is obtain the three 8101 determined short-range charge overlap but not exchange. The damped triple-dipole dispersion energy [33] is obtained as an analytical function of the interatomic distances and the geometry of the three atoms. The radial and angular dependence of the dispersion energy is determined. References [1]]. VanK [2113 R Co'r [3].\l Moor [4]] E Bo‘r [S]G_Bim‘r [6]K L C [7]L Gala: 1810 P (1. ML Silbc- [1011 Bar- IHM D 1' “le L 1I II3IAD| KLt [I4]K L [151K L “61PM 2932 References [1] J. Van Kranendonk, Physica 24, 347 (1958). [2] E. R. Cohen and G. Bimbaum, J. Chem. Phys. 66, 2443 (1977). [3] M. Moon and D. w. Oxtoby, J. Chem. Phys. 84, 3830 (1986). [4] J. E. Bohr and K. L. C. Hunt, J. Chem. Phys. 87, 3821 (1987). [5] G. Bimbaum, A. Borysow, and A. Buechele, J. Chem. Phys. 99, 3234 (1993). [6] K. L. C. Hunt, Chem. Phys. Lett. 70, 336 (1980). [7] L. Galatry and T. Gharbi, Chem. Phys. Lett. 75, 427 (1980). [8] D. P. Craig and T. Thirunamachandran, Chem. Phys. Lett. 80, 14 (1981). [9] L. Silberstein, Philos. Mag. 33, 92, 521 (1917). [10] T. Bancewicz, V. Teboul, and Y. Le Dufi‘, Phys. Rev. A 46, 1349 (1992). [11] A. D. Buckingham and G. C. Tabisz, Mol. Phys. 36, 583 (1978). [12] K. L. C. Hunt, Y. Q. Liang, and S. Sethuraman, J. Chem. Phys. 89, 7126 (1988). [13] A. D. Buckingham, Trans. Faraday Soc. 52, 1035 (1956); A. D. Buckingham and K. L. Clarke, Chem. Phys. Lett. 57, 321 (1978). [14] K. L. C. Hunt, B. A. Zilles, and J. E. Bohr, J. Chem. Phys. 75, 3079 (1981). [15] K. L. C. Hunt and J. E. Bohr, J. Chem. Phys. 83, 5198 (1985); 84, 6141 (1986). [16] P. W. Fowler, K. L. C. Hunt, H. M. Kelly, and A. J. Sadlej, J. Chem. Phys. 100, 2932 (1994). [17] K. L. C. Hunt, J. Chem. Phys. 80, 393 (1984). [18] A. Koide, J. Phys. B 9, 3173 (1976). [19] A. Koide, W. J. Meath, and A. R. Allnatt, Chem. Phys. 58, 105 (1981). [20] M. Karplus, J. Chem. Phys. 37, 2723 (1962). [21] M. Karplus and H. J. Kolker, J. Chem. Phys. 39, 1493, 2997 (1963); 41, 3955 (1964). [22] A. D. Buckingham, Proprie'tés optiques et acoustiques des fluids comprimés et actions intermoléculaires (Centre National de la Recherche Scientifique, Paris, 1959), p. 57. [23111111. [:4]? w ' [:5]? w [26] D. M 12713 M [28] 1'. ML [29]K L [30] K. L 52511 [31111. L [32111 P [33131 1 6 [23] W. Byers Brown and D. M. Whisnant, Mol. Phys. 25, 1385 (1973). [24] P. W. Fowler, Chem. Phys. 143, 447 (1990). [25] P. W. Fowler and E. Steiner, Mol. Phys. 70, 377 (1990). [26] D. M. Bishop and J. Pipin, J. Chem. Phys. 98, 4003 (1993). [27] B. M. Axilrod and E. Teller, J. Chem. Phys. 11, 299 (1943). [28] Y. Muto, Proc. Phys. Math. Soc. Japan 17, 629 (1943). [29] K. L. C. Hunt, J. Chem. Phys. 90, 4909 (1989). [30] K. L. C. Hunt, Y. Q. Liang, R. Nimalakirthi, and R. A. Harris, J. Chem. Phys. 91, 5251 (1989). [31] K. L. C. Hunt, J. Chem. Phys. 92, 1180 (1990). [32] R. P. Feynman, Phys. Rev. 56, 340 (1939). [33] S. F. O'Shea and W. J. Meath, Mol. Phys. 28, 1431 (1974); 31, 515 (1976). COM 2.] Intro Wh molecule dipoles a1 compress single-mc induced r needed in “st in terms ( axes rA E Summatic comPOTle: ClebSch-( the “bran Vah polarizaiir CHAPTER II COLLISION-INDUCED DIPOLES FOR PAIRS OF CENTROSYMMETRIC LINEAR MOLECULES AT LONG RANGE 2.1 Introduction When two molecules collide, transient dipole moments are induced within each molecule because of the distortion of the charge distributions. These collision-induced dipoles are responsible for infrared [1-3] and far-infrared [4-7] absorption observed in compressed gases and liquids composed of D001, molecules, though such absorption is single-molecule forbidden [8]. For quantum mechanical line shape analyses of interaction- induced rototranslational absorption, the net dipole of a pair of molecules A and B is needed in the symmetry adapted form [9, 10] 11111121441012 1.5 2 DtAiBttmwfie (01111118103th “'“(QRi x(‘I’0 |)(H0 — E0 )'1 (1 - | L1’0 )(‘I’O I) for the pair, and the complex conjugation operator C. The quantity its Tam (R)O‘§y in Eq. (4) is related to the direct product [11%) <8) (TO) 81 Off) )](0) by 11311111111911, = 435 / 2 1111;) ed“) 699531110). (5) where the relation between the spherical tensor components T1? )(R) (p = :3, i2, i1, or 0) and the Cartesian tensor components T6137 (R) is assumed to be identical to that of the first-order dipole hyperpolarizability B [21]. Equation (5) has been obtained following the observation that both sides of the above equation are scalar so that they can differ only by a constant. Substitution of Eq. (5) into Eq. (4) gives T9“ = J5 / 6 (1 +0011, mtg) ®G (pg) ®[T(R)(3) cat-of) 1“) )9) 104%) = J43 /6 (1 + (2)011? 11111;) @013 (pg) ®[T(R)(3) @6533 WW” 10)] 8’63). (6) In Eq. (6), ‘11? and GB denote the unperturbed wavefunction and reduced resolvent, respectively, for molecule B; (982,] is the permanent quadrupole moment of molecule A. Use of the it induced poi;- TF'H‘J. 1 where Tl ., 21.) nonx'anishi 111310). it “here Q of A bv 10 Use of the tensor-operator methods treated in Refs. 21-23 (and applied to collision- induced polarizabilities in Refs. 24-26) transforms Eq. (6) into 1 1 0 , T9“ = s/Z2/6ZII8{ }{[T(R)<3> e037,) 1“) ®(1+C) 8 1 1 g ><<‘1’1§3](HB®GBHB)(g)I‘Pd’>}“) 1 l 0 =sf42/6ZITS{1 1 Jame“)®®121D®a<§>19 8 3 2 1 =11? 1e $118,]A 1 A}<—1)’“{1®§§3em‘ge’i‘“131113133“). (7) 3.8). B where Hague = [(2a +1)(2b +1)---(20 +1)]/1/2/. For linear molecules, when the only nonvanishing component of the spherical tensor 79’) in the molecule-fixed frame is y(p, 0), the tensor components y(p, q) in the space-fixed frame are given by [23] 1m):r4n/(2p+1)11”2’1160111310). (8) Equations (7) and (8) imply 3 1 D9381, = 736/30(—1)1n,,{ A B AB 1 7&}® (2,0)01(}tB,0)T(3,0) l 3 _ 1+1 1/1 -4/ _ 1 . 3(1) (21+) {AB 1 A }o" 63(AB,0)R““, (9) where 01(0, 0) = —\/3’01, 01(2,0) = 2 / J6 (011T —01i), and T(3,0) = —3\/1_0R‘4. This result is consistent with earlier work [9-15]. The corresponding coefficients for the polarization of A by the permanent quadrupole of B are given by a Difzm = (‘1) +1 JOAB Diafgm- (10) Similar anal) KIN] D119}. uhere (1)A = satisfies 11- Did] . uith E121. Di'lf. 21,15 =1 C7 27‘ \' . .. “Iih a : D14 11 Similar analysis gives the coefficients for hexadecapolar induction term [12-15] in the form 5 l Digits=\/§(-1)}‘+1(24+1)i2{x8 1 )1 }¢AGB(XB,O)R_6, (11) where (DA 2 (D212, and Di‘fus = (—l)“1 goABfoBls. The E-tensor term [13-15] satisfies 7» a Diff,x5 = JEE(271+1)V2{ 5A 3 }EA(AA,O)®B R‘6 (12) with E(2,0) = —1/Jfi (352.222 - ataxm ), E(4,0) = 2N? (Ezm +2Exm ), and D3215 = (—1)“1 gaABDEfm. For back-induction [14], 1 2 1 DEA,“=157m]+(—1)*+'gaAB]Z(—1)’“+8+‘(2g+1)n,, 1 3 2 a’g AA L 8 a 1 a a a x{8 B HA B }(a200|aBo)(2300|Lo) a 1 2 1 L g x01A(}tA,0)01B(a,O)®BR'7 (13) with a = 0 or 2. Finally, the coefficients for the dispersion term are given by Dim“, =15Jfi h/ It R'7(2300|L0) [1+(—1)1+1 goAB]Z(-1)’“A+g+a 3’8 1 2 1 21n13zgi‘AlexAx + X” )1“ a 1 2 1 L g 413 L 8 °° (a) . B . Xlo doaB (AA,0;O,10))01 (33,030))- (14) In Eq. (14), a = O, 1, or 2, and a 8'1 {a Equafions mmhnbe quadrupof “d0x1> 0Perators 313131110 a- zero [27] 2.3 Sumn an, \ inlene] at POIafiZati 12 B(Z)(0,O;0,i0)) = 2 / J3_o [Bn,,,(o, 1(1)) + 23mm (0, 1(1) ) +2B,,,,,(o,i6) +13,“22 (0,10))+ 413,0,”(036 )], (15a) B(O)(2,0;0,ico) = -1/J§ [Bu,n(0,i0) ) +2B,,,,(0,im)], (15b) B“>(2,o;o,im) = 2 / J3 [13,W (0,160 ) — sz‘xz(0,iu))], (150) B(Z)(2,0; 0,16) = 2 / Jfi {-Bzm(o,i6) — sz,xz(0,i(o)— 13“,,(036) +3Bxx,ZZ (0,ico)+4Bmxx (0,160 )], (15d) 3(2)(4,o; 0,103) = 2/ J105 [3Bn’zz(0,i0))—4sz’xz(0,ico)—4Bzx,xz(0,ico) —2Bm(o,im)+23m,(o,im)]. (15e) Equations (13) and (14) for the back-induction and dispersion coefficients are structurally similar, because both involve a dipole-dipole interaction between A and B, a dipole- quadrupole interaction, and a dipole expectation value. In Eq. (13), the operators 11X, 11X , and OX (X = A or B) are coupled to produce 01x(a, O)Ox, while in Eq. (14) the same operators are coupled to yield B“) (1x00, 10)). The results given by Eqs. (9)-(14) also apply to atom-Dag,1 molecule interactions, with the index 21A for the atom always equal to zero [27]. 2.3 Summary and Discussion Angular momentum coupling algebra has been used to derive equations for the dipole coefficients Dixie“; in terms of 6-j and 9-j symbols. The results explain the interrelations among coefficients with different AA , AB, 71 , and L, for each of the polarization mechanisms, and yield compact expressions that simplify line shape analyses. Mechanism transitions 1 potential. consistent 1 a direct intt The requires 01] available fr order to dt however. i‘ a function componen dipole coc heafier 5p and 80h, dlSpersiOr B. and dis 10 estima1 Chapter) Bol pOIarilab HIMH‘, COefi‘li‘iet COHIn'bm Pairs, 13 Mechanisms giving nonvanishing coefficients D 11119114 can produce rototranslational transitions with AJ up to iAA for A and AJ up to ikB for B with an isotropic pair potential. The algebraic expressions for the coefficients given by Eqs. (9)-(14) above are consistent with Eqs. (9)-(12), (15)-(19), (24)-(29) and Tables I-IV in Ref. 14, obtained by a direct integration method. The numerical evaluation of the coefficients due to classical induction effects requires only the multipole moments and static multipole polarizabilities, which are available from ab initio calculations for a number of species including H2 and N2. In order to determine the dispersion contributions to the coefficients given by Eq. (14), however, it is necessary to obtain the B hyperpolarizability tensor (as well as 01 tensor) as a firnction of imaginary frequency. For H2 , values of the imaginary-frequency B tensor components have recently been computed with high accuracy [28], and the dispersion dipole coefficients have been evaluated numerically for pairs containing H2 [28]. For heavier species such as N2, where accurate values for B(0,ico )are not yet available , Hunt and Bohr [14, 27] have developed a constant ratio approximation that relates the dispersion dipole coefficients to the static multipole polarizability, static hyperpolarizability B, and dispersion energy coefficients (the constant ratio approximation has also been used to estimate dispersion contributions to pair polarizabilities; see Ref. 29 and the next chapter). Bohr and Hunt have used ab initio results for permanent multipole moments and polarizabilities to evaluate the classical induction contribution to DKAKBLL for pairs Hz-nHz, Hz-HNZ , and N2---N2 in Ref. 14. They have estimated the dispersion dipole coeflicients based on the constant ratio approximation and discussed the relative contributions of different polarization mechanisms to the collision-induced dipoles of these pairs. Reference [I] H Li Cram] (I950) [2]] W P Van K 111d 5 [3] L' 801 [4] J. A .~‘ Colpa [5]ZJI< Iél P. Dor (I983 [7]IR[ Can 1 [8] [726116 Xe“. \ edited 1911 var [1011 D, [IIIC 0 ME R [13111. x [14115 “SIG B “61H; [171K L [18“. G; I19] A D [30] D P_ 14 References [1] H. L. Welsh, M. F. Crawford, and J. L. Locke, Phys. Rev. 76, 580 (1949); M. F. Crawford, H. L. Welsh, J. C. F. MacDonald, and J. L. Locke, Phys. Rev. 80, 469 (1950) [2] J. W. MacTaggart, J. De Remigis, and H. L. Welsh, Can. J. Phys. 51, 1971 (1973); J. Van Kranendonk and D. M. Gass, ibid. 51, 2428 (1973); J. L. Hunt and J. D. Poll, ibid. 56, 950 (1978). [3] U. Buontempo, A. Filabozzi, and P. Maselli, Mol. Phys. 67, 517 (1989). [4] J. A. A. Ketelaar, J. P. Colpa, and F. N. Hooge, J. Chem. Phys. 23, 413 (1955); J. P. Colpa and J. A. A. Ketelaar, Mol. Phys. 1, 14 (1958). [5] Z. J. Kiss, H. P. Gush, and H. L. Welsh, Can. J. Phys. 37, 362 (1959). [6] P. Dore, L. Nencini, and G. Bimbaum, J. Quant. Spectrosc. Radiat. Transfer 30, 245 (1983). [7] I. R. Dagg, A. Anderson, M. C. Mooney, C. G. Joslin, W. Smith, and L. A. A. Read, Can. J. Phys. 68, 121 (1990). [8] Phenomena Induced by Intermolecular Interactions, edited by G. Bimbaum (Plenum, New York, 1985); Interaction-Induced Spectroscopy, Advances and Applications, edited by G. C. Tabisz and M. Neumann (Kluwer, Dordrecht, 1994). [9] J. Van Kranendonk, Physica 24, 347 (1958). [10] J. D. Poll and J. L. Hunt, Can. J. Phys. 54, 461 (1976); 59, 1448 (1981). [11] C. G. Gray, J. Phys. B 4, 1661 (1971). [12] E. R. Cohen and G. Bimbaum, J. Chem. Phys. 66, 2443 (1977). [13] M. Moon and D. W. Oxtoby, J. Chem. Phys. 84, 3830 (1986). [14] J. E. Bohr and K. L. C. Hunt, J. Chem. Phys. 87, 3821 (1987). [15] G. Bimbaum, A. Borysow, and A. Buechele, J. Chem. Phys. 99, 3234 (1993). [16] L. Galatry and T. Gharbi, Chem. Phys. Lett. 75, 427 (1980). [17] K. L. C. Hunt and J. E. Bohr, J. Chem. Phys. 83, 5198 (1985). [18] L. Galatry and A. Hardisson, J. Chem. Phys. 79, 1758 (1983). [19] A. D. Buckingham and A. J. C. Ladd, Can. J. Phys. 54, 611 (1976). [20] D. P. Craig and T. Thirunamachandran, Chem. Phys. Lett. 80, 14 (1981). 15 [21] C. G. Gray and B. W. N. Lo, Chem. Phys. 14, 73 (1976). [22] A. J. Stone, Mol. Phys. 29, 1461 (1975). [23] R. N. Zare, Angular Momentum (Wiley-Interscience, New York, 1988). [24] R. Samson and A. Ben-Reuven, J. Chem. Phys. 65, 3586 (1976). [25] T. Bancewicz, Mol. Phys. 50, 173 (1983); Chem. Phys. Lett. 213, 363 (1993). [26] X. Li and K. L. C. Hunt, J. Chem. Phys. 100, 7875 (1994). [27] J. E. Bohr and K. L. C. Hunt, J. Chem. Phys. 86, 5441 (1987). [28] D. M. Bishop and J. Pipin, J. Chem. Phys. 98, 4003 (1993). [29] K. L. C. Hunt, Y. Q. Liang and S. Sethuraman, J. Chem. Phys. 89, 7126 (1988). CENTR( AS 3.1 lntrod COIii scale The rototrans‘n; impulsive (IICIECIITC [ 131011118 ge CO"It‘ihutic Prmide flu because a “5091111111 Work-.ha\ The a Pair of L dipole‘lnd CHAPTER III COLLISION-INDUCED POLARIZABILITIES FOR PAIRS OF CENTROSYMMETRIC LINEAR MOLECULES AT LONG RANGE: THEORY AND NUMERICAL RESULTS FOR H2---H2, H2---N2, AND N,.--N2 3.1 Introduction Collision-induced changes A01 in polarizabilities occur on the subpicosecond time scale. These changes are detected experimentally in collision-induced Rayleigh and rototranslational Raman scattering [1-4], subpicosecond induced birefiingence [5-8], impulsive stimulated scattering [9-12], and measurements of the density dependence of dielectric properties [13, 14] and refractivity [15-17]. The purpose of this work is to provide general, symmetry-adapted equations for the induction and dispersion contributions to A01, complete to order R‘6 in the intermolecular interactions, and to provide numerical results for H2 - - -H2, H2 . - -N2 , and N2 - - -N2. These pairs were selected because collision-induced light scattering (CILS) spectra have been obtained in experiments on hydrogen and its isotopic variants [18-28] and nitrogen [29-35] over a wide range of densities and temperatures; and because the multipole moments, susceptibilities, and van der Waals energy coeflicients--needed as input parameters for the work--have been evaluated in ab initio calculations for these species [3 6-48]. The calculations are complete to order R'6 in the intermolecular separation between a pair of Dooh molecules. To this order, A01 is determined by first- and second-order dipole-induced-dipole (DID) interactions [49], higher-multipole induction by the laser field, dipole induction due to nonuniforrnities in the local field [3 5, 50, 51], hyperpolari- zation [52], and dispersion [52-56]. These polarization mechanisms suffice to predict types of scattering that are single-molecule forbidden, such as isotropic rototranslational 16 Raman sca transitions mechanisn molecules. In th been used. Kazmiercz to their trz intermolet‘ line shape This collision] has been ( energy 31 IW0~C€n1e Physica] 6 field and 1! comflatio each of II malitica] fieQUEnc) mumpliet and 1111.), approxjm 17 Raman scattering by centrosymmetric linear molecules during AJ = i2 and A] = i4 transitions, and depolarized rototranslational Raman scattering with AJ = i4. These mechanisms also yield simultaneous rotational transitions on each of the colliding molecules, with A] up i4 for one molecule and $2 for its collision partner. In this work spherical-tensor analysis and angular momentum coupling algebra have been used, as suggested by Samson and Ben-Reuven [57] and by Bancewicz, Glaz, Kazmierczak, and Kielich [58-61; see also 62, 63] to separate the terms in A01 according to their transformation properties under rotation of the molecular axes and the intermolecular vector R. This casts the results into the symmetry-adapted form needed for line shape analysis [64-67]. This work provides the first firll set of results for the dispersion terms A01diSp in the collision-induced polarizability of a pair of linear, centrosymmetric molecules. Here A01di5p has been determined by use of reaction-field theory to find the change in the dispersion energy at second order in an applied field [55], but identical results are obtained from a two-center perturbation treatment taken to fourth-order overall [54]. Two distinct physical effects contribute to Aotdis": (1) each molecule is hyperpolarized by the applied field and the fluctuating field of its neighbor [53], and (2) the applied field alters the correlations in the spontaneous quantum mechanical fluctuations of the charge density in each of the molecules, thus affecting the van der Waals interaction energy [55]. Exact analytical results are obtained for the dispersion effects, as integrals over imaginary frequency; terms in the integrands contain the polarizability 01(ico) of one molecule multiplied by the second hyperpolarizability 700), 0, 0) of the other. For H2 , both 01(ioo) and 760), 0, 0) have been determined with high accuracy using explicitly correlated wave functions [3 9]. For numerical applications to larger molecules, a constant ratio approximation [52, 54, 68, 69] is used to relate A01disP to the static polarizability, static y hyperpolarizability, and dispersion energy coeflicients, which are more widely available than 760), 0, 0). Comparisons of the approximation with ab initio results for H- "H2 , Hell} 1 approxima dispersion polarizahi‘. and 101011. the orient; range dist‘ and 30% 1 polarizab1 hwetpola' In a. calculatio- I | | | I DID interl contn'bur; I 35l 5610' 1581. Tit-I in molectl E.[[.| iMmaa the collisi. field indu. collision [1| accOttnts ; de11011112L and Le D. where the region of f 18 He- - -H2, and H2 - - - H2 [39, 70] show that the rms error in the approximation is approximately 20-25%. In the current work on qu-Hz, Hz-uNz , and N z-o-Nz , dispersion is shown to contribute significantly to the collision—induced change in scalar polarizability A018, which determines the spectra for isotropic collision-induced Rayleigh and rototranslational Raman scattering [24]. In fact, if A018 is averaged isotropically over the orientations of each of the interacting molecules and the intermolecular vector, at long range dispersion accounts for ~55% of the total value for H2 - --H2, 40% for H2---N2 , and 30% for N2 ~ - - N2 , based on current estimates. The dispersion terms in the anisotropic polarizability A018I are much smaller for the pairs studied in this work, because the y hyperpolarizability shows little anisotropy for H2 and N2 [39, 41]. In addition to dispersion tenns--which are purely quantum mechanical--the calculations include classical induction mechanisms treated in earlier work: First-order DID interactions [49, 71] give the leading R’3 terms in A01, and hence the dominant contribution to depolarized, collision-induced scattering by pairs of linear molecules [21- 35]. Second-order DID terms have been cast in symmetry-adapted form by Bancewicz [5 8]. Their effects on line shapes and depolarized scattering intensities have been analyzed in molecular dynamics simulations by Ladanyi and Geiger [72, 73]. E-tensor terms vary as R’5 [50], and two types contribute to A01: The applied field induces a dipole in each molecule; this creates a nonuniform field that induces a dipole in the collision partner, via its dipole-octopole polarizability E [74]. In addition, the applied field induces an octopole in each molecule via E, and the octopolar field polarizes its collision partner. Buckingham and Tabisz [50] have shown that E-tensor induction accounts in part for rotational Raman transitions observed in the far wings of the depolarized scattering spectrum of compressed SF6 in the gas phase. Bancewicz, Teboul, and Le Duff have analyzed scattering by N2 [35], with frequency shifts out to 700 cm’l, where the contribution from pressure-broadened allowed transitions is negligible. The region of the spectrum between 300 and 700 cm”1 cannot be fit within the DID model for do [35] I componer. | The hyper quadrupo‘: studied 111 the dipole permanen' I The overlap bfi exchange. effects ar WMRM and have results at Should bt limiting [ lmncatic Work ide reprOdUi COmpar inductio range ql 19 A01 [3 5]. Inclusion of the E terms in A01 gives the observed line shape, but the fourth-rank component of E determined by fitting the spectra is ~2.5 times the ab initio value [75]. The hyperpolarization terms, which depend to leading order on the dipole-dipole- quadrupole hyperpolarizability B [74], tend to be smaller than the E terms (for species studied thus far); but like the E terms, they vary as R's. The B-tensor terms account for the dipole induced by the external field, acting together with the field gradient of the permanent molecular quadrupole of a neighboring molecule. The results for A01 apply in the outer part of the van der Waals potential well, where overlap between the charge distributions is small. At shorter range, overlap damping, exchange, orbital distortion and charge transfer contribute significantly to A01. Short-range effects are not yet well characterized for H2---H2, H2---N2 , and N2---N2 since the ab initio studies to date [76, 77] have employed basis sets that are small by present standards, and have neglected correlation. Calculations on pairs of inert gas atoms show that the results are very sensitive to the size and quality of the basis [78]. The results given here should be useful in carrying out later ab initio work, because they provide the correct limiting form of A01 at intermolecular distances where numerical cancellation and Gaussian truncation error make it difficult to obtain accurate results ab initio [7 9]. Additionally, this work identifies single-molecule property tensors (01, O, E, B, and 7) that must be reproduced in a basis-set calculation, in order to obtain accurate results for A01. Comparison of the observed scattering spectra with the calculated form based on the induction and dispersion terms in A01 (given here) should yield predictions about the short- range quantum terms that can be tested in later ab initio work. The short-range quantum terms in A01 are expected to be important in determining scattering in high-fiequency spectral wings [18-35, 78], and in cases where the leading long-range contributions to A01 vanish by symmetry [78]. In other cases, the scattered intensity may depend predominantly on A01 at intermediate to long range; e. g., this applies to scattering by solid hydrogen and its isotopic variants [80-82], because the separation between H2 molecules in solid hydrogen i of the gas- phenomen This has been c CS: [90. ' range pole broad. as} In 11:, the (the frequ- Rayleigh s high-freq), CIT at ~30 transition for onh0_ temperatt lhe Presst In 1 field fact( the POIari l’Olaiion C as X21 0 allowed a in lime SC Scattering 20 hydrogen is slightly larger [80, 83] than R = 6.5 a.u., the isotropic van der Waals minimum of the gas-phase pair potential. Hence long-range models can predict interesting phenomena such as zero-phonon, double transitions in solid HD [27]. This work should be useful in analyzing the experimental light-scattering data that has been obtained for H2 [18-28], N2 [29-35], 02 [84, 85], C12 [86], CO2 [87-89], and C82 [90, 91], in order to obtain information on intermolecular dynamics, as well as short- range polarization effects. In the gas phase, collision-induced scattering contributes to a broad, asymmetric background superimposed on pressure-broadened, allowed lines [92]. In H2 , the first rotational lines are sufficiently well separated in frequency from the origin (the fiequency of the incident radiation) that purely translational, collision-induced Rayleigh scattering spectra can be obtained experimentally [18-28]. In this case, potential high-frequency contributions from short-range polarization mechanisms are effectively cut off at ~3 00 cm'l, because the pressure-broadened wing of the allowed J = 0 —> 2 transitions at y = 354.4 cm"1 appears in this region [25]. Nearer to the origin, the spectra for ortho-hydrogen show structure that is not found for para-hydrogen at low temperatures [21]: in ortho-hydrogen, the collision-induced scattering is superimposed on the pressure-broadened wing of the allowed Q0(1) rotational Raman transition [93]. In the liquid phase, the interaction-induced changes in polarizability determine local- field factors for allowed scattering [94], and also generate collision-induced components in the polarizability which transform differently from the single-molecule tensors under rotation of the molecular framework [94-97]. For liquids containing light molecules such as N 2, 02, C12, and C02, molecular dynamics simulations show [71-73, 98-101] that the allowed and collision-induced (CI) components of the polarizability are not well separated in time scale, with the result that significant interference occurs. Collision-induced scattering is predicted to increase the second moment of the depolarized Raman bands for liquid N 2 and Oz by ~20-30% [71-73], with larger effects expected for C12 and C02. EXPL’ probe .111 experiment|| and a wear polarizaticI In impulsi- pulses 01c measured C 52. on 11 and third 1 H; and S gases. an Set P01211128] compon Additio Usenet] eStirnat der W. Sectig sec, 3 21 Experiments using pulsed lasers with 100-femtosecond and shorter time resolution probe A01 directly in the time domain [5-12]. In subpicosecond induced birefiingence experiments [5-8], a high-intensity laser pulse induces optical anisotropy in the sample, and a weaker, time-delayed pulse probes differences in the refractive index for polarizations parallel or perpendicular to the pump field (the optical Kerr effect, or OKE). In impulsive stimulated scattering (ISS) experiments [9-12], two ultrashort excitation pulses overlap inside a sample, and typically the mean-square scattered probe field is measured. Both OKE and ISS experiments show collision-induced effects on 01 for liquid CS2, on the time scale t < 500 fs [12, 102]. Finally, collisional effects appear in the second and third refractivity virial coefficients, recently obtained in high-accuracy experiments on H2 and N2 [17]: A01 determines the second refractivity virial coefficients of compressed gases, and contribute to the second dielectric virial coefficients [103]. Sec. 3.2 of this chapter contains the symmetry analysis for the collision-induced polarizability. Sec. 3.3 gives equations for the coefficients appearing in the scalar component of A01, and Sec. 3.4 gives the coefficients for the anisotropic component. Additionally, the implications for rototranslational Raman spectra are discussed in Sec. 3.3 (isotropic scattering) and 3.4 (depolarized scattering). Section 3.5 describes a method of estimating the dispersion contributions to collision-induced polarizabilities in terms of van der Waals energy coefficients, static polarizabilities, and static hyperpolarizabilities. Section 3.6 contains a set of numerical results for H2---H2, Hz-s-Nz , and Nz-nNz pairs. Sec. 3.7 provides a brief summary and discussion. 22 3.2 Changes in Polarizability Induced by Long-Range Interactions Between Two Centrosymmetric Linear Molecules In this section, the collision-induced electronic polarizability A01 is determined for a pair of centrosymmetric linear molecules interacting at long range. The results are complete to order R’6 in the intermolecular separation R, and to this order A01 is a sum of induction and dispersion terms. To obtain the induction term A01ind , the effective multipole moments of two interacting Dooh molecules A and B in the external field F6 are determined self- consistently, and then Aamd is derived from the equation limF,_’06(u2 + [.12 ) / OPS = 012]} + 0125 + A0188l . (1) The static local field F polarizing A is related to F°,and to the dipole uB, quadrupole OB , and octopole QB of molecule B by F, = F; +T,,,,,(R)tt],3 +1/3 Tam (megY +1/15 T,,,[,,5(R)Qf,’,8 +--., (2) where R is the vector from the origin in molecule A to the origin in molecule B, Ta, (R) = VaVB(R"), Tam (R) = VQVBV7(R"1), and T,B,5(R) = V.,VBV,\75(R“l ). The Einstein convention of summation over repeated Greek suffixes is followed in Eq. (2) and below. For a molecule of Dooh symmetry in the local field F, the induced dipole moment is related to the field and field gradients at the molecular center of symmetry by [la 3 aaBFB + 1/6 YQBYSFBF'Y F8 '1‘ I / 15 EasBYSFé'YS + I / 3 Ba,B,78FBF‘):5 '1" ° °, (3) where 01 is the dipole polarizability of the isolated (unperturbed) molecule, 7 is the second hyperpolarizability, E is the dipole-octopole polarizability [74], and B is the dipole-dipole- quadrupole hyperpolarizability [74]. The quadrupole and octopole moments of the molecule satisfy 11151 (41 quadrupo‘. term [52] :10“ Where '1) first two ti noted in S arid nonui The 11131- terms as At 0f the A At.- ‘i‘CCOrdi mechan: Bi the 11 Change related 23 ®GB = @243 'i' CUB,75F‘}:5 +1/2 B‘Y,8,G-BF‘Y F5 ‘1" H, (4) QQBY = E59437 F3 +"'. (5) In Eq. (4), @843 denotes the permanent molecular quadrupole moment, and C is the quadrupole polarizability. The self-consistent solution of Eqs. (1)-(5) yields the induction term [52] 13an = (1 + gaAB)[0127 T,5(R)et§[, +613, T,5(R)a§8 T8,,(R)et[,‘B B B +1 / 15 613735,, (R) 13%,, +1/ 15 E2,758T758¢(R)01¢B —1 / 9 133,3,5858, (R)o‘,’f]. (6) where goAB perrnutes the labels of molecules A and B in the expression that follows. The first two terms in Eq. (6) give the first- and second-order dipole-induced-dipoles. As noted in Section 3.1, the E-tensor terms in Eq. (6) stem from higher-multipole induction and nonuniformity in the local field, while the B-tensor terms stem from hyperpolarization. The first-order DID terms vary as R'3 , E and B terms as R"5 , and second-order DID terms as R'6. At order R‘6, dispersion forces also contribute to A01, because the dispersion energy of the A---B pair changes in the field F“: disp _ __ - A0104} — 1""er oZEdiSPwe) / or; org. (7) According to the reaction-field theory for Edis" [104-106], spontaneous, quantum mechanical fluctuations in the charge density of molecule A produce a field that polarizes B; the induced polarization of B then creates a reaction field at A. The resulting energy change in A depends upon correlations of the fluctuating polarization within A, which are related to the imaginary part of the polarizability density [104] of A via the fluctuation- dissipation theorem [107]. Similarly, fluctuations in the charge density of B polarize A, leading affects fluctua the cor leading where imagin change and In (ma rotOtr; adapte Interm and B. I:unCtic 24 leading to a reaction field at B, and a second term in the energy change. The external field affects the dispersion energy because F° alters the response of each molecule to the fluctuating field of its neighbor, due to hyperpolarization effects [53]; and F° also alters the correlations of the spontaneous charge-density fluctuations in each molecule [55]. To leading order, the dispersion term in A01 is A8331 z Iz/ZTC (1 + ggAB) [Odes T,5(R)y§,a[,(i6,o,0)T,,(R)a:3, (im), (8) where 01(i10) is the polarizability at the imaginary frequency ico, and y(ico, 0, 0) is the imaginary-frequency hyperpolarizability. Through order R'6, the total interaction-induced change in polarizability is the sum of A01ind from Eq. (6) and AadiSp from Eq. (8). The polarizability of the interacting molecules A and B is a second-rank Cartesian tensor, with spherical tensor components of ranks O and 2. The components are related by [108, 109]: A618 =1/J3(A01XX+A01YY+A01ZZ), (9) Au? = 1/J6 (21161ZZ —A01xx —A01YY), (10) Act;l = $1/2[(A01XZ + Aazx)ii(A01YZ + AaZY)], (11) and A639- =1/2[(AetXX — Aayy)ii(A01xy + Aotyx)]. (12) In quantum mechanical line Shape analyses of collision-induced Rayleigh and rototranslational Raman scattering, the pair polarizability A01 is needed in a symmetry- adapted form in terms of the spherical harmonics of the orientation angle OR of the intermolecular vector R and the orientation angles (2A and QB of the axes of molecules A and B. Since A018 is a scalar, it must be obtained by scalar coupling of spherical tensor functions of QA, QB, and OR [71, 93, 110]: where the Ad with the CIehsci ranges interm. Values Separ' 59118.1 and in 1 Of Tc 25 A618 (rA, rB, R) = (471)3/5 /J'3 Z AWE,L (rA , rB, my]? (OA )Yf;B (OB) xYflm(QR)(AA AB mA mBIAmXALm —m|00), (13) where the summation runs over AA, AB, A, L, mA, m3, and m. Similarly [71, 93, 110] A6112“ (rA,rB,R) = (411)9’5 /J3 Z A21A[8u(rA,rB,R)Y}I:A(QA)Y;:;B(QB) xvii—WOW“ AB mA mBIAmXALm M—mIZM) (14) with the summation running over the same indices as in Eq. (13). In these equations, the Clebsch-Gordan coefficients are denoted by (21112 m1 m2 [A3 m3), and M in Eq. (14) ranges from -2 to 2. This work gives the dependence of AOKAKBKL and AZAAABAL on the intermolecular separation R, with the bond lengths held fixed at the vibrationally averaged values [11 1]. To cast A01 from Eqs. (6) and (8) into symmetry-adapted form, it is necessary to separate the multipoles and polarizabilities of A and B into components of different spherical tensor ranks [112, 113]. The polarizability and quadrupole satisfy 01a];=OEOQB+U3(an—ai)(3fafB—5a[3) (IS) and Oapzl/ZOOfafB—Oafi), (16) in terms of the direction cosine fa between the molecular symmetry axis 1 and the 01 axis of the space-fixed frame; 01“ is the polarizability for fields along the molecular axis f, 01] for fields perpendicular to the molecular axis, and a 2 (an + 201 i ) / 3. The susceptibilities E, B, and y are fourth-rank Cartesian tensors, and each has the form [52, 68] E has spl compont coefficie- indepen “1th 0 the to With 1101c \Vfi‘ 26 PaByS = P1504, 578 + P2 (6a, 5135 +6a5 Spy ) + P3 (3 fa fB —5a‘3)575 +P4 (3?, f5 -575)8a[3 +P5[(3’r‘a ’r‘y -—5ay)635 +(3?0L f5 ’5a6)5137] +P6[(3 fB fy _SBY)5a6 +(3 f5 f5 —6[35)5ay] +P7[35i‘*a q, f7 f5 — 50,1}, 578 Hi, i, 5B5 + fa is 5W +fB r7 60,5 +fB is 50w + i} is aafiflaafi 575 +5ay 5% +5,1L8 51371‘ (17) E has spherical tensor components of ranks 2 and 4 only, but B and y have spherical tensor components of ranks 0, 2, and 4. For 700) , O, O) of a linear molecule, the seven coefficients P1 2 71(im, O, 0) through P7 5 77(ia), O, 0) are linear combinations of the six independent components of the hyperpolarizability in the molecule-fixed frame: yj(ico,0,0) = aj ym(iw,0,0)+bj ym(iw,0,0)+cj yxxzz(ico,0,0) +dj 7W(iw,0,0)+ ej yxxyy (ico,0, O)+fj yxxxx(iw,0,0) (18) with expansion coefficients aj-fj listed in Table 3.1. Similarly, the seven coefficients P1 5 B1 through P7 5 B7 are linear combinations of the four independent components of the static B tensor in the molecule-fixed frame for linear molecules: Bj : aj BZZJZ + bj BX,Z,XZ + 0j BX,X,ZZ + dj Bx,x,xx (19) with expansion coefficients listed in Table 3.2. For both 7 and 8, P5 2 P6, but this does not hold for the dipole-oct0pole polarizability E. For linear molecules, E satisfies Eq. (17) With E1 = E2 = 0, E3 :13, =1/63(8Ex,m—3Ez,m), (20a) E4 = E6 = 5/126(3Ezm —81~:xm), (20b) lflkBl Table j \\ lJ\ ll \5 ‘3. 1% u..- BBBB .\._ LU, BB 3,, \ Table 3.]. Expansion coefficients for y-tensor components, Eq. (18). 27 Y1 a1 bj “1 dj 61' fr 7] 1/15 4/15 4/15 .4/15 1/3 1/15 72 1/15 -1/15 -1/15 2/5 -l/6 7/30 73 1/21 2/7 -1/21 -4/21 -1/3 1/21 74 1/21 -1/21 2/7 -4/21 -1/3 1/21 y, 1/21 -1/21 -1/21 1/7 1/6 -5/42 7, 1/21 -1/21 -1/21 1/7 1/6 -5/42 77 1/35 -1/35 -1/35 .4/35 0 1/35 Table 3.2. Expansion coefficients for B-tensor components, Eq. (19). Yj 31' hr 01 <11 31 -1/15 -4/15 -1/15 -4/15 132 1/10 2/5 1/10 2/5 B3 -2/21 4/21 2/7 8/21 134 1/14 -4/21 13/21 8/21 135 1/14 1/7 -3/14 -2/7 B6 1/14 1/7 -3/14 -2/7 B7 3/70 -4/35 -1/35 1/35 and En For among th algebra a for the Et between clarity in the multi the lth r2 28 and E7 =1/14(Ezm +2Exm). (20c) For each of the polarization mechanisms in AOL, this work has yielded relations among the terms in the coefficients AOlAlBu and AzxAkBu by use of angular momentum algebra, as in the work by Bancewicz [58] on DID interactions. The analysis is illustrated for the E-tensor term a2}, TYSQEEM’ which stems from the dipole-octopole interaction between molecules A and B, and interactions of both A and B with the external field. For clarity in Eqs. (21)-(23) below, only the spherical-tensor recoupling scheme is shown for the multipole operators of A and B, and the T tensor. The contribution of the a-E term to the Jth rank part of AOL transforms as TaE = [LIA-(1) ®{[HA(I) ®(T(4) @933))(1)](0) ® p'B(l)}(1)](1), (21) where (8 denotes the direct product. Then Tag = Emu—1)“ /3 [NW 69 (W) ®[(T“” @0305“) ® uB‘"1“‘>}“)1‘” (22) h with the notation Has-.2 = [(23 +1)(2b +1)---(22 + 1)]"1/2. Further analysis in terms ofthe 6-j and 9-j symbols gives 431 1 h Jul/m) ® WAG) ®[T(4) ® eB(k)](h)}(1)](J) TaE :Zznhkl/‘fj{ h k 4 3 1 1 1 j . :§Z§thk(_l)~l+h{l h k}{h J 1}{aA(J)®[T(4)®eB(k)](h)}(-I) J _ (2h+1)H' _1j43111jjk7t '3'}??? Jk’*()1hkth4Jh x{[aA(-i) ® 6300100 ® T(4)}(J) j 1 l :Zzznjfl(_l)j k 3 l {[aA(J)®eB(k)]O\-)®T(4)}(J), (23) 1' 1‘ ’~ A 4 J where 21‘ Eq. (:3 l. for the 0; reduced 1 account” Bib e her For Plk.q) a compont Pl “here D Fr. 29 where 3A”) denotes (uAm ® uA(1))(j), and e300 denotes (03(3) ‘8 LLB(1))(k). In deriving Eq. (23), Eqs. (4.3), (4.9), (4.19), and (4.24) of Ref. 109 have been used. Equation (23) for the operator structure corresponds to an expression for AOL obtained by inserting the reduced resolvent operators, the ground-state bra and ket, and a numerical prefactor, and accounting for different operator-ordering possibilities. In the equation for AOL, aAm and e300 become (1A0) and EB“), respectively. For species of Dooh symmetry, when the q = 0 components of the spherical tensors P(k, q) are the only nonzero components in the molecule-fixed frame, the spherical tensor components P(k,q') in the space-fixed fi'ame satisfy [see, e. g., Ref. 109] P(k,q') .—. D3,; P(k,q = 0) = [41: / (2k +1)]‘/5Y§'(e,¢)1>(k,q = 0), (24) t where D390 is the conjugate of the Wigner rotation matrix. From Eqs. (23) and (24), the a-E term in Aug“ satisfies j 1 1 T(a-E)=ZZZZZMnflcfi‘Eflfi—Djm k 3 laA(j,0)EB(k,0) j k 9» P1 P2 2» 4 J xT(4,0)Y,P‘(6A,¢A)Y,:’2(eB,¢B)Y}‘“PI+p2>(eR,¢R) ><(jkp1r>2l7~(p1+p2))(7»4(p1+pz)[M—(p1+p2)]|JM)- (25) In Eq. (25), (1A (0,0) 2 -J3 EA and (1A (2, 0) = 2 / Jam? - a: ). The relation for (1A (2, 0) is consistent with Eq. (10); but orA (O, O) differs from Eq. (9), where a8 is defined by 1/ J5 ass. This reflects the phase difference between the ordinary scalar product C(k) - D00 of two tensors C and D of rank k, vs. the rank 0 component of the direct product [109]: [C00 ®D(k)](0) = (—1)k(2k +1)"/2 C(k) - D“). (26) ln derin'r| antisymn contrihu The res Kielich results Equa “he 30 The components of the E-tensor in Eq. (25) satisfy EB(0,O) = O, EB(2, O) = BM 15133, and EB(4,O) = 4J7 E173. Also T(4,0) = 6J7O R‘S, and c313 = -c‘;E = 4% / 5. The ratio of —1 between cgE and cgE stems from the phase difference noted above. Then from Eq. (25), the (1A — EB contributions to the polarizability coefficients A3534 are given by j 1 1 A3334=4Jfi(2x+1)}’2(—1)‘+J’2 k 3 laA(j,0)EB(k,O)R'5. (27) J. 4 J In deriving Eq. (27), it has been assumed that j is even, for nonzero (1A (j, 0); i.e., the antisymmetric part of the polarizability tensor has been neglected. The BAA - (13 contributions are given by AkaM =( ”k 63ABAer-t4 (23) The results for the E-tensor terms are consistent with those of Bancewicz, Glaz, and Kielich [60]; the coefficients depending on the second-rank part of the E-tensor agree with results given by Borysow and Moraldi [51]. A similar analysis for the first-order DID terms gives j 1 1 ,Jm_6~/'(21.+1)/2(1)‘+J’2 k 1 1 aA(j,0)aB(k,0)R‘3. (29) 2. 2 J Equation (29) is identical to the result derived by Bancewicz [58]. The BA - 93 terms satisfy A333,,=_,/—2[(2;.+1)/(21+1)1%(_ 1)‘+“2{ x}BA(J;1,0)@BR-5, (3o) 4J2 where the B-tensor components BA (J; j, 0) are given by and Equau in BM safisfi; Expres COUp] billty . bility . 31 BA(O;2,0)=—2/J3(3B4A +413?) (31a) BA(2;0,0)=2J1_07§BA, (31b) BA(2;2,0)=—4/3JEBA, (31c) and BA(2;4,0)-—- MOM/105 B9. (31d) Equations (31a)-(31d) follow from the observation that the tensor operators which appear in BA(J;j,0) have the structure [(uA(1)® 11A“))(J ) ®®A(2)](j). The (DA —BB coefficients satisfy A%?M =("1)l SOABAfig4- (32) The second-order DID and dispersion terms are both derived fi'om perturbation expressions with an underlying tensor-operator structure exemplified by TAAB ={“A(1)®[{[“A(l) @(TQ) ®“B(1))(1)](0) ®[RA(1) @(TQ) ®“31))(1)](0)}(0) ®“A(1)](1)}(J). (33) Coupling the four dipole operators for molecule A to produce two factors of polariza- bility of A gives a second-order DID term, while coupling to produce the hyperpolariza- bility of A gives a dispersion term. B The coefficients for the second-order DID terms of the type aAa (1A are given by Afl = (—1)“‘*”2 ”£2229“ +1)n,b, (ab00|j0)(2200|Lo) a b n 1 1 2 1 1 a , x1 1 2 1 1 b{1{ 1; :}aA(a,O)aA(b,O)aB(k,0)R'6; (34) k n L n J ' A correspondingly, the coefficients for the aBa 0L8 terms satisfy The sec-g hr and £15ng 1 alpha: TESuhs fl 32 BAB 1. AB ABA AkaxL=(—1) p AijiL- (35) The second-order DID coefficients agree with the results of Bancewicz [58]. For the 7A - a8 dispersion coefficients, the result is 2 1 1 , 1J/2 J k 7» A}f‘m=(—1)+ 15J32(r1,,/rr,) 2 1 1{L J a}(2200|LO) a L k a xh/ nJ:yA(a,J;j,0|iw,0,O) aB(k,O;im) d0) 11:6, (36) where A . - _ A ~ A - Y (0,0,0,0|1(0,0,0)—3Y1(1CD,0,0)+2'YZ(1CD,0,0), (37a) JA(2,2;0,Oliw,0,0) = 2319mm), (37b) 7A(2,0;2,Oli0>,0.0)=- 2 [31?(iw,0,0)+41’s‘(iw,0,0)1, (37C) v"(o,2;2,orico,o,o> = 45 [37?(ico,0,0)+47§‘(iw,0,0)], (37d) yA(2,2;2,0|im,O,O) = —2Jfiyg‘(im,o,0), (37c) and v"(2,2;4,01im,o,0)=2J76v’7‘(im,o,0). (371) In Eq. (36), aB(k,0;iw) is the (k, 0) spherical component of aB(iw). The results in Eqs. (37a)-(3 7f) for W" (a, J; j, 0| 1w , 0, 0) have been obtained from the dipole operator coupling {[HA(1)® uA(')](a) ®[uA(l) ® uA(l)](J)}(j). The aA -yB dispersion terms satisfy Ail)“. = (-1)* BAH/315111.. (38) The coefficients in Eqs. (13) and (14) have been evaluated by direct integration usingMathematica [114], and by use of Eqs. (27)-(38), with identical results. In Sec. 3.3, CXplicit expressions for Aug are given and their spectroscopic implications are discussed; results for mg“ and their spectroscopic implications are given in Sec. 3.4. 33 Colli Th induced nonzero can OCCU Fo. the orier contnbu dispersir. Both D1 Coel’fidt- ls. PTOdUCe — — —' — coefficjc “"3 $61. S‘mUltan 33 3.3 Collision-Induced Changes in Scalar Polarizabilities This section provides results for the coefficients AOAAXBKL that determine the induced changes in scalar polarizabilities. Since Aug in Eq. (13) contains terms with nonzero AA and/or AB, isotropic rototranslational Raman scattering with A] = i2 and t4 can occur as a purely collision-induced phenomenon. For a pair at a fixed separation R, the change in effective polarizability averaged over the orientations of ii" , FE, and R is Aooooo/3- Through order R'6, there are two contributions to Aoooooa one from second-order DID interactions and the other fi'om dispersion Aooooo = (1+50AB){6EAEBEA +4/3(ai1‘ -ai)EB(aii -ai) +3h/1r Lida)680w)[3Yf‘(iw1020)+279(i®,0,0)]}R—6. (39) Both DID and dispersion effects increase the effective polarizability of the pair. The coefficient Aooooo accounts for isotropic, collision-induced Rayleigh scattering [24]. Isotropic scattering with a change in the rotational state of one or both molecules is produced by the polarization effects in nine other coefficients, through order R‘6. The coefficient A02022 is associated with transitions on molecule A only 14.02022 = 4J§/ 5 53mg —a1‘)R‘3 +2J§ / 5 53m? —ot':)(2aA +EB)R‘6 +2J§l15(afi‘ —afi)[aB(afi —a1‘)+2/3(afi —orfl3_)2]R'6 +Jgh/(51r) {face [3y{3(iw,0,0)+27§(i(o,0,0)][0t‘fi‘(ico)—0t3‘_(i(o)] +3 face [379001,0,0)+4y§‘(im,0,0)]EB(i(o)}R'6. (40) The set of coefficients A022LL with L=0, 2, or 4 can produce "double transitions", i.e., Simultaneous transitions in the rotational Raman spectrum [110]; for these coefficients, the with ctJ ‘1 7 l1 COCiilCit T0 ord». A1113; Spectra POlariz I and A and A. UlSOIr M3112: Iensi: nSor 34 associated selection rules are A] = :2 for each molecule (if the anisotropic terms in the interaction potential are neglected). The coefficients A022LL satisfy A022LL = (1 +50AB)l-\/17/ 15 51.2613 ‘01:)(0li)3 ‘0‘?)11‘3 + 2%” 8L4 [2/21(afi -ai)(3E§m —8EEm)—®AO(3B§ +4B,B)]R'5 +1:L {2/9(afi —a$)(afi —6E)[6 EA +(afi —or‘: )1 4.12/71]de [3y’3°‘(iw,0,0)+4y?(iw,0,0)][a11?(iw)—afl3_(i(o)]}R—6] (41) with CO =J§/25, 02 =J17/35, and c4 =18J76/175. The E-tensor polarization mechanisms give the only nonzero contributions to the coefficients A0413“: , through order R-6 A04044 = 8/3 518(an +2EQm)R‘5, (42) A04244 = —4J77‘ / 63 (61}? — a‘ixEQm + 25f,"m )R‘S. (43) To order R"6 there are three other nonvanishing coefficients in Equation (13) given by AB AB AB A00222 = 60 A02022, A00444 = 60 A04044, and A02444 = 50 A04244- First-order DID effects generally dominate in collision-induced light scattering spectra, unless the DID coefficients vanish due to symmetry. For changes in the scalar pOIanZability, fifSt-Ofder DID terms appear in Aozozz, Aozzzz, and A00222 only, A02022 and A00222 are associated with single transitions with A] = i2 for either molecule A or B and A02222 with double transitions having A] = :2 for both molecules (neglecting the anisotropy of the pair potential). Second-order DID and dispersion effects also appear in A02022a A0322, and A00222- Hence the net contribution of these terms to the scattering intensity is enhanced by the existence of cross-products with the first-order DID effect. E- tensor terms in Aug contribute significantly to scattering in the spectral wings [3 5], beca orhe effec cont DID rang 3.4 ( polar scan .~‘./.‘ scan lSOtrr with to [he and ,2- 35 because they produce transitions with A] = :4 for one molecule and AJ up to :2 for the other. These effects should be easier to distinguish in the isotropic scattering spectra than effects of the other R“5 or R'6 polarization mechanisms. The leading long-range contributions to isotropic, pure translational (Rayleigh) scattering stem from second-order DID and dispersion terms. Experimental spectra are expected to show substantial short- range overlap contributions to Aoooooa particularly for lighter, less polarizable species. 3.4 Collision-Induced Changes in Anisotropic Polarizabilities The collision-induced change in the second-rank tensor component of the polarizability A002"1 determines the spectra for depolarized rototranslational Raman scattering by A- - -B pairs. Through order R'6, A0112“ depends on a total of 38 coefficients A21. AthL- The collision-induced depolarized Rayleigh spectrum (pure translational light scattering) is determined by the collision-induced anisotropic polarizability A06", averaged isotropically over the orientations of molecules A and B. The averaging gives 1 Aal‘ = (4n / 3)/2 A2000: YthaR) (44) with A20002 = 64/10 / 5 #613 R'3 +(1+ goAB)[3J16/ 5 #536" + J1_o/ 75 x(a’fi —ai)6EB(afi‘ —a’:)+3\/I0/5 h/ 1: 3(de y9(im,o,0)aB(iw)]R*5. (45) From Eq. (45), the first-order DID interactions give the dominant long-range contribution to the polarizability anisotropy of a colliding pair, averaged over the orientations of FA and FE. T rules tic neglect 111th A A 4 With 1 and The ( i2 fC 36 The remaining coefficients in Aetg”I are categorized below according to the selection rules for rotational transitions they generate, if the anisotropy of the interaction potential is neglected. Three coefficients A2202L generate rotational transitions of molecule A only, with A] limited to :2. These are given by: A2202L = 4J7 ”51.2 a3 (at? —a’j)R‘3 +4J'3_5 /35 5L4[aB(3 133m 4133,“) — 7353 o“) ]R’5 + {(63 mi )aer‘orA +bL6EB +cL(afi‘ mi )] +dL(a1‘i-a3)(afi‘ —a’:)(a1‘?—aEJJR*‘ +eLh/ n I: do 7280a) ,o,0)[afi (ico)— a’:(iw)]R_6 +h/1tI:d(D[9\/l—O/5 amyf,‘(im,o,0)+fL y?(im,0,0)]6iB(ico)R'6, (46) with the coefficients aL - fL listed in Table 3.3. Two coefficients generate single-molecule transitions of A, with AJ up to i4 A24042 =[12J'33/175 63013—63)? +6JEh/(5n) A - —-B - —6 xJcho Y7 (rco,0,0)a (10))]R , (47) and A24044 = —2J154 /21 a3 (132”, + 2’3me )R’S. (48) The coefficients A222,,“ are associated double transitions, with selection rules of AJ up to i2 for molecule A and simultaneously AJ up to i2 for molecule B A2221.L = 31L (0‘11? “13W? — (1113.)R_3 +[(—1))‘ + f9AB1[bJ.LB?®BO — carat? —a1‘ X31327... 4152,... )1R‘5 +1(—1)’~ + W1 uuhc associ sari ‘ v . 7) 37 ><{(af? — A322LL = bLngoR—6U HOAB X313A +4'Y?)/(0L'fi\ ‘06), (61) with h0 = 45/50, h2 = -t/1—4'/70, and b4 = -9m/175. Similarly, the constant ratio approximation for the dispersion contributions to the remaining anisotropic polarizability coefficients with AA and AB £2 yields Agzozo = 413 /50 C§00R_6[Yi3 HuiB +15<3v2t High/(at? —at)1, (62) A3202 = 47 /35 CE‘X’Rih? EB — 217? Mai? —aj‘_)1, (63) A3202, = —18J'3_5/ 175 CéOOR‘6y2B /at'B, (64) A322“. = —1 / 2 C22°R*51(—1>’~ + #310111? with? your? mi), (65) with fo and gm given in Table 3.4. To estimate the dispersion terms with AA=4 or AB =4, the van der Waals coefficients C§ALBM are required. The coefficients Cg‘ALBO with L A or LB equal to 4 are related to the fourth-rank tensor invariant of the quadrupole polarizability C, C4 1, (66) and to E7(ic0 ), the fourth-rank part of the dipole-octopole polarizability E, by (3:00 = —20 rt / 1t [3" [3 c1: (i6)+ 2 B9 (10) )]a‘Bam )dco, (67) and Equai inure ennn Ct, 1411 5‘ . Tabk in ten .114. Coefi} cacui lKflar Coefi? and ( zero- 3330C to be Pretj dlSpe] aCCur 42 (3320 = -h/1t [:[88c9(io)+ 152/3 B? (ic0)][a11T3(ic0)—0tE(ic0)]d0). (68) Equations for CQZM with M ¢ 0 are given in Ref. 68; C340 and C3340 are obtained by interchanging the molecule labels A and B in Eqs. (67) and (68). The anisotropic dispersion coefficients with L A or LB equal to 4 have been estimated in terms of the anisotropic van der Waals interaction energy coefficients C300, €340, c320, and C3340, the fourth-rank part of the static y hyperpolarizability tensor from Table 3.1, and the static values of C 4 and E7 , with the results: #214042 = —3J3'5 / 100 CQOOR‘W’; /(3 Cf +2134 ), (69) Aé‘ott = 3/ 16 (—1)“‘C§2°tttR*t’-? /(33Cfi +1959), (70) in terms of the coefficients t (L given in Table 3.5. The remaining coefficients are given by d AB d d A AB d A20442 = 69 A24042 and A224J.L = (‘1) 50 A2421L‘ For small molecules, values of the permanent susceptibilities and dispersion energy coefficients appearing in Eqs. (54), (55), and (60)-(70) are available from ab initio calculations [3 8-48]. Table 3.6 gives the values of or and 7 used to estimate the dispersion polarizability coefficients for H2 and N2, and Table 3.7 gives the dispersion energy coefficients used. Coefficients not listed and not derivable by symmetry arguments (C340 and C§40 for Hzn-Nz, and C300, C340, C320, and C?” for N2---N2) have been set to zero in the calculations. This is equivalent to dropping the dispersion contributions associated with the fourth-rank part of the y hyperpolarizability for N2; these are expected to be small compared to the contributions from 7 components of ranks O and 2. Table 3.8 provides a test of the constant ratio approximation, by comparison of the estimates for the dispersion contributions to A JAAABAL for a pair of interacting H2 molecules with the accurate ab initio results recently obtained by Bishop and Pipin [3 9]. For H- - - H2, He- - -H2, and H2 - - - H2, the rms error in the dispersion polarizability coefficients with AA = 0 or AA = 2 for H2 is ~20% [39-70]. The rrns error for the pair H2---H2 with AA 43 Table 3.6. Molecular properties (in a.u.) used to calculate collision- induced polarizabilities. Property H2 N2 (9° 0.4828a -1. 1 1318 a 5.3966" 11.6756 ot1T -aJ_ 1.9793" 4.64se ym, 743.86c 1 172e 7m; 621.05c 639e ym, 229.616 319" Cu,ZZ 6.3926" 34.61‘3 mez 4.4441" 26.856 me 5.2032" 19.296 132 m 4.4424" 38.283 Em, -1.7740" -2200" d f Bzm -97.671 -174 d f me -63.417 -102 wa 36.746d 67f B,“x -71250‘1 -119.5f aRef. 36, value interpolated to r(H2) = 1.449a.u. in Ref. 37. bRef. 38. cRef. 39; values of the static y hyperpolarizability at r(H2) = 1. 449 a.u. were supplied by D. M. Bishop, personal communication. dRef. 40; static B values from D. M. Bishop, personal communication. eRef. 41; Ref. 42 gives a = 11.616 a.u. and an —oti = 4.654 a.u.; properties of N2 at SCF level are given in Refs. 43-45. fRef. 44; for B values, see also Ref. 45. 44 Table 3.7. Dispersion energy coefficients (in a.u.) used to calculate collision-induced polarizabilities. Coefficient H2 - --H2 H2 ~ --N2 N2 - - -N2 C20" 42.0583 -29.28° -738" C300 -1 .219a 288“ -782" C220 -1219" 3.59" -7.82° C330 -0390" -1. 12c -267" C30" -138" -322" — C§20 -0.57" -1.62° — aRef. 38. bRef. 46 with isotropic C8 coefficient for H2 pairs from Ref. 47. cRef. 46. "Ref 48. Table 3.8. Test of the constant ratio approximation for dispersion polariza- bility coefficients by comparison with accurate ab initio results (Ref. 39) for H2 - - ~H2. Results in a.u. for the coefficients of R'6. Coefficient Ab initio Constant ratio approximation ASOOOO 2960.8 2471.5 A32022 90.6 75.9 A‘Z‘0002 354.8 312.6 A‘Z‘2020 41.6 31.5 A‘Z‘2022 -234 2.99 A3202, 38.7 30.4 d A24042 -12 -O.6 l «184 1' approx depend im7-' more 31 l6Nu 45 or AB equal 4 ranges fi'om ~50% - 60%. This may be due to limitations of the approximation. On the other hand, the terms in the fourth-rank component of y for H2 nearly cancel, leading to a very small value for 77 and high sensitivity to its frequency dependence. In fact, the most recent ab initio calculation in Ref. 39 gives a different sign for 77 than found in Ref. 116, though both calculations are of high accuracy overall. For more anisotropic molecules, such near-cancellation is not expected. 3.6 Numerical RCSIIItS for H2°"H2, H2"'N2, and N2"'N2 Table 3.9 lists the contributions to the collision-induced polarizabilities Aug, Aug, and A062 from first- and second-order DID effects, E-tensor terms, B-tensor terms, and dispersion, for H2---H2, Hzn-Nz, and N2 ---N2 in collinear and T-shaped configurations. For each of the molecular configurations studied, the intermolecular vector points along the z axis. Molecule pairs are listed as A- --B. In the T configurations, molecule A points along the x axis and B along 2; hence results differ for H2 - - -N2 and N2- - -H2 in T shapes. As shown in Table 3.9, first-order DID interactions increase a8 and org for these configurations, but decrease or? for T-shaped pairs. Second-order DID and dispersion effects increase 018 and org , and or? in each case. E terms are positive in org"I for collinear pairs, but for T-shaped pairs the E terms are negative in Aug and nonuniform in sign for Aug". The signs of the B-tensor terms are nonuniform, because the quadrupoles of H2 and N2 are opposite in sign. The long-range polarization effects have been compared numerically for H2 - - - H2 at R=7.5 a.u. (approximately 1 a.u. outside the van der Waals minimum in the isotropic pair potential), for qu-Nz at 8.0 a.u., and for N2 ---N2 at 8.5 a.u. (0.5-0.7 a.u. outside the isotropic van der Waals minimum). The results are summarized below, first for the change in scalar polarizability Aug and then for the anisotropic polarizability A0112“. Act 40. 46 Table 3 .9. Long-range contributions to collision-induced polarizabilities Aug, A013 , and A031 for H2"'H2, H2"'N2, and N2"'N2 .8 AOL?" Configuration A---B DID-l DID-2 E tensor B tensor Dispersion A68 Collinear H2---H2 52.4 R'3 1645 R'6 120 R'5 81 R'5 1640 R’6 H2---N2 118.4 R'3 5747 R‘6 362 R'5 -26 R'5 3320 R'6 Nzu-Nz 267.3 R'3 17290 R"6 1110 R'5 -308 R'5 6420 R'" T, Attx qu-Hz 10.8 R'3 1175 R'6 -55 R'5 .40 R"5 1470 R'6 H2---N2 27.7 R'3 4226 R‘6 -349 R'5 13 R'5 2980 R'6 N2---H2 20.9 R'3 3845 R'6 -244 R'5 13 R'5 2680 R‘6 N2---N2 54.3 R'3 11950 R‘6 -1312 R'5 154 R'5 5230 R’6 hog Collinear 112-an2 184.0 R"3 1805 R"6 500 R‘5 636 R"5 631 R'6 Hz...N2 402.4 R'3 6382 R'6 2910 R"5 -163 R'5 1460 R'6 Nz-nNz 880.3 R'3 19370 R'6 10300 R'5 -2630 R“5 3200 R'6 T, Alix H2 ~H2 148.2 K3 954 R’6 93 R‘5 108 R'5 449 R'6 H2---N2 323.2 R'3 3701 R'6 1770 R'5 -901 R'5 1080 R'6 N2---H2 318.4 R'3 2892 R‘6 -834 R‘5 814 R'5 716 R’6 N2---N2 694.4 R’3 9797 R'6 1540 R‘5 —375 R'5 1730 R‘6 Act;2 T, Atlx qu-Hz -94 R'3 76 R’6 9 R'5 5 R'5 29 R‘6 qu-N2 -20.0 R‘3 216 R'6 -125 R‘5 89 R'5 53 R'6 N2---H2 -22.0 R'3 326 R‘6 135 R'5 -109 R'5 125 R'6 N2---N2 .471 R'3 824 R'6 -50 R‘5 19 R'5 234 R’6 aResults are tabulated in a.u. pan; und_ aresij anon“ I E-ten Tsha exceet still 51 for~1 heat'it larger Fore shapt exeet With less COn for Val 0b. f0 I $4 47 F irst-order DID interactions (DID-1) make the dominant contribution to Aug for the pairs and distances selected: for collinear pairs, the DID-l terms give 82%-84% of the total Aug, and for T-shaped pairs, 66%-82% of the total. Other polarization mechanisms are significant, however: for the T-shaped pairs, second-order DID interactions (DID-2) amount to 17%-29% of the total, and dispersion terms constitute 13%-21% of the total. E-tensor terms vary substantially in importance: they amount to less than 1% of Aug for T-shaped Hz- . - H2, but 27% for T-shaped N2 -- -N2. B terms are generally small, not exceeding 3%-4% of Aug. The corrections to DID-l are smaller for collinear pairs, but still significant: DID-2 terms account for 6%-9% of the total, E terms for 3%-5%, B terms for ~l%-2%, and dispersion for 3%-6%. Generally the DID-2 terms are larger for the heavier pairs, in both absolute and relative magnitude, while the dispersion terms are larger in absolute magnitude but smaller in relative magnitude for the heavier pairs. The first-order DID interactions give better approximation to A0112“ than to Aug. For collinear pairs, the DID-1 terms account for 86%-88% of the total, and for the T- shaped pairs studied, the errors in the first-order DID approximations for AmiVI do not exceed 6% in any case. DID-2 and dispersion interactions appear to be highly isotropic, with the result that DID-2 terms contribute ~1.5%-3% to A032“, and dispersion ~1% or less. The E- and B-tensor contributions to Aorg'1 are larger: E terms contribute 4%-14% to AOL? of collinear pairs (largest for N2 - - -N2), 1%-8% to AOL? for T-shaped pairs, and 1.5%-10% to A0112t2 for T-shaped pairs. B terms range from ~1% to 8% of A0112“. In the T configurations, both E and B terms are more important for the unlike molecule pairs than for the like pairs. As a general trend, the DID-1, DID-2, E, and dispersion terms increase in absolute value in the order H2-~H2 < [<01 mo 6(4) u: G(w)p(k')10> +<01 p00 6(a)) p(k'>° 9(0) no. 10> +(01uo 0(0) 900° G(w)P(k')|0)l (6) based on the general form for nonlinear response tensors given by Orr and Ward [28]. Damping has been neglected in Eq. (6); C(co —> -c0 ) denotes the operator for complex conjugation and replacement of to by -(0; 11a is the or component of the molecular dipole operator; p(k) is the Fourier component of the charge density operator: p(k) = Idr exp(ik-r)2qj 5(r — rj) = qu exp(ik-rj) (7) i 1' with q j the charge of particle j, and G(m)=(1—tao)(H—Eo —hco)“(1—tan), (8) where 500 is the ground-state projection operator IO) (0 ; also, in Eq. (6) p(k)° a p(k)—(O| p(k)|0), and similarly for p(k')° and [.13. The expression for the contracted susceptibility density BaB(k, k';(t), 0) is given by Eq. (6) with the dipole operator ua replaced by the molecular quadrupole operator @0113' Thus BaB(k, k';c0,0) and 13a(k, k';c0 ,0) are two separate contracted susceptibility densities, but they are different contractions of a single, underlying hyperpolarizability density B(r,r',r";0),0). Contraction of P(r") in B(r,r',r";c0 ,0) into the dipole operator 11a and then spatially Fourier transforming gives 80, (k, k';(0 ,0), while contraction of 61 P(r") into the quadrupole operator @0113 and Fourier transforming yields the density BaB(k, k';(o,0). To simplify Eq. (6), the density Ba (k, k’;(0, O) is expanded in terms of the spherical harmonics of the orientation angles of the vectors k and k' by substitution of Eq. (7) into Eq. (6) and use of the Rayleigh expansion for exp(ik - rj): (:0 L co A L' I . Ba(k,k';w,0)=z Z Z ZCLCL'YIV(9,¢)Y1¥(9'J¢') L=O M=—L L’=0 M’=—L' XBWMKMWO), where oL = (—i)L2LL!/(2L)! [41t/(2L+1)]‘/5, and 8m;(k,k';o,0)= [1+C(c0 —+ —o)] x [<01p1‘(k)G(m)u3 G(m)p£4'(k')‘ 10> +<01 p M(k)G(w>p1.4 (k')°G(o>ua10> +<01 no. G(OJpLM(k)° G(w>p14'(k')‘ 10)] with the generalized multipole moment operator p?‘(k) given by % m 21 11. PI m=(k) qu'(21:1) Y1 (9],9j)£3;fi)—Jz(krj), In Eq. (12), j ,(krj) denotes the lth spherical Bessel function. (9) (10) (11) (12) Bwak, k’;0),0) is a first-rank Cartesian tensor, with spherical tensor components of rank 1. The components are related by 62 "1.1.30 ( )(k, k'; (0 ,=0) BLUl(k, k', (0 ,,0) (13a) 13L"?I (f1)(k, k', 0) ,0:) +1/J‘[BLL x(1<,1r',o,0)+13LL ;,(k, k', (0 ,0)]. (13b) If the auxiliary functions i“(k,m)) = G(—co)pi“(k>10>. (14) then fi'om Eqs. (11) and (13), 1 )(91 (k'o)|°|<1>1‘(kw))) (15) BLL (r (k,k';o,0)= [1+C(w —+ —w)]((<1>1| p309” +<14'(k',o)|p1.‘(k)° with q equal to -1, 0, or 1. In Eq. (15), or are the spherical multipole moment operators given by of =quer14n/(2L+1)1%Yr§‘(9,-,¢j), (l6) 1' and the firnctions (DLM are defined by |¢1.‘)=G(0)p1410>. (17) The susceptibility density BOLL(k, k';(0, 0) has an expansion similar to Eq. (9), 00 L 00 L' Bae(k,k';w,0)= Z Z Z thcirYt:‘(e,¢)Y14(ez¢')‘ L=0 M=—L L'=0 M'=-L' xBL’tLrflLB(k,k';co,O), (18) where ngzflk, k';(0, O) are second-rank Cartesian tensor components, related to the spherical tensor components by BLL,(2 o)(k, k’, (0 ,=0) BLB’LJMfl(k, k'; 0) ,,0) (19a) BLMSJ,(+1)(k, k'; 0) ,=0) +,/2/ [/3BLL XZ(k, k'; 0) 0,):BLL MM;,L(k,k';00,0)], (19b) 63 and Bwégz)(k,k';m,0) = 1 /,/6[BL1§‘L¥§,,L(k,k';o,0)— Bfifl‘f'yy(k,k';o,0) aziBfihflymmgobn. (19c) The components Bflégfik, k';(0,0) (q = -2, -l, 0, 1, or 2) are given by the right side of Eq. (15) with (I):l replaced by og and p? replaced by pg. (1);" (k, (0) from Eq. (14) can be approximated by the function ‘1’ which makes the following functional minimum [16, 17-20], 1;“(41): (‘I’lHo — E0 + ’10)l‘Pl—(OIPWKY1‘1’)*(‘*’lpin(k)|0) (20) subject to the conditions (0| ‘1’) = 0 and 0) 2 0. In Eq. (20), H0 is the Hamiltonian of the unperturbed molecule, and E0 is the energy of its ground state. Eq. (15) and the corresponding expressions for the densities Big/{(3 )(k, k';(o , 0) are the principal results of this section. They express these densities in terms of the variationally determined firnctions (1)}“(k,01) that also fix the linear charge-density susceptibility [15, 16, 22]. In the next section, these results are employed to calculate the contracted B and B susceptibility densities for the hydrogen atom in the Is state. 4.3 Application to Hydrogen Atoms To illustrate the method given in Sec. 4.2, the z-component of the contracted B susceptibility density and zz-component of the contracted B susceptibility density are calculated in this section for the ground-state hydrogen atom. The results will be used to evaluate the damped dispersion-induced local dipole 142 and quadrupole On for a H- - -H pair in the next chapter. 64 Using a trial firnction of the form I‘P>= Mk,co)G(0)pi‘10> (21) and then applying the variational principle (20) to find A(k,(t) ), Koide [16] has determined the filnctions £4(k,01) for the hydrogen atom in the Is state: 1.‘(k,w)s(¢1“1 pt¥(k)10)[(<1>1‘| pi.“ IO)+9(1‘|1‘)[1 91.” (22) with 1/ —r (1)}:d = (2:1)" YL“(0,<1))[£L:1 +5367;— (23) Atomic units are used in Eqs. (22), (23) and below. For the z-component of the B density of the hydrogen atom in the Is state, only the elements BLWur are nonzero [29]. They are given by pml(k,k';o,0)=om. (1L0M|L:1M)(1L00|L:10)BLLL,(k,k';o,0), (24) where (L1 L2 M1M2|L M) is a Clebsch-Gordan coefficient, and 20) a a d BL L¢1(k,k';03,0)= 2 L 1‘; [fL(k)gL:(k')—M_fL(k)fL:l(k')] wL-w mL—mLfl +-2°°—Lfliuirfttr(k')htt(k)-—i—L31£~frl (DELL—(02 wL’thr (25) With L 2 1 for BL L+l and L 2 2 for BL L-l’ and 20) a Primates): 2 ' ‘2 f1(k') h0+(k), (26) 031 —0) 2(0 a 136(k,k';co,0)= ‘ ‘ f1(k)81—(k')- (27) ref—652 65 In equations (25)-(27), the functions 60 L , aL and IL (k) have been given by Koide [16] (91‘1pi‘l0): L(L+1)(L+2)(2L+1) (91.1914) L4+11L3+18L2+10L+2’ (0L: 2 K911418410) _ (2L)1(L+1)(L+2)2(2L+1)2 (oblwi‘l 22L+‘(2L“+11L3+18L2 +10L+2)’ 3L: and fL(k)= (oi MIMI” IO-> 2L/(2L+1) kLX,L+2(X-1-1/2L) (“WIMILTZ The firnctions new in this work are dLi, gLL(k), and hLi(k); they are given by , Pi l0?) pend 10) = —22L+3[(L +3)(2L +4)r]‘l (2L4 +17L3 + 46L2 +43L + 9), M <®L+l M, <®L+l dL,= [(1L0M|L+1M)(1L00|L+10)]‘l M PL+1 <¢L—1|Plo|®LM> -1 dL':(9>t-r|ptt|0)(¢1:‘|pi‘10)[<1LOMlL—1M><1LOO|L_IO>] = —22L [(2L —1)(L +1)(L + 2)(2L)r]'1 (2L4 + 9L3 + 7L2 - 6L — 3), 8L+(k)= <0<<1>MoMrM| 0) = —(L+3)/4 kL+1xL+3[2L(L+4)x2 +(3L+2)x+1], 0) [(1L0M|L+1M)(1L00|L+10)]‘1 I-fi 9° pitta)” |0> -1 gL_(k).-: (oMlpM10) [(1L0MlL—1M)(1L00|L—10)] L L (28) (29) (30) (31) (32) (33) = —1 / 2 kL‘1xL+2[(L +2)(2L -1)(2L +1)]‘l [4L(L +2)(L +3)(L +4)x3 —4(L + 2)(L + 3)(2L —1)x2 — (L + 2)(13L + 4)x — 3(L + 3)]+ 43 / 6 5L1x2, (34) 66 ®M+ pM(k)000 - hL+(k):<(Lt"ant‘n 0;)[(1LOMIL+1M)(1L00|L+10)] 1 = —1/2 kL xL+31(L +3)(2L +1021“ 14(L +1)(L +3)(L +4)(L + 5)x3 —4(L +3)(L +4)(2L +l)x2 —(L +2)(13L+17)x —3(L +4)]+43/186L0x2, (35) and n_(t)= 0(LiTL§)°|:)‘)I<1L00|L-10>1‘1 = —(L+2)(2L+1)/4(2L—1) it" xL+2[2(L-1)(L+3)x2 +(3L—1)x+1], (36) where x a 4 / (k2 +4). The zz-component of the B susceptibility density of the ground-state hydrogen atom has nonzero elements Bwfl and BIL/$4, which satisfy Bm'n(k,k';o,0) = 6m. (2L0M|L+n M)(2L00|L+n 0) XBLL+n(k,k';(D,O), (37) where n equals -2, 0 or 2, and 20) a a D , BL L:2(kak';0),0)= 2 L L2 [fL(k) ALL:2(k')_—Li—2—L£2"fL(k) fL:2(k )] (BL-(0 (BL-(015:2 20) a a D , +——_2Li2 L132 [fL:2(k')CLL:2(k)+—¢2—'f1](k)fLfl(k )] (38) ("Liz “"3 "3L _(DL:2 with L 2 1 for BLL+2 and L 2 3 for BLL_2, and 20) a B02(k,k';00,0)= 2 2 2, f2(k')C02(k), (39) (Dz-'03 203 a B20(k,k';03,0)= 2 2 2, f2(k)A20(k'), (40) (02 —CD 67 and 20) a BLL(k,k';®,0) = fig— [fL(k) ALL(k')+ fL(k')CLL (10] L _ 2 2 2 +2aLDLL(w‘a;2‘° )ft(k)ft(k'). (41) (mi—w In equations (38)-(41), the functions DLL+n, AL L+n(k) and CL L+n(k) (n = -2, O, or 2) satisfy D = ((DLM” 93W) [(2L0M|L+2 M)(2L00|L+2 0)]"1 ”+2 (on an |0)(91‘o are. 0) = 42143 (2L +3)[(L +4)(2L+ 5)(2L +4)r]‘l x(2L4 +23L3 +86L2 +110L +20), (42) M 0 M (QL—ZIPZIOL D = <‘DL I PL 'OX‘DL—zl PL—z 2LOM|L—2 M)(2L00|L—2 0)]-1 a)“ =(2L+1)/(2L-3) DL-ZL’ (43) ("WI I011?) —1 DLL= [(2L0M|LM)(2L00|L0)] p3 <41 410i = -22L+l (2L + 3)[(L +2)(2L +1)(2L +2)1]‘l x(2L4 +15L3 + 31L2 + 14L + 2), (44) <03 pLM+2(k).o '0?) [(2 LOM|L+2 M) (2 LOO|L+2 0)]—l ALL+2(k) : (WI Pi" '0) = —(L +3)(L +4)(2L +3)/ 24 kL+2XL+4 x[4L(L + 5)x2 +4(L + l)x +1], (45) 68 93' pifiztk)” 91") L ALL_2(k)= ("WM") [(2LOMlL—2 M) (2L00|L—2 0)]— = —1/6[(2L —1)(2L - 3)(L + 2)(2L +1)]'1 kL'ZxL+2[16L(L + 2)(L + 3) x(L +4)(L+ 5)x4 +16(1—8L)(L +2)(L +3)(L+4)x3 +48(L +2)(L+3) x(3L — 2)x2 +60(L + 2)(2L + 1)x +15(L + 4)]+107/12 5L2x2, (46) 9° PM(k)'° M ALL(k)== 2 L: M "BL > [(2 L 0M|L M) (2 L 0 0|L 0)]"1 (a, 1 m 10) = -1 / 12 (2L +1)‘1 kaL+3[8L(L +3)(L+4)(L + 5)x3 —4(L+3)(L +4) x(5L — 2)x2 —6(L +3)(3L + 2)x —3(L +4)], (47) 0&2 DIRK)“, I‘Di} CLL+2(k)-—= M M [(2L0M|L+2 M)(2L00|L+2 0)]-l (‘Duz PL+2 0) =(2L+l)/(2L+5) AL+2L, (48) CLL (k) = ALL(k)a (49) and ("iii-2 Pit/100‘o 0(2) CLL_2(k)= [(2L0M|L—2 M)(2L00|L—2 0)]-1 M M ("h—2| PL—z IO) = (2L +1)/(2L — 3) AL—ZL' (50) 69 4.4 Summary and Discussion This work has shown that the contracted susceptibility densities [3a (k, k';co , O) and 1304306, k';co , O) are related to the variationally determined fimctions <1>fid(k,co) that have been used to evaluate the linear charge-density susceptibility x(k, k';oo) [14, 15, 22]. For the hydrogen atom in the ls state, the work yields analytical expressions for these susceptibility densities. The results will be used in later computational work on the damped local dispersion dipole and quadrupole of a H- --H pair (see Chapter V). The method developed in this work can be extended to the S-state atoms with multiple electrons. In the extension [3a (k, k’;co , O) is still given by equations (9), (24)- (27); but the functions (0L, aL, dLi, fL(k), gLi(k), and hLi(k) appearing in Eqs. (24)- (27) cannot be evaluated analytically as in the case of the hydrogen atoms. Instead these fiJnctions are obtained from the one-electron transition density matrices DA and DB analogous to the matrices DA01 and DB01 introduced by Krauss and Neumann in the calculations of the charge-density susceptibilities x(k, k';(o) for inert gas atoms [23]: (0L 2 aL /{2 Tr[DA(L,L)S]}, (51) aL = afi /{4 Tr[DA(L,L)S]}, (52) fL(k) = 2/aL ZZDBJ-i (L)(i1 lpl‘ml w?) (53) i j gril lpfimk)” ¢})c“, (54) i j hLi(k) = 2/aLi1ZZDAji(1,Li1)<} |p?|<1>} ) c". (56) i ,- 70 In equations (51)-(56), aL is the static polarizability of order L; {(1)9} and {}), DA= M“p = Ap(I.)“(M‘1 )t where A is the density matrix for the ground state, and p (L) and M are defined by oil-(L) = (D? In? |§-’), and M,,- = (ch? H0 — E0 lo} > In Eqs. (54)-(S6), the coefficient c satisfies c:(lLOMlLilM)(1LOO|Li10). (57) (58) (59) (60) (61) (62) 7] References [1] W. J. A. Maaskant and L. J. Oosterhoff, Mol. Phys. 8, 319 (1964). [2] L. M. Haflcensheid and J. Vlieger, Physica 75, 57 (1974); 79, 517 (1975). [3] T. Keyes and B. M. Ladanyi, Mol. Phys. 33, 1271 ( 1977). [4] J. E. Sipe and J. Van Kranendonk, Mol. Phys. 35, 1579 (1978). [5] K. L. C. Hunt, J. Chem. Phys. 78, 6149 (1983). [6] K. L. C. Hunt, J. Chem. Phys. 80, 393 (1984). [7] K. L. C. Hunt and J. E. Bohr, J. Chem. Phys. 83, 5198 (1985). [8] K. L. C. Hunt and J. E. Bohr, J. Chem. Phys. 84, 6141 (1986). [9] J. F. Ward and I. J. Bigio, Phys. Rev. A 11, 60 (1975). [10] B. F. Levine and C. G. Bethea, J. Chem. Phys. 63, 2666 (1975). [11] E. M. Graham, V. M. Miskowski, J. W. Perry, D. R. Coulder, A. E. Stiegman, W. P. Schafer, and R. E. Marsh, J. Am. Chem. Soc. 111, 8771 (1989). [12] D. R. Kanis, M. A. Ratner, and T. J. Marks, Chem. Rev. 94, 195 (1994). [13] K. L. C. Hunt, Y. Q. Liang, R. Nimalakirthi, and R. A. Harris, J. Chem. Phys. 91, 5251 (1989). The co = 0 case of Eq. (1) has been proven early by Hunt, see K. L. C. Hunt, J. Chem. Phys. 90, 4909 (1989). [14] See, e.g., L. D. Barron, Molecular Light Scattering and Optical Activity (Cambridge University Press, Cambridge, 1982), p. 100. [15] A. Koide, W. J. Meath, and A. R. Allnatt, Chem. Phys. 58, 105 (1981). [16] A. Koide, J. Phys. B 9, 3173 (1976). [17] M. Karplus, J. Chem. Phys. 37, 2723 (1962). [18] M. Karplus and H. J. Kolker, J. Chem. Phys. 39, 1493, 2997 (1963); 41, 3955 (1964). [19] S. T. Epstein, J. Chem. Phys. 48, 4716 (1968). [20] P. W. Langhoff and M. Karplus, J. Chem. Phys. 52, 1435 (1970). [21] B. Linder, K. F. Lee, P. Malinowski, and A. C. Tanner, Chem. Phys. 52, 353 (1980) [22] P. Malinowski, A. C. Tanner, K. F. Lee, and B. Linder, Chem. Phys. 62, 423 (1981) 72 [23] M. Krauss and D. B. Neumann, J. Chem. Phys. 71, 107 (1979). [24] M. Krauss, D. B. Neumann, and W. J. Stevens, Chem. Phys. Lett. 66, 29 (1979). [25] M. Krauss, W. J. Stevens, and D. B. Neumann, Chem. Phys. Lett. 71, 500 (1980). [26] M. Krauss and W. 1. Stevens, Chem. Phys. Lett. 85, 423 (1982). [27] B. Linder and R. A. Kromhout, J. Chem. Phys. 84, 2753 (1986). [28] B. J. Orr and J. F. Ward, Mol. Phys. 20, 513 (1971). This reference gives general expressions for the frequency-dependent nonlinear susceptibilities (up to the second hyperpolarizability). [29] This can be shown by application of the Wigner-Eckart theorem to Eq. (15). CHAPTER V DISPERSION DIPOLES AND QUADRUPOLES FOR PAIRS OF ATOMS: EFFECTS OF OVERLAP DAMPING 5.1 Introduction Interactions between colliding molecules in gases and liquids distort the charge distributions of the collision partners, producing transient dipole moments in the molecular pair. Collision-induced changes in dipole moments give rise to the single-molecule forbidden infrared and far-infrared absorption by nondipolar species such as H2 [1-2], N2 [3-5], and inert-gas mixtures [2, 6], and contribute to allowed absorption. In line shape analyses of light absorption spectra, the collision-induced dipoles are needed as functions of intermolecular separation and relative orientation. For molecules interacting at long range, only classical induction and dispersion effects contribute to collision-induced dipoles; they are given in terms of permanent multipole moments, multipole polarizabilities, and hyperpolarizabilities of isolated molecules [7-12]. Long range dipoles of inert-gas heterodiatoms have been determined accurately in quantum perturbation calculations [9]. The net dipoles for pairs involving H, He, H2 , and N2 are known to a good approximation at long range [11, 12]. At shorter range, when overlap and exchange effects are significant, ab initio methods can be used to compute pair dipoles. Because large basis sets are often required in pair property calculations, accurate ab initio calculations (including correlation effects) for pair dipoles have been limited to small systems such as He- --H [13, .14], He- - - Ar [15], Hen-H2 [16, 17], and H2~--H2 [18, 19]. This prompts interest in developing long-range models that include overlap effects and yield good results in the regions near to the van der Waals minima. In these regions it is often diflicult to obtain accurate results ab initio 73 74 due to numerical cancellation and basis limitations. Therefore, long-range models, corrected for overlap damping, complement ab initio calculations at short range. Additionally, these models can be compared with experimental results or ab initio calculations to extract useful information on the short-range exchange effects on pair pnnxnfies A nonlocal polarizability density model developed by Hunt [20] serves this purpose. In the model the distribution of polarizable matter in the interacting molecules is represented by means of the linear response tensor a(r, r';0) ), the nonlinear response tensor B(r, r',r";oa ',m —co '), and higher-order tensors. These tensors give the a) - frequency component of the polarization induced at point r in a molecule by a perturbing field acting at other points. Earlier, Hunt [20] has used this model to derive equations for induction and dispersion contributions to collision-induced dipoles and polarizabilities, with overlap effects included. The equations have been applied to a H- - - H pair to evaluate analytically the lowest-order damped induction contributions to local dipoles, quadrupoles, and pair polarizabilities [20]. However, no calculations of damped dispersion dipoles have been reported so far because a method of obtaining the required [3 susceptibility density has not been available until recently. In Chapter IV, the B susceptibility density has been shown to be related to a set of variationally determined auxiliary functions (DLM(k,co ). For a ground-state hydrogen atom, analytical results have been obtained for the B susceptibility density. The purpose of this chapter is to derive equations for the damped dispersion dipoles for atomic pairs, and to provide numerical values for the atomic dipole and pair quadrupole of a H- --H pair, based on the results given in the last chapter. The dispersion contribution to the dipole for an A- - -B pair is related to the dispersion energy between A and B in the presence of a static, uniform applied field F by [21] An?" = ‘5AEdisp / 51:6 |F=o- (1) 75 The dispersion interaction results from the correlations in the fluctuating charge distributions of molecules A and B: The spontaneous, fluctuating charge density of molecule A polarizes B; the induced polarization in B gives rise to a reaction field at A, causing an energy change in A. The reaction field at B associated with the fluctuating charge distribution of B determines the energy change in B, the second term in the total energy change in the pair. Two distinct physical effects contribute to Audi”: (1) Each molecule is hyperpolarized by the simultaneous action of the applied field and the fluctuating field due to its neighbor, and (2) the applied field changes the correlations in the fluctuating charge distribution of each of the molecules, thus affecting the van der Waals interaction energy. The dispersion contribution to the pair dipole depends on the polarizability density of one molecule and the contracted B susceptibility density of the other, both taken at imaginary frequencies [22]: A633,” 2 h/(21t)7 [O dco [[dk dk' (1 +5.2“)(org,(k,k';iw)[3§w(—k,—k';i6,0) xexpri (6) Into with the generalized multipole moment operator p? (k) given by Eq. (4.12). Specialized to a pair of S-state atoms, equations (4) and (5) simplify because 77 BIIYEYfA(k,k';co,0)=6MM[5LL41Bwl,a(k,k';co,0)+6w_lamgmzwn, (7) and afi.m'(k,k';(o) = 5,1, 6mm. a,(k,k';co). (8) Substitution of Eqs. (4) and (5) into (3) and use of Eqs. (7) and (8) gives Aufitdisp =2 2 [Au+(I.L)+Au"(I.L)] (9) I L with Attia, L) = may? [6,6112 [dk exp(—ik-R)k'2 Idk'exp(ik'-R)k"2 m m r I * M F M r r XYI (9,¢)Y1 (9 db) YL (9,45) YL:I:1(e ,¢ ) x]: dco a?(k,k';im) Blbffi(l‘:)(k,k';ico,0), (10) where R s RAB, and the z axis is chosen to be along the interatomic vector R. It should be noted that the x and y components of the dispersion dipole vanish for a pair of spherically symmetric atoms. Equation (10) can further be simplified by substitution of the Rayleigh expansions for exp(—ik-R) and exp(ik'-R) exp(ik-R)=Z[4n(2L+1)1% ii Y?(e,¢)ji(kR>, (11) J and integration with respect to the polar angles (6,¢) and (9',¢’). This gives it L+1 Ltl+l ”I 2 co . . A” (1, L) 2 pm. 2 Z q,,, (2/n) [0 dk 1)(kR) J: dk' JJ.(k'R) J=|L-I| J'=[Ltl-l‘ xii/2n [:dm a?(k,k';im) BfLi1(k,k';im,O), (12) where 78 2th1 1 2 2 [211] L!(L:1)! (13) plLi:(21+1)(2L+l) (21)! (2L)!(2Li2)!’ qfi'i = (—i)’i‘iJ'(2J +1)(2J’+ 1)(1 L o OlLil o)(J lO 0|L O)(J' Li 1 o 0|10) xZXILOMlLilMXJlOMILM)(J'LilOMllM), (14) M and BLLi1(k,k';i(D,0) = pml(k,k';im,0)[(1 L o M|Li1MX1L o 0|Li10)]’1. (15) In Eqs. (14) and (15), (L1 L2 M1 M2 |L3 M 3) denotes a Clebsch-Gordan coefficient. Next the limiting expressions for Apia, L) are derived as the interatomic separation R approaches zero and infinity: Use of the expansion jJ(kR)=21J1/(21+1)!(kR)J+O(RJ+2), (16) gives Aui(1,L)~RJ+" as R——>O (17) with the consequence that the dispersion dipole vanishes at zero separation R. In the limit as R —) 00, Eq. (12) reduces to Apia,” : Di(l’L)R—(21+2L+2il)’ (18) after use of the relation 4/1:2 dejj(kR)[:dk'jJ(k'R)a,(k,k';ico)BLLfl(k,k';ico,0) = 6,,+L51.,+Li1 (21+2L)1(21+2L:2)1[22’+2Li‘(1+L)1(1+L :1)1]‘l xa,(i(o)BLLil(ico,O)R'(2l+2L+2fl) as R —> 00, (19) where 79 DiU, L) = p114: (I‘M) (”Lin (21 +2L)!(21+ 2L:2)![22'+2Lfl (1+ L)!(l+ Li 1):]—1 [Li xii/21c [:66 a,(im)BLLfl(ico,0), (20) (11(16) ) = limbo Iimk,_,0 a,(k, k';ico)k"k"’ , (21) and BLLi1(iO),O) = limk_,0 Iimk._,0 BLLfl (k, k';l(D , O)k‘Lk"(Lfl). (22) Equation (19) has been derived by an analysis similar to that given in Appendix 1 of reference 24. Eq. (18) gives the long-range form of the dispersion-induced dipole for a pair of distinct, spherically symmetric atoms, Ap=D7R‘7+D9R‘9+D“R‘“+-... (23) The coefficients D7, D9 , D11, and higher-order coefficients can be computed from Eq. (20) for Di(l, L). For example, the leading dispersion dipole coefficient D7 is given by D7 = Wu, 1) + D+(2,0) + D’(l,2)+ D'(2, 1). (24) The quadrupole moment induced in A by dispersion interactions with B satisfies Aofimisp = h/(81t)5 [dk exp(—ik~R)k"2 [dk' exp(ik'-R)k’_2 4: d6 xB(k,k';i6)1§QB(-k,—k';iw,0), (25) where 00 L 00 L' Baa 0. As R —) 00, an analysis similar to that leading to Eqs. (18) and (20) gives: AG), (1, L) = Ma(l,L)R‘(21+2L+2+°‘) (32) with 81 Ma(l, L) = s31tfi’IfL)(’+L+°‘)°‘(21+2L)1 (21+ 2L+2a)! [22’+2L+°‘(1+ L)!(l+ L+a)!]—l xii/27: [:ch 0t,(ico)BLL+a(ico,O), (33) where a = O,i 2, and BLLmUm, 0) = Irmkg0 Iimk,_,0 BLLm(k,k';i(1), O)k’Lk"(L+°‘). (34) Equation (32) gives the long-range form of the local dispersion quadrupole in an atomic pair: on =M6 R’6+M8 R"8+M10 R‘1°+..-, (35) where the coefficients M6, M8, can be obtained from Ma (l,L) given by Eq. (33). For example, the leading local dispersion quadrupole coefficient M6 satisfies M6 = M_2(1,2) + M_2(2,1)+M0(1,1)+M0(1,O). (36) 5.3 Application to Hydrogen Atoms For the hydrogen atom in the Is state, the z-component of the B susceptibility density and the zz-component of the B susceptibility density have been determined in Chapter IV. They satisfy , 20) a , a d , BL Li1(k,k .6,0)=——2L—‘3-[f1(k) grid )-—Lt‘—Li—fr(k)fw(k )1 mL“0 wL—wLfl 26) a a d +———2“‘ ”2‘ [fLi1(k')hth(k)_—‘I“—Li—'fL(k)fLir(k')] (37) mun“o mL—wLil WlthL Z 1 for BLL+1 311d L 2 2 for BLL-l’ and 26) a 2 I ‘2 f1(k')h0+(k), (38) C0] ‘0) [301(k3k';0>30)= 260 a 310(k,k';w,0)= (”2 ‘(0‘2 f1(k>g1-(k'), (39) 1- 82 and , 263 a , aLiZDLLiZ , BLLiZ(kak ;(‘020) z 2 L L2 [fL(k) ALLiZ(k )——fL(k) fLfl(k )] (0L —(0 03L _(°Li2 26) a , aLDLLiZ , + zLfl ”‘22 [fLiZ(k )CLL:2(k)+—— f1.(k) fLiz(k )] (Gun '0) col. _0)LiZ (40) with L 2 1 for BLL+2 and L 2 3 for BLL_2, and 20) a 802(k,k';co,0) = 2 2 22 f2(k')coz(k), (41) 032 —(0 20) a 820(k.k';w.0)= 2 2 22 f2(k)A20(k'), (42) (1)2 -(D and .. 2031. 3L , , BLL(k’k ,Q),O)= 2 2 [fL(k) ALL(k )+fL(k )CLL(k)] (DL —(D 23% DLL (wifioz) f (k)f (k'). (43) (mi _m2)2 1‘ L The linear charge—density susceptibility in Eqs. (12) and (28) has been obtained by Koide [24]: 20) a , a1(k,k';co)= 2 ’ ’2 W) W ). (44) CD] —(0 The fimc’tions (”Li 3L, fLU‘): dLi’ 813:“), hulk), DLL+na ALL+n(k)’ and CLL+n(k) (n= -2, O, or 2) appearing in Eqs. (3 7)-(44) have been evaluated analytically and they are given by Eqs. (4.28)-(4.36) and (4.42)-(4.50). From Eqs. (20) and (33), the leading three local dipole coefficients (D7, D9 , and D11) and the leading three local quadrupole coefficients (M6 , M8, and M10) have been obtained for a pair of ground-state hydrogen atoms. The results are listed in Table 5.1. 83 The ab initio values for D7, D9, and M6 [26-29] are also given in Table 5.1 for comparison. Table 5.1 shows that the relative error is 6% for D7, 5% for D9 , and 12% for M6. These errors stem from the approximations introduced in obtaining the function ¢[‘(k,m) from Eq. (4.22). Table 5.1. Local dispersion dipole coefficients and quadrupole coeffi- cients (in a.u.) for H(1s)- - -H(1s). Coefficients This work Ab initio calculations D7 -416.4 -394513 -394.5106b -393.5° D9 -13420 -12800d 1311 -544000 M6 -58.3 -522" 148 -5873 M10 -257000 aRef. [26]. bRef. [27]. cRef. [28]. dRef. [29]. 84 Substitution oqu. (37) for BLLfl(k, k';03,0) and Eq. (44) for or,(k, k';c0) into Eq. (12) gives the dipole induced in one hydrogen atom by dispersion interactions with the second, with direct charge-overlap effects included: L+I Lil+I 0° 40*(1,L)=4/nzpzri Z Z qii2ralar(co,+cor>“[0 dij(kR)f,(k)fL(k) J=[L—l| J'=|Ld:l—l| le dk'lJ'(k'R)f1(k')gLi(k')‘aL:1dLi((DL ‘09 Lid—1 xf dk'jyf1(k')fw(k')1 +2113132181)! “DMD—II: dk'jy(k'R)f1(k')fLfl(k') x1 If,” dkjl(kR)fz(k)hLi(k)+ardri(wL wow)" x1: dij(kR)f,(k)fL(k)1} (45) with/2 1, L 21 for u*(1,L), and I 2 1, L 2 2 for u‘(l,L), and [+1 Alv‘u,0)=-4/n2 p10 2 qlé’a1a1(o>1+co,>“[;° dk11(kR)f,(k)h0+(k) J’=Il-l| xEdk'j1.(k'R)fl(k')f,(k'), (46) [+1 00 All-(1,1):4/712P11— Z qi,’_a1a,(col+w,)“[0 dij(kR)f1(k)f,(k> 1:11-11 x [:dk'jz(k'R)t1(k')gl-(k'). (47) In deriving Eqs. (45)-(47), the intergral identity fdmab(a2+m2)'l(b2+m2)—l=1t/2(a+b)-1 (48) has been used for a, b > 0. The lowest-order contribution to the local dispersion dipole in the pair H- - - H is the sum of two terms, Au+(l,0) and Air-(1,1). From Eq. (18), both of these terms might be expected to lead an R’5 behavior as R ——) 00. Thus this contribution is denoted by p.5(R): 85 115(R) = Au+(l,0) + Al~l_(1,1)- (49) The long-range coeflicient D5 vanishes, so [.15 (R) is purely an overlap effect. This is consistent with the fact that the leading term of the long-range dispersion dipole for an atomic pair varies as R'7. u5(R) can be evaluated analytically as a function of the interatomic separation R, as shown in Appendix A. It is plotted against R in Figure 5.1. Similarly, the function p.7(R) that reduces to the leading R’7 term as R -> 00 is the sum of four terms: 117(13): 40*(1, 1) + Au+(2, 0) + Art—(1,2) + All—(2, 1) = x7(R)D7 R‘7 (50) with D7 listed in Table 5.1. u7(R) is also given analytically in Appendix A, and it is plotted in Figure 5.1. The damping fiinction x7(R) is shown in Figure 5.2. The local dispersion quadrupole of H---H pair, which approaches the leading R—6 term at long range, includes three terms: ®6(R) = A®0(1,1)+ A6920, 0) + A®_2(1, 2) = x6(R)M6 R‘6, (51) where M6 is listed in Table 5.1, and A90 (1,1), A®2(l,0), and A®_2 (1,2) are obtained by substitution of Eqs. (40)-(44) into Eq. (28) and use of the identity (48), as well as the identity I:dwm2(az+w2)-l(b2+m2)—2 =n/4 b"(a+b)‘2 (52) for a, b> O. This yields 46900.1):4/82 sf] 2 Z tlf'Orai/(zwmfi dij(kR)fE exp[—i(03 + 0) ')t]. (4) The term given explicitly in Eq. (4) represents the polarization route A —> B —> C —> A, and the term generated by 508C represents the route A —> C —> B —> A. Similar expressions for AES’) and ABE?) are obtained by finding the reaction fields acting on B and C. The average energy change AES’) depends on the correlation in the fluctuating polarization of molecule A at points r and r". According to the fluctuation-dissipation theorem [50, 51], the correlation is related to the imaginary part of the nonlocal polarizability density OLA" (r, r";c0) by 1/2&a(rv,m ')+ P&a(rv,00 ')P&,(r,m )) zit/21c ax(r,rv;0))8(0) +m') coth(hco /2kT). (5) To illustrate the quantum mechanical nature of the fluctuations given by Eq. (5), the limit as T —) 0 is considered, with infinitesimal damping; then ago, (r, r" ;c0) for the molecular ground state IO) is given by [52] 97 aga(r,rv;m) = NH: 2 [(0|P¢(r)|m)(m|Pa(rv)|0)6((0 —03m0) m¢0 —(0|P¢(r)|m)(m|Pa(rV)|0)5(m +com0)], (6) assuming for simplicity that the states lm) are real. As T —> 0 , coth(h0) /2kT) —) [0(03) —0(—c0)], where 0(0)) is the Heaviside step function. Thus Eq. (5) gives 1/2 (PM) (r,0) )PtLa (rv, (0 ') + Pflg (rv ,0) ')PflA) (r, 00 )> =1/2 Z (0|P¢ (r)|m)(m|Pa(rv)|0)[6(co —(0m0)+5((0 +com0)]8(co +0)') (7) m¢0 or equivalently l/2 = 1/2 2 (0qu> (r)|m)(m|Pa (rv)|0) {exp[—i(o m0(t—t')]+exp[i0) m0(t-t')]}. (8) man A direct computation of the dynamical correlations of the fluctuating polarization in the ground state yields 1/2 (PM (r,t)Pfl,a(rv ,t') + p212, (rv,t') Pfl,¢,(r,t)> =1/2(0|{exp(iI-lt/h)P¢(r)exp[-iI-I(t—t')/h]Pa(rv)exp(—th') +exp(th' / h) P02 (r”)exp[—iH(t' — t)/ h]P¢ (r)exp(—th)}| 0) -(0| P¢(r)|0)(0|Pa(rV)|0). (9) In equation (9), the static polarization product has been subtracted from the expression; static reaction field effects are treated separately in Sec. 6.3. Eqs. (8) and (9) are equivalent. The fluctuation correlations persist in the limit as T —> 0 , where they are determined by the intrinsic, quantum mechanical fluctuations of the polarization in the molecular ground state. Substitution of Eq. (5) into Eq. (4) gives 98 AB?) = -h/41t (1+ gaBC)‘l‘j’odaijdr---drv Tr[T(rV ,ri")-0tC (riv ,r'";0)) ~T(r"',r")~aB(r",r';(0)-T(r',r)-orA"(r,rv;0))]coth(hco/2kT), (10) where Tr[T(rV,riV)-ac(ri",r"';c0)-T(r"',r")-0LB(r",r’;(o)-T(r',r)-orA" (r,r";(0)] a T0250" ,riv )org1r (riv,r"';c0)T,,2~,(r"',r")or§,38 (r",r';0)) xT.¢(r',r)a£;(-,r";co), (11) and the Einstein convention of summation over repeated Greek subscripts is followed in Eq. (11) and below. Then in the limit T —) 0, AEEE) = —h/21t (1+ 5013C)RefdooIdru-drv Tr[T(r",ri")-orC (riv,r"’;0)) -T(r’",r")-0tB(r",r';0))-T(r',r)-0tA"(r,r";0))], (12) where Re denotes the real part of the expression that follows. Use of lims__,0[(x—(0)—ie]-l =P(x—w)’1+in5(x—0)) (13) and the Kramers-Kronig relation [53] between the real and imaginary parts of the polarizability density a'(r,r';co)= l/1t P_[1:0dxor"(r,r';x)(x—co)'l (14) (where P denotes the Cauchy principal value of the integral) gives a(r, r';00) = or'(r, r';(0)+ ia"(r, r';(0) = Iim8_,0 l/TE I: dxor"(r,r';x)(x -(o — ire)—l =1im,_,02/n [:dxxou'(r,r';x)[x2 -(CD +18)2 1“. (15) In the transformation between the second and third lines of Eq. (15), use has been made of the fact that or"(r, r';x) is an odd firnction of the frequency x. From Eqs. (12) and (15), 99 ABS) = —2h/1c3 (1+ 563C) 1im,_,0 [:dxfdyj: dzRe{x [x2 — (2+18)2 1“ xy [y2 — (z+ir»;)2 ]'1}J‘dr---drv Tr[T(rv,riV)-ac" (riv,r"'; x) B.. A» v -T(r'",r")-or (r",r';y)-T(r',r)-or (r,r ;z)] . (16) Adding the corresponding expressions for ABE?) and AE(C3) to Eq. (16) and taking the limit as 8 —> 0 yields Asggp = —2h/1t3 [:dx [5° dy [: dz (x +y + z)(x + y)“(y + z)“(z+ x)"1 x Idr-ndrv {Tr[T(rv,riv)-ac" (riv,r'";x)-T(r"',r") .613" (r",r';y)-T(r',r)-orA" (r,r”;z)] +Tr[T(r" , riv ) - orB" (riv , r'"; y) - T(r'", r") ~orC" (r",r’;x)-T(r', r)-orA" (r,r"; z)]}. (17) Use of the Born symmetry of the nonlocal polarizability density [28, 30], oraB(r,r';co)=orBa(r',r;0)), (18) and the symmetry of the T tensor, T(r, r’) = T(r', r), transforms Eq. (17) into ABS; 2 —4h/7r3 dejo dy‘[(<:)dz(x+y+z)(x+y)_1(y+z)-1(z+x)_1 x Idr- --drv Tr[T(rv,riv)-orc" (riv, r'"; x) - T(r'", r") B" n r r A" V «1 (r ,r ;y)-T(r ,r)-a (r,r ;2)]. (19) The frequency integrals over x, y, and 2 can be converted into independent quadratures using the identity (x + y + z)(x + y)"1 (y + z)_1(z + x)-1 : 2/1:[;°c16xyz(x2 +002)_1(y2 +62)“‘(z2 +62)*‘. (20) 100 From Eq. (6), which holds in the limit of infinitesimal damping, Edzz(22 +c02)_1a:é(r,rv;z) = TE/h 260mg ((03,,0 +002)'1(0|Pa(r)|m)(m|PB(rv)|0) m¢0 =n/2aQB(r,rV;im). (21) Equations ( l 9)-(21) imply AEES;p = —h/ n J: d0) [dru-dr" Tr[T(rV,riV)-ac(riv,r"';i(0).T(r"',r") -orB(r", r'; i0) ) - T(r',r) - 01A (r,r";i0) )] . (22) Equation (22) gives the principal result of this section; it expresses the nonadditive, three-body dispersion energy as a tensor product of the dipole propagators and the imaginary-frequency nonlocal polarizability densities of the three interacting molecules A, B, and C. Within the nonlocal response model, AEg’gp has a simple physical interpretation in terms of polarization fluctuations and the energy of polarization in the reaction field: A spontaneous fluctuation in the polarization on molecule A polarizes B, which in turn polarizes C; the induced polarization in C produces a reaction field acting at A. This polarization route e. g. A —> B —+ C —-) A gives one term in the energy shift of A, with the second term generated by the route A —> C —> B —) A. Similarly, there are energy shifts of B and C associated with the polarization fluctuations in these molecules. The net three- body dispersion energy is the sum of the energy shift of A, the energy shift of B, and the energy shift of C. 101 6.3 N onadditive Induction and Induction-Dispersion Energies Three-body nonadditivity appears at second order in the intermolecular interactions. At this order, the three-body energy is the sum of three induction terms, 2 2 2 2 AE( >=AEggB+AEggC+AEggA, (23) where AEEZAB represents the energy change in molecule A due to the static fields from the permanent polarization of B and C, AEgXB-t —Idr-- ~~dr'"P0C(r'") T(r'", r') 01A(r' ,r) T(r, r”)- P63 (r"), (24a) mac = ewe... <24)» and A533,, =pACAE8) (24c) Equivalently, AE(C2) can be viewed as the interaction energy between P8: (r’”) and the polarization induced in A by POB(r") (or similarly, with the roles of B and C interchanged). 1312533322 and AEggA represent the energy changes in B and C, respectively, in the fields due to the permanent polarization of their interaction partners. At third order, the total nonadditive three-body energy is the sum of the dispersion energy from Eq. (22), the classical induction energy, and a combined induction-dispersion term: 2113(3)=AE§§§p“mfg;+1359)d (25) The classical three-body induction terms can be fiirther categorized into three groups, according to their physical origins. These terms stem from (1) static reaction fields, (2) third-body fields, and (3) hyperpolarization: 3 3 AE§ 3, — _AE§,3 + A1133} + AE(hy)p . (26) 102 The static reaction-field effects correspond to the dynamic reaction field effects considered in Sec. 6.2, but they originate in the permanent molecular polarization, rather than the fluctuating polarization treated in Sec. 6.2. The static field due to the permanent polarization of molecule A polarizes B; the induced polarization of B sets up a field that polarizes C, and C in turn produces a reaction field at A, causing an energy shift that depends on the scalar product of the reaction field with the permanent polarization of A (and similarly, with the roles of A, B, and C interchanged). The static reaction field term associated with the permanent polarization of molecule A is ABS-2.4 = *ldr-"drv P6‘ (r)-T(r,r')-aB(r', r")-T(r".r"' C m iv iv v A v or (r ,r -T(r ,r )-P6 (r ). (27) The net contribution to AE‘” fiom static reaction-field effects is obtained by adding Angrif),A from Eq. (27) and the terms associated with the permanent polarization of B and C: 3 __ (3 (3) (3 458% - 41582.8 + ABM + 4135.36 (28) The quantity ABS,2 can be viewed as lowest-order screening term. From Eq. (27), this interpretation for AESr‘f),A holds as follows: the induction energy of molecule B, due to its polarization in the field from P6A (r), is altered by the presence of C, since C is also polarized by P6‘(r). The simultaneous action of the direct field from P6" (r) and the screened field from the polarization induced in C by A causes an energy change in B. The same interpretation holds with the roles of B and C interchanged, due to the symmetry of the T tensor. Equivalently, A5332 can be viewed as the interaction energy of the dipoles induced in B and C at first order, by P6“ (r). Third-body field and reaction field effects are related, but the polarization routes that contribute to the third-body field terms begin and end at different molecules: as an example, the route C —> A —> B —> A is considered; that is, C polarizes A, which polarizes 103 B, producing a field at A and changing the energy. The term in AEB) associated with this route is AEingAB = —I dr---drv P6C(r)- T(r,r')-01A (r',r")'T(r",r'" -orB(r"',riv -T(riv,rv)-P6" (rv). (29) The net contribution to AEO) from third-body field effects is the sum of 135131.083 from Eq. (29) and five additional terms from the remaining permutations of A, B, and C in the polarization route: (3) _ (3) (3) (3) (3) AEtbf — AEtbf,CAB + AEtbfABC + AEtbf,BCA + AEtbfACB (3) (3) +AEtbf,CBA + AEtbf,BAC' (3 0) In this equation, AEigixu denotes the right hand side of Eq. (29), after the label changes C—>X, A—+Y, and B—)Z. The third group of induction terms stem fi'om static hyperpolarization. For example, the hyperpolarization energy of A due to the concerted action of the fields from the permanent polarization of B and C is £13261 2 —(1 + 5'.9BC)1/2J'dr---drv BA(r,r',r")E[T(r,r"’)-P63(r"')] x1T(r',r“)-P1?(r“)1[T(r",r‘)-P§(r”)1. (31) The net hyperpolarization contribution is (3) _ (3) (3) (3) 111Ehyp — AEhmA + A1311“),B + AEhwfi, (32) and the fiill three-body, third-order classical induction energy is the sum of AESQ from Eq. (28), A513,} from Eq. (30), and 131313,), from Eq. (32). Nonadditive combined induction and dispersion effects also appear at third order. The dispersion energy between molecules A and B is altered by the presence of a third 104 molecule C, because the permanent polarization of C acts as the source of a static field that perturbs the A-B interaction. An applied static field F° affects the A-B dispersion interaction in two ways [31, 54]: First, each of the molecules A and B is hyperpolarized by the simultaneous action of Fe and the fluctuating field fi'om its partner (A or B). This effect is represented by use of an external-field dependent polarizability density 01(r, r';0), Fe) to describe the response of each molecule to the field from its neighbor. Second, the correlations of the spontaneous, quantum mechanical fluctuations in the polarization of A and B are changed by F°. For example, the application of an external field to a centrosymmetric molecule introduces correlations between dipolar and quadrupolar charge density fluctuations; these correlations vanish in the absence of the applied field. The applied field also alters the correlations of the fluctuating dipoles, at first order for non-centrosymmetric molecules and at second order for centrosymmetric molecules. To account for this effect, the field-dependence of the imaginary part of the polarizability density is included in the fluctuation-dissipation relation: 1/2 (r,0) )Pfifa (rv,0) ') + P6202 (rv,0) ')Pffid, (r,ro )>F° = h/21t012;(r,rv;co,F°)6((o +(1)')c0th(h(o /2kT). (33) Previously, Hunt and Bohr [54] have developed a theory for the dispersion dipole of an A-B pair, based on the change in the dispersion energy due to a uniform, static external field. After modification to allow for the nonuniformity of the field F6j due to the permanent charge distribution of C, the same analysis applies here, with the external field replaced by FOC. Then the nonadditive induction-dispersion energy associated with polarization fluctuations in A and B is AEgianc = —(1 +gaAB)h/21r I: dco‘fdrmdrv Bfiya(r’,r",r; 103,0)T75(r",r"' X 61,32 (r'", riv ;ico )Tep (riv , r')Ta¢ (r, r" )ngb (r“), (34) 105 and the nonadditive, three-body induction-dispersion energy at third order is (3) _ 3) (3) (3) . AEH-d — AEEA-nBy—C + AE(B---C)<—A + AE(C.~A)<—Ba (35) that is, A5931 is the sum of the change in the A- - - B dispersion energy due to the permanent polarization of C, the change in B- - -C dispersion energy due to the permanent polarization of A, and the change in C- - - A dispersion energy due to the permanent polarization of B. 6.4 Nonadditive Dispersion Dipoles, Classical Induction, and Induction-Dispersion Dipoles In this section, nonlocal response tensors and reaction field theory are used to derive the nonadditive three-body polarization induced in molecules A, B, and C. The method is illustrated with the calculation of the dispersion polarization; then the results for the classical induction and induction-dispersion polarization are summarized. The three-body dispersion polarization P295}, (r) is determined by the firnctional derivative of AE(3) with respect to a static external electric field F“, which may be disp spatially nonuniform: Pégs)p(r) = —6 AEggp /6Fe(r)|F2=0. (36) As noted in the previous section, application of an external field alters the dispersion energy via hyperpolarization and via field-induced fluctuation correlations. These effects are treated by allowing for the F° -dependence of both the real and the imaginary parts of the nonlocal polarizability densities. Then the same analysis that led to Eq. (22) gives 11133322,, = 4m: [:66 jar-“<11.v Tr[T(r",riV)-orC(riV,r"';i(0,F°)oT(r"',r" oaB(r",r'; i0) ,Fe)-T(r', r)oorA(r,rv;i0) ,Fe)]. (37) 106 The polarization 12633,, (r) is the sum of three terms, the polarization Pdisp (r)A"B’C induced in A by the dispersion interactions with B and C, and the polarization induced in B and C by dispersion: 3 3 3 3 Péigp“) ___ P((fis)p(r)A<—B,C +P§is)p(r)B+—A,C +1,((fis)p(r)C<—A,B_ (38) (r)A(—B,C The polarization P62, specific to molecule A is obtained by allowing for the external-field dependence of the properties of A alone. Thus, Péi)1,(r)A"B’C satisfies P(s) disp (r)A(—B.C : h/K I: do) Idr'...drVi Tr[T(rV'l ,rV).aC (rV’er;iw).T(riV,rm .aB(r"',r";iw)-T(r',r)-66A(r',r“;im,Fe)/5F6(r)|l,,:0] .(39) Expanding 01A (r', rVi;i0), Fe) as a series in powers of F°(r) gives [31] aA(r',rVi;i(0,Fe) zaA(r',rVi;i(0)+-[dr BA(r',rVi,r;i03,O)oFe(r)+---. (40) Therefore eaffiozrVHicore)/5F§(r)|,,=0=Bfifiaugrvtnimb), (41) and P(Egzfisp(r)’l“—B’C = h/tt [godm Idr'---dr"i T6Y(r"i,rv)a$5(rv,riv;i(0)T5€ (riv,r"') B m n.’ n r A r Vi .' xa8¢(r ,r ,10))T¢;2(r ,r)Bwa(r ,r ,r,1c0,0). (42) Equation (42) gives a key result: the polarization induced in molecule A by its dispersion interactions with B and C depends on the imaginary-frequency hyperpolarizability density BA(r, r', r";ico, 0) of A and the polarizability densities orB(r, r';ico) and are (r, r';ico) of B and C, integrated over frequency. The nonadditive three-body dispersion dipole is obtained fi'om Eqs. (38), (42), and the corresponding equations for the dispersion-induced polarization of B and C, by integrating P534, (r) over all space with respect to r. 107 The three-body, classical induction contribution to the polarization of A is obtained fi'om Eqs. (23 )-(32) and the analog of Eq. (36), by allowing for the F° -dependence of the polarization P6“ (r) and susceptibility densities 01’“ (r, r’) and B’“(r, r', r") of molecule A alone. At second order, the induction contribution is Pigg(r)A(—B,C : Idl’" . 'dl'iv BA (I',I",l'")I[T(I", rm) _ P§(I’"’)][T(l’",l‘iv)- P€(riv)] +(1+1(.9BC)J'dr'---driv 01’“ (r, r')-T(r', r") -orB(r",r'") .T(r"',r“’)-P§(ri"). (43) The first term in Eq. (43) gives the lowest order of the polarization induced in A due to the simultaneous action of the fields from the permanent polarization of B and C. The term given explicitly in the second line of Eq. (43) represents the polarization induced in A due to the field from B, which is polarized by the permanent charge density of C. The induction contribution from third-order effects is the sum of three terms: 1,1863%") = T1 + T2 + T3, (44) separated according to the highest order of the susceptibility of molecule A contained in the term. The T] term depends on the linear response tensor of molecule A; it is given by T1 = (1+ (013C)J‘dr'---drVi {01A (r,r')-T(r',r")-0tB(r",r"' °T(r"',riv) .aC(rIV,rV).T(rV,er).P0A(rVI) A r I n B n m "I 1V C 1V V V V1 B VI +01 (r,r)-T(r,r )-or (r ,r -T(r ,r )-or (r ,r )-T(r ,r )~P6 (r ) +aA(r,r')-T(r',r")'orB(r",r"' -T(r"’,ri")-01A (riv,rv)-T(rv,rVi)-P6C(r"i) A r I n B n M iv . m V A V 1" VI C V1 +a (r.r)-T(r,r )1) (r ,r ,r ).1T(r ,r )‘1’0(r )11T(r ,r )‘Po (r )1 +1/2 or’“(r,r')-T(r',r")-BB(r",r"',riv):[T(r"',rv)-P6:(rv)] x1T(r“',r"i)-P§(r"‘)1). <45) 108 The first three terms in Eq. (45) give the polarization induced in A due to sequential linear response to the permanent polarization of A, B or C. The polarization routes represented by the first three terms given explicitly in Eq. (45) are A —> C —> B —-> A, B —-) C —-> B —-) A, and C —-> A —> B —) A, respectively. The final two terms listed in Eq. (45) give the polarization induced in A by linear response to the hyperpolarization of B, either bilinear in P6“ and POC (fourth term) or quadratic in P6: (fifth term). The operator (08C perrnutes B and C in the five terms given explicitly, completing the set of induction mechanisms that involve linear response by molecule A. The field at A due to the permanent polarization of B (or C) and the field at A due to the induced polarization in B (or C) act together, to polarize A via its static [3 hyperpolari- zability density, BA(r, r’, r"). The T2 term represents this effect: r2 = (1+ 6%)] dr'---dr"i {BA(r,r',r”):[T(r',r"’)-P63(r"')] X [T(r", er ) .aC (rill, rV). T(rV, rVi ). POB(rVI )] A r n . r m B m n iV C iv V +(3 (r,r,r )-[T(r,r )-1’o(r )1[T(r ,r )‘a (r ,r) -T(r",r"i)-P5‘(r"‘)] A r n . r m B m n iv 13 iv V +13 (r,r,r ).[T(r,r )-1’o(r )][T(r ,r )‘a (r ,r ) -T(rv,r"‘)-P§(r)+ P332>(r)+ P.C,f,2>(r)], (60) 11 where P113330) denotes the classical polarization induced in B by interactions with A and C at second order. P3530) satisfies P3330) = Idr“ - - driv [orB(r, r')- T(r', r”)-01C (r", r'" -T(r'", riv)- P6“ (riv) +aB(r,r')-T(r',r")-a’“(r",r'" .T(r"',riV).P§(riV) +aB(r,r')-T(r',r")-aC(r",r"' -T(r"',riv)-P(?(riv)] +I dr'---driv BB(r,r',r"):[T(r",riv)-P6: (riv)] x [T(r',r"')-P6“(r"')+1/2T(r',r"')-P63(r"')]. (61) The nonadditive three-body force on nucleus K in A at second order is obtained from equations (23) and (51): “352) = zK [ drT(RK, r)-[Pi(n2d)(r)’“‘_B’C + 125,130?“ + piggoft-B], (62) where Pf;6(r)B"C is the polarization induced in B by the permanent polarization of C, at first order: Pfgg(r)B‘—C = [ dr' dr"a (r, r') - T(r', r")- P63 (r"). (63) Eqs. (57), (58), (60), and (62) provide an electrostatic interpretation of all of the nonadditive three-body forces on nuclei in interacting molecules A, B, and C. 113 6.6 Summary and Discussion Nonadditive three-body dispersion interactions appear at third order; they result from the correlations of the spontaneous, quantum mechanical fluctuations in the polarization of the three interacting molecules A, B and C. The three-body dispersion energy is given by Eq. (22) as a tensor product of the dipole propagators and imaginary- frequency polarizability densities of molecules A, B, and C, integrated over frequency. Unlike the three-body dispersion energy, which is a third-order effect, the classical nonadditive three-body induction energy includes a contribution from second order. The second-order induction energy is the sum of ABEZAB, [£ch , and AEgéA, where AEgiiz represents the lowest-order energy change in Y due to the static fields from the permanent polarization of X and Z. AEEZAB is given by Eq. (24a). At third order, the classical induction energy contains three types of terms: (1) a static reaction field term AB“), (2) a third-body field term ABS}, and (3) a hyperpolarization term AB(h3y)p. The mechanism that gives rise to AES} is related to the dynamic reaction field effects in the dispersion interaction, but it originates in the permanent polarization, rather than the fluctuating polarization. The static field due to the permanent charge density of molecule A polarizes B, which then polarizes C, and the polarization induced in C produces a reaction field at A. The resulting energy change in A depends upon the permanent polarization of A and the polarizability densities ofB and c. This term 181333,, is given by Eq. (27), and the total energy ABS]? fi'om this mechanism is obtained by adding ABS},A to additional two terms ABffr’f),B and ABSEC , the energy changes in B and C originating in the permanent charge distributions of B and C, respectively. The third-body field terms depend on the permanent polarization of two molecules, rather than one molecule as in the static reaction field 114 terms, because the polarization routes that contribute to ABE}? begin and end at different molecules. One representative term is defile“, given by Eq. (29). The polarization route associated with this term is C —> A —> B —> A; that is, C polarizes A, polarizing B, producing a field at A and changing the energy. ABg} includes five additional terms obtained from the remaining permutations of A, B, and C. In addition to the static reaction-field and third-body field effects, hyperpolarization also contributes to the pure induction energy at third order. In this mechanism, the concerted action of the fields due to the permanent polarization of molecules B and C produces an energy change in A via the B hyperpolarizability density of A. The hyperpolarization energy of A £13618 satisfies Eq. (31), and the total hyperpolarization energy is the sum of AE‘QP’A, £13;th and Anyhc. Nonadditive effects of induction and dispersion also occur at third order. For example, the dispersion energy between molecules A and B is changed by the static field from the permanent polarization of C. The induction-dispersion energy AEE‘S",,‘)NB)(_C depends on the scalar product of the static field from POC (r) and the dispersion-induced polarization in each of the molecules A and B. The expression for ABEQUBFC is given by equation (34), and the net contribution from the induction-dispersion effects is obtained by adding AEiiiu-By—C to additional two terms AEEQCFA and AEE%)A)2_B. The three-body polarization 15(3) (r) is derived based on the change in the three-body energy due to a static external electric field F°, which may be spatially nonuniform: P(3)(r) is obtained from the fiinctional derivative of AB“) with respect to F‘. The three- body polarization 15(3) ( r) can also be categorized into dispersion, classical induction, and induction-dispersion terms, depending on the term in the interaction energy with which P(3)(r) is associated. The dispersion polarization P621: )(r) induced in molecule A by its interaction with B and C is given by Eq. (42), which depends on the imaginary-frequency B hyperpolarizability density of A and the imaginary-frequency polarizability densities of B 115 and C. Two distinct physical effects contribute to Pfigkr): (1) the applied field changes the response of a molecule to the local fields from the neighboring molecules, due to hyperpolarization effects, and (2) the external field also alters the correlations of the spontaneous polarization fluctuations in the molecules because the imaginary part of the polarizability density depends on the applied field. The three-body, classical induction contribution to the polarization of A is obtained by allowing for the Fc -dependence of the permanent polarization P6“ (r), the polarizability density 01’“ (r, r'), and the hyperpolarizability density B’“( r, r', r"). This yields a total of 9 terms, given by Eqs. (45)-(47). Finally, the induction-dispersion contribution to the polarization of A satisfies Eq. (49). The three-body dispersion force acting on nucleus K in molecule A is given by Eq. (56) or equivalently by Eq. (57). Eq. (57) shows that this force can be understood as the electrostatic attraction of the nucleus K to the three-body dispersion-induced polarization of the electrons in molecule A itself. This provides the generalization of Hunt's proof [48] of Feynman's conjecture [49] about the origin of two-body dispersion forces to three-body dispersion forces. In the next chapter, time-independent perturbation theory is used to analyze nonadditive three-body energies and polarization through third order in the intermolecular interactions. By proving the equivalence with the results given in this chapter, the reaction-field method and the perturbation analysis are unified. This work should prove useful in later computational work on the long-range contributions to nonadditive three-body potentials. From equations (22)-(35) and (42)- (49), it is easy to derive the corresponding long-range expressions for three-body interaction energies and dipoles. The results are obtained in terms of single-molecule properties such as permanent multipole moments, polarizabilities, and hyperpolarizabilities. Given ab initio values for these properties, the long-range model should yield accurate three-body potentials and dipoles at large intermolecular distances 116 where numerical cancellation and basis limitations make it difficult to obtain accurate results from an ab initio approach. Additionally, usefirl information on short-range exchange effects [SS-60] may be obtained by comparison of long-range models and ab initio calculations [61-66] or experimental data. Experiments that are relevant to three- body interactions include measurements of third virial coefficients of compressed gases [3- 8], binding energies of rare-gas crystals at low temperature [8-11], collision-induced far- infrared absorption by dense gases [12-15], and rotational and vibrational spectra of van der Waals trimers [16-27]. Referen [1113. 1s [2] Y. 1 [311. K [413. 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Kearsley, third edition (Pergamon, Oxford, 1980), Chapter XII. [52] Equation (6) is obtained by analogy with Eq. (124.8) in Ref. 51. [53] See, for example, Ref. 51. [54] K. L. C. Hunt and J. E. Bohr, J. Chem. Phys. 83, 5198 (1985). [55] L. Jansen, Phys. Rev. 125, 1798 (1962). [56] L. Jansen, Adv. Quantum Chem. 2, 119 (1965). [57] E. Lombardi and L. Jansen, Phys. Rev. 136, A 1011 (1964). [58] L. Jansen and E. Lombardi, Phys. Rev. Lett. 12, 11 ( 1964). [59] A. E. Sherwood and J. M. Prausnitz, J. Chem. Phys. 41, 429 ( 1964). [60] A. E. Sherwood, A. G. De Rocco and E. A. Mason, J. Chem. Phys. 44, 2984 (1966) [61] J. E. De] Bene and J. A. Pople, J. Chem. Phys. 55, 2296 (1971). [62] A. Karpfen, A. Beyer, and P. Schuster, Chem. Phys. Lett. 102, 289 (1983). [63] Z. Latajka and S. Scheiner, Chem. Phys. 122, 413 (1988). [64] G. Chalasinski, S. M. Cybulski, M. M. Szczesniak, and S. Scheiner, J. Chem. Phys. 91, 7048 (1989). [65] G. Chalasinski, M. M. Szczesniak, and S. Scheiner, J. Chem. Phys. 94, 2807 (1991). [66] M. M. Szczesniak, G. Chalasinski, and P. Piecuch, J. Chem. Phys. 99, 6732 (1993). CHAPTER VH NONADDITIVE THREE-BODY INTERACTION ENERGIES AND DIPOLES: PERTURBATION ANALYSIS 7.1 Nonadditive Three-body Interaction Energy at Second Order For three interacting molecules A, B, and C, the Hamiltonian is HzHA+HB+HC+VAB+VBC+VCA:H0+V’ (1) where HX is the Hamiltonian for molecule X when isolated, the unperturbed Hamiltonian H6 is the sum H A + HB + HC , VAB is the perturbation due to interaction between A and B, (similarly for VBC and VCA), and V = VAB + VBC + VCA. Here the intermolecular exchange effects are neglected, and thus the eigenstates of H6 can be written as direct products |klm) ofeigenstates |k) of HA, |l) of HB, and |m) of HC. At second order, the change in the ground state energy of the interacting molecules A, B, and C is [l] A130) =—(‘P6|VGV|‘P6). (2) In Eq. (2) I‘I’O) denotes the three-body ground state |000), and G is the reduced resolvent operator, given by G=(1—1aooo)(Ho —Eo)“‘(I—eooo), (3) where 59000 is the ground-state projection operator, 69000 = |000)(000|, and B6 is the unperturbed ground state energy. For a set of interacting molecules A, B, and C, G can be split into 7 terms, to reflect different possible molecular excitation patterns: 120 121 G = GArooBtofi + 68898508 + Genital? +GA$BQS + GAEBCgog + GBeBCgJS‘ + GAeBec. (4) In Eq. (4), G‘“ is the reduced resolvent for an isolated molecule A, and 50‘6“ is its ground- GAGE state projection operator. 506: contains the terms in the sum-over-states expression of Eq. (3) with both molecules A and B in excited states, but C in the ground state, while GA®B€+3C contains the terms with all three molecules excited. Substituting V = VAB + VBC + VGA and then expanding the sum in Eq. (2) gives 9 terms. Of these, there are three additive two-body terms, — (W6 | VAB G VAB I‘I’O), —(‘I’6 I VBC G VBC Nb), and -(‘P6 | VCA GVCA I‘Po). The remaining 6 terms give the nonadditive three-body energy. All of the nonadditive terms are treated here. First the two terms involving the perturbation operators VAB and VCA are analyzed: ‘(TOIVAB GVCA lw0)_<‘PO|VCA GVABiqu) = _<\PO|VAB GASOgJOgVCAIT01‘<‘P0|VCA GAME)3 698 VABWO) = —I dr'- --dr'" a, (r, r”)P6§Y (r")TBS(r', r”')P6CS(r"') ><1:(r)ra[,(r, r')PBB(r') (6) with Pa(r) denoting the a-component of the polarization operator and TaB(r,r') = VaVB(|r — r'l'1 ), the dipole propagator. In Eq. (5), P6a(r) a (OlPa(r)|0), and the nonlocal polarizability density (1360, r') is defined by 122 aésw') = (OIP:(r)GA $0010) +<01Pi“(r') 6" P3010). (7) Eq. (5) is equivalent to Eq. (6.24a) for AEng because of the symmetry of the T-tensor Tag (r, r') = T130: (r', r) and the Born symmetry [2, 4] of the polarizability density “are“: r') = a6a(r’, r). It gives the lowest-order energy change in molecule A due to the fields from the permanent polarization of B and C. Equivalently, equation (5) can be viewed as the interaction energy between P62 (r’") and the polarization induced in A by P630") (or similarly, with the roles of B and C interchanged). The sum of the two terms containing operators VAB and VBC is identical to Eq. (6.24b) for ABSQC , and the sum of the two terms containing operators VBC and VGA is equivalent to Bq.(6.24c) for AngA. 7.2 Nonadditive Three-body Energy at Third order: "Circuit" Terms At third order, the change in the ground state energy of the interacting molecules A, B, and C is given by [l] AE‘3)=(‘I’6|VGV°GV|‘P6), (8) where V° = V — (000|V|000). Substituting V = VAB + VBC + VCA and then expanding Eq. (8) gives 27 terms. Of these, there are three additive two-body terms, in which a single perturbation operator such as VAB appears three times; the remaining 24 represent the nonadditive three-body terms. All of these three-body terms are considered here; but in this section, attention is focused on the six terms with interactions of "circuit" type. These terms contain the operators VAB , VBC , and VCA, each appearing once. A representative term of this set is (W6 | VAB G Vfic G VCA I‘I’O). From Eq. (4) for G, retaining only the nonzero terms yields 123 (‘1’6|VAB GVB°C GvCA [810) 2 (W0 IVAB GA6903 $90: VBC GAeC 5008 VCA We) +(‘POIVAB 58503 690C V130 GC 98 903 VCA 1W0) +(‘P0 l VAB GAGBB 690: VBC GA 691113 693 VCA W0) +0110 lVAB GAGBB {Joc Vrgc GA®C 6903 VCA ITO) (9) The sum of the "circuit" terms in ABC” is ABS} = (1 +5.9AB +5.913C +5.;AC +goABC +60ACB)<"P6 |vAB 6ng GVCA [810), (10) where goAB interchanges the molecule labels A and B, pABC permutes the labels A —-) B —) C —> A, and similarly for the remaining permutation operators. In ABE? , induction and dispersion effects are separate and additive. The first three terms in Eq. (9) and the terms into which they transform under the permutations in Eq. (10) represent induction, while the fourth term and its transforms represent dispersion. This follows from an expansion of Eq. (10) into its component matrix elements, using Eq. (6) for VAB , its analogs for VBC and VCA, and a sum-over-states representation of G from Eq. (4). From Eq. (10), the terms that contain ground-state matrix elements of the polarization operator for molecule A, and transition matrix elements for molecules B and C are selected, and they are denoted by AESBA. For simplicity, it is assumed that the molecular eigenstates may be taken as real. Then AE‘ci’n = -[dr---dr"<0IP:(r)10><0IPt(rV)I0) x Z '(0|PBB(r')| k)(k| 1>,’3(r")|0)(0|1>2~,C (r'") k,m x12 4134;} +2Ai‘mr + Am)‘1 m) +2 A305,, + Am)‘1]TaB(r,r')T75(r",r'")T2¢(r'V,rV). (11) 124 The prime on the summation in Eq. (11) implies that k = 0 and m = 0 are excluded from the sum; Ak denotes Bk - B6, the difference in the energies of the unperturbed states k and 0 for molecule B (and similarly for Am ). Following algebraic simplification and using Eq. (7) for the definition of the nonlocal polarizability density, Eq. (1 l) transforms to ABQA = —Jdr-- -drv P6: (1') 1160;301:337 (r', r")TY5 (r", r'" C I" IV 1V V A V xor522(r ,r )T8¢(r ,r )P6¢(r ). (12) Equation (12) shows that the three-body "circuit" induction can be understood as a static reaction-field effect, where the permanent polarization of A polarizes B, polarizing C, giving a reaction field back at A, with an energy shift that depends on the permanent polarization of A. Eq. (12) is identical to Eq. (6.27) from the reaction-field method. The net contribution to the induction energy from the "circuit" terms is given by ABS:A + AEQ’)B + AE(3) which is equivalent to Eq. (6.28). on , cir,C ’ The final component of ABE? comes from the fourth term in Eq. (9) and the corresponding terms generated by the permutations in Eq. (10). It represents the nonadditive three-body dispersion effects. This component is denoted by AEEfi’S)p ; in matrix element form Anggp = —J'dr---drv x 2'(0|P:(r)lj>|0> j,k,m x<0|PBB(r') m)<1|P;‘(r")|0> J. 0) +ll>5-’j,(r“’ )pg, (rv )ras (r, r"')r[,2(r', ri" )r,¢(r", rv ). (20) From the definition of the B hyperpolarizability density [3], Bum“, r',r";—0)o;(01,0)2) =[1+C(c01—>-001,0)2 —)—002,0)O —>—(00)] x [(Ol Pa (r)G((0 c,)1’(§’(l")G((JD 2)Py("")10> 0) +(0l Py (r")G’ (—0>2)P§(r)G( —031,032 -—) «02,002I —) —coo) denotes the operator for complex conjugation and replacement of (01 by —(0 1, 0) 2 by —(02, and (0 o by — co c,; the frequency-dependent reduced resolvent operator G((0) is given by 6(a)) = (1—I0><0|PsB(r'") k) =-1 dr-«w P86):'1IJ> j,k 128 x(k|P2B(r‘V)|0)][2A;‘A;‘ +2(Aj +15.)1 111115360”) xTa5 (r,r'")TBS(r',r"’)TY¢(r",rv). (24) The S2 terms firrther separate into two sets. In the first, designated 82,1216, the field from the unperturbed charge density of C polarizes A, which in turn polarizes B; the induced polarization in B produces a field at A, giving rise to an energy shift that depends on the scalar product of this field and the permanent polarization of A. 82,ind satisfies SZ,ind : —J dr. . .drv P6: (r) , T(r, r!) '(XA (r', r"). T(r',, r", -orB(r"',riv -T(r'v,rV)-P6“ (rv). (25) The second set of 82 terms gives one part of the contribution to the induction-dispersion energy, as discussed below. This part is 82 — Sum]; in matrix element form 82 — s2,ind = Idr-udr" P5), (02 '(0|P6‘ (r')| j)( j|1>$(r")|0) (0|P53(r'") j,k k) x(k|Pg3(r‘V)|0) [2133Mj +Ak)"]1>(§,(r”) x Tas (r, r"')T[38 (r',riv )TY¢(r",rV). (26) 83 contains the remaining terms, 53 = (“P0 I VAB GAEBB fore) VXB GASOiJB 600: VAC I‘P0> +(‘i’oIVAC GAME); Pi)2 VXB GAeB p63 VAB I‘Po) +0110 I VAB GAGE $00: VXC GAEBB 608 VAB I‘Po) (27) The matrix element form of S 3 is given by s3 = —[dr---derZ’;"[2(o|P:(r)|j)(j|1>§(r')|1)(1|1>;‘(r") +<0|Pi?(r)|1>(i|1>;’A (r") 0) (Aj +Ak)"A7‘ I>(42 +4..)“(Ar +40") 129 X(OIP§3("")Ik> 0. Earlier, Hunt [4, 5] has shown that the two-body dispersion interactions between A and B produce a change in the polarization of A given by Pas-182(r): h/ZTCI: dCDJ'dr" 'vdr BBYa(r" r" r; —i0) i0) O)TBO(r' rm) xaB 52(r'",r iv;ico)T,,,2(r",ri"); (32) A<—B that iS,P Pa disp (r) depends on the imaginary-frequency hyperpolarizability density of A and the imaginary-frequency polarizability density of B. A comparison of Eq. (3 2) and (29) gives S—821nd+s3:__[drdrPéd‘TsBp(r)Taa'B(rr)POB(r) (33) 130 The right side of Eq. (33) represents the electrostatic interaction between the unperturbed polarization of C and the dispersion-induced change in polarization of A, which is due to the two-body interactions between A and B. This energy term is denoted by Sim. Then from Eq. (33), S2 + 83 2 S2,,“ + SM. (34) The quantity AE(3)(AB, AB, AC) is the sum of S1 from Eq. (22), Sum from Eq. (25), and Sim from Eq. (29). The 18 third-body perturbation terms covered in this section sum to give mag; = AE‘3)(AB, AB, AC) + AE(3)(AB, AB,BC) + AB<3>(AC, AC, AB) +AB<3>(AC, AC, BC) + AE(3)(BC,BC, AB) + AE(3)(BC,BC, AC), (3 5) which is identical to the results for ABS} + AEIEJP + AEEiZ, from the reaction-field analysis in Sec. 6.3 of Chapter VI. The fill] third-order interaction energy is 3 3 3 AE( ) = 13ng + A136,; (36) with ABS!) given in Sec. 7.2. 7.4. Three-body Polarization at Second Order In this section, the nonadditive components of the three-body polarization at second order are evaluated. The second-order term in an arbitrary property 9 is obtained from 9 : (‘1’6 +‘I’, +‘I’2+---IS|‘I’6 +‘l’1+‘l’2+-~)/ (‘1’6+‘I’l+‘P2+---|‘P6+‘P1+‘I’2+m) (37) as 9(2) = (WOI,9IW2)+<‘P2 I9I‘I’o)+<‘*’1ISI‘I’rI'i‘PoISI‘i’oIi‘PrI‘m, (38) 131 where ‘1’] and ‘13 are first- and second-order corrections to ‘1’6 due to the perturbation V, ‘11] = —GV‘PO, (39) and ‘112 = Gv°va0. (40) In equation (40) v° e v (110 |V|‘I’6). From Eqs. (38)-(40), 9(2) =(‘I’OISGV°GV|‘I’6)+(‘I’6|VGV°G3|‘I’6) +0110|vcscv|w0)—(\PO|s|\1'0)()1'0|v0 GVI‘PO) =(‘I’OISGV°GV|‘I’6)+(‘I’6|VGV°GSI‘I’O)+(‘I’6|VGS°GVI‘PO). (41) With substitution V = VAB + VBC + VCA, equation (41) expands to yield 27 terms. Of these, there are 9 additive two-body terms, in which the same perturbation operators such as VAB appears twice. The remaining 18 terms represent the nonadditive three-body interactions. All these 18 terms are considered here. The operator 9 is taken as P6“ (r) and P6“ (r) is abbreviated by PA in the following equations. First the sum of the six terms containing operators VAB and VCA is calculated, Q1 -_-(\110|PA GVXBGVCA [)110)+(\1I0|PA 0ng GVABI‘I’O) +(‘I’6IVABG ng GP’“ |\110)+()r'0|vCA GVXB GPA I810) +(\1(2|v,,2,GP°A GvCA |)1(,)+()110|vCA (31>0A GVAB m2) = (‘POIPA GAfi’iiffioC VXB (#8903500 VCA Iq’oi +(‘I’o IPA 6A9? 10% V84 GAsot‘i 508 Van I‘I’o) +(‘I’o IVABGAa-JE :98 V8a 6%? 108 P“ I‘I’o) +(‘I’o IVcn 6A1??? 108 VXB GAB? 1:28 P" I‘I’o> 132 + ],n )0) )n (nIP:(r)|0) + 0|P§(r)|1'i)( |P°A(r")| n)gA (r)|n)(n|P6A (r')|0)]A}1A;1. (43) Equation (43) is equivalent to Q1 2 Idr’mdri" [33670,r',r")T65(r',r”')P6%(r"')TYS(r",r “’)P02(r”) (44) where 83137 (r, r', r”) -=- 33137 (r, r',r"; 0; 0,0). The Q term represents the lowest-order hyperpolarization of molecule A due to the simultaneous action of the static fields from the permanent charge densities of molecules B and C. The sum of the six terms containing operators VAB and VBC is given by Q2 = ()1!0 IPA vaB (ivBC |\r(,)+()110|r’A (3ng GVABI‘I’O) +(‘I’OIVABGVfic GP‘“ |W0)+()110|VBC GVXB GP’“ my) +(lrlo|vAB (31>0A GvBC |l110)+(\1'0|v23C GP°A GVAB | l1'0) 133 = (‘POIPA 6%.)? 508 VXB GAgo63 gag VBC I‘PO) +(‘1onPA GAP? 505 VBC GMBB 5.28 VAB I‘POI +0110 IVAB GAGBB £93 VBC GA“? 93 PA I W0) + +0110 I VAB GA®B 598 POA GB 803 £915 VBC IWOI +(‘1’0IVBC GB 600 659i): POA GAeB 690: VAB ITO), (45) or in a matrix element form, Q2 = I dr'u-driv TB,(r',r")r2~,,2(r"',riv )P(§f,(riv) XZ '<0|Pi‘ (r)l1>Ik>(k| PoB(r"')|0) AE'AI' (46) i k Equation (46) can be recast into the form Q2 = Idr'- --dr'v a36(r,r')T67 (r',r")a?5 (r",r"’)Tg>8 (r"’,r"’)P&3 (riv) (47) in terms of the polarizability densities of molecules A and B. The Q2 term gives the lowest-order three-body component of the polarization induced in A due to the field from the permanent charge distribution of C . The fiJll second- order, three-body polarization of A is given by < P6? (r) >(2) = Q] + (1 + goBc)Q2. 134 7.5 Three-body Polarization: "Circuit" Terms In this section, the "circuit" contributions (third-order effects) to the three-body polarization are calculated; they are analogous to the "circuit" terms in the energy analyzed in Sec. 7.2. The third-order term in an arbitrary property 9 is obtained from S : (‘1’6 +‘I’1+‘I’2 +‘I’3+---ISI‘I’6 +‘I’1+‘I’2 +‘I’3+---)/ (8’0qu+W2+W2+-..|w0+\r1+W2+l113+-.-) (48) as 9(3) = (1+C)[<‘P0I9I‘P3)+(‘Pr ISI‘P21“<‘P1I‘P1><‘P0I9I‘P1> -<‘I’i I‘I’2)<‘I’ol9|‘1’o)] (49) assuming that all of the corrections ‘13- to the wavefirnction are orthogonal to T6. The corrections ‘1’] and )112 are given by Eqs. (39) and (40), respectively, and w, is given by \112 =—GV°GV°GV‘I’6+(‘P6IVGVI‘I’6)GGV‘P6. (50) Then 8‘” =(1+C)[—(‘P6ISGV°GV°GVI‘I’O)—(‘1’6IVGS°GV°GV I‘I’O) +(wolvcv I‘I’OX‘I’OISGGVI‘I’6)+(‘P6IVGGVI‘I’OX‘POISGVI‘I’Ofl. (51) With the substitution V = VAB + VBC + VCA , each of the terms in Eq. (51) expands to give 27 terms. In this section, the sets of 6 of "circuit" type, which contain VAB , VBC , and VGA, each appearing once, are analyzed. As used in the last section, the a-component of the polarization operator for molecule A, P: (r), is taken as 8 , and P6“ (r) is abbreviated as P’“ in the following equations. 135 A representative circuit term from Eq. (51) is (‘I’O IPA GVXB GVB°C GVCA I‘I’O) With Eq. (4) for G, a direct expansion of (‘PO IPA G VXB GV§C G VCA I‘I’O) generates 7 3 terms; but of these, only 4 are nonzero and (r0 |r>A GVXB GVB°C (3vCA I110) =(‘1’oIPA GA 891113108 VXBGA @3103 Vfic GAGDCQEVCA I'l’o) + (“’0 I PA GA 601113608 VXBGBQQ £93 VBC GC 603 £953 VCA I W0) + ("’0 I PA GA 5963 506C VXB GMBB poo V36 GA 5.363 5.98 VCA I‘PO) + j,l,m 0) +(0lP£(r)lj>(1'lPr°A(r")I1><1|Pt‘s“(r')|0)l ‘ +<0|Pi lj><1‘,,,,(r",r"i)otg2,(r"i ,rV)r22(rV,r‘V)POE(r‘V). (54) 136 These terms represent the hyperpolarization of A by the direct field from the unperturbed charge density of B, acting together with the field from the polarization induced in C by P6B (riv). The analogous hyperpolarization contribution associated with the unperturbed BC T], charge density of C is given by 5) where goBC perrnutes the molecule labels B and C. In Appendix B, the linear induction contribution Tlmd is separated out from T3; here Tlmd satisfies T3,ind = (l + 598C )I dr'- - - drVi (1213 (r, r') T135 (r', r"')oc§‘a (r'", riv x T.2(riv,r")a$i(rv,r”‘ m2 (r:i ,r")Pi*2 (r"), (55) or in matrix element form T3,ind = 2(1 +5230) 2......228 2'<01P5(r)|i>PoA20") . k B J, ,m x (OIPsB(r"')I k><0IP1CIm> x 4 Ag‘Ai‘ A}; 1125(r', r'")r2,, (riv ,r" )rY2 (r", r"i . (56) The dispersion contribution to P:(r) is given by Tdisp = T4 + (T3 — T3316). It satisfies rdisp = 2Idr'...dr‘" . 121: {[(OIP: (r)Ij)(jIPI”“ (r")|1)(1|1>g‘ (r')|0) j, , .m +Ii><1lPiA 1)(1|1>;‘(r") 0)]f4a(Aj,A,,Ak,Am) +(0IP6“(r') i)fa(A,-.Ai,Ai,Am)) x(0|P2,B(r"') k)(k|P,B(r‘V)|0)(0|Pf(r")|m)(m|1>f (r‘“)Io) xr,,2,(r',r'")r24,(riv,r")r,2(r",rvi , (57) where f4a(Aj,A1,Ak,Am) iS given by EC]. (814), and 137 f4c(Aj)AlvAkaAm) = (Aj + Ak)_1(Al + Ak)—](Al + Am)—l +(Aj +4..)“(41 +Ai)“(Ai +4...)-l +(Aj +Ak)"(Aj +A,,,)"(AI +Am)'1 +(Aj+Am)'1(Ak +Am)“(A,+Am)“. (58) Eq. (57) can be recast in terms of the B hyperpolarizability density of A and the polarizability densities of B and C, taken at imaginary frequencies. To prove this, complex contour integration methods are used to write f48(Ai’AIrAkrAm) = Maj: dxA}‘A,(A2,+x2)“1Ak(A2k+x2)—1Am(A2m “(2)—r, (59) and f4c(Aj,A,,Ak,Am)=4/7tI:dx[(Aj+ix)(A,+ix)+(Aj —ix)(A,—ix)] xAk(A2k +x2)"Am(A2m +x2)“‘. (60) Then Eq. (5 7) transforms to Tdisp = h/rt Igodm _Idr'mdr‘" TBg(r',r"')or63£(r"',riv;ico)T8¢(r"’,r") x0126 (r",rVi ;i0))T;2y (r",r”) B$6a(r",r’,r;ioo,0), (61) which is equivalent to Eq. (6.42) for Pfizfis}p (r)'““3’C obtained by the reaction-field method in Sec. 6.4 of Chapter VI. 138 7.6 Three-body Polarization: "Noncircuit" Terms This section presents the analysis of the remaining 18 nonadditive terms in the polarization induced in A by its interactions with B and C. In these terms, one of the three perturbations VAB , VBC , and VGA appears twice, the second appears once, and the third does not appear. These can firrther be categorized into three distinct sets, according to the perturbation operators involved. The first set contains operators VAB , VAB , and VBC , the second contains VBC, VBC, and VAB, and the third contains VAB,VAB, and VAC. The first set of terms are first analyzed. A representative term in this set is — (‘11, I P’“ G VXB G VXB G VBC I‘I’O). Substitution of Eq. (4) for G into this yields three nonzero terms -()110|1>A GVXBGVXBGVBCI‘I’O) = ‘(Wo IPA GA 3903 608 VAB GA 508691? VAB GB 693 £03 VBC IWO) -(‘I’oIPA GA @3108 V23 GB 526‘ (08 V23 GB (08108 Vacl‘Po> "(WOIPA GA 100350th GAGBB 608 VXB GB 600“ 508 VBC I‘Po) (62) The firll expansion in Eq. (51) gives 27 terms of this type. These can be grouped into three sets U1 - U3 so that 3 (P:(r)>( ) = Ul + U2 +U3, (63) where the terms U1, U2, and U 3 are analyzed in Appendix C. The matrix element form of U1 is given by Eq. (C2). The U1 term contains an induction term Uhmd, which satisfies Ul,ind : IdI'" ' -drVior2I3 (1')“)pr (r', “313123580", I'm, riv) xTSAUWJV)P$t(rv)Ten(riv,TVi)chIO'Vi)~ (64) In matrix element form, 139 Una = 4 I dr'---dr"i Priitr") x1<0IPiBIm> +<0IPf(r"> k> +(011230")II<>sBI0)] x P6Cn(r"')A'JTIA}1A;I TBy (r',r")r52 (r'", rV)T8n(r'V,rVi ). (65) In ULmd, the fields fi'om the permanent polarization of A and C hyperpolarize B; the induced polarization in B gives rise to a field that polarizes A. The dispersion contribution to the polarization from the U1 term is U1 — Ulmd. The U2 term given by Eq. (C6) contains the permanent polarization of B and C. It reflects the hyperpolarization of A by the simultaneous action of the direct field from P630) and the field fi'om the polarization induced in B by P6: (r). This effect is designated by UZJnd, which is given by U2,ind = Idr'.--dr"i 133520, r', rV)TBY (r', mp6: (r”) x Tm(rv,r"')or§e(r"', riv )Tsn(r"’,r"' )1>OC,,(r"i ) = 4Idr'---dr"‘ Izk' [(01119(r)li>1 x (0| 1>g3(r'")|1r)(1t|1>2'3(riv )|0)P5’, (r")P6Cn(rVi) x A-j‘AilAi‘ TB, (r', r")'1‘2~,2 (r'",r" )"r2,](riv , rVi ). (66) The dispersion contribution to the polarization from the U2 term is U 2 — Uzmd. 140 The dispersion component in the polarization Udisp is U1 — Upmd + U2 —' U2,ind ‘1‘ U3, and Udisp SfltlSfiCS uni = I dr'---dr" j IZKZEKOIPi‘ (mlPeB(r")|0)(Ai +Ar)"(Ar +Ai)"A;l 0) x<<42 +4..)“(Ar +41)"‘(Ar + Am)" +(A, +4..)“(Aj mm)" +(0I P§3(r"')I k>(kIP2‘2’B(riv )I m)(mI P330") x(A,+Am)-1) +(Aj + Ak )-1(A1 + Ak )_1 A—ni 1} x p§n(rvi)rfi,(r',r")r82(r"',rv)r2n(r”,rV‘). (67) Equation (67) can be recast in terms of the B hyperpolarizability densities of A and B, both taken at imaginary frequencies: Udisp = h/27t 'I‘godoxIdr'ondrVi B63622 (r",r’,r;i(0,0)T6Y (r',r") X 1338“,", r", er ;10.) , 0) T8}, (rm, rV ) Ten (riV ’rvi )P§n(rVi ), (68) after use of the integral identity Eq. (31) and 141 I:dx[(a+ix)—1(b +ix)‘l +(a—ix)"(b—ix)“] x[(c+ix)“(d+ix)-'+(c—ix)‘1(d-ix)“] = 27: [(a + d)"(b + c)"(b + d)" +(a + c)"(a + d)"(b + c)“], (69) where a, b, c, d > 0. Equation (68) for Udisp represents the effect of the perturbation of the two-bony dispersion interactions between A and B due to the presence of the third body C; specifically, Udisp gives the change in the dispersion-induced polarization of A in the A- - - B pair due to the lowest-order static field from C acting on B. Next, the set of terms that contain the perturbation operators VAB , VBC , and VBC are evaluated. A representative term of this set from Eq. (51) is — (‘1’6 I PA G VXB G Vfic G VBC I‘I’O). Substitution of Eq. (4) for G into this yields three nonzero terms -(\i'0|1>A (3v);B G vgc GVBCI‘I’O) = -<‘I’o IPA GA (063198 VXB GB 506* 508 Vfic 08108198 Vac I‘I’o) "(WOIPA GA 591113 690C VAB GB 600 630: VBC GC 690A 6903 VBC I‘Vo) —(LP0IPA GA {903 @0C VAB GB 690A 570: VBOC Gm 898 VBC I‘POI- (70) The full expansion in Eq. (51) gives 34 terms of this type. These can be grouped into four sets W1 — W4 so that (PA (3) a 0)) =wr+wa+ws+w4, (71) where the terms W] — W4 are analyzed in Appendix D. The W, term is given by Eqs. (D2) and (D3), w] = 2Idr'mdr‘" Z'(0IP§(r)|j)(j|P,§‘(r') 0)(0|P2,B(r"')|k)(k|P,B(r")|0) j,k,m x 1361M)(0|1>,,C(rV)|m)(m|1>f(ri")|0)[2A3'A;,I(Ak + Am)" 142 +2 AglAijAjgnBy (r',r")'r2,,(r"',riv )Twrhrvi ). (72) The W1 terms reflect the linear polarization of molecule A due to the field from the permanent charge density of B via the polarization route B —-> C —> B ——) A; that is, the permanent charge density of B polarizes C, which then polarizes B, producing a change in the polarization in A. This effect is denoted by Wund, which satisfies Wund = Idr'- - - drVi 0126 (r, 1")TI3y (r', r")01;35(r", r'” XTse (r'", riv )agb (riv, r")T¢k (rv, I,vi )P&(r"') = 8 I dr'---dr"i @1011): (r)li> (01880")!k> 1, .m x H30" )(OI Pf (r'v)I m)(mIP§(rV)I0) AEIALIA'HI XTBY (r',r")Tg€(r"',r'V)T¢l(rv,rVi). (73) W] also contains one part of the dispersion contribution to the polarization; it is given by W1 ' W1,ind- Equations (D5)-(D11) give the W2 term in the form: W2 : 2‘Idr'mdrVi Z '(OIP: (r)Ij)(jIP6“ (r') 0) j,k,m ><[2(0|1’1§3(r'")|l<)(k|1’i’B(r"i )lm)(m|PI3(r")|0) +(0Ir’2~,B(r"')I1r)(1<|1>;’B(r")|m)(m|r>{3(rvi )|0)] A3‘A11Aj; x ng:3 (riv ) P6C¢ (rV)Tfiy (r', r")T2~,e (r'", riv )Tw' (rV , rVi ). (74) Equation (65) can be converted into an equivalent form in terms of the static polariza- bility density of A and the static B hyperpolarizability density of B, _ I VI A I I II B II III Vl W2—1/2Idr-udr aaB(r,r)TfiY(r,r )Bygfir ,r ,r ) xr52(r"',r“’)POC2(r‘V)r2,(rV‘,rV)P§,(rV). (75) 143 The W2 term represents the three-body component in the induced polarization in A due to the permanent charge density of C; specifically, the field from the permanent polarization of C hyperpolarizes B, and the induced polarization in B gives rise to a static field that polarizes A. Terms that contain the matrix elements P0138 (r'"), P6310"), and P63Y (r") are grouped into W3. These terms cancel out so that W3 = 0, as shown in Appendix D. The remaining dispersion terms are grouped into W4, and it satisfies Eq. (D16). The dispersion contribution to the polarization desp is W4 + W1 — Wmd, given by Wasp =2Idr'mdr‘" 2' (0|P:(r)|j)(j|1>§(r') 0) j’k9m7l x12<0lPsB(r'") k>(mlPiB(r")|0>A‘,-'A:.1(Ai +41)“ +(0IP53(r"')|k)(k|P;’B(r") m)(m|P2B(rV‘)|0) x 43‘ (4.. +41)" (4... + Air‘l x(0|Pf(rV)|1)(l|Pf(r")|0)rl,,(r',r")r2,2(r"',ri")r¢2(r",r“). (76) Equation (76) is equivalent to Wdispz h/21t‘Idr'u-dr" a26(r,r')I:dco B632», (r"',r"i,r";i0), O) ag¢(r'v,rv;im) xrBy (r', r")T&(r"',riv )T,2 (r”, r"i ); (77) Wdisp gives the polarization induced in A due to the static field from the polarization induced in B by the two-body dispersion interactions between B and C. Finally, the set of terms that contain the operators VAB , VAB , and VAC are calculated. A representative term of this set from Eq. (51) is — (‘16 I P’“ G VXB G VXB x G vAC pro). Use of Eq. (4) for G yields 144 -()r(2|1>A GVXBGVXBGVAC Ira) = -(‘I’o IPA GA 106823 VXB GA (053108 VXB GA 825108 Vacl‘l’o) -(‘I’oIPA GA (01113590: VXB GB 196‘ 108 VXB GA (953 108 Vac I‘I’o) -(‘I’oIPA GA 195’ 198 VXB 0“” 108 VXB GA 193898 Vac I‘I’o) (78) The firll expansion in Eq. (51) gives 33 terms of this type. These can be grouped into four sets X1 — X 4 , which are given explicitly in Appendix B, so that A (3) (P2, (r)> = x1 + x2 + x3 + x4. (79) The matrix element form of X, is given by Eq. (E2). It can be recast in terms of the static y hyperpolarizability density of A X1 =1/2‘Idr'mdrVi yQBYg(r,r',r",r"';0,0,0)Tm(r',riv)P£(riv) me(r", rV)P63n(rv )T& (r'", rvi )POC, (rvi ), (80) assuming all the eigenstates can be taken as real. The 7 hyperpolarizability density is defined by [3] 7.26250,r',r",r"';-wc;w1,wz,O) =[1+C((01—> -(l)],c02 —-)-(1)2,(1)0 —> —to2,)] >< {(OlPa(r)G(wo) X{P§ (r')G(2S,’(r"')G(o2)1>y (r")]IO) 145 +<0I P7 (I"')(‘I"(-(D 2)P§(r)G(a)G(0)Ps(r'")lo) ><[(0|1>I,(r')G(to2)PY (r")I0)+(0IP7 (r")G(c01)P6(r’)IO)] —(OIPa(r)G((01)PB(r')IO) x[(0|P, (r")G‘(—c)2)G(0)P5(r"') 0) ><[(0IPp(I")G'(—+(0IPg(r'")G(O)G(co1)P6(r')I0>] —(0IPa(r)G(0)P8(r"')I0) x[(0IP6(r')G'(—0)1)G((02)P7 (r")I0)+(0IP7(r")G'(—0)2)G(c01)P6(r')I0)]}, (81) where coo —:— 031+coz, and C(m1—) —(01,0)2 —) —coz,(06 —> —0)6) denotes the operator for complex conjugation and replacement of (01 by —co 1, (02 by —(02, and 0025 by —0)o, and G(m)=(1-too)(H—Eo —hco)"(1—roo). (82) 146 Specializing to the case co, = 032 = 0, Eq. (81) yields YorByS (r,r',r”, r"';0,0,0). The X1 terms represent the hyperpolarization effects due to the fields from the permanent charge density of molecules B and C: The fields from the permanent polariza- tion of molecule B and C hyperpolarize A to produce a change in the polarization of A via 721311503 r',r", r"';0,0, 0); the field from P63 (r) acts twice at A, and the field from POC (r) acts once at A. The matrix element form of X2 is given by Eq. (E4). In part, it reflects the hyperpolarization of molecule A due to the fields from the permanent polarization of A and C: The field from the polarization induced in B by the permanent charge density of A acting together with the field directly from the permanent polarization of C hyperpolarizes A to produce a polarization via BA( r, r', r"'). This term is labeled by szd: X2,“ = Idr'---dr"i BQBg(r,r',r'")T6;_(r',riv)a;l?n(riv,rv) x an (r‘“,r")1>6‘7 (r")rg, (r'", rvi )1>(§E,(rvi ) = 4Idr'n-dr‘" Z'[(0|1>2;‘}(r)|j)(j|1>§A (r')In)(nIP6“ (r'") j,n,k +(0111ii"‘ 0) +<0|P1§‘(r') 1)(J'l1’o'2A (r)ln>(nIPr§‘(r"') 0)] XPA II (OIPB 1V k k PB V O AflA-lA-l Mr) is )1 ><12. .1. x P620" )Tm(r',r'v)Tm(r",rv)T58(r’",rVi ). (83) X2 also contains one part of the dispersion contribution to the polarization. Terms containing the permanent polarization of molecule C alone are grouped into X3. These terms reflect the linear polarization of molecule A due to the field from the permanent polarization of C: The permanent polarization of C polarizes A, which in turn 147 polarizes B; the induced polarization in B gives rise to a field that polarizes A. This induction term is given by Ximd = Idr’mdrwaa ’“ B(r, r ’)T 1310'" r"’)agnh”, r")Tfly (rv,r") X 0'98 (r", r!!!)Tsc (rfllr Vi )Poe (er) :81 1.4.111 Z'<0|Poi(r)|1“><1'lPé'(r'>10><0|Pr"(r")|n> j,n,k XIO'Pfl'W”k><1181 (1)1><1|Pi’(r)|><1“|Pi’A (r'") x A;‘A;‘(An +Ak)" +(OIP6' n)(n|P;‘(r") x14]- +4..)“(Ar + 4..)“(An +41)“ ') 0011):"(r)|I>l1><1|PiA< 11in») x A;,‘(Aj +Ak)"(A, +Ak)" 148 + (OI Pg" (r'") 0) 1><11P3A x A;‘A7‘(An +Ak)‘1 —(0|P§(r)|j)(j|1>g"(rm) 0)(0|P§(r')|n)(n|1>;‘(r") o) x[A;?-(An + Ak)“+A}1(An + Ak)’2] 0) n)(nIP6“ (r’”) -<0|P;‘.‘ (0100111? (r') o><011>2A (r") x A'J.‘A;,‘[(An + Ak )‘1 +(Aj + Ak )-1 +(A, +4..)“(An +4)-‘01,- +An +4151) x (0|P2?(r‘V)|k)(k|1>,',3(r”)|0)1>§,(r"i) x Tar.(r'.r‘V)Tn2 (rvrr")Toe(r'"rl'Vi ), (85) or equivalently, Xdisp = h/ 21: Idr'mdr‘" Igodor 7375a (r',r",r"’,r;ior,0,0) T133» (r',ri") x (133110", r";i(0)Tfly (rv, r")T&(r"', rVi )POC8 (rVi ), (86) where the following integral identities have been used I:dx2a(a2 +18)" [(b —ix)‘2 +(b+ix)'2] = 2n (a+b)‘2, (37) and I:dx2a(a2 +x2)’1[(b—ix)'1(c—ix)‘l(d—ix)'1+(b+ix)"1(c+ix)’1(d+ix)'l] =2n(a+b)"'(a+c)“‘(a+d)“, (88) and the y hyperpolarizability density 767569,, r", r"', r; i0) , 0, 0) is obtained fi'om Eq. (81) with or] replaced by —ico and co 2 replaced by its . The Xdisp term represents the change in the dispersion-induced polarization of molecule A in the A- - - B pair due to the presence of the third body C. Here C acts as the 149 source of a static field that perturbs the dispersion interactions between A and B. Eq. (86) gives the lowest-order change in the dispersion polarization of A of the A- - - B pair due to the static field from C acting on A. The total contribution from the "noncircuit" terms to the third-order polarization induced in A is if =(1+g.)BC)(U1+U2 +U3+W1+W2 +W4 (3) none +X1+X2+X3+X4), (89) and the full third-order polarization of A satisfies P3‘3)(r)=(P:(r)):f +(P:(r))‘3' (90) noncir with (P: (0):) given by Eq. (44). The results fiom the perturbation analysis in this section are identical to the corresponding results obtained from the reaction field method in Sec. 6.4 of Chapter VI. This establishes the validity of the reaction-field results, subject to the assumptions of the perturbation analysis. 150 References [1] See, for example, E. Merzbacher, Quantum Mechanics (Wiley, New York, 1970), Chapter 16. [2] W. J. A. Maaskant and L. J. Oosterhoff, Mol. Phys. 8, 319 (1964). [3] Equation (21) for B04” (r, r', r";—c0(,;(01,602) and equation (81) for I(011375(r’r" r", r"';—03 0301,60 2, 0) are obtained by analogy with Eqs. (43b) and (43c) in Ref. 6. [4] K. L. C. Hunt, J. Chem. Phys. 80, 393 (1984). [5] K. L. C. Hunt, J. Chem. Phys. 92, 1180 (1990). [6] B. J. Orr and J. F. Ward, Mol. Phys. 20, 513 (1971). CHAPTER VIII EFFECTS OF OVERLAP DAMPING ON THREE-BODY DISPERSION ENERGIES 8.1 Introduction In Chapters VI and VII, both reaction-field methods and perturbation theory are used to analyze nonadditive three-body energies and dipoles. The analysis identifies different polarization mechanisms that contribute to three-body interactions. These include induction, dispersion, and combined induction-dispersion effects. Induction effects are classical, resulting from the polarization of a molecule by the fields fiom the permanent molecular charge distributions. For molecules interacting at long range, induction effects are determined simply by the permanent molecular multipole moments, the static polarizabilities, and the static hyperpolarizabilities. Dispersion interactions, however, are purely quantum mechanical in origin, stemming from correlations between the spontaneously fluctuating charge distributions. Dispersion effects are important, because they are present in a variety of interactions, including molecule-molecule, molecule-atom, atom-atom, and molecule-surface interactions [1-9]. Moreover, in such cases as interactions of three spherically symmetric atoms at long range, only the dispersion effects survive at third order. The three-body dispersion energy is the subject of this chapter. The leading term in the three-body dispersion energy for S-state atoms A, B, and C interacting at long range was first derived by Axilrod and Teller [1], and independently by Muto [2]: AEE,I(1,1,1) = (3 coseA coseB cos0C + 1)C111 (RAB RBC RCA)‘3, (1) where R AB denotes the distance between atom A and atom B, 0 A is the angle between R AB and RCA, and C] 11 is a coefficient related to the first-order imaginary-frequency 151 152 polarizabilities 01100)) of A, B, and C by C111 =3h/71I:aI“(i(0)aI3(im)a1C(ioo)d(0. (2) Eq. (1) is valid at long range when the multipole expansion of the interaction potential holds. At shorter range, the dispersion energy is damped due to modifications of charge-overlap effects. The damped dispersion energy can be derived within the nonlocal response theory that uses polarizability densities to describe the nonlocal response of a molecule to the fields from its interaction partners. The theory gives the three-body dispersion energy as an integral of imaginary-frequency polarizability densities of the interacting molecules [see Eq. (22) of Chapter VI]. The purpose of this chapter is to study how overlap modifies three-body dispersion energies. In Sec. 8.2, the damped three-body dispersion energy for interacting S-state atoms is analyzed. It is shown that at long-range the equations for damped dispersion energies reduce to a multipole series, in which the leading term is the triple-dipole energy fiom Eq. (1). Sec. 8.3 contains a numerical application to a model system, interacting ground-state hydrogen atoms. An analytical expression for the damped triple-dipole dispersion energy is derived, and the radial and angular dependence of the dispersion energy is investigated. 8.2. Dispersion Energy for Three Interacting S-state Atoms In this section, the nonadditive three-body dispersion energy is calculated for interacting S-state atoms. The calculation includes the effects of direct charge-overlap on the dispersion energy, but not the effects due to intermolecular exchange or charge transfer. In Chapter VI, a nonlocal response theory is used to derive an expression for the nonadditive three-body dispersion energy of interacting molecules A, B, and C. The theory yields a simple physical interpretation for the dispersion energy in terms of the induced 153 polarization and the energy of polarization in a reaction field. The spontaneous, fluctuating polarization in molecule A polarizes B, which in turn polarizes C. The induced polarization in C produces a reaction field at A. This polarization route A—)B—)C—>A gives one term in the energy shift of A, with the second term generated by A—>C—-)B—)A. Both terms depend upon the correlation in the polarization fluctuations at two points within A, which are related to the imaginary part of the nonlocal polarizability density of A via the fluctuation-dissipation theorem. Similar energy shifts occur on molecules B and C, giving the second and third terms in the total energy change of the three interacting molecules. The theory gives the nonadditive three-body dispersion energy AEffizp = -h / it I: don Idr dr' dr" dr'" driv dr" 01360, rV;i(0 ) T131 (rv , r") x 01550" , r"'; i0) ) T3,c (r"’, r0013 (r", r'; im)T¢a (r', r), (3) in terms of the tensor product of the imaginary-frequency nonlocal polarizability densities 013,550, r';ico) (X = A, B or C) and the dipole propagators TaB (r, r') Ta6(r,r') = raga — r') = VaV6(Ir — r' ) 5)-471:/3 501135“ — r') : [3(r0L -r(;)(rB — r6)—5a6Ir—r'I2]/(Ir—r' (4) The Einstein convention of summation over repeated Greek subscripts has been used in Eq. (3) and below. An expression equivalent to equation (3) can be obtained by first using the standard perturbation theory to find the interaction energy for three molecules A, B, and C at third order, then selecting terms that are purely of dispersion origin [see Sec. 7.1 of Chapter VII]. The result can be cast into the computationally useful form 154 AB“) asp = Ir/(8n7) deo I dkexp(ik-RAB)k‘2 I dk'exp(ik'-RBC)(k')'2 xIdk"exp(ik"-RCA)(k")_2 x’“(k,k";ico)xB(k',k;ico)xC(k",k';i00), (5) where RXY is the vector from the origin in molecule X to the origin in Y (X, Y = A, B or C) and 3((Ig k'; 1(0) is the imaginary-fiequency charge-density susceptibility, x(k,k';i, (6) nan In Eq. (6), a) n0 is the transition fiequency between the ground state I0) and the excited state In), and p(k) is the k Fourier component of the charge density operator: p(k) = Idrexp(ik~r)2qj 6(r— rj) = qu exp(ik-rj) (7) r 1 with q J- the charge of particle j. To evaluate the dispersion energy given by Eq. (5), the charge-density susceptibility x(k,k’;co) is needed as a function of k, k', and 0). Given an accurate ground state wavefunction, x(k, k';(0) can be determined from Koide's method [10]. In this method, x(k, k';co) is expanded in terms of the spherical harmonics of the orientation angles of the vectors k and k' by substitution of Eq. (7) into (6) and use of the Rayleigh expansion for exp(i k - rj): co 1 co 1' , 1101,1211) = 1:0 15:1 12.0 “Er c7. Yl‘“(e,¢) Yrt'(9',¢')‘a;‘;f" (k. k';w), (8) where c, : (—i)1 2’ 11/(21)1,I4n/(21+ 1), (9) and [10] ai‘if“'1¥" (10) natO with the generalized multipole moment operator pI“ (k) given by 155 (21+1)! 2l 1! m 47: ’1/2 m . PI (k)=ZqJ-( I Y1 (9139]) 110(5), (11) j 21+1 where j,(krj) denotes the 1th spherical Bessel function. If the auxiliary functions =l/hzmno+00)-1In);?'(k',—o)). (13) (1);“ (k, (0) can be approximated by the function ‘1’ which minimizes the fiJnctional [11, 12] JI“(‘I’) = (‘I’IHO - E6 +1161|\P)—(0|p;“(k)‘|)11)—()P|p;“(k)|0) (14) subject to the conditions (0| ‘1’) = 0 and (0 2 0. In Eq. (14), H6 is the Hamiltonian of the unperturbed molecule, and E6 is the energy of its ground state. For spherically symmetric atoms, orW‘Yk, k';co) takes a simple form [13]: ai‘im’lk, kw ) = 611 6mm «10430). (15) Substitution of Eq. (8) into (5) and use of Eq. (15) yields xIdk"exp(ik"-RCA)(k")’22 Z Z Z Z Z Ic, c1. c," I2 l m l I ml 1" m" x6819.4)“Y,m(e,¢)Y;£‘"(e',¢')' YIP’ (9',¢')Y,m(e",¢")‘Y1?" (6".11") xor?(k,1r";io)otI?(k',k;io)or[5.(k",train). (16) The value of AE(3) must be independent of the choice of the coordinate frame because it is a scalar. For convenience in the subsequent analysis, a coordinate (X, Y, Z) 156 is used in which atom A is located at the origin, B on the positive Z-axis, and C in the X- Z half-plane with nonnegative X coordinate. Then use of the Rayleigh expansion for exp(iktR) yields exp(ik‘RAB) = 2141(2L+1)1% iLYB(e,¢)ii(kRAB). (17) L exp(ik-REC) = Z Z(—i)L'4nYL/(0',¢')Y{4' (03,1t)' jL. (k'RBC), (18) L' M' and exp(ik - RCA) = Z Z(-i)L"4rt Y5!" (0",¢")Y1'ji" (0 A,0)“ [LunchC A ), (19) L" M" where 0 A is the angle between R AB and RCA, and 0B is the angle between R AB and RBC . Inserting Eqs. (17)-(19) into (16) and integrating with respect to the polar angles (0,4)), (0',d)') and (0",¢)") gives the dispersion energy in the form: AE(3) = Z Z ZAE(3)(I,I',1"), (20) l I' l" where AE<3>(1,1',1")=h/(2n6)§§§afifinjo dorI: de: dk'IO dk"ot;‘(k,lr";io) xal?(k'.k;ico)afi(k",kuic)irlkRABnplk'RBo)ip.(k"RoA) (21) . III I" . wrth an,” glven by I II . . I II ' H I); 2 _ I 311.11» =1L(-1)L +1: (2L+1)1(2L +1)(2L +1)1-crch1»l ZZZYL'M (98,“) m m" MI I" m") ”51(9),,0)(L10M|1'm)(L100II'0)(L'1'M'm x(L'l'00 l"0)(L" 1" —M'm"Ilm) (L" 1"00|10), (22) 157 where (1'1" m' m" I] m) denotes a Clebsch-Gordan coefficient. In order to simplify the notation, the subscript "disp" is dropped in Eqs. (20) and (21) and below. AE(3)(I, l', 1") represents the energy of interactions among the 21-pole of A, the 2" - pole of B, and the 2’" -pole of C, with the inclusion of the short-range overlap effects. This can be shown by examining the asymptotic behavior of Eq. (21) in the long-range limit. Use of the theorem on integrals involving spherical Bessel firnctions given in Appendix 1 of Ref. 10 gives I: dk for I: dk"orf“(k,k";i(0) or}?(1rt,k,im ) ag,(k~,kr;iw) X jL(kRAB )JL'(k'RBC)JL"(k"RCA) = n32'2<’+”+’">+3 (21+ 21')1(21'+ 21")! (21+ 21~)1[(1+1')1(1'+1")1 (1+1")1]'1 X51 M 511141" 5L~1+1" 01;“ (i03 )ai? (ice )are" (iO) ) 132-($1M) Rg‘c’"’"“’ 12314141) (23) for R AB —> 90, RBC —+ co, and RCA —> oo. With this, Eq. (21) yields the three-body dispersion energy at long range in the form AESIUJ'J") = b , ,. ,1 [3° do 01;“ (io)otI?(io )6?" (ico ) 1129"“) Rggt’"“) 1135“”, (24) where 2 l-I-II-I” 11'1" I I II II x [(1+1')1 (1' +1")! (1 + 1")1]“. (25) In Eqs. (23) and (24), 01,000) is the imaginary-frequency multipole polarizability of order 1, related to or,(k, k';i(0) by ot,(io) =11mk_,011mk._,O or,(k,k';i0) ) k” dr')" . (26) 158 Specializing to I = 1' =1" = 1, Eq. (24) recovers the long-range triple-dipole dispersion energy given by Eq. (1). The dipole-dipole-quadrupole dispersion term is obtained by setting I: 1' =1, and l" = 2 in Eq. (24), 451310.12) = 311(1611)vnafr(k)1'i(kRAB)de:frvtk')f1~(k')1'o(k'RBo)dk' x]:liar")f1~(k")1'r.~(k"RoA)dk", (34) where the integral identity [3"(a2 +602)"](b2 +1112)"(c2 +c02)"1d0) =1t/2 (a+b+c)[abc(a+b)(b+c)(a+c)]‘l (35) has been used for a, b, c > 0. The integrals in Eq. (34) can be evaluated analytically using contour integration techniques. For example, from Eqs. (30)-(34) the damped triple-dipole dispersion energy (I = 1' =1" = 1 in Eq. (34)) has been obtained as an analytical firnction of the interatomic distances R AB, REC, and RCA: AB<3>(1, 1,1) = 21871344 [f(RAB)f(RBC)f(RCA)+(3c082 0C — 1) Xf(RAB)8(RBC)g(RCA)+(3 COSZGA —1)g(RAB)f(RBo)g(RoA) +(3 cosz0B — 1) g(R AB) g(RBC)f(RC A ) + (3 0030 A coseB cos0C +1) Xg(RAB)8(RBC)8(RCA )1, (36) where the functions f (x) and g(x) are defined by 160 f(x) = e‘z"[89/1152+(89/576)x+(119/864)x2 +(5/72) x3 +(1 11540) x4 +(1 / 324) x5 +(1 / 5670) x6], (37) g(x) = 3/(2x3)—e"2"[2+3/(2x3)+3/x2 +3/x+x+(l699/4320) x2 +(259/216O)x3+(197/7560)x4+(19/5670)x5+(l/5670)x6]. (38) For RAB —> oo, REC —) 00, and RCA —3 00, Eq. (36) reduces to ABI31>2(1,1,1)= cm (3 cos0A c0503 cos0C + 1) R33 Rg3c RE?A (39) with the coefficient c1 11 = 59049 / 2752 E 21. 4568, in good agreement with the accurate value 21.6425 [14]. To illustrate how overlap modifies the dispersion energy, the results from Eq. (36) are compared with those from Eq. (3 9). Figure 8.1 gives the dispersion energies as firnctions of R for the geometry of an equilateral triangle, where R denotes the separation between the hydrogen atoms. The damping function x111 is plotted against R in Figure 8.2.; here x1” is given by 11111 = AE(3)(1,1,1)/AEIJ3&(1,1,1). (40) Figure 8.1 and Figure 8.2 show that including the charge-overlap effects appreciably reduces the rate of increase of the dispersion energy with decrease of the interatomic distance R. In fact, at vanishing R, Eq. (36) gives a finite value for the dispersion energy, while’Eq. (39) goes to the limit of infinity. The same effects are present in two-body interactions, which have been studied extensively [10, 13, 15-21]. The inclusion of charge-overlap effects also modifies the angular dependence of the three-body dispersion energy. To illustrate this, three interacting ground-state hydrogen atoms in the geometry of an isosceles triangle are considered. In Figure 8.3, the damped and undamped dispersion energies are plotted as functions of the angle 0 between the two 161 equal sides (R) of the isosceles triangle, for fixed values of R. Two values R = 3, and R = 4 have been selected. For all values of R, the long-range three-body dispersion energy is repulsive for 0 < 117.2 , while it is attractive for 0 > 117.2. With R = 4, the damped three- body dispersion energy from Eq. (36) is only attractive for 0 > 156.7, and it is repulsive for all values of 0 when R = 3. These results agree qualitatively with those obtained by O'Shea and Meath, using a formal partial wave analysis and pseudo-state techniques [22, 23]. 162 A P F 0 v O to.) 5 5.5 6 6.5 7 7.5 8 R(a.u.) Figure 8.1 The triple-dipole dispersion energy of interacting ground-state hydrogen atoms in the geometry of an equilateral triangle with R the length of a side. (1) the damped dispersion energy from Eq. (36), and (2) the undamped form from Eq. (39). 163 1111 R(a.u.) Figure 8.2 The damping function x111 for the triple-dipole dispersion energy of interacting ground-state hydrogen atoms in the geometry of an equilateral triangle. 164 60 00 100 120 140 160 Figure 8.3 The triple-dipole dispersion energy for interacting ground-state hydrogen atoms in the geometry of an isosceles triangle, as a function of the angle 0 between the two equal sides R. (1) AE(3)(1,1, 1) with R = 3, (2) AB<3>(1,1, 1) with R = 4, and (3) ABfigm, 1) with R = 4 [24]. 1 65 References [1] B. M. Axilrod and E. Teller, J. Chem. Phys. 11, 299 (1943). [2] Y. Muto, Proc. Phys. Math. Soc. Japan 17, 629 (1943). [3] A. D. McLachlan, Mol. Phys. 6, 423 (1963). [4] B. Linder and D. Hoemschemeyer, J. Chem. Phys. 40, 622 (1964). [5] B. Linder, Adv. Chem. Phys. 12, 225 (1967). [6] R. J. Bell, Proc. Phys. Soc. 86, 519 (1965). [7] J. D. Johnson and M. L. Klein, Trans. Faraday Soc. 60, 1964 (1964). [8] A. D. McLachlan, Mol. Phys. 7, 381 (1964). [9] H. Margenau and J. Stamper, Adv. Quan. Chem. 3, 129 (1967). [10] A. Koide, J. Phys. B 9, 3173 (1976). [11] M. Karplus and H. J. Kolker, J. Chem. Phys. 39, 1493, 2997 (1963). [12] P. W. Langhoff and M. Karplus, J. Chem. Phys. 52, 1435 (1970). [13] K. L. C. Hunt, J. Chem. Phys. 78, 6149 (1983). [14] Y. M. Chan and A. Dalgarno, Mol. Phys. 9, 525 (1965). [15] A. Koide, W. J. Meath, and A. R. Allnatt, Chem. Phys. 58, 105 (1981). [16] B. Linder, Adv. Chem. Phys. 12, 225 (1967). [17] B. Linder, K. F. Lee, P. Malinowski, and A. C. Tanner, Chem. Phys. 52, 353 (1980). [18] M. Krauss and D. B. Neumann, J. Chem. Phys. 71, 107 (1979). [19] M. Krauss, D. B. Neumann, and W. J. Stevens, Chem. Phys. Lett. 66, 29 (1979). [20] M. Krauss, W. J. Stevens, and D. B. Neumann, Chem. Phys. Lett. 71, 500 (1980). [21] M. Krauss, and W. J. Stevens, Chem. Phys. Lett. 85, 423 (1982). [22] S. F. O' Shea and W. J. Meath, Mol. Phys. 28, 1431 (1974). [23] S. F. O' Shea and W. J. Meath, Mol. Phys. 31, 515 (1976). [24] The undamped dispersion energy at an arbitrary distance R = R' is given by AES§(1,1.1)IR=R1 = (4111')9 4583.0, 1.011141. Appendix A By means of contour integration techniques, the functions u5(R), u7(R), and (96(R) given by Eqs. (49), (50), and (51) of Chapter V, respectively, are evaluated analytically: 115(k) ——- 410062" - tame“, (A1) 117(R)=97 R”+<13(R)e‘2R —¢4(R)e‘4R, (A2) and (96(R) = M6 R‘6 +11),(R)e‘2R —¢6(R)e"4R. (A3) In Eqs. (Al)-(A3), _ _ 107571969 D 7 — 258344 ’ (A4) _ 40095 M6 ‘ T 688 1 (A5) __ 124101 28431 —2 28431 —1 29331 3267 2 ¢1(R)‘ 110080+44032R +22016R + 55040R+ 24080R 1923 3 257 4 1 5 + 96320 R + 144480 R + 14448 R , (A6) _. 275283 28431 —2 2843] -1 176070249 279868959 2 ¢2(R)‘ 55040 +44032R +11008R + 28180480 R+ 49315840 R 1188219827 3 56994281 4 204673009 5 61303057 6 + 295895040 R +24657920R +184934400R +138700800R + 181250911 R7+ 11949733 R8+ 955321 R9+ 4009973 R10 12483072000 312076800 121363200 . 3276806400 + 456077 R” + 143 R12 + 2._1_2 R13 +__1 R14 (A7) 3276806400 13003200 409600800 81920160 , -11219260329 107571969 —7 107571969 —6 107571969 —5 ¢3(R)‘ 578690560 + 129172 R + 64586 R + 64586 R 35857323 —4 35857323 —3 73127134197 -2 11930636277 -1 + 32293 R + 64586 R + 330680320 R + 165340160 R 153760947 5402517 2 998061 3 808739 4 6163 5 + 36168160 R+7233632R +10333760R +108504480R +27126120R , (A8) 166 and 167 :311621667759 107571969 —7 1075719691 —6 107571969 —5 ¢4(R) 231476224 + 258344 R + 64586 R + 32293 R 143429 92 -4 4 4 9 9 -3 1174664096757 —2 + 32293 R + 32293 R + 330680320 R 195520130037 R" + 296563134494651 R + 32416131852059 R2 82670080 444434350080 1 1 1 108587520 63098078603249 R3 + 16520765448167 R4 + 3466717350289 R5 555542937600 416657203200 277771468800 + 148048034867 R6 + 2999482556923 R7 + 346045385663 R8 41665720320 3281 175475200 1640587737600 84623425049 9 18626834567 10 7894399 11 + 1968705285120 R + 2460881606400 R + 7031090304 R +42$8233§J_R12+_2§242LR13+ £854.31 R14 9228306024000 214611768000 4614153012000 1217 15 16 + 26826471000 R + 958088250 R 2 (A9) _ 1414961361 40095 -6 40095 -5 40095 —4 337949577 -3 d’5(R)‘ 82670080 + 344 R + 172 R + 172 R + 2066752 R _ _ 7 + 97058817 R 2 + 7485399441 R 1+ 268721181 R + 6273129 R- 1033376 165340160 57869056 7233632 4302§§7 3 2495753 4 1178543 5 11271 6 + 36168160 R + 180840800 R + 994624400 R + 248656100 R ’ (A10) ¢6(R) : 16944372081+ 40095 R—6 + 40095 R-5 + 4080695 R“4 + 1301512617 R—3 41335040 688 172 20667 52 337959577 —2 92616153201 —1 150493937751 15957391370551 2 + 516688 R + 165340160 R + 578690560 R+ 111108587520 R + 1921075957543 R3 + 8085289847027 R4 + 24607739770651 R5 + 15948903098831 R6 27777146880 277771468800 2291614617600 4583229235200 + 1135044714739 R7 +17828O75669207R8+ 4835951926759 R9 + 905293545841 R10 1 145807308800 72185860544400 90232325568000 90232325568000 + 108448476469 Rll-i- 14581137517 R12+ 404737297 R13+ 1712983 R14 67674244176000 67674244176000 16918561044000 805645764000 12211 15 211 16 1 17 + 85447278000 R + 32787909000 R + 7025980500 R ' (A1 1) Appendix B Fully expanded, Eq. (51) gives 68 nonzero "circuit" terms. These terms can be grouped into four sets T1 — T4 according to the types of matrix elements appearing in each. TI is given by Tl =(1+C){—(‘I’0|PA GA 52598 X [VXB GA 6953 floc (V130 GMBC $53 VCA + VgA GC $8 6053 VBC) + Vé’A GAGBC to? (Vfic GA 5953 508 VAB + VXB GC 596‘ £053 Vec) GAEBC 5053(V8A GA to? 598 VAB + VXB (Gctoé pt? +V§C +6Aec JOEWCAHI‘PO) —<%IVAB GAtoEtoS P°A GAtJEtofiwge GAEBCfJgVCA + VSAGC 503‘ ngBcWI’o) -<‘1’o|VBcGCsa€ta§ P°A GA®C to}? (VSA GAtaS’loS VAB +VXBGC 98 5053 VCA + VXBGAGBC @gVCA W110) —<% I veA (60:28 to? Mime?) P"A GA®C to}? (V130 GAtJEtoS VAB +VXBGC608 £9lll3VBc)l‘{’0> +0110IPAGAKJgSOOCGAngSVABI‘P0)(1+C)(W0|VCAGC693608VBclq'o) +(‘I’o IPA GA {053695 VAB ITO) (1 +C)<‘I’0 I VBCGCSOOA 59(1)3 GC 608 §)(E)3VCA|\PO>}° (Bl) The T1 terms reflect the hyperpolarization of A by the simultaneous action of the direct field from P530) and the field from the polarization induced in c by P§(r). The 168 169 corresponding set of 20 fully expanded terms, with A hyperpolarized by P6: (r), is contained in T2: T2 = 5ch T1, (132) where pBC perrnutes the labels A and B. Terms containing the permanent polarization of A are grouped into T3: T3 : (1 +C)(1 +£JBC){-(‘1’0|PA GA 6053 :28 Will; GB 98 p8 (V130 0% to}? Vet + VBAGWtas‘ VBC) +V§C GMBBEBC VXB GC 5.98 595’ VCA ]|'~I’0) - (‘I’o IVAB GA698 508 P°A GBm‘ raocwfic Get??? ngCA +V8AGB€BC 693 VBC )| Yo) -(‘1’o|VCA GC 506‘ p3 P°A GAec tat? (VBC GA®B£00C VAB +VXBGB€BC 693 VBC)|‘I’0> " (“’0 I VBCGB€BC 500A PM GAQBGBC VXB GC 93‘ 5953 VCA I ‘I’ofi- (B3) T3 subsumes the static induction effects that involve sequential linear response. Specifically, the permanent polarization of A induces a polarization in B, which polarizes C, giving a reaction field back at A, thus polarizing A; and the same polarization mechanism applies with B and C interchanged. T3 also contains one part of the dispersion contribution to the polarization. The remaining dispersion terms are contained in T4: T4 = (1+C)(1+50I3(3){—(‘I’(,|PA GA 5053 508 X [VSAGAec K353 (VlgC GA®B 69g VAB + VXB 086C 698 VBC) + VlgC GA®B®C VSAGAeBiJOC VAB 1|W0> 170 —(‘P0 I VAB GA®B MC) POA OWE“? (VIgC GAEBC 6953 VCA + VgAGBGC 698 VBC )I‘Po) ‘(q’oIVBCGBecWQ POA GMBBGBC VSA GAGBBSOOC VABIWOBt (B4) To analyze these contributions, T1 — T4 are converted into matrix element form. For T1, T1 = 2 j dr'-~drvi lj‘lgm' «(oil’s (r)|j) +ll>l1>lo>lrt +(0|P: (r)|j)(j|P;‘(r")|1)(1|1>§(r')|o)f1b(Aj,A,,Am)} +jzt1n'<0IP:(r)lj>P5:(r")flctAJ-Amn xPéumlPéio-‘V)(0IPf(rV)Im> ”950-3r"')"r,,4,(r“’,r")r,,(r",r"i (135) assuming all the molecular eigenstates can be taken as real. In Eq. (B5), fla(AJ-,A,,Am)= A}‘A‘;‘(A, +Am)’1 +A31A71Ajg +A;‘(Aj + Ana-RA, +Am)“ +A31<<0IPeBtr"')Ik>10><01Pf(r”)lm>10) xf3(Aj,Ak,Am)T55(r',r”')T8¢(riv,rV)Tyx(r",rVi (311) with f3(AJ-,Ak,Am)= Ag‘Af/i; +A}‘A}‘(Ak +Am)’l +A}‘A;,}(Aj +Ak mm)“ +A;,}(Aj+Ak)“(Aj+Am)"+A;‘A;,1(Aj+Ak)-1 +A:,1(A,- +Am)“Il><1IP:Al1>l0> Xf4b(Aj,A1,Ak,Am)] B "I B W C V C Vi ><(Oll’t (r )|k><0IP¢. (r >lm>= A;‘<(AJ-+A,(+A,,,)‘1 = 2A;‘(Ai +At + AmXA, + Akr‘mt +Am)"(A1+ Am)", (814) and f4b(Aj,A,,Ak,Am) = (Aj +Ak)‘1(A, +Ak)"(A, +Am)‘l +(Aj+Ak)’1(A,+Ak)"(Ak +Am)"1. (315) The linear induction contribution to the polarization P: (r) is denoted by Thad. The quantity T3,ind is identical to the right side of Eq. (B11), except that f3(AJ-, Ak, Am) is replaced by f3,tnd(A,-,A1,Am) = 4A}‘A}‘A},}. (816) T3,,“ is related to the nonlocal polarizability densities of molecules A, B, and C by Eq. (55) in Chapter VII. The dispersion contribution to the polarization PO? (r) is given by Tdisp 2 T4 + (T3 — T3,ind)- The effect of adding (T3 — Tim) to T4 from Eq. (313) is to replace (“mt-om by §(r')|o) j,l.k,m xf48(AJ-,A1,A1,Am> +(o|3;‘(r') l>II> 0) x f4b(Aj,A,,Ak,Am)] ><<0|PeB(r”')l1<>(0lP50”)!nr1)(nrlle(rVi )l0> xTB5(r’,r'")T8¢(riv,rv)Tyl(r",rVi . (B18) After label changes j <—> l, k (—-> m, B <—> y, e <—> d), 6 <——> X, r' <—-) r", r’" (-) r“', and rv <——> rVi in the term generated by 503C in Eq. (B18) and summation with the expression given explicitly in the same equation, Eq. (57) of Chapter VII is obtained. Appendix C The full expansion in Eq. (51) yields 27 terms of (VAB , VAB, VBC) type. They can be grouped into three sets U1 — U3, according to the different types of matrix elements appearing in these terms. U1 is given by U1 =(1+C){-(‘P0|PA GA 506’ tog x [VXB GB £26 508 (VXB GB 506‘ 500C V3C + Vfie GB (26‘ :98 VAB) + Vlgc GAGEB 698 VXB GB 6’93 60:): VAB]I W0) -(‘Po I VAB (GB 506 508 P°A GA613 (08 +(3A6313 508 P°A GB 596‘ 598) X V2.3 GB 898 600C VBC ”0) —(\110 I VA3 GB 649854951)“ GAGBB 608 Vigc (GA 5963 608 + GWB gag )vAB I‘i’o) —<‘POI(VAB GAGBB 93 PM GB 608 503 VlgC + VBC GB .698 5031)“ x cm e8 32ch e3 e8 vie 1%) +0110 IPA GA @133 508 VABI‘Po) (1+C) x(‘1’oI(VAB GB 506‘ :08 GB 506(98Vecl‘1’o». (C1) In matrix element form, Ul =2jdr'u-dr"i Z'(O|P:(r)|j)53(r'") k)(k|1>;’B(r") mIImIPEO'iV)IO>81a(AthkiAm)} x 3,2,0“)pr(r',r")r&(r'",t-Vyrmu“,r‘“) (C2) with glaA GA 353 gig X [ V723 GA 60133 60% (V23 GB 600A 600C V3c + V130 GAGE 608 VAB) +(va GA®B gag v"BC +v§C GA®B gag v,‘§B)GA go}? 565 VAB]|‘P0) ‘(q’o I VA3 GA 6053593: POA GA JOE'SOOCWXB (38508 590C V3C + V130 GAEBBJ‘JOC VA3)I‘P0) ’ (“’0 I (VAB GAGBBQOC POA GMBB 693 V130 + VBC (38693 593 x P°A GAQBrJoC VXB)GAta€ta8 VABI‘POI +(‘1’oIPA GA 506398 GA (963508 VAB I‘Po)(1+C) X (910 I VAB (38898 898 V3C I‘Yo» (C5) 176 The matrix form of U2 is I)I0> U2 —_— 2Idr'---drvi '{[(o|1>;}(t-)|j)(j|1>[§’A (r') j,l,k +<0IP:(r)I1)ll>Pi%(rV)gzc(AJ-,At)) x(OIPtsB(r"') k>Po‘3(r")PSntn-V‘) x TB,(r',r")r5,(r"',r")rm(r‘“,rV‘) (C6) with 823(Aj1 Ala Ak ) = AEIAIIAEI + AEIAIIMI + Alt )4, (C7) gem-Amt) = AEIAI'A1‘+A]‘A7‘(A1+ At)“ +A3‘(Aj +Ak)“(A, +Ak)'l, (C8) and gzc(Aj,At)=A;‘A1‘(Aj +At)“. (C9) U 3 contains the remaining dispersion terms, U3 =(1+C){—(‘I’6IPA GA 5063 500C X [VXB GAEBB 690C (VXB GB 503 693 V3C + V130 GAGE £98 VA3) +V§e 6"“ (:28 Vie Gm 508 VABII We) — (“'0 I VAB GAGBB 500C POA GAGBB 59% (V23 GB 598 pg V3C + Vigc GAeBB 593 VA3)I‘¥0> +16 I V3C GB (at? tag P°A GAeB 508 Vie GAG?B :98 VAB I ‘16)). (C10) 177 In matrix element form, the U3 terms reduce to U3 = zjdr'...dr"‘j IZRZJ;3(r") k)(k|1>53(r"') m)(m|1),‘3(hiv )|o) xA]1(A,+Ak)'1A_n} k)(k|Pe°B(r“’)lm)Il)I0)<0IPeBIk)Io) x P§,(r')P§,(r")A1‘(Aj+Ak)‘2 +<0IP6(r') J> -(‘I’oI(VAB GMB (00C P°A GB 506‘ 500C Vfic + Vee GBE‘13C (6 x P°" Gm”C VXB)GC tot? 600 Vac I‘Po>} (D1) or in matrix element form, k>|0> w1 = zjdr'u-dr"i Z '(oIP:(r)|j)(j|P§(r')|o)(o|1>53(r"') j,k,m xPain")(0|Pf(r")|m)(m|1>§(r“)|o)h1(Aj,Ak,Am) xTB,(r',r")rs,,(r'",ri")r¢,(r",r“) (D2) with h1(AJ-,Ak,Am)=A;‘(Aj+A,,,)“(AJ.+Ak)"+A;‘(AJ.+A,,,)“(Ak+Am)‘1 —l -l -l —1 -l —l +Aj Am(Aj +Ak +Am) +AJ- AkAm +A:(Aj+Ak)‘l(AJ-+Am)"+A;1(Aj+Am)‘1(Ak+Am)‘l +A“ m(Ak+Am) (A +Ak+Am)‘1+(A +Ak)‘A'1A';n =2A71A’1M(AR+A )"+2A"A"A;,,l (D3) 178 179 In the W1 terms, the field from the permanent polarization of molecule B polarizes C, which then polarizes B, and produces a field that polarizes A. W2 is given by w2 = (1+C) {—(‘I’OIPA GA gag gag x [Vt‘éc GA6313 (08 Vfic (GA (063 (08 + GA®B (06C) VAB +(V13C GAGEB £08 VXB + VXB GB 533 595 V§C)GB 598 £98 VBCII‘P0> -(‘Po I V3C GB (06‘ (08 P°A GA613 (03 V1‘§c(GA (963508 +GA®B 60%)VA3I‘POI —(‘PoI(VBc GB (96‘ (00C P°A GA6313 598 VXB + VAB GAGBB (08 x P°A GB 536‘ (98 Vt‘ic)GB (06(98V3cl‘1’o> +(‘I’o IPA GA 5053 500C GA 5053 98 VABI‘POX‘PO IVBC GB 598 608 V3C I‘I'o) +(‘F0IPA GA 953 500C VAB I‘I’OX‘POIVBC GB $3 603 GB 603 600C VBC IWOI- (D4) The matrix element form of the W2 term satisfies w. = zldr'---dr" ,kz'wlpmelioIPitr') 0) 1. .m x [(o|I>g3(r'")|lt)(lt|r>f‘3(rvi )| m)(m| 3,90") O>h2a(AjaAltaAm) +<0IPsB(r"'>Ik) +(‘1onPA GA 6353608 GA 6053608 VABI‘Po) XIWOIVBC GC 603 6053 VBCI‘Po) ‘(q’o I V3c GO 698 5053 POA GAGBC 39(1)3 x (Vfic GA (063 (08 VAB + VXB GC (96 (063 Vec)l‘1’o) +(‘Po IPA GA 5953 600C VABI‘POX‘PO I V30 GC 598 698 X GC (9'3 6053 V3C ITO»- (D6) (D7) (D3) (D9) (D10) (D11) (1)12) 181 or in matrix element form w3 = ZIdr'mdrVi Z '(o|1>3(r)| j)(j|1>[;‘(r')|o)1>035(r"')1>(§(r“)Pgi (r") Jim x(oIPf(rV)|m)(m|Pf(r‘V)|o)h3(Aj,Am) xTBY(r',r")T&(r"',riV)T¢;V(rv,rVi), (D13) where _ -2 -1 -1 -1 -1 —2 —l h3(Aj,Am)—Aj (Aj+Am) +AjAm(AJ-+Am) —Aj Am _2 —1 —l -l —1 —1 -2 +Am(Aj+Am) +Aj Am(Aj+Am) -Aj Am. (314) The W3 term vanishes because h3(AJ-, Am) = 0. W4 contains the remaining dispersion terms, W4 =(1+C){-<‘P6IPA GA to}? (08 x [Vie Gm”C Vie (GA (25' (.28 +GA$B (98mg +(Vl‘3c OMB“ VXB + VXB GB (16‘ 598 VEC)GB®C (26‘ VBCII‘Po) —(‘*'o I V3C GB®C 603 POA GAGBBEBC Vrgc (GA 5053 690C +GA®B $98 ) VA3 I‘I’OI -(‘P0 |(vBC G363C 503‘ 1>°A GA$BGBC v38 + vAB GA6313 33 x P°A GB 526‘ 506C Vficmm (06‘ V3c I‘I’o) +(‘Po IPA GA 6053 («"3 GA 50(1)3 695 VABIWOX‘PoIVBC GBEBC 608 V3c I LP0> +(‘Po IPA GA 8953 603 VAB I‘POX‘I’OIVBC GBEBC £08 GBEBC 500A V3c I “10»- (D15) 182 The matrix element form of Eq. (D15) is W4 = ZJ'dr'u-drVi - k2 'I(OIP: (r)|j)(j|Pé‘ (r')|0) J! rm, x[(0|1>53(r'") k)(k|Pf(rV‘)|m)(m|P,B(t-") 0) h4,(Aj,Ak,Am,A,) m)133? (r")hewI-AWI x(0|Pf(rv)|1)(I|P8C(riv)|O)Tfiy(r',r")T58(r'",riv)T¢x(rv,r"i), (D16) where the fact has been used that simultaneously interchanging the operators P8C (r”) and Pf (rv ), and P? (r’") and P? (r“) leaves the result unchanged, and h4a(Aj,Ak,Am,A,)= A}‘(Aj +Ak +A,)’1(Aj+Am)‘1+A]1(Ak +A,)“A;,} +(Aj+Ak +A,)“(Ak +A,)"(Aj+Am)" +(Ak +A,)‘1A:,}(Aj+Am)‘1 =2A‘J."A;,}(Ak +A,)", (1)17) h4b(Aj,Ak,Am,A,)=A}1(Aj +Ak +A,)“(Am +A,)'1 +(Aj+Ak +A,)“(Ak +A,)“(Am +A,)'1 = A;‘(Ak +A,)"(Am +A,)", (D18) and h4c(Aj,Ak,A,)= A;2(Aj +Ak +A,)‘1 —A}2(Ak +A,)'1 +A}‘(Aj + Ak +A,)“(Ak +A,)‘1 — A‘j'(Ak +A,)'2 =—A}‘(Ak +A,)‘2. (319) Appendix E The full expansion in Eq. (51) gives 33 terms of (VAB , VAB , VAC) type. These can be split into four sets X] — X4 , according to the types of the matrix elements appearing in each. X1 is given by X1=(1+C){—(‘I’oIP"GA 60535-98 leXB GA (063 :38 (VXB GA (063 (08 VAC + VXc GA (963 (98 VAB) +VXc GA (063 (08 V23 GA (063 (28 VABII‘I’o) —<% I VAB GA (063 (00C P°A GA (963 508 (VXB GA (063 (.28 VAC +VXC GA 6053 600C VA3)I‘I’0> (110 | vAC GA 505350813“ GA 5053 500C vXB GA 353 5.28 vAB |‘P0) +(‘I’oIPA GA (063 508 GA 506’ 500C VABI‘I’o) X (‘1’0 I(VAB GA 6053 398 VAC + VAC GA 5053 698 VA3)I‘1’0) +(‘I’oIPA GA (063 (08 GA 508508 VAC I‘I’o><‘1’oIVAl_>,GA (063508 VABI‘I’O) +0110 |1>A GA 5e? 3,0 VABI‘PO)(‘I’0 I(VAB GA (05’ (08 GA (063 (08 VAC +th G" 506’ 508 GA (06' 508 VAB)I‘Po) “LIWOIPA GA Qt? SOS VAC ILPOXWOIVAB GA 6063 503 GA 5.953 KJoC VABI‘i’o), (El) or in matrix element form, 0) x1 = zjdr'mdr"i z ' {[(0|P:(r)| j)(j|1>(;A (r')|l)(l|Py°A (r")|n)(n|Pg‘ (r'") J,l,n +<0IP:(r)Ij>II>,A (r”) 183 184 n) (anf‘ (r")|0> 0) n)(n|Pf‘ (r")IO) +<0IP:(r)|j),;’A (r')|n)(nIPf(r")Io)] AglAylAj,‘ ~21 12 (OII’6‘(-')li)(i|Pt’sA (r')I0><0|1’4A (r")|n> Ln +<0I1’6‘(r)ll'>(J'lPt‘sA (r"')IOXOIPt’i‘(1")ln)321(r”)Poi(r"i )(AfAi‘ + ATM?» x Tm(r',riV)Tm(r”,rv)T5€(r'", rvi ). (E2) The X2 terms are given by X2 = (1+C){—(‘P0|PA GA 5053 600C x (Vie GAaaB (08 Vie + Vie GA (2%? (.28 VXB)GB (06‘ (8 VABI‘Po) “(9’0 I VAB GB 508 508 POA GAQB 600C (V133 GA 59:13 (Joe VAC + VXC GAeBB 600C VAB)I To) —<‘I’oI(VAe GA (1338 1>°A GA ()5 (a8 Vie + VAB GA6913 (8 X POA G'WB 608 VXC )GB 600A 608 VABI‘Po», (E3) or in matrix element form x2 = 2Idr'u-dr“ Pg; (r") Z'{(o|1>:(r)|j)(j|P§(r')|n)(n|Pg‘ (r"')|o) j,n,k x[A;‘A;‘(A,, + Ak)‘1 + A;,‘A;J(Aj +Ak)“] +l<1IP(°A(r'")In>I0> +<0IPi(r'>I1> 0)] 135 x[A;‘A;,‘A;‘ +A;‘(Aj +Ak)'1(An + Ak)'1]} (<(o|1>{3(riv )|k)(k|3§(rv)|o)3&(r") xTm(r’,ri")Tm(r",r")T5€(r'",r“ ). (E4) Terms containing the permanent polarization of molecule C alone are grouped into X3: X3 = (1+ C) {-(‘I’o I(PA GA (063 (98 V23 GB (96‘ (98 + VXB GAQB (98 x P°A GB (06‘ (90C)(VXB GA (063 (08 VAC + VXc GAQB (08 VAB)I‘P0)] = 2Jdr'~~dr" Zk'<0lP£(-')IJ)|0><0|P(A(r">|n> 1,", x(0|Pi3(r“)Ik>(k|Pf(rV)I0)P&(r“) x[A3‘A;,‘A;‘ +A;1 11(An +Ak)‘1+A],1A](1(AJ-+Ak)‘l +A1‘(Aj +A..)“(A,. +At)"1 x Tm(r',riv)Tm(r",rv)T58(r"',rVi ). (35) X 4 contains the remaining dispersion terms, X4 = (1+C)(—<‘I’6|PAGA (263 (8 x [V2.3 GAGE (98 (V23 GA (063 (08 VAC + VXc GAEBB (soC VAB) + VXc G" (a?)3 (08 V23 GA6313 (08 VABII‘I’o) +0110 IPA GA 6053 595 GA 60g (of): VAC ITO) X10110 I VAB GB 633 500C VAB I To>+(‘P0IVA3 GAGBB 698 VAB I‘P0)] -(‘Po I VAB GAEBB (98 P°A GA$B (28 (VXB GA (963 (98 VAC +VXC GAeB $08: VA3)I‘P0> 186 —<‘P0 I VAB GB (93 £98 POA GAEBB 600C VXC GB 508 605 VAB I lPo) -(‘I’o I VAC GA (95’ (06C P°A GA (063 (08 V23 GA6913 (08 VABI‘PO) +0110 IPA GA 6053 600C VAC I‘llo) [(WOIVAB GB (JOA 60%)3 GB 698 895 VAB [1110) + (W0 IVAB GAeB 508 GAGBB 690C VAB IWO)” , (E6) or in matrix element form, X4 = ZJ‘dr'n-drVi ‘ IZ;{(O|P§‘ (r)|j)(j|P§ (r')|l)(l|P;A (r")|n)(n|P§(r"')|O) 1, (n, xA31A‘,,1(A, +Ak)‘1 0) +<0|Poi|1><1IP§A(r'")II>In> x A;‘A;'(An +Ak)" 0) n)(n| P5: (r'") +<0|P(§‘(r')I1>li><1IPt§’A (r’") n><1IP:A(r>II)In),§3(r")|o)1>(§(r"‘) x Tm(r',riv)Tm (r",r")r6,(r"',r"i ). (37) Adding X2 — Xund and X3 — X 3,ind to X4 and grouping the terms according to the types of the matrix elements yields X4 + X2 — x2,ind + X3 — x3,ind = Zjdr“ - 0dr“ Tm(r', riV)Tm (rv, r"')Tg,c (r"', r“) x Z'(oIPf(r‘V)|lt)(k|P,’,3(r")|o)POC,(rVi) j,l,n,k X[Y1+Y2+Y3+Y4+Y5+Y6+Y7+Y8], (E8) where Y1 = <0IP: (r)I1>A}‘A:.‘(A( + A )—1 n)(n|1>g‘(r"') mpg; (r")A]'A‘,,‘(An + Ak)‘1 —<0IP:(r)IJ> 40ng (">|i>P66(r')A3‘A:.‘(A,- +AI.)" +(o|1>;‘ (r)| j)(j|P5A (r'") o)P§B(t-')Pg; (r") [Ang‘k1 — A‘j'A‘kz +A;2(Aj+Ak)“] = (Oll’o’?(r)|1‘>ln> x A;‘A;,‘(A, +Ak)‘1, (39) 0) Y2 = (oil’s?(r>IJ>II> x A}‘(A, +Ak)"1(An +Ak)’l om, (r") —<0IP:(r)Ij>: (r)| j)(j| PgA (r"')|n)(n|P,g‘(r')|o)Pg; (r") xA‘j‘Aj,‘(An +Ak)"1, (310) 3 = (01?: (r)|j> x(Aj +Ak)’1(A, +Ak)‘1(An +Ak)“, (312) Y5 = (OIP€(r')ll>Ii><1IP3A(r>In> 0380") x A'JT'AI,1(AJ-+Ak)"(An +Ak)“(Aj +An +Ak) = (OlPé‘(r'>IJ>(1|P(°A|n>g‘(r')|j)(j|P§A(r)|n)(n|P§(r'") o)1>(§§(r") >g‘(r"') 1><1IP:A(r>II> n>I0> xA‘j‘A;‘(An +Ak)'1, (314) 189 0) >([A;2(An +Ak)‘1+A}‘(An +Ak)"2], (315) Y7 = —(o|1>:(r)|j)(j|1>g‘(r"') o)(o|P§(r')|n)(n|Pf(r") and Y3 =—Ij><0IP4‘(r">ln>Io> x A‘lej,1[(An + Ak)’1+(AJ-+ Ak)’1+(Aj+ A!)1 ((An +Ak)"1(Aj +An +Ak)]. (E16) Adding the second term of Eq. (E10) to Eq. (El 1), and adding the second term of Eq. (E13) to Eq. (E14) converts the matrix element (1| P6“ (r')| n) in Eqs. (E11) and (E14) to (1| PEA (r')| n). This transforms Eq. (E8) into Eq. (85) of Chapter VII. "IIIIIIIIIIIIIIIIIIII