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'r‘; ‘ {f}! 1% "‘v “A“. a" «‘21. . macaw. ’1)? n J7: .. a. fi...’ 1‘? ‘r” 4' . ‘x‘w 5 0L4!“ q, ‘f “£5. 2-“ k x“- I I M, W “'«n. a-us‘gt‘fi.‘ ‘31“. m, 1. “‘2'“ u Val f ‘\ \Q. I" \ V ”c. 04-“, . L, ‘31. (“‘r " 3‘? i .1 5:} f. '1 f rgr’; if}; {'5 fit ‘5' 3. ~ Elm “1523132: @«LQ-M. \ ‘13 ': “'31:: \. -. KAN-Q" - ..¢ . 3 t ‘r . 5% {an 34¢ i836: ;, - ,gflffgf I]: ‘ I tr ‘ (- .. ’r r.‘ Q .~ ,9; R." r)“ ., [if ( _‘.~r , f f 5" (at: 1 x , / 477,; .‘(4 "r; I 1y I - r .4 5m f .31! rap, “53:8! 1": I ,5 '3» This is to certify that the dissertation entitled FIELD—INDUCED FLUCTUATION CORRELATIONS 1 AND THE EFFECTS OF VAN DER WAALS INTERACTIONS 1 ON THE PROPERTIES OF PAIRS OF ATOMS AND MOLECULES AT LONG RANGE presented by James E. Bohr has been accepted towards fulfillment of the requirements for Ph.D . degree in Chemistry ; é Major professor. Date 2/27/87 “man“, ,. .. "1 m ~ . - - 0-12771 MSU LlBRARlES RETURNING MATERIALS: Place in book drop to remove this checkout from your record. Elflg§ will be charged if book is returned after the date stamped below. II- FIELD-INDUCED FLUCTUATION CORRELATIONS AND THE EFFECTS OF VAN DER WAALS INTERACTIONS ON THE PROPERTIES OF PAIRS OF ATOMS AND MOLECULES AT LONG RANGE By James E. Bohr A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Chemistry 1987 ABSTRACT FIELD-INDUCED FLUCTUATION CORRELATIONS AND THE EFFECTS OF VAN DER WAALS INTERACTIONS ON THE PROPERTIES OF PAIRS OF ATOMS AND MOLECULES AT LONG RANGE By James E . Bohr One contribution to pair properties during molecular collisions comes from the van der Waals interactions between the fluctuating charge distributions of the collision partners. Application of an external field to a molecular pair changes the van der Waals interaction energy in two ways. First, the field alters the response of each molecule to the nonuniform, fluctuating field of its neighbor. Second, the applied field induces new correlations between the fluctuating charge moments on each molecular center. Such field-induced fluctuation correlations have not been included in earlier models of the van der Waals contribution to pair dipoles and pair polarizabilities. Using a reaction field model that includes these effects, general equations are derived for the van der Waals dipole and polarizability of a molecular pair interacting at long range, where overlap and exchange effects are negligible. The van der Waals dipole and polarizability each depend on an imaginary-frequency integral consisting of two or more terms; each 33m H fir:- " "“‘jfl‘ m Fit: .. .i' ’H YE" term involves the product of either a linear or nonlinear response tensor for one molecule and a nonlinear response tensor for the neighboring molecule. Symmetry-adapted expressions for the van der Waals contribution to the dipole moments of some specific systems are derived in this model. In particular, the long-range van der Waals dipole of a pair of dissimilar atoms, an atom-centrosymmetric linear molecule pair, and a pair of centrosymmetric linear molecules are investigated. The leading contributions vary as R_7 in the intermolecular separation R and depend upon products of the dipole polarizability aa8(im) of one molecule with the dipole-quadrupole hyperpolarizability B (0,iw) of the other, integrated over a8,Y6 imaginary frequencies. Because the B tensor is not well known as a function of frequency, approximations are developed for the integrals in terms of the static aaB polarizability, the static BaB,Y6 hyperpolarizability, and the van der Waals energy coefficients C6 and C8 (both isotropic and anisotropic components for atom-molecule and molecule-molecule pairs). These approximations agree well with accurate perturbation results for the model systems H...HC (where HC is a hydrogen-like atom scaled by a factor a) and H...He. Applied to He...H2, He...N2, H2...H2, and N2...N2, leading van der Waals contribution to the dipole moment of each of the approximations provide the first direct results for the these systems. For some symmetry components of the long-range dipoles, van der Waals effects are greater than induction contributions; both need to be included in fitting collision-induced rototranslational spectra. . I .2: ETI'lfi-i'a- Eli-.1." Eff.- - .'.-" H . If'-— - .E - 91W15m&._m-.WWLIaaIfi-N-fiu I 10 650.33 mm 1 eaiuoelom nasal! sienna-teenage 'l'o «Esq s has .1lsq alumina «mil an: n; T'51 as 115v aflOIJIIdI'I-‘IHU'J- _nl.'.;-E".~l an? .bsnsaiaaavni 515 r—l’oai'fi mi; "in; a'i'mbc'iq roqu ban-get- '. 5- 2‘. r"i‘- "$qu ..—:.! -ebmrwrsJ ACKNOWLEDGMENTS I gratefully acknowledge receipt of a Graduate Research Fellowship from the National Science Foundation and a Continuing Doctoral Fellowship from the Michigan State University College of Natural Science. I am also grateful for a grant from the Foundation for Science and the Handicapped. I am very grateful for the guidance and direction of my research preceptor, Dr. Katharine Hunt. This dissertation depends greatly on numerous discussions we have had and the many helpful suggestions she has given me. I hope that I might approach her standard of excellence in my future endeavors. I acknowledge also the members of my Guidance Committee. Dr. James Harrison, who served as second reader, helped to resolve questions concerning the effects of temperature on van der Waals interactions. Serving also on the Guidance Committee were Dr. Daniel Nocera, Dr. Donald Farnum, Dr. Harold Eick, and Dr. Steven Tanis. I thank Ms. Debbie Robbins for her patience and expertise in typing this manuscript. I acknowledge the assistance of countless friends and associates, including the faculty and staff of the Michigan State University Department of Chemistry. The concern and support shown by everyone is very much appreciated. I thank Dr. John Eulenberg, Dr. George Leroi, Dr. Paul Hunt, Dr. Tom Atkinson, and Steve Blosser for their development and production of an adaptive computer input system. It will continue to be of value to me in future work. no:3sohhui'sd5vnbni Ffléii i 451 Tfliiviiifelii II I .IIHIIII’I‘iiQii .beauzo:bnsfl and has sonaroa 161 ,5- .o wish-xi: hns sown!“- '-:ia ".21 rJ'I‘S'I. 1S: ."es-‘r 2:5 I -* :-'~.-‘.-.~r -‘:-' ‘Il'hf-‘r' .- .21? .fir.Z-E "-'..'-'.:‘.".-.“. ."J' .- '.-e'_\'_1':.. :‘F’-3'J". . . . (L - 1 l . . I am very grateful for the love and support of my father and mother, Delbert and Lucille Bohr. You provided a stable environment in which to develop and grow, and were so very helpful during traumatic times. Know that I love you both and will always appreciate you. The warmth and companionship of a close family cannot be overestimated. Kris, Paul, Stacy and Bryan, Joe, Peg, Jamie and Danny, Andy, and Eric: You all have been there, in bad times as well as good times. Though distance may separate us, I hope we will stay near to each other in our hearts and in our thoughts. My father- and mother-in-law, Gerald and Colleen Francis, have also been very supportive. I am grateful to you, Linda, Kevin, Lance and Jane, Lisa, Jim, Bert, and Jerry, Jane and Doug, Gail, and Margaret (Grandma) McCarthy for accepting me into your family. I give special thanks to Laura, the woman who chose to marry me, for her constant love and understanding throughout the period of this work. You have bouyed me up during many trying times, and deserve as much credit as anyone for the successful completion of this dissertation. I will always be grateful for your support, and you can always be certain of my love. ' .:'. '_I., _-_-' ' 1-H"- ” aonnso umn «nib. a to “mama-o. In lid-11¢ g“; bflB eimfil- .891 .901: .nsv-as bns tease .LL-I‘I .enfl .hssulJIQ'l’vo 3“ ”1'” b5” n!.~1:fi* need sasfl [is ucv :9:": the .ank .Ynnsfi !. -- : wfi E .:H .‘-“539& gen -?“21:': 'g-J‘E "="’ .o:: -t -iau -. . -. - .-. . _ ._ . _ _ _ . _ '-."=. 15):. U: l." '-' i. ' u' n - .'."' ~' - .. "-n '4' ' a . hr' ‘ ' 1». Chapter 1. Chapter 2. Chapter 3. Chapter A. TABLE OF CONTENTS Introduction A. Foundations of Molecular Interactions B. Collision-induced Properties C. Overview of Dissertation The Effects of Field-induced Fluctuation Correlations A. van der Waals B. van der Waals C. van der Waals Collision-induced Specific Systems A. van der Waals B. van der Waals Interaction Energy Pair Dipole Pair Polarizability van der Waals Dipoles for Dipole for a Heteroatom Pair Dipole for an Atom and a Centrosymmetric Linear Molecule C. van der Waals Dipole for a Pair of Centrosymmetric Linear Molecules Application of a Constant-ratio Approximation A. van der Waals B. van der Waals C. van der Waals Dipoles of H...HC and H...He Dipoles of He...H and He...N 2 2 Dipoles of H2...H2 and N2...N2 vi Page 36 36 38 A8 6” 64 7O 81 'fl TAM-q 1" :x-z-aui.-:-.i~mn:!m.’.‘- .6 --.H- -- ;-- 1r Chapter 5. Appendix A. Appendix B. Appendix C. Appendix D. Discussion and Conclusions Equivalence of Reaction Field and Perturbation Results for Long-range Dipole Moment of Two S State Atoms Frequency-dependent B and E Tensors for Linear Molecules Expanding a Function in Spherical Harmonics Clebsch-Gordan Coefficients 91 95 97 112 117 Table 1.1 Table 4.1 Table “.2 Table 4.3 1 9 .m" LIST OF TABLES Page Units of Measurement 13 The long-range dipole coefficients D7 for a hydrogen atom interacting with a hydrogen-like atom scaled by a factor a. The system is n + _ polarized H HC . 69 Molecular properties and van der Waals energy coefficients used in calculating DAL for the He...l-l2 system. The internuclear distance in the H2 molecule is held fixed at its equilibrium value of 1.4 a.u. All values are given in atomic units. 79 Molecular properties and van der Waals energy coefficients used in calculating DAL for the He...N2 system. The internuclear distance in the N2 molecule is held fixed at its vibrationally averaged value of 2.07 a.u. All values are given in atomic units. 80 viii l 1! I'. P E F. . *’.I' 5-4.- -w a . I!” ~J‘--- . “91 éG-ifiéilwim‘eMnW-m mum. ' " 5: ain-nsfiodbgu 5 Asia fini395fi93nl moss upgoqbtn Hi msdeta an? .3 ~n3nsfi s to baits: moss ‘ 9° .' * “ ~ Eifinzr; Table A.A Table “.5 Table A.6 Molecular properties and van der Waals energy coefficients used in calculating D for A1A2AL the H2...H2 system. The internuclear distance in each H2 molecule is held fixed at its ground-state vibrationally averaged value of 1.AA9 a.u. All values are given in atomic units. 87 Comparison of van der Waals and induction contributions to the D coefficients for A1A2AL the H2...H2 system. The internuclear distance in each H2 molecule is held fixed at its vibrationally average value of 1.4A9 a.u. All values are given in atomic units. 88 Molecular properties and van der Waals energy coefficients used in calculating DA1A2AL for the N2...N2 system. The internuclear distance in each N2 molecule is held fixed at its vibrationally averaged value of 2.07 a.u. All values are given in atomic units. 89 II :olauuhni his = -?‘ r ’r- is 'oei~nnmnfi 8.J aldsT airsifii‘ --' - . 'JL '- . Table A.7 Table D.1 Comparison of van der Waals and induction contributions to the DA1A2AL coefficients for the N2...N2 system. The internuclear distance in each N2 molecule is held fixed at its vibrationally averaged value of 2.07 a.u. All values are given in atomic units. 90 Values of Clebsch-Gordan coefficients 119 .‘oflh ii .u.- i'o.s 13 sufisi bean-vs vilmlw' '.'__ "’ :9 .eflnu claims :11 newts an souls: III. '~.- a. . _.__.. i." “:'-.. CHAPTER 1. INTRODUCTION A. Foundations .of Molecular Interactions Forces of attraction and repulsion are known to exist between molecular systems. These stem from electromagnetic interactions and, at small intermolecular separations, electron exchange effects. Although gravitational forces are also present, these are extremely weak and may be ignored. For the purposes of this dissertation, magnetic effects will be neglected, and the focus will be entirely on the electrical forces of interaction between molecules. It is useful to separate the electrical forces of interaction into short-range and long-range contributions. When the distance- between centers is small, overlap of the electronic wavefunctions of the individual molecules is significant. The interaction force in this region may be either attractive or repulsive, but at very small separations the force becomes entirely repulsive and behaves exponentially. Long-range forces vary as R_n in the separation R, where n is a positive integer. Electron exchange is negligible at long range, so for the purposes of numerical calculation the electrons may be assigned exclusively to one or the other of the interacting molecules. This means that for calculation purposes the total system wavefunction does not need to be antisymmetrized with respect to exchange of electrons, and a perturbation scheme may be employed in which the unperturbed wavefunction is a simple product of the wavefunctions of the isolated molecules. At long range then, the forces between a pair of molecules can be related to the I'a. . ,,.. anolfizswssni sifonSF“°1359:° “911 993‘ 9""? .ealflltl 1.1”5.‘.. .E-ag «since-anemia} 118%: :fi. ,bns . q q qfinfifl_p noqioeis .ecct3= .' ..-- _--. lg!) : '.._ -- _ .". ..' ‘-'. flan": -.-:.l.l- _--"_1 :' "i. . . “’ a . 1... .VI '3" 155. {‘11:}.- ..-._ . 1.‘ E?- _I l I 4'3 'v'i-t' '- ' $.13“! 2 properties of the individual molecules. At relating short-range interactions to properties, because in this region electron and the total wavefunction must be fully range. present no theory exists individual molecular exchange is significant, antisymmetrized at short The long-range interaction energy of a molecular pair can be divided further into several contributing types. When the free (0) (0) molecules are in states wn and T% gives as the energy of the pair , standard perturbation theory AB = E(o) + E(o) + ”1 n2 “1 11T2 I<¢;:)¢ ¢(é)iH'iw(O)¢ (o)> >l2 _ z! + . . . (l-l) (3(0) 0)) + (E(o o) _ E(o)) J1 1 J2 “ where H' is the perturbed part of the Hamiltonian and where 2' (0) indicates a summation over all states w. The first-order term in Eq. (1.1) is the electrostatic energy AE except wn n 2 e1 w(o) (o) W(O)- J2n1 and results from the interaction between the permanent electric moments (charge, dipole, quadrupole, etc.) of both molecules. While it is true that the charge distribution of each molecule is modified by the presence of the other, this is get included in the electrostatic energy; only the permanent moments of the free The second-order term in Eq. (1.1) molecules contribute to AE includes both the induction energy and the dispersion or van der 3 Waals energy. The induction energy AEind arises from the interaction between the induced electric moments of each molecule and the permanent moments of its partner. It is produced from the matrix elements of H' that are diagonal in n or n 1 2’ (o) ¢(O)IH' (o) w(o)> >|2 _ d 1|2 _ . 11.2) J2;“2 (ESQ) — E(O)) J2 “2 Both the electrostatic and induction effects can be explained through the use of classical electrodynamics. The remainder of the second-order term in Eq. (1.1) constitutes the van der Waals energy vdW_ AE o o o 2 ll n J J AEvdW = _ g 1 2 (1 3) J1n1 (E(o)_ 3(0)) + (E310) - E(o)) #n J1 n2 J2 2 AEde results from the correlation in the fluctuating charge distributions of the interacting molecules. It is a purely quantum mechanical effect: A region of space at a particular temperature T is permeated with a quantized radiation field characteristic of that temperature. This photon field will interact with any molecule that is present, inducing instantaneous multipoles in the molecule. The h instantaneous multipoles arise from electron-photon interactions in which excited electronic states are mixed into the ground electronic state, with simultaneous changes in the state of the photon field. For sufficiently low temperatures, the photon field can be considered to be in a vacuum state in which virtual photons may be produced; for atoms and molecules, the electron-photon interaction at room temperature is essentially the same as the interaction at zero temperature. In other words, at ordinary temperatures the van der Waals interaction between molecular systems results when the electrons interact with virtual photons that are produced from the vacuum state; effects of the real photons are negligible, because their energy is too low to induce electronic transitions. Using second-order quantum-mechanical perturbation theory, London was able to describe the attractive long-range van der Waals force between two atoms, showing that it arises from a correlation in the fluctuations of the electronic coordinates [1,2]. He based his theory on the assumption that an electron in a particular atom perceives the instantaneous rather than the average position of the electrons in a neighboring atom. Modeling the neighboring atom as an instantaneous dipole, London solved the Schrodinger equation to second order in the perturbation, expressing the energy reduction caused by two-electron correlations in terms of one-electron excitations. The exact theoretical treatment of van der Waals interactions requires the quantization of both matter and electromagnetic fields. The coupled electron-photon system should thus be treated using quantum electrodynamics. Then the total Hamiltonian is comprised of " ' . ' ' . -' - - - " r: -‘ ':.":r.;-:..'-":m: an electron contribution He a photon contribution H and an 1! phi electron-photon interaction Hin The van der Waals energy between t' two molecules becomes a fourth-order perturbation, due to the interaction of two electrons with two photons (two electron-photon interactions on each center). Quantum electrodynamics gives directly the proper effects of retardation on the van der Waals pair energy at very large separations. The field of a fluctuating multipole requires a finite propagation time R/c (where R is the distance between molecules and c is the speed of light) before it reaches and polarizes a neighboring molecule. At very long range this propagation time is nonnegligible; this leads to weaker correlations and a smaller energy change. Both the retarded and nonretarded behaviors arise naturally from a quantum electrodynamic treatment of interacting molecules. The nonrelativistic London formalism assumes a static Coulomb interaction potential between the electrons. It does not account for field propagation, and therefore cannot explain retardation effects at very large separations. Retardation can be incorporated by using time-dependent perturbation theory. Casimir and Polder used this approach in their investigation of retardation [3]. They introduced the randomly fluctuating vector potential of the electromagnetic radiation into the Schrodinger equation and calculated the energy of interaction between two atoms to fourth order in the perturbation. The interaction energy was found to vary as R—7 for very large separations R, instead of following the usual R-6 behavior. . .1. II__I ‘1. ." e: I- "'1'-"_. '. ',_.'.'. ' ' ' ' " -..\'i 'mummnwgaubmmw Jam ‘0."- nuue ,_ 135g slant web 115': an: nonunion-.11 to use“. «ago-IQ ’“.M5 a'Hrquui'i s '1'.- hlsi'! 911'” .an:_.°s-Is'1=: 9211.1. new :5 131cm. . 9:1: .91 i ~194in mi. -.':-2: n:- :-':-'..' r" 63.1126: evil-m" alcqnim .- - rune-1. -" -. ' " ' -' . ': —-'- .' '=-.;- 3-". ' ”J l l _.u Quantum electrodynamics replaces electrons and photons by quasi-particles. Following an analysis by Langbein [73], if one assumes the electron system to be essentially in its ground state and averaged over all instantaneous excitations, electric and magnetic susceptibilities may be defined and Maxwell's equations retrieved. This assumption not only reduces the quantum theoretical effort but also allows the van der Waals energy to be expressed in terms of the susceptibilities of the interacting molecules. Assuming instead that the photon system is essentially in its ground state, the electric and magnetic interaction potentials may be introduced and Schrodinger's equation for the interacting molecules retrieved. In this dissertation, the first step away from the quantum electrodynamic procedure is taken, but not the second. The electrons are assumed to be essentially in the ground state; fluctuating multipoles arise from the absorption and emission of photons by the electrons. It is the correlation of these fluctuating multipoles that gives rise to the van der Waals interaction between molecules. The quantized photon field is not treated explicity here, but it should be understood that the fluctuations in the electronic charge distributions are due to electron-photon interactions. How are the fluctuating multipoles correlated? The parameters that couple photons and molecules are the molecular susceptibilities. The susceptibilities are complex functions whose real parts describe the polarization of the molecule, and whose imaginary parts describe the energy dissipation from the molecule to '9: .1“ . . 7 the photon field and from the photon field to the molecule. The fluctuation-dissipation theorem as derived by Callen and Welton [A] states that the correlation function of the fluctuating multipoles is proportional to the imaginary part of the susceptibility. The model for molecular interactions that is formulated in this dissertation is essentially a reaction field model based on the concept of fluctuating molecular charge distributions. These fluctuations are due to the interaction of electrons with a photon field, and give rise to instantaneous multipole moments. The instantaneous moments of one molecule set up a field and field gradients which propagate to a second molecule, inducing multipoles in it. These induced moments produce a reaction field and reaction field gradients that act back at the first molecule, lowering its energy. This energy change is determined by taking a time average over the coupled fluctuations, as computed by use of the fluctuation-dissipation theorem. The same scenario holds for the second molecule as well, and the total van der Waals energy for the pair is obtained by adding the energy reductions for each molecule. Treatments of the van der Waals interaction energy which employ the reaction field model may be found in Refs. [71-73]. The intermolecular separation, though large enough that overlap and exchange can be ignored, is assumed to be small compared to the characteristic wavelengths of the radiation associated with the fields. Under these conditions a multipole expansion in the fluctuating and induced moments can be used in describing the fields and field gradients. so: no beesd Eaton biev1 no 12. ' '1'». :,:‘.'..-l.-.-..-'.-. ‘-- 1:. " '-:I'-" _.;1r.tv--c'_"..1a.i’:- 91:- :3 115.1- 91' aka: n1.bezsiumn03 e1 JEN‘ enclasefliihl istbatihl id! ICIII ..-. yuan-in. u «mum-ah r 3n15ouailfi 1o Jqsonoo ,h ." :" :-‘.L".C)_’ .‘MJJ'JUI'I B. Collision-induced Properties Collisions between molecules in compressed gases and liquids cause distortions in the charge distributions of the.colliding partners, giving rise to changes in molecular dipole moments [5]. These transient dipoles are responsible for the collision induced absorption and emission spectra of such nondipolar species as H2 [6], N2 [6—9], 02 [6-8], CH“ [10], and SF6 [11,12], and the absorption by mixtures of inert gases [13,14], H with He [15-18], 2 and H2 with N2 [19]. The spectra can provide information on the pair and cluster dynamics of these systems, if the interaction- induced dipoles are known as functions of intermolecular separation and relative orientation. High-resolution, gas-phase spectroscopic measurements have recently become accurate enough [16-18] that it is necessary to include the van der Waals contribution to collision- induced dipoles in order that theoretical and experimental results can reach agreement. The polarizabilities of molecules in compressed gases and liquids are also affected by intermolecular interactions, as evidenced by the dielectric and optical properties of bulk samples. Dielectric virial coefficients [20—2A], virial coefficients for the DC Kerr effect [25,26], birefringent response of fluids on the subpicosecond time scale [27,28], and intensities of collision- induced Rayleigh and Raman light scattering [29—41] all depend on the transient changes in polarizabilities that occur when molecules collide. Calculations of these changes are needed to evaluate local field factors or effective polarizabilities of molecules in dense hcsubn! noiai:?co an: 101 oidranoqeswm Q' . -. ‘ _. .te] 3.1m slams 131-9591011!‘ mt cm 1: an mi. .0 F a. 2;iceqt wsjtqltnan ”spa in maoioqib claim “If I s1°csce noiactma bns noiiqfioads ‘6' .K‘a-j H eta] S ‘T :5 ncZJnefiedr 9 media [AZ-45]. Also, information about intermolecular dynamics in liquids can be obtained from line shape analyses of light scattering spectra, if the polarizabilities of the interacting molecules are known [46]. When the separation between a pair of molecules is large enough that overlap and exchange effects are negligible, the collision- induced dipole and polarizability come entirely from classical polarization (induction effects) and from van der Waals interactions. Previous reaction field models have attributed the van der Waals contribution to polarizabilities [N7-51] and dipole moments [52—5A] to hyperpolarization of each molecule by the field and field gradients arising from the fluctuating charge distribution of the other. The van der Waals dipole has also been evaluated by considering the changes in the reaction field at one molecule due to the application of an external field to the second molecule [55]. These changes result from the nonlinear polarization of the second molecule by the simultaneous action of the external field and the nonuniform field due to the fluctuating charge distribution of the first molecule. Hunt and Bohr have shown that an additional physical effect contributes to van der Waals dipoles [56] and polarizabilities [57]. The external field not only combines with the fluctuating molecular field to produce nonlinear polarization, it also alters the correlations between the fluctuating moments of each molecule taken singly. For example, application of an external field to a centrosymmetric molecule introduces correlations between the fluctuating dipole and quadrupole moments that are absent in the ‘- . '-. " .. iwwhfi‘flil‘rm: iL} P . I- r “L _ p15 13‘. -.'..- -_~_ -- 5;. -- k 1‘ “r :1E245321e 'oq b; a szuqib Deanna: ; . . *Jur: nar'rr‘w'luq .-..,I.': " .e:1--' ‘-’. .' 'i-z'fli . l . lO unperturbed molecule. Such field-induced fluctuation correlations were not included in earlier models, though the effects are implicitly present in calculations of the van der Waals contribution to dipole moments using two-center, third-order perturbation theory [58-60] and to polarizabilities using two-center, fourth-order perturbation theory [61,62]. - This dissertation will focus on the long-range van der Waals contribution to dipoles and polarizabilities as formulated in a reaction field model which includes field-induced fluctuation correlations. The van der Waals interaction energy for two nonoverlapping molecules in the presence of a uniform, static external field is derived and then differentiated once with respect to the field to determine the van der Waals dipole, and differentiated twice to give the van der Waals polarizability. At short range, overlap and exchange contributions to pair dipoles and polarizabilities are significant [51,63-68], and the van der Waals contribution is damped by overlap [69,70]. The present model does not include these effects. The long-range induction and van der Waals contributions to dipoles and polarizabilities are additive to second order in the molecular interaction. To a first approximation, the van der Waals and overlap/exchange contributions to pair dipoles and polarizabilities may also be treated as additive. a r5 .:.-1-'.-:'; r' 2-“. Mao-mm} mam-ow 15m manmum.«- Id. .50.») won: «013.3%! 32351! "1:95: f-E" agar-Mane: 91.1 no 51-5-51 Ilia meanness.” ain't .-.--.-.. :-.: a ‘11- '2- c! -:i.3ur::‘1:'noo -_.' r9" ll C. Overview of Dissertation Chapter 2 of the dissertation is devoted to the derivation of general equations for the van der Waals contribution to the long- range energy, dipole moment, and polarizability of a pair of molecules. In Chapter 3, the collision-induced dipole moment is examined in detail for a number of different interacting systems. For a pair of dissimilar S state atoms, the only source of a long- range dipole is the van der Waals interaction, giving a dipole which varies as R.7 in the interatomic separation R. When an atom interacts with a diatomic molecule, or when two diatomics interact, there is an induction contribution to the collision-induced dipole as well. Induction and van der Waals interactions are both studied' in Chapter 3 for these systems. The resulting equations for the van der Waals contribution to the dipoles of all of these pairs depend on integrals over imaginary frequencies of products of certain susceptibility tensors for the individual atoms or molecules. Since these tensors are not readily known functions of frequency, an approximation technique is needed to find numerical results for real systems. A constant-ratio approximation is used in Chapter A to relate these integrals to static susceptibilities and the van der Waals energy coefficients Cn' Then, values are found for the collision-induced van der Waals dipole moment for the pairs H...H , C H...He, He...H He...N2, H ...H and N ...N For the latter four 2’ 2 2 2 2' systems, the van der Waals effect is compared with the induction contribution to the total dipole. Finally, Chapter 5 provides a discussion of results with recommendations for further work. =1 enema aindaéi ' P I: 5.1. 5": r lb benubLi-rsicITIca In: ,E 103qsd$ n! a no! «5106* inla- . m-aul- DIOR" I o'm‘M .— . "twin! 1:539: n: bonzllxo “» filaJ n 10! ”fl; ..‘". .a.:- ." '5. 9mm ..iv OJ! (a.u. ) "Inn ;gir'sin211fies____ oompujitignal work, i__1'.__ is 3931;- plunge” “W... :'.._..:....-.= ”'4 ::- mania 1:4 pravidas vaiuésrmuztyaien-fi'fi. i ' atomic units for‘the SI system (if-measurement and for other~comon1g i l used units. , .iii '. iii- . ='- s ..w “h“aiflmfl was nan-Id --..- _ , 'F:fi ‘ 901011.: -o site an: ion El :1 .al'lw tantalum mam-u :' 0: main spa asuzev active-'11: V.’ 951.151 statistic-110m 101- ,_ ,. «9-131— -.:‘ t..-. '51:“. 'masar' ‘1‘. 121332.? '3 9.1.1 to? sum; aim:- .e..ir._- been Energy Charge Dipole Quadrupole 1 1 hartree . ' [7" £1 '1 31- . . " “0.529177 3 5.A8580 x 10'” unified amu (u)_ 219.u7u.5u cm" ~627.5 kcal/mol n.803zu x 10‘10 statcoul ~2.5H Debye (D) 10.5%??? 940719.11. 9.10953 x 10'31kg 1.359828 x 10‘1 ‘2.6 x 106 J/mol 9c ~1.6 x 10‘1 ~8.us x 10‘3OCm ~n.5 x 1o‘"° OmZ 8J CHAPTER 2. THE EFFECTS OF FIELD-INDUCED FLUCTUATION CORRELATIONS As discussed in the Introduction, pair properties are induced and/or changed during collisions between molecules. When the intermolecular separation is large enough that overlap and exchange effects can be neglected, these collision-induced properties result entirely from classical polarization and van der Waals interactions. In previous models of pair dipoles [52-55] and pair polarizabilities [A7-51], the van der Waals contribution has been attributed to the hyperpolarization of each molecule by an applied field and by the field and field gradients due to the fluctuating charge distributions of the other molecule. In this chapter field-induced fluctuation correlations are shown to have an effect on van der Waals pair properties. These effects have not been included in earlier models. The derivations which follow are carried out under the assumption that the interacting molecules are well separated, so that overlap and exchange effects are negligible. This means that a multipole expansion can be used to characterize the molecular electric fields. Retardation effects are not considered in this work; the molecules are sufficiently close together that the signal propagation between them is essentially instantaneous. IA . n 3-,” _ '-a We» knew-1m pan-II: an. ..ynz u «annular some: “use?! sen-yiqowq bseuuni-notsui-n 0.39.1.1 Juana-n 96 use mm - .4‘ : .anaueaasflnl Lia-5‘11 ~:-t rev 3:1:- 119. 3.1:. :sit: harass-Tn 1.10.. ..ewfla. A " ' '1‘- Z ' V " ' r..""’" ‘ "- -"'.|"|q 1:1 :15." 55.. a “u' n“: _ . ._ _ , - - g __ ’_-:11 ‘ «r- ’0- ’ . l" 1 15 A. van der Waals Interaction Energy In this section, an expression for the van der Waals interaction energy AEAB of two molecules A and B in the presence of a uniform, static applied electric field E will be developed. This interaction energy is attributable to correlations of the fluctuating charge distributions of the two molecules. The fluctuating field of one molecule polarizes the other; this induced polarization gives rise to a reaction field which acts back on the first molecule, resulting in an energy shift for the system. The effect of the external field is incorporated in the susceptibility tensors of each molecule. As the theory is developed and the dependence of these tensors on the external field is explicitly determined, the manner in which the field induces correlations in the molecular charge fluctuations will become apparent. The fluctuating charge distribution of molecule A induces time dependent moments in molecule B. When the separation between the molecules is sufficiently large that the multipole expansion is valid, the w-frequency component of the dipole moment induced in B by A (in the presence of the static applied field E) is [74-76] 1 (w) = fly...) {Bun + g Eq..) i B B ' BIND (w) H“) + %§ EB(E,w) E E"B(w) + (i) g'B<§,w> @301.) + (—> sz<§,w> ; Ev where §B(w) is the w-frequency component of the field at B due to the fluctuating multipoles of A, E'B(w) is the field gradient, and rfluflarfg' “ 1'. —_.'.I It _ ‘1“ - ...h.¢§IWQb:-_Sd [111-1 1 #1311 yhmz. ”:m'ol3m '- ' _ . 5’”: '5'1""5£'35' ""4 35-3 ’3: :- .'..":r13'1:r:h and-r13 Rhiannon "3.“.- 9.1-4 1v eroitsiz 1130 0.1 efflefudl'itn e1 mum norms-{3'21“} v _'|- -_ ._ . ,' . -..q . . ..- . . . - - - 5‘ '31.. ch“: . 1:213: ='l- .‘. ..a" .-- ' -' .' '-' .'. .5 F"? :51. ' 5330?:1 _ _ . | r . _ v ' - G I I' _. 16 B H (w) is the gradient of the field gradient. §B(w) is the w- ""1 frequency component of the time derivative of the magnetic field at B due to A, and E'B(w) is the gradient of BB(w). The susceptibility tensor gB(§,w) is the dipole polarizability of B in the presence of the static applied field E, AB(§,m) is the dipole-quadrupole polarizability, and §B(§,w) is the dipole-octopole polarizability. g'B(§,w) is a gyration polarizability that determines the optical rotation of an isotropic medium [7H]. Similarly, the quadrupole induced in B by the fluctuating charge distribution of A is =8 B B(§.w> 3 3%) H 2 U | CU IIO — (5 g'B<§,w) ~§ (w) + . (2.2) In this equation gB(§,w) is the quadrupole polarizability of B in the presence of the static applied field E, The octopole induced in B by the fluctuating charge distribution of A is B B 8 2mm) - :2 (EM) -: (w) + . (2.3) Provided that the intermolecular separation is small compared to the characteristic wavelengths of the radiation associated with {8(m), the field and field gradients at B are related to the A fluctuating dipole EQL’ fluctuating quadrupole 2FL’ and fluctuating octopole 2A of molecule A by FL 5 (w) = 3mg) Liam) + 33- 3(3)”) 39:} (w) 1 (LI) - A + T? I (3) : QFL(w) + "' , (2.4) 13%) = — T(3)(R) 1315..) —l3g('”<3) ngLhn - . (2.5) and 9%) = 3“”(3) 125..) + . (2.6) In Eqs. (2.4) through (2.6), 3 is the vector from an origin in molecule B to an origin in molecule A; the propagator tensors are (2) _ TaB (5) — Va Vs (a ), T(3)(R)=V v v (R ), dBY —- a B Y and (u) _ —1 TaBY6® ‘ Va VB VY V5 (R ) ’ where Greek subscripts designate the vector or tensor components x, y, and 2. At this level of approximation, §B(w) and its gradient vanish, simplifying Eqs. (2.1) and (2.2). The moments induced in molecule B by molecule A give rise to a reaction field which acts on A: l8 1 (u) . B - oo- +—53 (5) ; 2mm.) , (2.7) a reaction field gradient: ,A _ (3) . _ _ l.(M) ; .,, E (w) — Z (5) ”IND(m) (5) g BND(w) + , (2.8) and a gradient of the reaction field gradient: M B (m) = g‘ )(5) . Emu”) + . (2.9) The resulting change in the energy of molecule A is found by averaging the instantaneous energy shift over the configurations of the charge distribution of A: A 1 A A - ,A AB = - —2— < uFL(t) ' E (t) >3 l6< 9F LUZ) 'E (t) )5 1 A - "A — 30 < gFL(t) :g (t) >s - ~~- , (2.10) where the subscript s on the angular brackets denotes the average of the symmetrized contracted product of the terms within. Substituting frequency Fourier representations for EA(t), F'A(t), §"A(t), £§L(t)’ g:L(t) and g:L(t), and using Eqs. (2.1) - (2.9) gives the shift in energy as a sum of terms AB = E AE: (2.11) 19 where AEfi varies as R-n in the intermolecular separation. The contributions through order R_8 are A32 = — %.f° dw If” dw' exp ['i(w + w')t] <5§L(w)-g‘2)(5)-[5B(5.w) + gB(5,wv)J-g(2)(5)-5§L(w')> , (2.12) AF? = ‘ % If. a.) If. dw' exp t—irw + w'm . [<£:L(w)'l(2)(§)'[g8(fi,w) + B(5, w] 5(3)(5)3A gF Lw )>s (3 OJ "9 — <5§L(w)-5(2)(5)-[5B(5,w) + A (F,u) )Jog W5) “W )> s], (2.13) (3) (11) and AEg=—gfm dwf dw' exp[1(w+u))t] x [% <5§L(m)-5(3)(5>3[gB(_F_,w) + (B (F _,u) ')l 135‘ )(5)-5§L(wv)> + s A 11 (Fun) + 5 B(_F_.w')]-'£ (5);c;),f.‘L(m')>s > A E V "a A m ”T, x v 0 fl u> a A Im a V + ll3> w (5.m')]' ||’—] A I (5);5FL((» )>s 1)) Cl> A N v U) A ((3)-5 (5mg (5,...) + gB(§,w')] . ""3 - A ' (5);gFL(w )>S A W v U3 A «3" (w )1; (5mg (5...) + gB(5,w')iog 3)(5);9‘F‘L(mv)>]. (2.1a) CAI—a ml—a w]—-w|—- ml—a A /\ IT: '11 r In Eqs. (2.13) and (2.1”), the subscript 5 denotes the symmetrized contracted tensor product. For example, (”(5) [9B (F, (u) + q B(F, u) ')l 'r‘3)(5)1A (3F Lw )>s AW»). 1(2)(5) [gB (F,w) + a B(F, u) '-I)] (”F (3) (2) 4. <0; Lm:(')T (R) u: L(m)>. (2.15) MIA mid (fl)'[gB (53w) + g B(E,w')]°: a. - (F . .-'_ -'.' . 'ST. A. ':-..),.,§"('§_) )_-'1-".r..3' ff " ('.._"' :_‘\i) l '(J) ._.':_.'_‘I} 1- , . . , _. A is. I . TI. . -. ‘ _ ‘ '-".‘. _ ' 'I; {"1“ 121‘ i LIB. -\ | . 20 If the applied field is absent, and if molecule B is centrosymmetric, then §B(§=O,w) = O and the second term of Eq. (2.13) will vanish. If molecule A is centrosymmetric, the first term of Eq. (2.13) also vanishes in the absence of an applied field, because the fluctuating dipole moment of A is then uncorrelated with its fluctuating quadrupole moment. If A and B are 8 state atoms, then in the absence of the applied field the only nonzero terms of Eq. (2.1M) are the first and sixth terms. This is due to the fact that for isotropic systems both §(E=0,m) and A(§=O,w) vanish, eliminating terms two through four, and also because octopolar fluctuations are uncorrelated with dipolar fluctuations for isotropic systems, eliminating the fifth term. Application of the external field E distorts the charge distributions of molecules A and B, introducing additional interaction effects. One effect arises from changes in the response of molecule B to the nonuniform local field of A. For example, the applied field alters the susceptibility tensors of B such that AB(§,m) is no longer equal to zero for a centrosymmetric molecule B. Thus the field gradient from the fluctuating dipole of A induces a dipole in B, and the field from the fluctuating dipole of A induces a quadrupole in B. Both of these induced moments produce reaction 7 (see fields at A, with resulting energy shifts proportional to R— the second term of Eq. (2.13)). Earlier models of the van der Waals contributions to collision-induced dipoles [52-55] and polarizabilities [N7-51] have included these types of effects as a W46.) um husky-mom mm.) a: 1 310 am album yawn Hi. Li. Jute-std moqmbwp gunman =13 --'_e 4-. .LS “LI '1: ram .:-..F:.‘a 53"‘21 945 a has A 11 25-1)“ -,|' -- ..‘i ’:-I .,.‘T"= ';' ?-'.-'-. = '3‘ '.-_"_'.:- 9d: 8151‘; f‘9lqu5 .‘J.’ "‘ ‘.')'3 ’. A 1.," 21 net hyperpolarization of molecule B by the simultaneous action of the applied field E and the fluctuating local field of molecule A. In addition, the applied field modifies the correlations between the spontaneous quantum mechanical charge fluctuations on each center. By the fluctuation-dissipation theorem [H.771 % A225 (F ”HCOBMETT] (2'23) for the van der Waals energy shift of the AB pair through order R-8. AB The permutation operator P interchanges the molecule labels A and B, while leaving the sign of B unchanged. AEAB should be computed as the Cauchy principal value of the integrals in Eq. (2.23), since the field-dependent susceptibilities vanish at m=0, and zero-frequency fluctuations cannot contribute to the van der Waals energy shift. The poles of the susceptibilities g(F ,w), A(F, m), C (E, w) and E(F,m) all lie in the lower half- plane in w, by causality [73,77]. Therefore the integrands in Eq. (2.23) are as . - 5? .31 .35)! M" eis'xaaani 911.1 0:! 93.182130! ”Joli! ' .[ET] '..I. " 4 I. .. F“ . r-" 5 ..- 4Fi.ufi .8 ainfrlnm ?o :5 9755a [assay on: anlhbl . i“ . I... . f ' . ‘ ‘ ‘.h5. :7 '_f : “ 2- ‘ . ., (H-i(u’T [n3 (fiJBS’T I _ _ -. a —. I ._ ._ -'-1. .5 _ 5;-.. :.--a . Jun; 2t;‘)n ., -'. ... A . ‘LBE: [ITOSQES'I fl. 2h analytic functions of w in the upper half-plane, except at the poles of cothfng] lying on the imaginary axis at wn = 2winkT/h. Integrating along a contour running from -R to -e on the negative real axis, around a small semicircle r = e exp(ie) in the upper half-plane to +5, along the positive real axis to +R, and then around a large semicircle r = R exp(ie) yields in the limits R + m and e + O AEAB = — kT 1:;1 T5525)(5) aBY (5.1.1n ) TT2)(5) a§5(5.a5) + 3§—T (1 — FAB) 2' :T2)(5) A5 Y5(5,an ) T835) aTT5 (5.1.15) — 5;: (1 + FAB) :1 [T FT§;(5) c5Y 55(5,n (1) ) F535(5) a55(5.n ,w) + 3TT2)(5) E5 Y55(5.an ) TT5)5(5)A A5a5 (5.1.15) -: T3535) A5Y 5(5,n u) ) T535(5)TT A 555 5(5,a5) -%T TT2)(5) A5 Y5(5,an) TT5;5(5) A 2555(5.a5)] , (2.211) where the sum over n runs from O to m, with the prime on the summation indicating that the n = 0 term is multiplied by % The sum in Eq. (2.2”) can be used directly to compute the van der Waals energies of interacting rigid rotors or oscillators. If the spacing between the poles wn is small relative to the frequency range over which the susceptibilities change appreciably for imaginary m, then the infinite summations in the above equation can be converted to integrals [73]. This requirement is fulfilled for atomic and molecular systems at ordinary temperatures, and Eq. (2.2“) becomes A — '. I. " . "' ,' :. V 7- ' i " l - . _ 5: . :5' _‘. . ,'_ ' , i " ‘ . _ . . -. 5 I r I: . . I : " I I Ia. a. * ' . ” — -" =- '. .:.. r3... nuts bus .910 o: m: is” 99mm .6199 in}! :5! at.“ V @- '- fl uni; m1: n1 able” git-Hue a - —. sis-1:01:93 owl I .I.' h, (loath-I21"L 25 AEAB = - é— fwdm TT2)(R)Ba5Y(5,1m) TT2)(5)Aa55(5,1w) 1T 0 C18 — + %%5 (1 - FAB) Ide TT2)(5)B A5 5(5, iw) 6TT3)(5)A a5 5(531w) _ h w (3) . (3) A . 55 (1 + PA B) fodw [T T55Y(5) C5Y 55 (5,1m) T555(5) a55(5,1m) 5 T(2) (u) . 5 (R) E55 Y55(§_iw) TY5E 5(§)A5 aA 5(Eglm) T(3) (3) . —T55Y(§) ABY, 5(F, iw) T555(§)5AA555(§31w) _555 (2) . (u ) . 3 (5) A55 Y5(5,1m) TY55 5(5) AA 555 5(5,1w)] . (2.25) Application of the external field E shifts the poles of the susceptibilities from their zero-field locations, causing large changes in them near resonant frequencies. Along the imaginary w axis the changes are smaller and g(§,im), 5(§,iw), g(§,iw), and §(_, iw) may be expanded in Taylor series [74]. Specializing to centrosymmetric systems, these expansions are a55(5.1w) = a55(im) + é-Y55Y5(im,0,0) FY F5 + ... , (2.26) A555Y(5,im) = B5555Y(0,1m) F5 + ... , (2.27) C555 Y5(§_,1w) = c555Y5(im) + é-P555555Y5(0,0,iw) F5F5 + ... , (2.28) and E55 5Y5(5_,1m) = E555Y5(1m) + é-Q55555Y5(0,O,im) F5F5 + ... , (2.29) to second-order in the applied field E. The expressions on the right-hand sides of Eqs. (2.26) — (2.29) are given in terms of the response tensors in the absence of the applied field. In Eq. 26 (2.26), the dipole indices Y and 6 are associated with the static field (w = 0) and the index 8 with the frequency im; in Eq. (2.27), the dipole index 6 is associated with the static field and the quadrupole indices BY with the frequency im; in Eq. (2.28), the dipole indices 5 and ¢ are associated with the static field and the quadrupole indices Y5 with the frequency iw; and in Eq. (2.29), the dipole indices 5 and ¢ are associated with the static field and the octopole indices 8Y6 with the frequency iw. When the Taylor series are substituted into Eq. (2.25), the van der Waals energy for a pair of centrosymmetric molecules A and B interacting at long range in the presence of a static applied field is AB h m (2) (2) AE =_ZTOdMTaB ' %— (1+PA B) fm Odw Ta n (R) a: a(iw) (2) . (5) YEYE :(iw, ,0, 0) TY (5) a: a(1m) FE 5¢ (R) a: Y(iw) TY (2)Y + %; (T-PA B) IZdw TT2)(5) 38¢ Y6(o,1w) TT§E(5) a :a(iw) F¢ — (1+pA B) 1:11 [TEEY(5) ch “(111) 15:35): a (111) 1% T;:)(B) EEYYGEUw) TT::¢(5)A11 (111)] _ *6)? (11pm) fmdu) [% T5335) Rm BY “(0, 0,111) TéiiQ) agafiw) +1E T;ZY(5)BY c ’86 (111) Té3;(fl):a 1 (1,11 0,0) + %T 1T2)(5) anY YéeTO’O’iw) TY6::(5):aa(1w) +A§T TT2)(5) E8 mum) TT::¢(5) Y¢W(1w,o,o) J? TT§Y(5) 1:?” BY(o, 1w) Té2;(fl) BM 51mm”) —l3T 1T2)(5) Ben YY5(o,111) TT::¢(5) a; 16¢(0'im)]Fn FY , (2.30) -8 . . to second-order in E and to order R in the intermolecular separation R. \ J’ .' m .:as- -.s3 .9: m .ul. «new! «a fill " 3:33 hrs 1:19-. suns-a an: cut-a mum-s fin “at a noun- |I J .6 . . ..m [@532 .ni-I H: mm uni Ysmups'fl 9.13 any 6,? «all»! BI“U‘H’._, - 9*? "'3 -:=‘“ i " adf fiaih :n'a: :11: ewfi Q hrs 3 avazbnl slant. ”a: ~- A P+L3 .:i\uqu. .- - ~ :2! ” usufbrl cloqadao n: .' w'. ‘-i?:fioue ?fln i ,H. a .I u .u ..u n u .u 28 B. van der Waals Pair Dipole The van der Waals contribution to the dipole induced by the interaction of two molecules A and B is AB _ BAEAB Evdw ‘ a: §_= o (2.31) where AEAB is the van der Waals interaction energy for the AB pair in the presence of a static, uniform, externally applied electric field E. This energy was derived in the previous section and given by Eq. (2.25) for interacting molecules of unspecified symmetry; our attention will be restricted to the first two terms of that equation, which gives the van der Waals energy through order R_7. As in Section A, aa8(§}iw) and Aa,BY(E’lw) can be expanded in Taylor series. To first-order in the applied field 5 these expansions for general noncentrosymmetric molecules are aaB(E}im) = aa8(iw) + BaBY(iw’O) FY + ... (2.32) A .8Y(E’lw) = Aa,BY(lw) + B (O,im) F + ... , (2.33) o 6 a6,BY where g(iw). §(im,0), §(iw), and §(O,iw) are the molecular response tensors in the absence of the applied field. These expansions differ from Eqs. (2.26) and (2.27) in that the latter equations are 29 specialized to centrosymmetric systems, for which E and 5 vanish. In Eq. (2.32), the dipole index Y is associated with the (w = 0) static field and the index 8 with the frequency it»); in Eq. (2.33) , the dipole index 6 is associated with the static field and the quadrupole indices BY with the frequency 1w. When these expansions .are substituted into the first two terms of Eq. (2.25), application of Eq. (2.31) results in AB h w [T (2) (2) u5Yde =-§; f0 dw <53 BBY5(im, ,0) TY (5) a? (1w) +T;:)(:)a: Y(iw) T(2)(§)m855(1w.0) "éT T<2>(R)B 35¢ Y5(o, iw) T(3)(5)A a a(iw) --§T T(2)(5) AE Y5(1w) T$§:(R)A B i¢(i“'°) +-§T T‘2)<5) BEY5(iw, ,0) T(3’(5>A A5 a for the van der Waals contribution to the collision-induced dipole through order R-7. This equation holds for any pair of molecules of arbitrary symmetry interacting at long range. The molecular systems investigated in this dissertation possess centrosymmetric or greater symmetry. Then the tensors 5 and B vanish, and Eq. (2.3”) simplifies to AB h w (2) (3) . “¢,vdw =.§; f0 dw [T5 (5) a: Y(iw) TY 8(5) BA 55 5 (0.1w) -T(2)<5> BB5 Y5 12 Q(0 15) [ (5&8 5Y6 555 + 6a8 6Y5 555 + 6&8 6Y¢ 665 + 6 + 6 aY 686 65¢ aY 688 66¢ + 5aY 68¢ 658 + 6&6 68Y 66¢ + Gas 6BY 56¢ + 60¢ SBY 665) + 5 (6&6 686 6Y¢ + 6&6 68¢ 6Y5 + due 686 6Y¢ + 6&9 68¢ 6Y6 3h + 5a¢ 686 5Ye + 5a¢ 5BE 6Y6)] (2.u3) When Eqs. (2.38) - (2.“3) are used in Eq. (2.37), together with the propagator products T(2) T(2) 2 -8 a8 aY = (3 RB RY + 68Y R ) R . (2.”h) (3) (3) _ 2 —1o TaBY Tags - (36 RY R5 + 18 5Y6 R ) R , (2.u5) and (2) (u) _ _ 2 —10 Tag TamS — (108 RY R5 36 5Y6 R ) R , (2.46) the van der Waals contributions to the xx and zz components of the pair polarizability for a pair of S state atoms separated along the z axis by a distance R become AB h AB —6 m AaXX,VdW — E (1+1) )R fodw [BY + __(1+PAB 1T B . B . A . XXZZ(im,0,0)+Yzzzz(iw,O,O)] a (1w) —8 m B . B . A )R fodm [[12Yxxzz(lm,0,0)+3Yzzzz(lw,o,0)] 0 (1w) + g; PB(0,O,iw) aA(iw) - 6 aB(iw) QA(O,0,iw) +.% BB(O,im) BA(O,iw)] , (2.u7) 35 AB 71 AB -6 m B . B . A . A —' (1+P )R fodw [ZYXXZZ(1w’O'O)+uYZZZZ(lw,O’O)J Cl. (1(1)) azz,vdw = 2n - m B . . . + 2-(1+PAB)R 8 fodm [[5YXXZZ(15,0,o)+9vfzzz<1w,o,o)i CA(1w) 153 11 PB(O,O,iw) 5A(16) + 12 58(16) QA(O,O,im) _6_38 2 B + (0,16) BA(0,iw)] . (2.u8) These equations are identical through order R_6 to the expressions derived by Hunt, Zilles, and Bohr [U9]. Eqs. (2.47) and (2.”8) are thus equivalent to the results of two-center, fourth- order perturbation theory for Au through order R_6 [62]. vdw Although a proof has not yet been formulated, indications are that the R-8 terms are equivalent to the perturbation theory results as well. The effects of field-induced fluctuation correlations are present at order R_6 in Eqs. (2.47) and (2.48), but become particularly clear at order R_8. Specifically, the terms involving products of the dipole-quadrupole susceptibilities BA(O,iw) and BB(O,iw) cannot be interpreted as changes in the polarizability of atom A or atom B, nor can they be explained by earlier hyperpolarization models. These terms represent the concerted effect of the external field acting at the two centers. The external field induces a correlation between the fluctuating dipole and quadrupole of one atom, and combines with the fluctuating field of that atom to produce a net nonlinear polarization of the other atom. 1WD: : r .-‘:,-3:. [9&1 '1 “mi-i ' Ema. .-:-' ’H" 4 . . ‘ a ' if J *" - "‘-' '-"u.- :L' .,,.- . . , .,. _ I. _ . . CHAPTER 3. COLLISION-INDUCED VAN DER WAALS DIPOLES FOR SPECIFIC SYSTEMS In Chapter 2, an expression was derived for the long-range van der Waals contribution to the dipole moment of two interacting molecules, using a new reaction field model which includes field- induced fluctuation correlations. This expression, specialized to molecules of centrosymmetric or greater symmetry, is given by Eq. (2.35). In the present chapter, Eq. (2.35) will be examined in detail for some specific systems. First, in Section A we consider the long-range van der Waals dipole for a pair of dissimilar S state atoms. Then, in Section B, the case of an S state atom interacting with a centrosymmetric linear molecule will be investigated.‘ Section C covers the interaction of two centrosymmetric linear molecules. For the latter two systems, long-range induction also contributes to the collision-induced dipole, because a centrosymmetric linear molecule possesses nonzero permanent quadrupole and hexadecapole moments. These induction effects are also presented in Sections B and C. A. van der Waals Dipole for a Heteroatom Pair Consider two dissimilar S state atoms labeled A and B, separated by a distance R, with the vector 3 pointing from atom B to atom A. The dipole polarizability g(iw) and the dipole-quadrupole hyperpolarizability §(O,iw) of an S state atom are isotropic tensors given by 36 m 3 r) 37 a (im) = a(im) 6GB . (3.1) dB (0,111)) = B(0,iu)) [§( 1 BaB,Y(S u 56Y685 + GQGGBY) 25685Y5] ' (3'2) Substitution of Eqs. (3.1) and (3.2) into Eq. (2.35) leads to AB ’6» u¢,vdw _ 2? fo d“ [T _ (2) TaB (2) aY (3) aY¢ (3) aA(iw)] . (3.3) (5) cam») T (3) BA(O.iw) (3) B . (B) B (0,1w) Ta8¢ The propagator products in the expression above take the form (2) (3) _ _ '8 TaB (5) TGBY(§) - 18 RY R . (3.“) Using Eq. (3.“) in Eq. (3.3), and specifying the interatomic separation R to be along the z axis (R = R2 and RX = Ry = 0) gives to order R-7 “:?vdw = :5 R'7 I: dw [aA(iw) BB(0,iw) — BA(O,iw) aB(iw)] . (3.5) This expression is shown in Appendix A to be equivalent to the results of Craig and Thirunamachandran [60] from two-center, third- order perturbation theory. From Eq. (3.5) the dipole moment coefficient D is 7 D = Eh I: dw [aA(iw) BB(O,iw) — BA(O.iw) aB(iw)] . (3.6) I 'I‘._m _ '4- '1') w ‘ “ ' . ‘ . w aura-.6” .-, I It: .I:, .=| j ‘ 1 38 B. van der Waals Dipole for an Atom and a Centrosymmetric Linear Molecule Consider an S state atom A interacting at long range with a centrosymmetric linear molecule B. Define a vector r_that lies along the symmetry axis of the molecule B (taken to be the z axis of a body fixed frame), and a vector_3 originating from the midpoint of molecule B and pointing to the nucleus of atom A; a and B are the corresponding unit vectors. The dipole polarizability 2(im) of molecule B can be expressed in terms of the orientation of its symmetry axis with respect to the laboratory frame as [7D] aa6(iw) = a,(im) 6&8 + [5"(16) — q,(iw)] 3a 38 , (3.7) where 3a is the direction cosine between the a axis of the laboratory frame and the symmetry axis of the molecular frame, and where the subscripts H and l_denote components along and at right angles to the symmetry axis: a"(iw) = azz(iw) . (3.8) qL(iw) = axx(iw) = ayy(iw) . (3.9) 39 Similarly, the dipole-quadrupole hyperpolarizability B(O,iw) of molecule B is BaB’YG(0,iw) = B(o,im) [% (GGYGBG + 55553Y) - %-5GB5YG] + 82a(0,iw) (3 rues - Gas) 6Y6 + Bab(0,iw) (3 rYrG - 6Y6) 6a8 + 820(0,iw) [3 BaeY — 5a,) 585 + (3 ra“6 555) 58Y] + 82d(0.im) [3 FBFY - 58Y) 555 + (3 r835 686) 557] + Bu(0,im) [35 353393? - 5 (3&3 6Y6 + BaeY 586 + Bard 53v + BBBY 555 + 9395 day + 9$35 6&8) + 5636v5 + 6aY685 + 56558Y] ' (3'10) where B, B and B14 are linear combinations of the five 2a-d’ independent components of B(O,iw) for molecule B: §(0.iw) = $§ (B , (3.11) +28 +28 +8 +UB ) zz,zz xz,xz zx,xz xx,zz xx,xx 82a(0.iw) = - —- (B . (3.12) +8 +8 -38 -HB ) zz,zz xz,xz zx,xz xx,zz xx,xx 26B 16B ) , (3.13) B2b(0,1w) = -— (3B xx,zz+ xx,xx —AB -AB + zz,zz xz,xz zx,xz B20(O’iw) = fl? (3Bzz,zz_quz,xz+1OBzx,xz_9Bxx,zz-1ZBxx,xx) ’ (3‘1”) 1 - = __./ - - ‘ 82d(0,1w) “2 ‘3Bzz,z:+1OBX2,XZ “Bzx,xz ngx,zz 128xx,xx ) , (3.15) M3331 1"“ (Mm aw... ’ ' 3*.-.M ”M! "i“ an? via-Tb - 5‘5" t) “0100)6“ .: '. .1 -- ,~. in - ..." .35 - ,‘zn'v S] («r-hmfifi ' . ' ‘ I I I ' I ' 5. " - .' ' " ' "I.- :"_' ..uIJ‘J, -5 '. .3'_ -_ .- tJ ‘ \ I 3"- \ ' ‘ l A. ‘ ' I .‘ Y i, MO . 1 Bu(0,iw) = 76 (3B ) . (3.16) -U8 -UB -28 +28 zz,zz xz,xz zx,xz xx,zz xx,xx Each tensor component Be on the right-hand side of Eqs. (3.11) - B,Y6 (3.16) is understood to represent B (O,iw). The frequency- a8,Y6 dependent B tensor lacks the symmetry of the static B tensor with respect to interchange of the two dipole indices, and so differs in form from the B tensor as given in Ref. [7A]; it has one additional independent component. Appendix B provides a detailed analysis of the frequency—dependent B tensor for a linear molecule. Use of Eqs. (3.1) and (3.2) for atom A and Eqs. (3.7) and (3.10) for molecule B in Eq. (2.35), together with the propagator products (2) (3) _ _ -8 Tea (3) TaBYCB) - 18 RY R (3.17) and (2) (3) _ _ -10 Tm6 (5) Toma) — 12 RBRYRé R 3 -8 + [6 138%S - 3 RY685 — 3 R568Y] R . (3.18) and noting that aura = 1 leads to AB i1 .. _ B . A . ~8 u¢,vdw = ? f0 dw [ 9 qL(iw) 8 (0,1m) R¢ R B . B . A . A A -10 _ 2 [a“(iw) - al(im)] B (0,1w) ra rB Ra RB R¢ R + g [55(16) — qE(iw)] BA(0.16) 2a r¢ Ra 3'8 8 _.% [55(16) - qE(im)] BA(O,iw) R¢ R- 8 + 9 8B(O.iw) aA(iw) R¢ R" ..-i 'i ‘ . . ‘1. . a'. bl + u ng(0,iw) aA(iw) (3 fig 98 - 5a8) Ra R8 R¢ R—10 + 2 ng(o,iw) aA(iw) (3 rd r¢ — 5a¢) Ra R'8 + 8 Bgc(0,im) aA(iw) (3 fig 98 - 5&8) Ra RB R¢ R_10 - 2 B§C(o,im) aA(iw) (3 Ba e¢ - 555) Ru 3‘8 + 12 Bgd(0,iw) aA(iw) (3 ea r¢ - 5a¢) Ra R—8 + 1A0 BE(O,im) aA(im) fig 28 rY r¢ Ra RB RY R_1O - 6O BE(O,iw) aA(iw) ra rs Ru R8 R¢ R_10 — 60 BE(o,1m) aA(im) 3a B¢ Ru 3‘8 + 12 Bi(o,iw) aA(iw) R¢ R_8] (3.19) 7 for the van der Waals dipole moment to order R_ . The form of Eq. (3.19) is not convenient for analysis. Alternatively, the pair dipole for the interaction of an atom and a centrosymmetric linear molecule may be written as a first-rank spherical tensor by coupling spherical harmonic functions of r and R in the following way [9,16,78]: NW W313) = —1/2 2 D (r,R) vm(3) YM_m(R) C(AL1;m,M-m) , (3.20) (3) A.L.m ‘L A L where r'is the vector connecting the atoms of the diatomic molecule and R is the vector from the center of the diatomic to the atom, with 3 and R the corresponding unit vectors. The functions DAL(r’R) are expansion coefficients which must be determined and C(AL1;m,M-m) is a Clebsch-Gordan coefficient; M is the spherical tensor index of E_in the laboratory frame, and can take the values M = — 1, 0, 1 (the laboratory z axis corresponds to M = O). Spectroscopic “'3“ 4: ‘ 3C .:-. 8 e.‘ ’51,: 1,, 3 3:) 1.3333133.“- 11'- ? - °"‘:- .11 a 3 3 3 fi(u1)‘a(ui.0):l0lfo 18:15an N- . ._ A a ' . . .' i. .. - I . , .q. 9‘.- if. D" a. ... (I01) I 1.3.3.0)‘8 03 “'7 ”a _= -3 (3:. ‘An (61.6)?! 06 “'. 3:1“; 23.0)fie TE - ‘ - ‘ - I ' ‘:J "5": 1:4: f #2 lineshape analyses [16-18] for collision-induced absorption generally employ functions 3 of the form in Eq. (3.20). We now wish to determine the van der Waals contribution to the DAL coefficients. In so doing the focus will be on the dependence of these functions on the separation R, with r held fixed at its equilibrium or vibrationally averaged value. Let VDWz designate the entire right hand side of Eq. (3.19) with ¢ = z. Specifying the undetermined coefficients to be the long-range van der Waals contribution to the AB interaction dipole, and equating VDWz to Eq. (3.20) with M = O we have An vdW m A -m ‘ -—-T7§ Z DAL (r,R) YA(r) YL (R) C(AL1;m,-m) = VDWz (3.21) (3) A.L.m 01" 1111 vdW _—_T72 DAL (r,R) C(AL1;m,-m) (3) m - -m ‘ * = f dnr I 59R VDwZ [YA(r) YL (R)] . (3.22) A The factors that depend on r and R in VDWZ can be expanded in spherical harmonics; Appendix C presents the method. When these vdW expansions are used in Eq. (3.22) the DAL coefficients that survive are D3?” = 9 2 3‘7 I: dw [aA(iw) 8B(O,iw) — BA(0,iw) &B(im)] , (3.23) DZ?” = - 41/223” 1;... 5(2) M3 x [A aA(iw) [3ng(0,iw) + B: c(o, iw) + 1052 01(0, 16)] — BA(O,iw) £5E(1w) — q:(iw)]] , (3.2u) 1/2 DE§W= ”(3) 5371: dw [2 aA(iw) [B§b(o,iw) + ZBgc(0,im)] - BA (0,15) [53(15) - afi(iw)i] . (3.25) and ngw= - 16 §»B'7 f: dw aA(iw) BE(0.iw) . (3.26) In Eq. (3.23) we have introduced for the first time the isotropically averaged part of the molecular dipole polarizability, given by &(m) = % [6”(16) + 2555113)] . (3.27) Because the molecule B has nonvanishing permanent quadrupole and hexadecapole moments, induction effects also contribute to the long-range dipole of the AB pair. The permanent multipoles of B produce a field that polarizes A, and the moments induced in A give rise to a static reaction field that acts back at B (back- induction). The induction dipole that results from these effects can be expressed to order R_7 as AB _ —la T(3) B Lla,ind _ 3a ATaBY(B) 0BY _1_ (5) 105 “A TaBYGe(-) °sv5e (38.8) ..I - n‘u% 'II‘ lull-135:1). l- ' ' {1.1.0355 ‘3 3 (3.11.111 :7 MW‘;% .--'-'it(»x)§n - (en‘s: (can't-1. .- ' uh ) B (2 A (3) B - a'LTaB (5) a TBY5 (R) 0Y5 l 3 — % (51,3, - 531 Té§)(3)A a 1(3)(11)B 06 13a 2 . (3.28) The quadrupole and hexadecapole moments of B appearing in Eq. (3.28) can be written as [7M] 9 on NI—- 9 m p m (3.29) and l—-¢ [105 r r r F aBYG 2A a B Y 5 - 15 (9 96 6Y6 ra Y 86 + ra r(s GBY B Y 565 re ”5 5GY I PY r5 6&8) + 3 (5&8 6Y6 + 50LY 586 + 5&6 5BY)] , (3.30) where the center of symmetry of the B molecule is taken as the origin for the charge moments, with e = O and ¢ = ¢ . zz zzzz Using Eqs. (3.29) and (3.30) in Eq. (3.28) gives u:?ind = --1§AaT(3)(R)BGB’r"B rY 21$T§Z§55]* , (3.36) = I er f dQR INDz [Y1 where the DiEd coefficients need to be determined. When INDz is expanded in spherical harmonics (see Appendix C), use of Eq. (3.36) gives for the induction coefficients to order R—7 ind _ 6 B _ B A B —7 D01 —-§ [an dL] a O R , (3.37) ind _ _ 3 1/2 B B A B -7 D21 - -§ (2) [ZaH + BL} a O R , (3.38) ind _ 1/2 A B ‘4 N 1/2 B B A B '7 D23 — (3) a O R + 35 (3) [3a” + ”3L1 a O R . (3.39) 147 ind__2_u B_B AB-7 Du3 — 35 Ca)! a1} a O R . (3.40) and Digd = (5)”2 aA @B R_6 . (3.41) The polarizabilities aA, a3, and aE_in Eqs. (3.37) - (3.41) refer to the static, zero-frequency values. Adding the van der Waals coefficients in Eqs. (3.23) - (3.26) to the corresponding induction coefficients from Eqs. (3.37) - (3.”1) gives the exact long-range dipole coefficients through order R-7 for an S state atom interacting with a centrosymmetric linear molecule in a fixed configuration. L18 C. van der Waals Dipole for a Pair of Centrosymmetric Linear Molecules Consider the interaction between two centrosymmetric linear molecules A and B. Let_r_1 denote the vector that lies along the symmetry axis of molecule A, and let :2 be the vector that lies along the symmetry axis of the B molecule. Let R be the vector from the midpoint of B to the midpoint of A. The dipole polarizability g and the dipole-quadrupole hyperpolarizability g for a centrosymmetric linear molecule have been given in Section B by Eqs. (3.7) and (3.10). First using Eq. (3.10) for both-A and B in Eq. (2.35) results in utidw = g; f: dm [g- T(2)(R)B aB Y(im) T(3)(R) BA(o, iw) + 1:2)(R): a Y(iw) Tig:(§): BA b(o, im) (3 R R1E - 55$) + 2 T(2)(R_)BB(1B Y(iw) T§21(R)3BC(O, im) (3 RYa R16 - 5&6) + 2T T32)(R)B aB Y(iw) T;3§(R)23Ad(o,1a) (3 R16 R1¢ — 15¢) 35 TE2A(R)B aB Y(1a) T(3)(R) BA (0, im) R R R R T(2) + (R) aB Y(1a) TA3)(R) B A(o, iw) R R 15¢ —— 1a 15 - 1010T32A(R)BaB Y(iw) T;3§(R) B A(o, iw) R16 RY¢ - 5 Té:A(R)B a Y(1a) T§§:(R)A B (0, im) R R1E + 2T TA2)(R)B a8 Y(iw) 1:3;(R) B A(o, iw) - MA2)(R)B B (0,1w) T(3)(R)A a (1w) - TA2)(R)B132 b(0,111) (3 RZY 926 - Y) T(3)(R)A a (iw) 2TA2)(R)2B 32 (0 im) (3 R2 BR2Y— Y) TA3)(R)A a (iw) — 2 TE: 8)(R)B 82 d(0,111)) (3 r2¢ BZY - ::Y)T T(3)(R)A ae a(iw) T(2) A A (3 ) . — 35 (R) B B(O, iw) r28 r2Y r26r 2¢ TY6€(R) a: a(lw) 1m 1"! akin-um f " 3 2 1’1 1111:. 3111.13 ~91! 31.1.1 13.19911 0.13 am» ,‘2 m .I'UB ." .l.‘ - . fi-u" .. 1‘" in W _ "I . .h 'I’ I'I «fl: fins M9 T(3) (2) B . A . + 10 Tue (R) BMAO’lw) r28 r2Y TYE ¢(R)€ aA a(lw) + 10 TA2A(R) RB(o, iw) R2Y R2¢ TéY:(R)A a (im) + 5TA2A(R) BB (0, 1a)R wfi25 TA3A(R)A a (im) - 2T TA2 2A(R)B B (0,1w) TA3A(R) aA (iw)] . (3.12) Next, using Eq. (3.7) in Eq. (3.H2) leads to AB _ R I: w[3 T(2) AA¢,vdw3 ' 3T 2 Tue T(2) 2 Tue (R) [a”(iw) - qi(iw)] Ta ¢(R) BA (0, iw) FZBr 2Y TA2A(R) aB(1a) TA3A(R)AB2b(o,1a) (3 A916? 16 - 166) TE:)(R) [aH(iw) - a3(1a)1 bTfig;(R)2 BA b(0,111) 2 A3 r1621e ' 552A 232 T(2) B . (3) (R) qL(1w) T 86¢(R) B T(2) (R) af(1a) TA3A(R)A B (0.1m) (3)¢ + + + + (0,101)) (31:1022‘1‘S ‘ 5‘25) (3) (R) B: c(0,1(11) F2Y A 2c AT 15¢ — (R) [aBg1a) - a3(iw)T + A A3 r1ar15 ' Baa) W2sr T(2) B . (3) A A (R) al(1w) Ta86(R)2 B d(O, im) (3 Ar16r1¢ - 56¢) T(2) (R) [a“(iw) — af(1a)1dT;3§(R) 82d(0.im) + + 2 A3 215:1a ' ¢A 223221 T(2) (3) . A A A A (R) aE(iw) T865(R) B 2(0, 1w) rYarYdrYEr14> (3) A . Y6€ Bu<0yl€0> + 35 STABNR)Tafiua)-ag1an T + “ 21a215:1221¢223221 T(2) (3) A . A 1 ‘ (R) aE(im) T85¢(B) Bu(0,1w) r1a216 0TA2A(R) [a BR1a) - afi(1a)] TA3A(R)A B (0, im) R Y5¢— 1(13316 r328 FZY T(2) (3) aBGAB) B 3(0, iw) rYGrA1¢ (3) B(R)aL(111)T - 1O OTA2 8A(R) [afi(1a) — aE(1a)J TaY6(R) B A(011w) 91591¢R2892Y T(2) (3) TB¢ (R) aléiw) T86€(R) B 3(0, iw) rYdr1E T(2) (3) (R) [aB ,(1a) — af(1a)] TY5€(R) Bfi(o,1a) R15R16R28R2Y 50 + (2) B . (3) A . TaB (R) ql(1m) Ta8¢(3) Bu(0,1w) + (2) B . B . (3) A 2 TGB (R) [al“lw) - aiflw)] TaY¢(R) BA “(0, 1m) r28r2Y - 2TA§A(R)BB (o, 11) TA3A(R) af51m) - 2 T;:A(R) B B(O,iw) TE3:(R) [a”(iw) — afig1w)1 RTaR1E - T;:A(R) 82b(0,iw) (3 R2YR26 — m) T<3>(R) “1AANA RENE) ngAo'i‘AA (3 223226 - 15A BEBEABA x £afi(1:) - aA(iw)] FTGRTE - 2TA2A(R)B B2 c(o, iw) (3 R2BR 23 - BY) TA3)(R) BJEiw) - 2TA2A(R) BB (0,11) (3 R2BR2Y - M1 TZZEAR) x [aH(i:) — Rfi§1w)3 RmR1e - 2TA2A(R)B B2 d(0,111) (3 F2¢F2Y - 6¢Y)T T(3)(R) BLAiw) _ 2T(2)(R)BB d(0,111,) (3;. f. _6 )T T(3)(R) 2¢ 2Y $7 BYE - x [0”(iw) - qfi(iw)] B10? T.(2) 16 (R) B B(0, iw) r28r2Yr25r2¢ TaY6(B) qlfim) - 35 3S'r::A(R) BBT(0,111) R2BR2YR26R2¢T T3§:(R) x [13(11) - 1A51R)J R R16 1 0TA2A(R)B B ”(0, im) R2BR 2T T;3;(R) afg1w) 1 oTA2A(R)B B (0,1111)R2BR2Y T32;(R) [13(11) - 13311)] RTaR1E + 10 T::)(R) Bu(0,iw) 9232 2¢T TA3 3A(R) 1J51m) + 10 T;:)(R) Rfi(o,1m) R2YR2¢ TE3:(R) [aA(1w) — 22(1R)J FTaRTe + 5 T;:)(R) Bfi(o,1w) R2TR26 Té3§(R) qfig1w) 1 5 T;:)(R) Bfi(o,1m) R2YR25T TY§:(R) [1”(11) — 1A51w)1 RTaR1e - 2 T;:)(R) Bfi(o,1m) TA3A(R) qf(1w) - 2 T;:)(R) Bfi(o,1w) T:::(R) £a,(1w) - Rff1w)1 RTaRTE] . (3.u3) The propagator products in the above expression take the forms: Sl T(2) (3) —8 Tue QR) T aBYAB) = — 18 RY R , (3.44) T(2) (3) _ _ -10 (R) T ems(R) — 12 R2 RY R6 R -8 + [6 R8 536 - 3 RY 626 — 3 R6 523] R . (3.u5> and T(2) (3) _ -12 (R) TY 6(R)= 45 Ba RB RY R6 R8 R + [1: BY R6 Re 6a8 + 9 Rd RB RY G58 —10 + 9 Ra R8 R6 6Y5 + 9 Ra R8 R8 6Y6] R -8 _ [3 RY 6&8656 + 3 R6 SaBGYe + 3 Re dadeG] R ' (3.46) Using Eqs. (3.44) - (3.46) in Eq. (3.43) and noting that fi1afi1a = FZaFZa = 1 gives as the van der Waals dipole for this system “:Bvdw = E-IZ dw [6 {a3_ng + EB B20} R¢ R_8 — [a3 + SqEJ [% BA - 2 82d + 2 B3] R¢ R-8 - 6 {BED afi_+ Bic éA} R¢ R‘8 + % BB — 2 Bid + 2 Bi] [afi + 5&2; R¢ 3‘8 - 6 aE_[B2b — 320] RTaR1¢Ra R'8 — 2 [afi + Sci} [3 32d — 5 Bfi] B1ar1¢Ra R 8 1 R R '8 2a 2¢ a , — [g-BB - 3 Bib - 2 Bic — 2 Bid + 7 Bi] [afl- 2:2 x BTGFT¢RQ R—8 B A . 2b ' B2cJ “l-rzaA2¢Ba B B B A A A A + 2 [3 82d - 5 54] [an + Sal} r2ar2¢Ra R 52 B _ B A A _ A A A A A ‘8 2 [“11 all [3 225 + 3 320 A0 Bu] 21a21322a22¢Be B B B A _ A A A A A A _ [01” 01L] [6 B20 6 32d + 35 B4] r1ar1¢22a22833 R B B A A A A A A - 2 [an “13 [3 B2c 5 B4] ’1a21522a2232¢ B B -8 + + + + + + B B A A A A A A 2 [3 32b + 3 B20 10 B4] [an “1.3 r1ar1¢r2a228R8 R B _ B B A _ A . 2 2 2 —8 20 6 B2d A 35 Bu] [2” “13 r1J1522Jz¢BB 2 [3 3‘20 — 5 RE] [113, - [6 B GA] R R R R R R‘ J. 1a 18 2a 28 ¢ B B A 1 A . . A - - 7o [“11 “13 B4 21a21521121¢22a2232Y R 70 BB [BB ' “:3 21a21sM2a 25 22V 22¢BY R 8 12 aE_[B2b + 2 820- 5 Bu] RTGR ATBRaRBR¢ R 10 - [a?T- RE} [6 BA - 9 BAD — 8 B20 — 8 82d + 23 Bfi] * 22a 228Ba282¢ B 10 [6 BB — 9 BB - 8 330 - 8 32d + 23 BE] [afi — a3} x 21a218RaRBR¢ R ‘0 12 [82b + 2 520- 5 Bi] q: 22a228RaR8R¢ R 30 [afi — a3; [18 B20 - 24 BAd + 115 82] 2 21a 21¢22B22YRaRBRYA R 10 [ca — a3; [15 32b + 18 B2 - 55 Bfi] x R R R R R R A0 12101 18 l22Y 2¢ a 8 Y 2 [aTT- RE; [9 B2B + 2n B2c - 55 Bfi] x R R R R A0 M1a 18 22a f:2Y B Y ¢ + 18 B20 - 55 82] [a?.- afij -10 B 2b ‘ M1a 1¢ 223 22YBaBsBY B B A A [18 B20 — 24 B + 115 Bu] [an — RT} [15 B 2d x FTaBTsfzyfi2¢RaRBRY R 10 2 [9 ng + 24 Bic - 55 Bu] [afi - afiJ M1a 1B f2201222YRBRYR¢> R 30 140aiBAF RR R R10 4 M1a 1B M1Y 1¢ a 8 Y 53 -10 + A u “1 22J 28 22Y22¢BaBsBY B -10 385 [a,,- qEJ BA M1“ 1BR1YR1¢R20R25RBRYR6 R —10 1H0 BB + 1aflmzaa2zs2mfim%2 F A 2 5 BU] -12 21J 1B2 2Y 2252aRBRY25:¢ R A A5 [22b + 2 B2c ' 5 Bu] [“11‘ “L1 —12 385 Bi [a,,- “l3 r R5 [a2 — a3; [32b + 2 B c 4.. 1J 1B 2Y2 A25 a B Y 5 ¢ 525 [“II‘ 2L2 B: 21J 1B21Y21¢226 22s2aRBRYBaBe B B A A2 525 BR [an “LA 21J 1B 22Y 225 22s 22¢BaBsBYBsBe 2 x F R R R R R R B -12 ... A2] (3.u7) to order R-7. The frequency dependence of the d and B tensor components on the right-hand side of Eq. (3.”7) is to be understood. The dipole moment for a pair of linear molecules may also be expressed in terms of spherical harmonics as 3/2 - (NR) M-m M(r‘ 1_ IR) = __ X D (22 )r2 9R) Ym :(1‘1 ) Yr: 2(22 2) Y (R) —1 2 (3)1/2 A1A2AL 1 A2 L x C(A1121;m1m2m) C(AL1;m,M-m) , (3.M8) where the summation is carried out over the indices A1, A2, A, L, m m and m. The vectors r ,_§2, and_R are as previously defined, 1’ 2’ —1 C(A1A2A;m1m2m) and C(AL1;m,M-m) are Clebsch-Gordan coefficients, and M is the spherical tensor index of E in the lab-fixed frame. The functions DA1A2ALA21’22’R) are undetermined expansion coefficients; they will be found from Eq. (3.H7). Fixing r1 and r2 at their equilibrium or vibrationally averaged values allows us to focus on the R dependence of the DA A AL 1 2 5h coefficients. Designate by VDwz the right hand side of Eq. (3.47) with ¢ = z. Equating VDWZ to the M = O instance of Eq. (3.“8) gives )3/2 vdW (Mn 1/2 2 DA1A AL <—?;;——— A 2 (r1,r2,R) C(A1A2A;m1m2m) C(AL1;m,-m) = I an I an I an VDw [Ym‘(2 ) Ym2(9 > i'm(§)]* (3 A9) r r R z A 1 A 2 L ’ ' 1 2 1 2 where the coefficients have been specialized to the van der Waals contribution to the interaction dipole. For a pair of interacting centrosymmetric linear molecules, the DA A AL coefficients possess the following symmetry relation upon 1 2 interchange of the indices A1 and A2: A+1 AB D = (-1) P D , (3.50) A2A1AL A1A2AL where FAB interchanges the molecule labels A and B. When VDWz is expressed in terms of spherical harmonics (see Appendix C) and substituted into-Eq. (3.“9), the following coefficients result: vdW h _ _ w —B —A _ —A -B —7 90001 - 9 n f0 dw [B a B a J R , (3.51) vdW _ g_ 1/2 3. w A A A —B 02021 - 1o (2) w f0 dw [u (3B2b + 82c + 1OB2d) a ‘ (“fi ‘ Bf) éB] R_7 . (3.52) vdW 1/2 h w A A B B IO dw [(a,,— aL) (32c + B ) ) 2d u 1 D2201 "§ Cg 'F vdW 2211 vdW 2221 vdW D11221 DvdW D2023 vdW 2223 vdW 2233 vdW 22u3 vdw uou3 vdW u223 Dvdw DUZ33 55 A A B B -7 — (B2c + 82d) (a,,— aJ)] R . 6 1 1/2 h w A 0—) E'fo dm [(an al) 82d A B -7 * 82a (“n ' “12] R . 2<-—)1/2 fi-x: dw [ 5 35 H 1- 2b 2c 2d B B -7 — (632b - 232 0+ 732d) (an - ql)] R . 8 1 1/2 h w A B _ B -7 §'(79 E-fo dw Bu (a|‘ aJ) R , u ( )1/2 E fw d [( A _ A) 2B 5 3 ? o w “U %L _ A A -B -7 2 (82b + 2320) a ] R , 8 2 1/2 h w A A A B B §'(76§) F f0 d“ [(sz _ 2820 + 282d) (an ‘ “4) _ A _ A B _ B B -7 (an “1) (B2B 2B20 + 22d)] R ’ --3 <—3)‘/2 h-x“ dw [(53A + 832 d) (a5 - a8) 5 15 n 0 2b .L A A B B2 -7 + (an - 94) (5132b + 882d)] R , 3 (—3)”2‘E f: dm [(513A + ABA + 2AB2 d) (a8 — B) 5 35 2b 20 11 9L _ A _ B B d-7 (an 3L) (5B2b + uBZc + 2M82d)] R , 16 E-f dw B a8 R 7 , —5 <—3)”2 3 f d BA ( B - B) 3‘7 15 21 w u “H %L ' — 3 (111/2 2 f d BA < B - B) R'7 3 3 w u “n 91. ’ .55) .58) .61) .63) 56 0:223 =%(12)”21Th I: don 52 (01?.- 01:) 13—7 , (3.611) 132225 = 11 (3)”2— h I: du) [(012, — a2) (ng + 252C) — (132b + 21320) (6?, — 62)] R‘7 , (3.65) 1312225 2 (7—5)”2 h I“ did 32 (a, (If) 3'7 , (3.66) 9:225 — -§ (2)”2 :1 I“; a... 62 (6%- (1:) 3‘7 , (3.67) and 0:225 = i2 (3241/23? d... 132 (a?! — 6:)11'7 . (3.68) Again, the frequency dependence of the a and B tensor components in Eqs. (3.51) — (3.68) is to be understood. Induction effects also contribute to the long-range dipole moment of a pair of centrosymmetric linear molecules. The permanent quadrupole and hexadecapole moments of each molecule produce fields which polarize the other molecule. Additionally, the permanent quadrupole of each molecule sets up a gradient of a field gradient that polarizes its partner. The moments that are induced in the molecules from these fields and field gradients themselves give rise to static reaction fields which produce back-induction effects at each center. The combination of all these effects results in an induction dipole that may be written as B Y6 AB A T(3) 1 “a,ind ' §aaeT316(R) OY 1. B T(3) + 3 (1018 TBY5(E) 915 -..l. A T(5) 105 “as Tsvae¢(-j ¢Y66¢ + _1_ dB T(5) A 1 5 a8 BYGe¢ —- Y6€¢ _ .1 A (5) U5 Ea,8Y6 TBY6£¢(B)5 08¢ .1 B (5) A¢ + AS Ea,BY6 Tsvss¢(5) 96¢ l A T(2) T(3) + 3 aaBT (R) dB T65 ¢(§) 0A ;¢ .1 B (2) A: T(3) 3 “GB TBY (3) aY H¢(§) , (3.69) through order R_7. Expressions for the dipole polarizability, the quadrupole moment and the hexadecapole moment have been given by Eqs. (3.7), (3.29) and (3.30) of the previous section; the propagator tensors and tensor products that occur in the above equation were also given in that section by Eqs. (3.32) - (3.3“). A quantity that appears for the first time in Eq. (3.69) is the dipole-octopole polarizability Ea 8Y6(1m). For a linear molecule, the E tensor has two independent components and can be written (see Appendix B) . _ _l . . (1w) - 1M [Ez,zzz(lw) + 2 Ex,xxx(lw)] [35 r103.813 Y 136 - 5 (rarB 6Y6 + rarY 686 + rmr(S 68Y + rBrY 6&6 + rBr6 GaY + rYr(S 6&8) + 6&8 6Y6 + 561 586 + 6&6 68Y] Ea,BY6 1 . u A A - 63 [3 Ez,zzz(1w) - 8 Ex,xxx(lw)] [(3 1”(1138 _ Gas) 6Y6 + (3 Igozf‘Y _ GaY) 686 + (3 rafié _ 6&6) GBY] 5 . . A A _ + 126 [3 Ez,zzz(lw) _ 8 E:x, xxx(1w)] [(3 r'Br‘Y éBY) Gad + (3 92rd — 625) GaY + (3 rYrG - 6Y2) 6&8] . a“ 1: «31:01:13 .... h :. J:‘.L'::I£u-‘ " .‘. :rgq 58 Making use of Eqs. (3.7). (3.29), (3.30). (3.32) — (3.3”), and (3.70) in Eq. (3.69) leads ultimately to “:Bind =-2 [6 ai_d§_OA R-B — 6 61262.03 R 8 + 5 E: XXX 0A R‘7 — 5 E2 XXX OB R'7 ai_:B5 R 7 - 1% aB 6A R 7 — 3BLZB R + 3aiOAR W + [3 qB_OA R 15 B A A U1 + :1 R a :2 [a f - Bl] OB R 5 R7 +‘%AEGH'GJB_] (98151—7 7 + 5 EB GA R" Q P > p r.a e x + Nk»N Q ‘— l_1 a» 1 9 Vi? C) I 1 a: I—l 1 5 B B A x B + E [2 Bx,xxx + 3 Bz,zzz:I B B ‘ 3 aJ_ai_OB R- +33 2“j_[aBl' of} GA B_B] 15 B A -7 105 as 2A —9 [ ' ‘n—a rzarzsfis _ §§ A eB R-9 35 EB OA R-9 2 x, xxx 2B x, xxx A B A ‘10 A [a A B ‘10 +6aLaLO R +§aB 1,—2Q0 R ] B1B B1YRaBBRY B -7 _ 105 “A B -9 T “.l l—I _s NU] @fi.9 R 6 R + £2 EA GB R-9 _ 35 B A -9 2 x,xxx 2 X,xxx + 6 013 0119B R-1o BZBBZYBGBBRY . ' [3 [0% - qu GB R-B - lB-[afi - B2] 6B R-7 X, XXX BZ, ZZZ X,XXX 2 2 8 - 3 dB [al(_ a1} OB R i 2 A _8 A A A A [BM 0:] Ca” a1} 9 R ] B1aB1BB28B2YBY ' 3 0111MB - 01:] 0A R 3 B B A + 2 [01” - 033 [01“ - [ A B ‘7 B - 5 9 R - E x,xxx x, + [1—3 [012) - 01A] GB RB7 -35 [2 EA X, XXX 15 ' ‘2 “l_[“ll' “13 9 A A B B A + E'Eall' Bl; [G'l' cl} 0 R xf'r XXX _ 105 +3 BA A B afiOR '8] 6 6 2 2 R 26 2sr1sr11 Y A —7 2 A 2 2 R ] r'1BA"11(A"28‘"21R61 A _ A B -9 [01“ all 6 R 1 0B R'9 - 35 EB Z, ZZZ X, XXX GB R R R R -10 -10] 1a 18A BZY 1B726 8 Y 6 15 B 7_ - [—2 [a,,- a3} 0A R A B - 35 Bx,xxx 0 R _ 15 A E B _ B] '_2'a—La“ (1“— 9 B B A A B +§[a“-aJ_J [GH'aJ—JO R X 9 _ + [70 Ex, xxxOB B 7 + 9aiEaBH-ai] A X 105 -9_ GA R 35EaH-aJ—JBA R-g —5[2 Bx, xxx B 3 Bz,zzz] -10 -10] 2a :28 11 16BBBYB6 0 EB BA R‘9 x,xxx A '10 B A A B OR -901J_[a”-01'_JGR R R R M18 16 B2Y :26 a B Y - B—g [012: [ai- 0133] 9A R_12 - aiEaBl- 01:] GB R-12 + 7 E1:,xxx BB B417— 7 BB, xxx BA B-H] BB1B 1YB 26 BZEBOLBBBYZ6B€ - ['4 [10 EAx, xxx + B E:z,zzz:l BB B 9 - BB {BAA R 9 + 6 [65A, - 01:] aBLOA 13—10] M1a 1BB1YB16BBBYR6A + [B [10 BE, xxx + B Bizzz]A B BB9 ——3—% aAd>BR 9 + 6 [BR -ai] 0110B B_AO] X R R R B26 B28 lB2Y B26 8 Y 6 GA R -10] -9 60 [g [10 E:,xxx + 5 E::,zzz] OB R—Y - 3 [a?.- afij [a?l- a?} GA R_8] x I21019189131(sf‘zyfizsfie [% [1O Eixxx + 5 E2,222J GA R_7 ~ 3 [a?1- a3} [afi - cf} OB B_8] X F2u92892Yfi26F1YF16RB [21 [1O E:,xxx + 5 E:,zzzJ OB R-g -— [01.- ail 11,3- a1; 91 R101 x I“mm13131119159259238111155 [$1 [10 Eixxx + 5 Elizzz:l GA R_9 ‘ §% [02 - 6?] [mi - afiJ OB R_1O] “ FZaFZBBZYfi2€F16fi1eRBRYR6 Lg; [1O E:,xxx + 5 E:,zzzJ OB R-11 - 5% [afl - 9:] [ca — BE} GA R‘12] x F1afi1891YF1692892¢R8RYR6R8R¢ [%§-[1o E1:,xxx + 5 EB,ZZZ] GA R_11 - 5% [a?,- a5} [afi - afi] OB R_12] x rA‘2111f‘213921r’q2es’21e’qwfisfivRsHERA 3% [GA ' “:3 ¢B “‘9 l“111191sf‘zgsfizv’azcs’A‘zeRYRcsRe i; [“5 ' “EA ¢A ““9 I1201921961391111215’A‘1eRYRsRe EA: 9: ¢A R—11 F1Bfi1Yfi16fi1ERaRBRYR6R€ §%§ Q: ¢B R—11 $2892Y92692€RaRBRYR6R€ 3&2 [afi — qfi] ¢B R_11 ‘ 91a$1892Y92692562¢R8RYR6R6R¢ 2%? [a3 - 63] 6A R‘11 x F2QFZBF1Y91661EF1¢RBRYR6RER¢ (3.71) 61 for the induction contribution to the long-range dipole moment to order R_7. The induction dipole can also be expressed in terms of spherical harmonics, as in Eq. (3.118). Letting INDz denote the right-hand side of Eq. (3.71) with a = z, we can write in direct analogy with Eq. (3.119) (4w)3/2 ind 1/2 A1A2AL -(-3)—- A (r1,r2,R) C(A1A2A;m1m2m) C(AL1;m,-m) m 1 - = f d9r f dflrz f an INDz [YA1(r "12 A -m A * 1 ) 152012) XL (12)] . (3.72) 1 where the coefficients have now been specialized to the induction contribution to the long-range dipole moment. Expressing INDz in terms of spherical harmonics (see Appendix C) and substituting into Eq. (3.72) results in ind _ _ 6 A _ A -B A _ B _ B -A B —7 D0001 - g [[a“ a‘L] a O [01H 01] a G) ] R . (3.73) ind 1 1/2 ' A A —B A 132021 = g (2) [3 [201” + aJ_ a e 1 B B A A B —7 - E [01” - 01] [an- 01] O ] R 1 (3.714) ind _ _.3_ .1 1/2 A _ A B _ B A D2201 - 35 (5) [[au aL} [a,, 9L] 0 _ B _ B A _ A B —7 [an 011—] [01“ 1.] O ] R , (3.75) ind _ §_ .1 1/2 A _ A B _ B A 2211 ‘ 35 (5) [[“H Bl} [an “L; o B B A A B -7 + [an - a1} Ea” — a1; a 1 R , (3.76) ind 2221 ind D11221 ind 2023 ind 2223 ind D2233 ind 2243 ind £043 ind D21223 ind DA233 ind Du2u3 62 1/2 [[11afi + Blafi] [ca - dig 6A _ B B A _ A B -7 [11“11+ 31ai} [aH a1} 9 ] R , 1 1 ‘35" (3—5) 12 1 1/2 A A B B A ‘7 T—g (7) [an - a4} [all_ QLJ O R , 1/2 -B A 'U A A A 'B A ‘7 - (3) [a O R + §§ [3al.+ Roi} a e R 8 B B A A B '7 ' 7g [an - qL] [an - cl; 9 R ] . 2 3 1/2 A _ A B -u _ B _ B A —u _§(76) [bu QJ 0 R [w, ad 0 R A A A B B A '7 --§§ Ea“ + 6al} [all- alj G R A B B A _ A B -7 + §§'[a|l+ 6alJ [all 8L} 0 R ] . 2 3 1/2 A _ A B -u B _ B A —u 'g (T6) [[0], $1} 9 R + [all qu 9 R 1 A A B B A '7 ‘ §§ [13all+ 22ai3 [all_ GLJ e R _ _l B B A _ A B -7 35 [1301“ + 2201”“11 all] e R ], 2 5 1/2 A A B -A B B A -4 _-§(1—u) [3 [UH—C11} 9 R ‘3 [a"-G_L] e R 1 A A B B A -7 ' gg [33aH + 2&1} Edi]- a1} 6 R 1 B B A _ A B —7 + gg C33a“ + 2aig [aH a1} 9 R ] . 2A A _ A —B A -7 E [an (1.1-.1 a 9 R , 2 2 1/2 A A B _ B A -7 77—5 (a) [“11‘ “l3 [“11 all 9 R 1 1 1 1/2 A _ A B _ B A -7 " '3—5 (g) Ea“ (1.1—:1 [an (1.1.1 G R I 1 11 1/2 A _ A B _ B A —7 ii; 0—7) [a(( qu [all di; 9 R , (3.77) (3.78) (3.79) (3.80) (3.81) (3.82) (3.83) (3.8“) (3.85) (3.86) 63 ind _ 21/2 A _ A B —6 D2245 - (7) [[3Ez,zzz 8Ex,xxxJ O R B B A -6 _ [3Ez,zzz - 8Ex,xxxJ O R _ 2 A A B _ B A -7 7 [301” + ROLL] [01H 01] O R 2 B B A A B - + 7 [301” + 11011 £01,, — “l3 e R 7] , (3.87) ' d 1/2 A -B —6 933% - — (5) <1 a R , (3.88) ind 71/2 1 A B _ B -6 ”112115 ' 2 (E) [3 ° [“N “1.3 R 1 A A B —6 7 [EZ,ZZZ + 2EX,XXXJ 9 R 2 A A B B A -7 + fi§ Ea“ - a1} [all_ a1} 9 R ] , (3.89) ind _3;1/2 A B_B —6 “255 — 3 (5) [¢ [a], a1} R + [EA + 23“ ] SE B‘6 z,zzz x,xxx _ 2 A _ A B _ B A -7 7 [an “1.3 [an 011-] e R ] , (3.90) and ind _ 131/2 A B _ B —6 Du265 - 2 (R) [<1> ta“ “1.3 R - 2 [EA A 198 B'6 z,zzz X,XXX + % [afi - afifl Cafi - qEJ GA R_7] . (3.91) In Eqs. (3.73) - (3.91), the u and E susceptibility tensor components take their static values. The exact long-range dipole coefficients through order R_7 for the interaction of two centrosymmetric linear molecules are obtained by adding the van der Waals coefficients given by Eqs. (3.51) - (3.68) to the corresponding induction coefficients from Eqs. (3.73) - (3.91). CHAPTER A. APPLICATION OF A CONSTANT—RATIO APPROXIMATION In order to determine dipole coefficients directly from the equations derived in Chapter 3, it is necessary to have accurate values of the B tensors as functions of imaginary frequencies. Calculations of E(0,iw) are now practicable, but results are not yet available. In this chapter, simple approximation methods are described which provide estimates of the van der Waals effects in Eqs. (3.6), (3.23) - (3.26), and (3.51) — (3.68). Numerical results are found in Section A for the atom-atom systems H...H; and H...He, in Section B for the atom-diatom systems He...H and He...N and in 2 2’ 2...H2 and N2...N2. The estimates for the van der Waals dipole coefficients of the latter Section C for the diatom-diatom systems H four systems are also compared with accurate values of their induction coefficients in Sections B and C. A. van der Waals Dipoles of H...HC and H...He The dipole coefficient D for two interacting S state atoms is 7 given by Eq. (3.6). If the ratio BX(O,im)/ax(im) is approximated by a frequency-independent quantity IX, then Eq. (3.6) can be expressed in terms of the C6 van der Waals energy coefficient [79] c6 = 351” dw 01A(iw) aB(iw) (11.1) n o as 6h 65 D 2 3 c [13 - IA] . (u.2) As a first approximation, we choose for Ix the ratio of the integrals of BX(O,im) and ax(iw) over all frequencies: IX = I: dm BX(O,iw) / I: dw ax(iw) . (u.3) We assume that IX has the same relation to the static a and B values as in the Unsbld approximation. For an atom or centrosymmetric molecule in the ground state, the Unsbld approximation gives 2 2 2 ua8(im) - E <0Iuau8|0> fl/(Q + m ) (A.A) and . _ 2_ 2 2 2 2 2 BaB’Y6(0.lw) — 52 (3n + m )/(9 + w ) [ - <0|“a“slo> (0'9Y6|O>] , (u.5) where Q is the average excitation frequency of the species. Using these expressions in Eq. (A.3) and carrying out the integrations leads to xx = 28X/3ax , (u.6) where Bx and ax are respectively the static dipole-quadrupole hyperpolarizability and static dipole polarizability of X. Using Eq. (u.6) in Eq. (u.2) yields D = 2 c [BB/a8 — BA/aA] , (4.7) 7 6 in agreement with the earlier result of Galatry and Gharbi [53,5A]. Accurate values for both D7 and the B tensor are not available for many systems. In a series of large—basis perturbation calculations, Byers Brown and Whisnant evaluated D7 for a model system consisting of a hydrogren atom A and an atom B with a hydrogenic wavefunction scaled by a factor c [58]. The wavefunctions of these two systems can be expressed in atomic units as 11A = (91/2 exp E-r] , (11.8) and 3 WE = (fr—W2 exp {—ch . (4.9) The accurate results for D7 are listed in Table 4.1 for comparison with the approximations. The polarizability a; and hyperpolarizability BC of the scaled atom HC are related to a and B of the hydrogen atom by a = g a (4.10) 67 B =C B . ((4-11) Eq. (U.7) thus becomes —u D7 = 2 C6 B (c - 1) / a . (4.12) For the hydrogen atom, a = g and B = - 2%; in atomic units. Values of D as computed using Eq. (H.12) are given in Table ”.1 for 7 c = 1.0 - 2.0; the error in this approximation ranges from 15% for small ; to 12% for large c. For large c, Eq. (4.12) gives a slight improvement over the Unsbld approximation of Byers Brown and Whisnant [58]. The approximation of Ix in Eq. (A.3) does not take into account the frequency-dependent weighting of B tensor values for one atom by the a polarizability of the other, as found in the exact expression of Eq. (3.6). This suggests that Ix might be estimated more closely by the Unsold approximation for the ratio on B 00 IA = f0 dw BA(O,iw) a (it) / f0 dw aA(im) aB(iw) . (n.13) Employing the Unsold approximations given by Eqs. (4.“) and (u.5) in Eq. (H.13) and carrying out the integrations leads to [2 +-————13———] . (u.1u) 68 . . . B . A B . with a Similar result for I . USing I and I together in Eq. (4.2) yields A 1- -B— [2 + —‘———11. (21.15) BB 1 (1 + QA/QB) GA (1 + QB/QA) _B[2+ (1 D7 = 66 [ Specializing to the pair H...HC, Eq. (4.15) becomes D7 = c6 B (§_u - 1) [2 + ;2(1 + c2)-2] / a . (4.16) As Table 4.1 shows, Eq. (4.16) provides good estimates for D7, underestimating the accurate results by only 4.3 - 4.5% over the entire range of c values. As a second test of the approximations, consider the interaction of hydrogen with helium. The accurate value of D7 is 120 a.u. for H+He_ [59]. With the values (all in a.u.) eHe = 1.383 [80], BHe = — 6.587 [81], and C6(H...He) = 2.82 e 0.01 [82], Eq. (4.7) yields D7 = 107 a.u., while a direct Unsold approximation gives D7 = 108 a.u. [59]. An improved estimate may be based on Eq. (4.15). Substituting the ratio of ionization potentials IH/IHe = 0.553 for QH/QHe in Eq. (4.15) yields D7 = 122 a.u. (~2% error). Table 4.1 The long—range dipole coefficients D interacting with a hydrogen-like atom scaled by a factor Q. system is polarized H+H - for a hydrogen atom The C’ ; C6 [58] D Perturbation Unsbld Eq. Eq. results [58] approx. (4.12 (4.16 [58] 1.0 6.499027 0 O O 1.1 4.862103 85.65 72.84 72.95 81.99 1.2 3.703157 106.29 90.31 90.75 101.73 1.3 2.865143 102.87 87.29 88.13 98.43 1.4 2.247960 91.48 77.51 78.71 87.51 1.5 1.785978 78.49 66.39 67.84 75.06 1.6 1.435108 66.29 55.97 57.56 63.38 1.7 1.165113 55.64 46.90 48.55 53.18 1.8 0.9548598 46.65 39.25 40.89 44.58 1.9 0.7893366 39.17 32.91 34.50 37 42 2.0 0.6577167 33.00 27.68 29.19 31.52 70 B. van der Waals Dipoles of He...H2 and He...N2 As in the atom-atom case, a constant-ratio approximation may be vdW used to connect the DAL coefficients given by Eqs. (3.23) - (3.26) to van der Waals energy coefficients for the interaction of an 8 state atom with a centrosymmetric linear molecule. For such a system the van der Waals energy can be expanded to order R-8 as d1: 2 —6 AE" (mar) = - [02 + 06 P2(coser)] R 2 4 -8 - [03 + C8 P2(coser) + 08 Pu(coser)] R , (4.17) where R is the vector from the midpoint of the molecule to the atom, Or is the angle between this vector and the molecular symmetry axis (coser = F ' R), and P2 is the 2th Legendre polynomial. The energy . . 0 2 O 2 4 . coeffiCients C6, C6, C8’ 08, and C8 are expressed in terms of the susceptibilities of the separated species as 02 = 3 g I: du) 01A(iu)) 5’30...) . (4.18) c2 = 5-1“ dm aA(iw) [aB(iw) - aB(iw)] (A 19) 6 n o 11 l— ' ' c2 = 15 %-f: dw [CA(iw) &B(iw) + 03(iw) aA(iw)] , (4.20) c2 547” d [7 CA(i ) [aB(i ) - 0113(1)] 8 711 o w “’ 11 “’ .L ‘*’ B , B , B . A . + 3 [5C22,zz(lw) + ucxz,xz(lw) - 8CXX,XX(1w)] a (1w) B . B . A . + 3 [3Ez,zzz(lw) — 8Ex,xxx(lw)] a (1w)] , (4.21) 71 and u L; h a) B . _ B . B . A . C8 = 7-;~f0 dw [3 [2C2z,zz(lw) 4CXZ’XZ(1w) + Cxx,xx(1w)] a (1m) + 5123mm.) + zai’xxxum Jam] . (4.22) A susceptibility that appears for the first time in these expressions is the quadrupole polarizability C (iw). For the a8,Y6 atom A A . A . 1 1 = — + -— Gamma...) c (14)) [2 ( <019a310><0101510>1 —(92 + (”2) , (4.25) and . 2 (2 Emmsu‘”) ‘ 11 (0140191311510) 2—2' (4°26) (Q + w ) 72 We assume that the component of the B hyperpolarizability that transforms as a spherical tensor of rank 2 will be best approximated in terms of components of g, g, and E of the same spherical tensor vdw . . A A rank. If DAL can be approx1mated in terms of either C6 or C8' the former is preferred, because more reliable values of the cg energy coefficients are available. The resulting approximations are B A vdW ; o -7 B_ _ §_ D01 — 2 c6 R [_B A , (4.27) (1 a vdw - 1 1/2 2 —7 BA 1321 = -5— (2) c6 R [Cl—A B B B (38 + B + 10B ) 4 2b B 208 2d 1 , (4.28) (a - a ) H .L B B (8 2B ) A vdw : 8_ 1/2 2 —7 2b 2c _ _B_ 1323 —15(3) 0611 [2 B B A] , (4.29) (an _ (1L) 0. and vdw : _ 2‘2 4 —7 B Du3 — 3 Ca R Bu B B B B B -1 x [3 (zczz,zz — xz,xz + xx,xx) + 5 (£2,222 + x,xxx ] (”'30) All susceptibilities appearing in Eqs. (4.27) - (4.30) take their . B B static values; at zero frequency B = . xz,xz zx,xz The susceptibilities and van der Waals energy coefficients are known for the He...H2 system and are given in Table 4.2, so we can dW determine the DIL coefficients from Eqs. (4.27) ' (4.30). Fixing - _. 4 44.1%. 1 “.711. Dgfiqn-JQJSFRTWT, uh- r. «sn‘flizml‘ :91qu 1M ‘ 1 than '9: 23-5-1“. . _.. . .,._ . _1 _ ‘n‘. 2'" .__ '3' . ‘ 02;" - . 1.46 0'7 , -. _ -. (4.33) and vdW -7 D43 0.01 R . (4.34) A positive value for the net dipole moment corresponds to the 2 C For comparison, the long-range induction dipole coefficients polarity He+H for the He...H2 system are 03’1“ - 1.37 3‘7 , (4.35) 0:?d - - 9.29 3'7 , (4.36) D230 - 1.094 R'” + 4.68 3'7 , (4.33) 0,13“ = — 0.78 3'7 , (4.38) and 74 Bind = o 87 R‘ (4 39) “5 . . . These dipole coefficients are expected to increase by ~ 10% if the internuclear separation in the H molecule is increased to its 2 ground-state vibrational average of 1.449 a.u.; an exception is Dfigd, which increases 25% when the vibrationally averaged value of the hexadecapole is used in place of the value in Table 4.2. The constant-ratio approximations suggest that van der Waals interactions affect D and D significantly for He. ..H 01 21 at long 2 vdW range. In fact, D01 is roughly fifty times Dad in magnitude. The D23 coefficient is dominated by the quadrupole-induced dipole, which is proportional to B_u. The van der Waals contribution to D43 is very small compared with the induction contribution, though both vary as R-7 at lowest order. Dispersion effects on the total collision-induced dipole moment are small relative to induction effects, but still appreciable. When the intermolecular vector R points along the z axis of the H molecule, the long-range van der 2 Waals term in the dipole “z is 13% of the induction term for collinear He...H2 at R = 7 a.u., and is 19% of the induction term for the T-shaped geometry at this distance. At R = 5 a.u., the van der Waals contribution increases to 33% of the induction term in “z for the collinear configuration and 50% for the T-shape. However, overlap damping of both the induction and dispersion dipoles reduces DXEW and Dizd below their asymptotic forms at this intermolecular distance, and short-range exchange effects predominate in the _ag initio collision-induced dipole for R less than the collision diameter of 5.7 a.u. [68,90]. We may conclude that van der Waals 75 effects on the dipole moment of this system are most significant when the D01 term can be distinguished from the D23 term, as in spectroscopic lineshape studies. Selection rules for transitions of the atom-diatom complex differ for the D01 and D23 terms [16]. Given the accuracy of the approximations in Section A to the D7 coefficient for H...H; and H...He pairs, together with the accuracy of a similar approximation for the van der Waals contributions to the pair polarizabilities of H and He atoms [49], the error in D317) (the coefficient of R—7 25%. The value D61!) ; — 71 a.u. determined here agrees with the result of a valence-bond calculation by Berns e_t a1. [90] within this error margin. The valence-bond result, D617) ; - 62 a.u., was in D01) is expected to fall between 15% and obtained from a numerical fit of the calculated dispersion dipole between R = 7 a.u. and 10 a.u. For comparison, we may assume that the van der Waals dipole of He...H2 corresponds to a shift in the center of the electronic charge distribution of each system by a distance (5 = 28' Then, if (S is determined by balancing the dispersion energy gain vs. the energy required for polarization, the computed dispersion dipole coefficient for He.. .H is D(7)= — 2 01 48.5 a.u. [17,18], or - 44.5 a.u. using the molecular properties from Table 4.2. Meyer and Frommhold fit the collision-induced roto- translational absorption spectra of He...HZ, yielding an effective value of D317) = - 81 a.u. [17,18]; but their fitting procedure incorporates not only the leading R_7 terms, but also all higher- order induction and dispersion effects in the range R = 7 a.u. to 10 a.u. A direct ab initio calculation of DEW based on Eq. (3.23) 76 would be valuable both in assessing the different numerical results and in providing a test of approximations analogous to Eq. (4.27). . vdW vdW vdW . The estimates of D21 , 023 , and D43 in terms of the anisotropic C2 and cg energy coefficients are probably susceptible to larger errors than the isotropically averaged ngw. However, the charge shift model described above gives Dé:) = 8 a.u. [17,18] or 9.9 a.u. using the properties in Table 4.2, in good agreement with the value of 10.53 a.u. from Eq. (4.32). Experimental results for these coefficients are not available; numerical noise prevented the recovery of an effective value of D;:) from the most recent fit of the collision-induced absorption spectra [17,18], and the fit of D23 reflects the quadrupole-induced dipole predominantly. The susceptibilities and van der Waals energy coefficients are known for the He...N2 system as well and are given in Table 4.3, so we can also determine the DAL coefficients for this pair. However, as can be seen from the table, a discrepancy in both sign and order of magnitude occurs for the susceptibility component B4 as computed using the ab initio results of Jameson and Fowler [87] and of Dykstra [86]. It is therefore pointless to compute ngw (its vdW contribution is small in any case). The remaining DAL coefficients are, in the constant-ratio approximation, 3‘?” = — 130.0 (— 113.96) R”, (4.40) 02‘1”“ = 17.41 (28.28) R_7 , (4.41) and flu? ssmparisbn. the langrrange.1ngucsion dipolsheberfteiants Eon the He...N2 system are 03',” g — 8.42 3‘7 , -- (4.43) 0:?“ = 50.53 8'7 , (4.44) 0:533“ = — 2.61 13‘” - 25.19 6‘7 , (4.45) D1130 = 4.81 3‘7 , (4.46) and Digd = — 23.10 2'6. (4.47) The approximation results show a significant contribution from the van der Waals interaction to the coefficients D and D2 for 1 33W is about fifteen times D;?d in 01 He...N at long range; D 2 magnitude. Again, the D23 coefficient is dominated by the quadrupole—induced dipole, which is proportional to B_u. We note that here the van der Waals and induction contributions to each dipole coefficient enter with the same sign and so enhance one 78 another, while these contributions were of opposite sign in the He...H2 system. (7) As with the earlier approximations, the error in D01 should fall between 15% and 25% if the static B tensor is accurately known. However, the two calculations of Dde 01 using values from Refs. [86] and [87] for the B tensor components give results that differ by nearly 15%, and this increases the likely error. The difference between the two results for ngw is greater, but the values should indicate the magnitude of this coefficient. The two results for Dggw differ so much that the usefulness of the estimate is questionable. More reliable values of the static B tensor components would improve the estimates based on Eqs. (4.27) — (4.30) for He...N direct_ab initio calculations based on Eqs. (3.23) - 23 (3.26) would be even more valuable. 79 Table 4.2 Molecular properties and van der Waals energy coefficients used in calculating D for the He...H system. The internuclear distance AL 2 in the H2 molecule is held fixed at its equilibrium value of 1.4 a.u. All values are given in atomic units. Property Value Property Value 3 8 He a 1.383 b sz’zz 5.927 B —6.587 c 4.2423 xz,xz c 4.944g c xx,xx h H 0 0.4568 E 3.93 2 d z,zzz h 6 0.2826 E 1.76 e x,xxx a” 6.380 a_,_ 4.578e He... 02 3.9043 _ f 2 i Bzz,zz 92.12 C6 0.445 B -56.92f c“ 0.3081 xz,xz 8 f Bxx,zz 30.54 f -62.24 xx,xx a) Ref. 80; b) Ref. 81; c) Ref. 83; d) Ref. 84; e) Ref. 85; f) Refs. 86 and 87; g) Ref. 88; h) Ref. 89; i) Computed using isotropic Cn coefficients from Ref. 88 and anisotropy factors from Ref. 89. . _ . V I. 'l:"' 1 ‘_ a. n I_. I' f gin-If- H. "5 .. I seamim-héetmw-HM-m M6: -: A; a: emis- ...;3161'1131- --...‘: 3| IO!!! Ofelia-'4 . I'I '- .‘-.v .:, are. nevi; I'D WM" 4 1 ‘ '1 .0' . " 4"“ . “'91 Table 4.3 80 Molecular properties and van der Waals energy coefficients used in calculating D for the He...N2 system. The internuclear distance in the N2 molecule is held fixed at its vibrationally averaged value of 2.07 a.u. Property He a Q 9 F : zz,zz xz,xz xx,zz XX,XX UJUJUJCUUJIWCDCDUJ NNN 000’ 1:: a) Ref. 80; b) Ref. 81; 0) Value 1.383a N -6.587b —1.o9C He...N -7.47d 14.718e 10.065e -174f (—17o.55g) f 8 —102 (—105.35 ) 67f (61.233) —119.5f (-97.768) —132.4f (-122.9og) f 8 2.95 (8.54 ) -7.21f (—12.428) —7.21f (—12.428) -1.13f (0.1873) 87; g) Ref. 86; h) Ref. 93. All values are given in atomic units. Property E z,zzz X,XXX O O O\I\)O’\O Value 22.04h 16.67h 9.795h 1.103h Ref. 94; d) Ref. 95; e) Ref. 96; f) Ref. 81 C. van der Waals Dipoles of H2...H2 and N2...N2 We now apply the constant-ratio approximation technique to the vdW )1AZAL interaction of two centrosymmetric linear molecules. The dipole coefficients given by Eqs. (3.51) - (3.68) for the coefficients will be connected to the van der Waals energy coefficients for this system; the energy can be expanded as A142m vdW -n m m AEn (R,01,¢1,02,¢2) ~ R Z Cn PA1(cose1) PA2(c0502) x ccs[m(¢1-¢2)] . (4.48) where the summation is over )1, )2, and m. The particular energy coefficients to be used in the approximation are 0200 = 3 2 f: 0.1 5’41...) £13111.) , (4.49) 0200 —4 f” d [A(i ) — A(1 )3 73(1) (4 50) 6 - '11 O U.) a“ (1) OJ— (L) (1 (1) y . 0220 = 2— 7: 01. [82(16) — 63316)] [65(11.) — ai(iw)] , (4.51) cum—347% [3t20A (1)—4cA (- )+cA (' )1 8 — 7 n o w zz,zz m xz,xz 1w xx,xx 1m + 5 [3222204) + 2E:,xxx(iw)]] 850.1) , (4.52) 6m "40': (88.8)“ — 'i (refs-16111111116678 . ._.. yansns aissw «19".- r- . 5111 0:1 b533"-‘fi09 ed 1116: ataolfil ,. 4.541. anqu edT .aeiuaeiom menu shaman-tam «I 16. as bebnsqxv 01 has (an-us 9r. :rsJegw min: 101 63n089I11l08' - . .- OTI . .‘ . , . ' " . . . ..' (-0901) ‘ ( Gare} 9 3 i H ' (, 4.,u..L.E1 3A 1 82 420 _ 4 h w A . _ A . A . 08 — 165'? f0 dw [66 [zczz’zz(1w) "sz,xz(1w) + Cxx,xx(1w)] + 95 [E zz(1..) + 22“ (16)J] [65(16) — 62516)] . (4.53) A 2,2 x,xxx The Unsbld approximations for the a, B, C, and E tensors were given earlier by Eqs. (4.4), (4.5), (4.25), and (4.26). Using these equations together with Eqs. (4.49) - (4.53) results in the approximations de : 000 -7 1. _ E— a a (38 + B + 10BA ) ‘B Dvdw ;.l (2)1/2 C200 R-7 [4 20 20 2d -.§_] (u 55) 2021 5 6 A A _B ’ (01“ — (11L) (1 B B A A DVGW :‘_§ [1)1/2 C220 3-7 [(B20 + 2d) _ (820 20)] (4 56) 2201 - 15 5 6 ( B - B) ( A _ A) . a” aL “11 BI BB BA WW :5. 11/2 220 -7 20 2d 2211 ' 5 (5) C6 R [( B _ ) + ( A _ )1 . (4 57) a” a; 0'“ (IX—L B B B Dde ; 11_ (1‘4” 022° 4'7 [W 2221 15 35 6 ( B B a” al) A A A (6B2b 2820 + 7B2d)1 (4 58) (“11‘ 8i) vdW ~ 1/2 420 —7 A D4221 - 4 (7) 08 R Bu x [66 (204 A . CA ) . 95 (EA A )]_1 zz,zz xz,xz xx,xx z,zzz x,xxx ’ (4.59) 83 (BA + 23A ..B ) vdw ; 8 1/2 200 -7 §_ _ 2b 2c D2023 "T5 (3) C6 R [-3 2 “"A"“K“‘] . (4.60) °‘ (“11‘ ‘1) A A Dvdw ; lg (_2_)1/2 C22o R-7 [(82b 2320 + 252d) 2223 15 105 6 ——?_K':—7f—‘-_—' B B a)! “4) _ (82b ' 2820 + 282d)] (4 61) ( B — B) ’ ' “11 “J. A A Dvdw ; _._4 (_2)1/2 C22o R—7 [(5326 + 832d) 2233 15 15 6 ( A _ A) B B a" a4- (SB2b + 832d) * "'—-—-‘—---] . (4.62) (GB - (18) ll .L A A A DVdW ; _B.(_§)1/2 C22o R—7 [(5B2b + 4320 + 2”132(1) 2243 15 35 6 A A (a - a ) B B Q ‘L _ (5132b + 4520 + 2482d)] (4 63) ( B - B) ' ' “11 G; vdW : B_B 400 -7 A Du0u3 — 3 08 R Bu A A A A A -1 x [3 (”322,22 - 4cm,xz + cxx’xx) + 5 (£22,222 + 2Ex,xxx)] , (4.64) vdw ; 2 14 1/2 420 —7 A D4223 - g (7) C8 R Bu A A A A A -1 x [66 (ZCZZ'ZZ - uCXZ-XZ + CXX.XX) + 95 (Ez,zzz + 2Ex,xxx)] ’ 84 vdw : _fi _1_1/2 420 —7 A D4233 ‘ 3 (3) C8 A B4 A A A A A —1 x [66 (zczz,zz sz,xz + xx,xx + 95 (Ez,zzz + 2Ex,xxx)] ’ (4.66) vdW ; 5 1/2 420 —7 A Du2u3 — —3- (77) C8 R Bu x [66 (2c - A + A + 95 (EA A )]"1 . zz,zz xz,xz xx,xx z,zzz x,xxx (4.67) B B A A DvdW : .8. (3)1/2 C22o R—7 [(B2b + 2B2c) _ (B2b + 2B20)] (4 68) 2245'37 6 B_B (A_A ' ‘ (a,l qL) a” a4) vdW ; 2o g1/2 420 —7 A D4245 '_3' (11) C8 A B4 x [66 (2cA - 4cA + cA ) + 95 (EA + 25A )]_1 . zz,zz xz,xz xx,xx z,zzz x,xxx (4.69) vdW ; _140 51/2 420 —7 A 4255 ' 3 (3) Ca A B4 x [66 (2cA — 4 A + A + 95 (EA + 213A )]‘1 , zz,zz xz,xz xx,xx z,zzz x,xxx (4.70) and vdW ; 700 261/2 420 —7 A D4265 ‘ 3 (33) Ce A Bu x [66 (2cA - A + A )+ 95 (EA + 215A )]‘1 zz,zz xz,xz xx,xx z,zzz x,xxx (4.71) The susceptibilities that appear in Eqs. (4.54) - (4.71) are l t d t f n where BB — BB eva ua e a zero reque 0y. xz,xz ' zx,xz' J -.‘) “ ° ...»- EQ’QC‘ s"; 85 Susceptibilities and van der Waals energy coefficients are given in Table 4.4 for the H2...H2 system, so the dipole coefficients can be determined from Eqs. (4.54) — (4.71) . We use the vibrationally averaged separation = 1.449 a.u. in the ground state as the representative value for the internuclear distance in the H2 molecule; the values in Table 4.4 are given for this distance. The B tensor components were interpolated graphically to r = from the data of Dykstra [86]. who gives values over a range of internuclear separations. Values for the E tensor components are unavailable at the vibrationally averaged distance, but since they only affect terms of A = 4 symmetry (which are small), they may be estimated from their values at the equilibrium separation without introducing significant error. The change in the B and C tensor components upon increasing r from req to averages to 7.05%. Using this as a correction factor to the Mulder gt El; values [89] gives €2,222 = 4.21 and Ex,xxx = 1.88. Results for the van der Waals dipole coefficients are compared with the induction coefficients in Table 4.5 for H2...H2. The van der Waals contribution to D2211 is four times the induction is about 60% of D1nd 2021. Van der Waals effects . . vdW contribution, D2021 are very slight for the remaining coefficients. The induction coefficients show an appreciable contribution from back-induction, 7 (to lowest order). even though these enter only at order R- Interpretations of collision-induced spectra typically include only the effects of quadrupole and hexadecapole induced dipole moments; addition of back-induction terms may improve the agreement between theory and experiment. 86 In Table 4.6 are listed the susceptibilities and van der Waals energy coefficients for the N2...N2 system. As discussed in Section B of this chapter, the large discrepancy in the B4 values from Refs. [86] and [87] precludes computation of the dipole coefficients that depend on them. The remaining van der Waals dipole coefficients are listed in Table 4.7 (values in parentheses were determined using the results from Ref. [86] for the B tensor components), along with all of the induction dipole coefficients. Inclusion of back-induction effects is again seen to make an appreciable contribution. Lastly, using the values in Table 4.4 for H and in Table 4.6 2 for N2, along with the result C200(H2...N2) = 29.28 a.u. [93], we may determine Dggg1 for the H2...N2 system from Eq. (4.54). We find vdW _ '7 D0001 — 169.4 (217.3) R , (4.72) where the result in parentheses is based on the B tensor values from Ref. [86]. For comparison, the induction dipole coefficient is 7 And = - 46.20 R_ . (4.73) D0001 From these results it is clear the the van der Waals contribution to the D dipole coefficient is much greater than 0001 the induction contribution. 87 Table 4.4 Molecular properties and van der Waals energy coefficients used in calculating DA1A2AL for the H2...H2 system. The internuclear distance in each H2 molecule is held fixed at its ground-state vibrationally averaged value of 1.449 a.u. All values are given in atomic units. Property Value Property Value a e H2 9 0.4847b H2 sz,zz 6.357e 0 0.3532 sz,xz 4.512e a" 6.713 cxx xx 5.227 6L 4.736c Ez ZZZ 4.21f B —1oo.31d E 1.88f zz,zz d X, XXX sz xz —61.44 ' d 000 g Bxx'zz 32.63 H2...H2 06 12.078 d 200 h Bxx'xx 66.20 06 1.233 B -77.10d 0:20 0.398h d 400 h B2b 0.482 08 1.38 d 420 h B20 4.02 ca 0.57 _ d 32d 4.02 d Bu ~o.1o1 a) Ref. 83; b) Ref. 84; c) Ref. 91; d) interpolated for r = from data in Ref. 86; e) Ref. 88; f) estimated for r = from values in Ref. 89 (see text); g Ref. 92; h) Ref. 93. V Table 4.5 Comparison of van der Waals and induction contributions to the D A1A2AL coefficients for the H2...H2 system. The internuclear distance in each H2 molecule is held fixed at its vibrationally average value of 1.449 a.u. All values are given in atomic units. A1 A2 A Xfrzit DA::2)L 2 0 2 1 — 27.24 R'7 40.192 R_7 2 2 1 1 — 0.58 R_7 0.145 R"7 4 2 2 1 — 8.12 x 10‘Ll R-7 0.049 R_7 2 0 2 3 — 6.46 R‘7 — 4.529 R'A — 19.88 8'7 2 2 3 3 1.36 R-7 0.700 B'A — 3.828 R’7 4 0 4 3 — 0.07 3‘7 3.545 R‘7 4 2 2 3 - 1.11 x 10‘” R‘7 0.007 R_7 4 2 3 3 5.17 x 10'” B‘7 ~ 0.031 3‘7 4 2 4 3 — 1.12 x 10‘3 8‘7 0.068 R_7 4 o 4 5 o — 4.258 3'6 4 2 4 5 - 9.13 x 10'A 3'7 - 0.23 R_6 + 0.055 3‘7 4 2 5 5 4.62 x 10’3 B'7 2.36 R‘6 - 0.280 R'7 4 2 6 5 — 0.02 B'7 — 6.24 R'6 + 0.961 R"7 89 Table 4.6 Molecular properties and van der Waals energy coefficients used in calculating DA1A2AL for the N2...N2 system. The internuclear distance in each N2 molecule is held fixed at its vibrationally averaged value of 2.07 a.u. All values are given in atomic units. Property Value Property Value N2 0 -1.09a N2...N2 0200 73.88 ¢ —7.47b 0:00 7.828 a“ 14.718C 0:20 2.678 @l 10.065c . Bzz’zz —174d (~170.559) sz,xz —102d (—105.35e) Bxx,zz 67d (61.239) Bxx xx -119.5d (—97.76e) é ’ -132.4d (-122.9oe) 32b 2.95d (8.549) B2c -7.21: (-12.42e) 52d —7.21 (—12.429) Bu —1.13d (0.1876) Ez'zzz 22.04: EX'XXX 16.67 a) Ref. 94; b) Ref. 95; 0) Ref. 96; d) Ref. 87; e) Ref. 86; f) Ref. 93; 8) Ref. 97. 90 Table 4.7 Comparison of van der Waals and induction contributions to the DA1A21L coefficients for the N2...N2 system. The lnternuclear distance in each N2 molecule is held fixed at its vibrationally averaged value of 2.07 a.u. All values are given in atomic units. A1 A 2 A A” D: 211:2“. D232” 2 0 2 1 - 108.73 (- 187.61) 8‘7 - 423.05 8‘7 2 2 1 1 - 2.96 (- 5.10) 8'7 - 1.81 3'7 4 2 2 1 — - 0.61 3’7 2 o 2 3 — 46.73 (— 25.83) R'7 21.93 R'” + 207.21 8'7 2 2 3 3 4.80 (6.33) 8'7 - 3.70 13‘“ + 43.69 8‘7 A 0 4 3 — - 40.41 R_7 4 2 2 3 - - 0.08 8'7 4 2 3 3 - 0.39 8'7 4 2 4 3 - - 0.85 8‘7 4 0 4 5 0 194.03 8'6 4 2 4 5 - — 2.12 8'6 — 0.69 8'7 4 2 5 5 — - 49.12 8‘6 + 3.48 8‘7 4 2 6 5 - 76.30 3'6 — 11.97 R'7 CHAPTER 5. DISCUSSION AND CONCLUSIONS One contribution to the collision-induced changes in the energy, dipole moment, and polarizability of a molecular pair comes from the van der Waals interactions between the colliding partners. The van der Waals interaction arises from the correlations in the fluctuating charge distributions of the two molecules, which at long range can be characterized by a multipole expansion. The correlations are then connected to the susceptibilities of the individual molecules by use of the fluctuation-dissipation theorem. Application of an external static field alters the response of each molecule to the fluctuating field of its neighbor, and induces new correlations between the fluctuating charge distributions on each molecule. A reaction field model incorporating these effects was developed in Chapter 2, and expressions for the long-range van der Waals dipole and polarizability of a molecular pair were derived. Field-induced fluctuation correlations have not been included in earlier models of the van der Waals contribution to pair dipoles and pair polarizabilities. Previous methods of calculating dispersion dipoles [53-55] have been based on use of the fluctuation-dissipation theorem. One method [55] accounts for the polarization of molecule A bilinear in the field due to the fluctuating multipoles of molecule B and the external field E. The polarization of A sets up a reaction field at B, producing a shift in energy that depends on the magnitude and direction of E, The energy shift at A is found similarly, and the van der Waals dipole is obtained by differentiation of the total 91 92 interaction energy, as in Eq. (2.31). However, in calculating the energy shift, the external field is restricted to a region around molecule A and vanishes at molecule B. Thus the field-induced fluctuation correlations included in this dissertation are not present in the method of Ref. [55]. In a second method [53-55], the van der Waals dipole induced in each molecule is computed directly from its nonlinear polarization by the nonuniform field of the fluctuating multipoles of the neighboring molecule, in the absence of any external field. For a pair of S state atoms A and B, the dipole induced in A is expressed in terms of an integral over the frequency w of the product aB(w) BA(w,-w). But the hyper- polarizability BA(w,-w) has poles in both the upper and lower w half planes, which prevents a straightforward application of the residue theorem to evaluate the integral (see Eqs. (2.23) - (2.25)). This problem is avoided here, because the model gives an expression for the interaction energy that depends upon the linear response of each molecule, as modified by the presence of the external field. For a pair of S state atoms, the van der Waals dipole given in Refs. [53-55] is equivalent to Eq. (3.5) of the dissertation, but the derivations differ in physical content. Symmetry-adapted equations were derived in Sections B and C of Chapter 3 for the leading induced dipole coefficients of collision systems comprising an atom and a centrosymmetric linear molecule or a pair of centrosymmetric linear molecules. These results are new, and provide a basis for comparison of the van der Waals and induction contributions to collision-induced dipole moments. Application of an approximation technique, in which the dipole coefficients are 93 expressed in terms of static susceptibilities and van der Waals energy coefficients, was carried out in Chapter 4. Results for the model systems H...H; and H...He show this approximation method gives agreement with accurate perturbation calculations to within 15%. Analogous approximations give numerical estimates to the van der ...H Waals dipole for the systems He...HZ, He...N2, H 2, H N and 2... 2, N2...N2. When compared with the induction dipole, the dispersion 2 contribution is seen to be significant for certain symmetry components, particularly D (for He...HZ) and D (for H2...N2). O1 0001 Given the high accuracy now possible in measurements of the roto- translational absorption spectra of atom-diatom and diatom-diatom complexes, it is necessary to account for van der Waals effects in order to obtain agreement between theoretical and experimental lineshape analyses. This dissertation opens many avenues for future studies. The van der Waals dipole moments of more systems may also be estimated, as values for their susceptibility tensors and van der Waals energies become available. Improved approximations could also be developed, possibly using Pade approximants for the susceptibil- ities, or combination rules to estimate the dipole of a system based on the values for related systems. Ideally, direct ab initio calculations based on the integral equations would give the best results. Eq. (2.37) gives results for the van der Waals contribution to the polarizability of a pair of centrosymmetric molecules; Eqs. (2.47) and (2.48) specialize this to an atom pair. The same techniques that were used in deriving the van der Waals dipoles of APPENDICES Appendix A. Equivalence of Reaction Field and Perturbation Results for Long-range Dipole Moment of Two S State Atoms Craig and Thirunamachandran [60] have shown directly from two- center, third-order perturbation theory that the van der Waals dipole of interacting S state atoms A and B at long range is A 048 .018 AB uvdW _.§ 5 =l|:U) 3‘7 f: dm [BB (16) a$Y(iw)] (A.1) . A . _ — aB,aB(lw) aYY(1w) B in atomic units (h = 1). The dipole-quadrupole hyperpolarizability E(iw) is defined as 60m mn no 60m mn no B (16) — z [ “A “Y “A + “A “Y “A ‘ ———T‘ _T aB,Y6 m,n EnO(EmO 1 w) EnOAEmO + 1 w) 60m mn n0 + a8 “6 Y (EnO - 1hw)(EmO - ihw) 60m mn n0 + a8 “6 “Y (Bno + iRw)(BmO + £80) uOm ~mn Lln0 uOm émn un0 + (E6 -a§5m)E + (E(3 +a§hw)E ] (A'2) n0 m0 n0 m0 The matrix element 022 is [58-60] émn = < w I —-l 2e (r. r. - 1-6 r?) I w > (A-3) as m 2 = 30 JB 3 as J n 95 96 for an atom with ne electrons. Matrix elements of the quadrupole mn . . . . . ~mn operator 9&8 used in this dissertation are three times Qas' Ignoring damping, which is negligible at imaginary frequencies, Ba8,Y6(O’lw) 18 [98-100] (0,16) = 3 B (m) . (A.4) BaB,Y6 Y6,a8 Then from the relations 13 (on—£13m» (A) 68.03 , w — 2 ,1w .5 (A.6) aYY(im) = 3 a(iw) , it follows that Eq. (A.1) is equivalent to Eq. (3.5). 97 Appendix B. Frequency-dependent B and E Tensors for Linear Molecules We seek expressions for the frequency-dependent B and E tensors for a linear molecule, in terms of the orientation of the molecule with respect to the laboratory frame of reference. Beginning with the B tensor, we can in general write BaB,Y6(o’iw) = a0!i aBj aYk a51 Bij,kl(o’im) , (8.1) where a, B, Y, and 6 are the space-fixed laboratory axes, and i, j, k, and l are molecular axes; the aai are direction cosines between the two frames. g is a fourth-rank tensor with 81 cartesian components (for simplicity only the subscripts ij,kl are written): xx,xx xx,xy xx,xz xx,yx xx,yy xx,yz xx,zx xx,zy xx,zz xy,xx xy,xy xy,xz xy,yx xy,yy xy,yz xy,zx xy,zy xy,zz xz,xx xz,xy xz,xz xz,yx xz,yy xz,yz xz,zx xz,zy xz,zz yx,xx yx,xy yx,xz yx,yx yx,yy yx,yz yx,zx yx,zy yx,zz yy,xx YY3XY YY:XZ yy,yx YY’YY YY»YZ YY:ZX YY:ZY YY,ZZ yz,xx yz,xy yz,xz yz,yx yz,yy yz,yz yz,zx yz,zy yz,zz zx,xx zx,xy zx,xz zx,yx zx,yy zx,yz zx,zx zx,zy zx,zz zy,xx zy,xy zy,xz zy,yx zy,yy zy,yz zy,zx zy,zy zy,zz zz,xx zz,xy zz,xz zz,yx zz,yy zz,yz zz,zx zz,zy zz,zz For an axially symmetric molecule, components with a single x, y, or z vanish, leaving xx,xx xx,yy xx,zz xy,xy xy,yx 98 xz,xz xz,zx yx,xy yx,yx yy,xx YYvyy yy,zz yz,yz yz,zy zx,xz zx,zx zy,yz zy,zy zz,xx zz,yy zz,zz The B tensor is symmetric with respect to interchange of quadrupole indices, so xy,xy = eryX . yx,xy = yx,yx , xz,xz = xz,zx , zx,xz = zx,zx , (B.2) yz,yz = yz,zy , zy,yz = ZY.ZY 9 leaving the following representative components: xx,xx yy,xx zz,xx xy,xy yx,xy xz,xz zx,xz xx,yy yy,yy zz,yy yz,yz ZY.YZ xx,zz yy,zz zz,zz Because the molecule is symmetric about its 2 axis, rotating the x axis onto the y axis and the y axis onto the -x axis leaves 2 unchanged. Thus xy,xy = yx,yx , xx,xx = YY3YY : xz,xz = yz,yz , zx,xz = zy,yz , yy,xx = xx,yy , zz,xx = zz,yy , (B.3) xx,zz = yy,zz , leaving the following representative components: xx,xx (= YY,YY) zz,xx (= zz,yy) xy,xy (= xy,yx = yx,xy = yx,yx) xx,yy (= xz,xz (= zx,xz (= xx,zz (= zz,zz We note that B yy,xx ) xz,zx = zx,zx yy,zz ) 88,77 = zz,xx + zz,yy + zz,zz and xx,xx + xx,yy + xx,zz 99 y2.zy = yz,yz) zy,zy = zy,yZ) 0. So = 0 => zz,xx = - %~zz,zz , (8.4) = 0 => xx,yy = - xx,xx - xx,zz . (8.5) We consider now an arbitrary rotation of the x and y axes about the z axis. The unit vectors in the new primed coordinate system can be expressed as X' y! z! where 0 = x 0030 + y sino = - x sin¢ + y 0050 =Z is the rotation angle. We have from Eq. (8.1) xy,xy = xx' yx' xx' yx' x'x',x'x' + xx' yx' xx' yx' xx' yx' XX' yy' XX' yy' XX' yy' XX' YY' xy' yx' xy' yx' xy' yx' xx' xy' xy' xx' xx' xy' xy' xx' xx' xy' yy' YX' yy' YX' YY' yX' yy' yX' yy' yX' x'x' x'x' .X'y' ’yix' ’ylx' V ’yly' ,x'x' ,X‘y' ,yVXI + xy' YX' + xy' yy' + xy' yy' + xy' yy' + xy' yy' Just as in the y' vanish, leaving yy' yX' yy' YX' yy' unprimed system, those components with a single x' or xy,xy = xx' yx' xx' yx' x'x',x'x' + xx' yx' + XX' YY' + XX' YY' + xy' YX' + xy' YX' + xy' yy' + xy' yy' xy' xx' X'X' x'y' x'y' .y'y' .x'y' ’y'x' The same rotation and index interchange properties hold, so we have stxy = [XXI yx' XX' YX' + XY' yyv xy' ny] x'x',x'x' + [xx' yx' XY' yy' + XY' yy' xx' yX'J x'x'.y'y' + [xx‘ yy' xx' 77' + XX' yy' xy' yx' + xy' YX' XXI yyv + xy' YX' xy' yx'] vav’xvyv . Now the direction cosines are xx' = 0030 , xy' YX' = 3109 . yy' Thus 2 xy,xy = [2 cos 0 = -sin¢ , = 0030 . 2 2 sin 0] x'x',x'x' — [2 cos 0 sin2¢] x'x',y'y' + [cosuo - 2 00520 sin2¢ + sinuo] x'y',x'y' lOl Setting 6 = 45° (any value of 0 leads to the same result, as it must) gives xy,xy - é-x'x',x'x' -‘% X'X',Y'Y' = x'x',x'x' + % x'x',z'z' = xx,xx + é—xx,zz . (8.6) Using the 21 nonzero components in Eq. (8.1) gives aB,Yd = ax Bx Yx dx xx,xx + ay By Yx dx yy,xx + az Bz Yx dx zz,xx + ax By Yx dy xy,xy + ay Bx Yx dy yx,xy + ax Bz Yx dz xz,xz + az Bx Yx dz zx,xz + ax By Yy dx xy,yx + ay Bx Yy 6x yx,yx + ax BX YY 5y xx,yy + ay BY YY 5y YY,YY + dz 82 YY 5V zz,yy + ay 82 Yy dz yz,yz + az By Yy dz zy,yz + ax Bz Yz dx xz,zx + oz Bx Yz dx zx,zx + ay 82 Yz dy yz,zy + az By Yz dy zy,zy + ax Bx Yz dz xx,zz + ay By Yz dz yy,zz + az Bz Yz dz zz,zz . Making use of Eqs. (8.2) through (8.6) gives aB,Yd = ax Bx Yx dx xx,xx - ay By Yx dx (xx,xx + xx,zz) 1 -'§ az Bz Yx dx zz,zz + ax By Yx dy (xx,xx +-% xx,zz) + my Bx Yx dy (xx,xx + é-xx,zz) + ax Bz Yx dz xz,xz + az Bx Yx dz zx,xz + ax By Yy dx (xx,xx + % xx,zz) + ay Bx Yy dx (xx,xx + é—xx,zz) - ax Bx Yy dy (xx,xx + xx,zz) + ay By Yy dy xx,xx - é-az Bz Yy dy zz,zz + ay 82 Yy dz xz,xz + az By Yy dz zx,xz + ax Bz Yz dx xz,xz + az Bx Yz dx zx,xz 102 + oy Bz Yz dy xz,xz + oz By Yz dy zx,xz + ox Bx Yz dz xx,zz + oy By Yz dz xx,zz + oz Bz Yz dz zz,zz xx,xx [ox Bx Yx dx — oy By Yx dx - ox Bx Yy dy + oy By Yy dy + ox By Yx dy + oy Bx Yx dy + ox By Yy dx + oy Bx Yy dx] + xx,zz [ox Bx Yz dz + oy By Yz dz - oy By Yx dx - ox Bx Yy dy + é-(ox By Yx dy + oy Bx Yx dy + ox By Yy dx + oy Bx Yy dx)] + xz,xz [ox Bz Yx dz + oy Bz Yy dz + ox Bz Yz dx + oy Bz Yz dy] + zx,xz [oz Bx Yx dz + oz By Yy dz + oz Bx Yz dx + oz By Yz dy] + zz,zz [oz Bz Yz dz - % oz Bz Yx dx - é-oz Bz Yy dy] . (8.7) We must now cast this equation into the same form as Buckingham [74]. Working in reverse from Buckingham's expression for BoB,Yd one can easily verify that the terms involving the components xx,xx and xx,zz and zz,zz are in agreement. (Note that oz = Fo etc.) Thus Buckingham's equation remains unchanged for those components. The affected components are xz,xz and zx,xz. In the static case these are equal to each other; in the frequency dependent case they are not. Looking at the xz,xz term of Eq. (8.7), we can write xz,xz [Bz Yz (ox dx + oy dy) + 82 dz (ox Yx + oy Yy)] = xz,xz [- 2 oz 82 Yz dz + 82 Yz (ox dx + oy dy + oz dz) + 82 dz (ox Yx + oy Yy + oz Yz)] = xz,xz [- 2 oz 82 Yz dz + 82 Yz 6o + 82 dz 6o . .1 = xz’xz [5 (667 688 + 666 68)!) — T: (SOLE 6Y6 — 2—12£(3 oz Bz — 8&8) dYG + (3 Y2 dz - dYé) sag] - 5% [(3 oz Yz — 6oY) (586 + (3 oz dz — dod) GBY] + 2—15 [(3 82 72 — 8m) 8&5 + (3 82 dz - (586) 8a,] 103 . - 2 oz 82 Yz dz 2 + 7—(oz 82 6Yd + oz Yz 586 + oz dz dBY + 82 Yz 6od + 82 dz 5oY + Yz dz 6&8) 6 6 + 6 (608 6Y6 + oY Bd od 587)] _ _E 35 Similarly the zx,xz term can be expressed as zx,xz [oz Y2 (Bx dx + By dy) + oz dz (BX YX + By YY)] = zx,xz [ - 2 oz 82 Yz dz + oz Yz (Bx dx + By dy + Bz dz) + oz dz (Bx Yx + By Yy + 82 Yz)] = zx,xz [ - 2 oz 82 Y2 62 + oz Yz 686 + oz dz dBY] = zx’xz [5 (567 686 + 6&6 587) ' 1% 6GB 578 - 5% [(3 oz 82 - dag) 6Yd + (3 Y2 dz - 6Y6) 6GB] +-§% [(3 oz Yz - daY) 6Bd + (3 oz dz - God) dBY] — 5% [(3 82 Yz - dBY) God + (3 82 dz — 586) daY] — 2 oz Bz Yz dz + é-(oz 82 6Yd + oz Yz d86 + oz dz dBY + 82 Yz daé + B2 dz doY + Yz dz 5oB) 2 35 (6o8 6Yd + 5oY 686 + 6od 687)] Putting these results together gives finally the following equation for the frequency-dependent 8 tensor of a linear molecule: _3 (B + 28 + 28 + B + 48 ) zz,zz xz,xz zx,xz xx,zz xx,xx + 6 8&8’86(0,iw) = ...; 15 3 _ _ x [4 (587585 66587) 2 6&86Y6] —-—g (B + B + B — 3B - 4B 21 zz,zz xz,xz zx,xz xx,zz xx,xx x (3 Po rB- dag) dY6 + —1-(3B — 4B — 48 + 268 + 168 ) 42 zz,zz xz,xz zx,xz xx,zz xx,xx x (3 r7 r6- 6Y6) 6&8 1 + 42 (3822,22 quz,xz + 10Bzx,xz 98xx,zz 128xx,xx) ‘A‘va u’ A “A" I" . as bones-Ion u 136' m 1! {{\'f Y3 "(318) 35 SD " (261. ‘ 3. a, 1: - s7: 32‘ 3i :1) S - 1 32.31" ' 1 a 1- -.i -... - r.‘ .:8‘. ST S!) + . . . + . 4- .4". x8) 3?. .:v- + -‘.' ‘3 - J "x,xs : ' 'J) L1 ..'? 1 -_ (J l i '- - ' l I 104 x [(3 I86) 187 _ 567) 686 + (3 [a 66 — 6&6) 687] + 42 (3Bzz,zz + 10sz,xz — uBzx,xz — 9Bxx,zz _ 128xx,xx) x [(3 98 [Y — 687) 6&6 + (3 F8 F6 ' 686) GoY] + 7% (3Bzz,zz - quz,xz — uBzx,xz - 2Bxx,zz + 2Bxx,xx) x [35 1“"61 rq8 1[7 [8 ' 5 (9o IQ8 576 + r16 r81 686 A A A A + A A + ra rd dBY + rB rY God r8 r6 GoY + rY r5 6&6) + daBdYG + 6aY685 + 566587] , (8.8) where the frequency dependence of each 8 component on the right side of the equation is understood. We now consider the E tensor. Just as in Eq. (8.1) we can write (10)) = a a (iw) . (8.9) Eo,BY6 oi 83 37k adl Ei,jkl E is also a fourth-rank tensor with 81 cartesian components. For an axially symmetric molecule, components with a single x, y, or z vanish, leaving x,xxx x,xyy x,xzz x,yxy x,yyx x,zxz x,zzx Y.XXY Y,XYX y.YXX Y»YYY y,yzz y,zyz Y,ZZY z,xxz z,xzx z,yyz z,yzy z,zxx z,zyy z,zzz The E tensor is symmetric with respect to interchange of any two octopole indices, so -'.':-:. ’-.-'=J 50 . lOS x,xyy = x,yxy = x,yyx v x,xzz = x,zxz = x,zzx y,xxy = y,xyx = y,yxx , (8.10) Y»YZZ = y,zyz = y,zzy . z,xxz = z,xzx = z,zxx , Zuyyz = Z,YZY = Z,zyy , leaving the following representative components: x,xxx x,xyy x,xzz Y:XXY y,yyy y,yzz z,xxz z,yyz z,zzz Because the molecule is symmetric about its 2 axis, rotating the x axis onto the y axis and the y axis onto the -x axis leaves E unchanged. Thus x,xxx = y,yyy . x,xyy = y,yxx (= y,xxy) , (13.11) x,xzz = y,yzz , z,xxz = z,yyz , leaving the following representative components: x,xxx (= Y.YYY) x,xyy (= x,yxy = x,yyx = y,xxy = y,xyx = y,yxx) x,xzz (= x,zxz = x,zzx = y,zzy = y,zyz = y,yzz) z,xxz (= z,xzx = z,zxx = z,zyy = z,yzy = z,yyz) z,zzz I We now consider an arbitrary rotation of the x and y axes about the z axis. The unit vectors in the new, primed coordinate system can be written x' = x cos¢ + y sin¢ y! - x 31 no + Y co 5¢ 2! Z 106 wh er e ¢ is th e an gl e of ro ta tion . Us' in 8 E q . (B .9) , an d n ot' in g th at co m po ne nt 3 w‘ 1th a 31 n gl e x v, y! or z' va n' is h ju st as in th e un PP' 1m ed 3 ys te m , we ha v e X ,X W xx' X X! YX' yX' X' ,X' Xv x! + x X' + x xx' x' YY' + X xx' x' yz' + x xy' x' yx' + x xy' x' YY' xz' YX' 4. xx' xz' y2' + xz' + xz xx' y V + x xx' yx' 2' Z + x xy' y ' z! + x xy' yy' 2' z + X xz' y ' z' x + x xz' y ' z! xz' yy' Z! w e no ti ce th at xz' YY' YZ' H“ x! .X' y! y! .X' z! z! .Y' x! y! .y' y! X' .Z' X' Z' .Z' Z' X! .X' X! y' ,X' y! X! .y' X! X! .y' y! y! .Y' z! Z! .2' y! 2' .2' z! y! .X' x! Z' .X' z! x! .Y' y! Z' .y' Z! Y' ,Z' X! X' ,2' y! y! .Z' 2'2 1 = O . Th e ab 0 ve r ed uc es to 107 x,xyy = xx' xx' yx' yx' x',x'x'x' + xx' xx' yy' yy' x',x'y'y' + xx' xy' YX' YY' x',y'X'y' + xx' xy' yy' yx' x',y'y'x‘ + xy' xx' yx' yy' y‘,x'x'y' + xy' xx' yy' yx' y',x'y'x' + xy' xy' yx' YX' Y'.Y'X'x' + xy' xy' yy' yy' y1,y'y'Y' x,xyy = [xx' xx' yx' yx' + xy' xy' yy' yy'] x',x'x'x' + [XX' xx' yy' yy' + XX' xy' yx' yy' + xx' XY' yy' yx' + xy' xx' yx' yy' + xy' xx' yy' yx' + xy' xy' yx' yx'] x',x'y'y' . The direction cosines are xx' = cos¢ , xy' = — sin¢ , yx' = sin¢ , yy' cos¢ . Thus x,xyy = 2 cosz¢ sin2¢ x',x'x'x' + [cosuo - 4 coszo sin2¢ + sinu¢1 x',x'y'y' . Letting ¢ = 45°, we have x,xyy = % x',x'xlxl _ % x',x'y'y' 01" 1 x,xyy =-§ x,xxx, (8.12) leaving as representative components: x,xxx (= Y»YYY = 3 x,xyy = 3 x,yxy = 3 x,yyx = 3 y,xxy = 3 y,xyx = 3 y.yxx) x,xzz (= x,zxz = x,zzx = y,zzy = y,zyz = y,yzz) z,xxz (= z,xzx = z,zxx = z,zyy = z,yzy = z,yyz) 108 z,zzz Now, Ea,BYY = 0. So z,xxz + z,yyz + z,zzz = 0 => z,xxz = - % z,zzz , (8.13) and 11 x,xxx + x,xyy + x,xzz = 0 => x,xzz = - g-x,xxx . (8.14) Using the 21 nonzero components in Eq. (8.9) gives a,BYd = ax Bx Yx dx x,xxx + ax Bx Yy dy x,xyy + ax Bx Yz dz x,xzz + ax By Yx dy x,yxy + ax By Yy dx x,yyx + ax Bz Yx dz x,zxz + ax Bz Yz dx x,zzx + ay Bx Yx dy y,xxy + ay Bx Yy dx y,xyx + my By Yx dx y,yxx + ay By Yy dy y,yyy + ay By Yz dz y,yzz + ay Bz Yy dz y,zyz + ay 82 Yz dy y,zzy + az Bx Yx dz z,xxz + az Bx Yz dx z,xzx + az By Yy dz z,yyz + az By Yz dy z,yzy + a2 82 Yx dx z,zxx + oz 82 Yy dy z,zyy + a2 82 Y2 dz z,zzz . Making use of Eqs. (8.10) - (8.1”) gives a,BYd = ax Bx Yx dx x,xxx +~l ax Bx Yy dy x,xxx - B-ax Bx Yz dz 3 3 x,xxx 1 1 +‘g ax By Yx dy x,xxx +-§ ax By Yy dx x,xxx -'% ax Bz Yx dz x,xxx - % ax Bz Yz dx x,xxx +'% ay Bx Yx dy x,xxx + % ay 8x Yy dx x,xxx +.% ay By Yx dx x,xxx + ay By Yy dy x,xxx - g-ay By Y2 dz x,xxx — g-ay Bz Yy dz x,xxx - % ay 82 Yz dy x,xxx - é—az Bx Yx dz z,zzz - %-a2 Bx Yz dx 2,222 1 — E'GZ By Yy dz z,zzz - é-az By Yz dy z,zzz — é-az Bz Yx dx z,zzz - % a2 82 Yy dy z,zzz + a2 62 Yz dz z,zzz 109 a,BYd = x,xxx [ax Bx Yx dx + % (ax Bx Yy dy + ax By Yx dy + ax By Yy dx + ay Bx Yx dy + my Bx Yy dx + my By Yx dx) —-% (ax Bx Yz dz + ax Bz Yx dz + ax Bz Yz dx + ay By Yz dz + ay 82 Yy dz + ay Bz Yz dy) + ay By Yy dy] + z,zzz [az Bz Yz dz - é-(az Bx Yx dz + a2 Bx Yz dx + az By Yy dz + az By Yz dy + az Bz Yx dx + a2 82 Yy dy)] . (8.15) This equation must now be expressed in terms of the orientation of the molecular symmetry 2 axis with respect to the lab frame. Note that Fa = az, etc. Looking first at the z,zzz term in Eq. (8.15), we have z,zzz [az Bz Yz dz - é-(az Bx Yx dz + a2 Bx Yz dx + az By Yy dz + a2 By Y2 dy + az Bz Yx dx + az Bz Yy dy)] = z,zzz [% a2 Bz Yz dz - % (az Bx Yx dz + az By Yy dz + a2 Bz Yz dz) -‘% (az Bx Yz dx + az By Yz dy + a2 82 Y2 dz) - % (a2 82 Yx dx + az Bz Yy dy + a2 Bz Yz dz)] =zzzz[§rarBrYrd-lrarBd -lf"arYd ~lrardd] ’ 2 2 Y6 2 Bd 2 BY (8.16) Next, the x,xxx term of Eq. (8.15) is x,xxx [ax Bx Yx dx + é-(ax Bx Yy dy + ax By Yx dy + ax By Yy dx + ay Bx Yx dy + my Bx Yy dx + ay By Yx dx) --5 (ax Bx Y2 dz + ax Bz Yx dz + ax Bz Yz dx 3 + ay By Yz dz + ay 82 Yy dz + ay 82 Yz dy) + my By Yy dy] 110 = x,xxx [g—(ax Bx Yx dx + ax Bx Yy dy + ax Bx Yz dz) 4. Bx Yx dx + ax By Yx dy + ax Bz Yx dz) A Q X + (ax Bx Yx dx + ax By Yy dx + ax Bz Yz dx) Bx Yx dy + ay By Yy dy + ay Bz Yz dy) A Q ‘<: Bx Yy dx + ay By Yy dy + ay Bz Yy dz) 6‘ '~< By Yx dx + ay By Yy dy + ay By Yz dz) A Q ‘< A Q N Bx Yz dz + my By Yz dz + az Bz Yz dz) Bz Yx dz + ay 82 Yy dz + oz 82 Yz dz) A Q X 82 Yz dx + ay 82 Yz dy + az Bz Yz dz) A Q N + m wlm wlm (.0an w|—- w|—- bot—d wig (”'_. az Bz Yz dz] 1 d 3 36+?“x‘5x‘ssv + | Q X .< X o; = x,xxx [l-ax Bx dY 3 1 1 1 +§ay5y561r+3aywdsd+3o‘ygy6Yd 5 _ 2 _ 2 E—Yz dz 6aB 3 82 dz 6aY 3 82 Y2 Gad + 5 dz Bz Yz dz] 1 1 1 — x,xxx [g-éaB dY5 + 3 6ad 686 + 3 add dBY _ 2 - 2 _ 2 3 Y2 dz 6aB 3 82 dz 6aY 3 82 Yz 5ad 1 1 _ l g az dz dBY g-az Yz d86 3 a2 82 6Yd + 5 oz Bz Yz dz] 1 1 1 =X'XXX [§ 6018 6Yd+§6aY GBd+§6ad BY -2» ~ -5» . _2- ~ 3 rY rd 6aB 3 r8 rd 6aY 3 r6 rY dad — g-ra rd dBY 3 re rY 68d 3 rd rB dY6 +590. 913m 95]. (13.17) Finally, we can express the frequency-dependent E tensor for a linear molecule in terms of the orientations of its symmetry axis as . _ l . - . . . Ea’smum) — 3 Ehxxxm) [15 rd rB rY rd 111 dB Yd + GGY 686 a BYd(iw) = 1? [Ez,zzz(iw) + 2 Ex,xxx(iw)] [35 1"‘01 1$8 FY F‘d ' 5(901 PB 6Y6 ’"a 91! dd J" P01 96 68Y + 98 9Y ad 9B 96 6dY l’AY f:6 6&8) + 6a8 6Yd + 6(W 686 + 5ad 6BY] _ %§ [3 Ez,zzz(iw) — 8 Ex,xxx(iw)] [(3 fa faB — 608) 6Yd + (3 901 r11r ' 6w) 686 + (3 Fa f‘d _ 601d) 68v] + 126 [3 Ez,zzz(iw) _ 8 Ex,xxx(iw)] [(3 1QB l’AY — SBY) 6ad + (3 98 96 - 586) 5w + (3 FY 96 — 5195013] . (8.18) 112 Appendix C. Expanding a Function in Spherical Harmonics We want to determine the symmetry-adapted expressions for the dipole coefficients using the Cartesian expansions in Eqs. (3.19) and (3.u7) for VDWZ, and in Eqs. (3.35) and (3.71) for INDZ. These depend upon several different types of products of the unit vectors 9 and R. This appendix will show a method to express these factors in terms of spherical harmonics; we then can easily find DISH and D1nd from Eqs. (3.22) and (3.36), and Dde and Dind from Eqs. AL )1A2AL )1AZAL (3.“9) and (3.72). One type of Cartesian term is simply 82. We can write R2 = R coseR , where OR is the angle that the vector B_makes with the space—fixed z axis. Now 10(9) = (un)‘1/2 O o A _ §_ 1/2 Y1(R) — (Mn coseR , SO Mn Rz ' 1/2 0 o A R Y (F) Y (R) . (0.1) (3) O 1 ll3 A more complex example is the factor 98 az RB' This can be expanded in spherical harmonics as a(A, L) Ym(r) Y; m(R) C(AL1; m, -m) . (C.2) where the a(),L) must be determined. We consider first A = 2 and L = 1 (A > 2 or L > 1 need not be considered). We integrate Eq. (C.2) " X- with [yg(?) Y?(R)] , where o . 1/2 2 Y2(r) = (m) (3 cos Or, - 1) o‘_31/2 2 Y1(R) — (Mn) cos OR . The right hand side of Eq. (C.2) is just a(2,1) C(211;00), while the left hand side becomes 0. 0‘ * fan fdQRr Br ZRB [Y2(r) 11102)] 1/2 _ (15) - - 2 _ — -—§;—-— f er f dQR r8 r2 R8 (3 cos er 1) coseR —iflfanse(3fir~—1)Idafifia (c3) ‘ 8n r B 2 z 2 R B z ' ' The first integral in the above, I er rBrzrzrz, is the 8222 component of an isotropic fourth rank tensor, with all indices interchangeable: 11” f d9 r r r r = A (dasdY6 + 6aYéBd + GadéBY) We compute A from A A A A _ 2n fl . U f er rzrzrzr — 3 A — f0 d¢ f0 Sine cos 0 d0 _ 2 _ UN — 2n 5 — 5 So I an 1 a a a - 3 A 5 — 51 5 (c u) r PB 2 z z - Bz _ 5 Bz ' ' The next integral is A A = ' f er rBrz A GBZ with 2w fl . 2 2 4n ' = = I — = _ A f0 dd f0 San cos 9 d0 2n 3 3 Thus I d9 6 9 = 31-5 = I an fi i (c 5) r B z 3 82 R B z ' ' Using Eqs. (C.u) and (C.5) in Eq. (C.3) results in A A o . o “ * f amp I dQR r8 r2 RB [Y2 (r) Y1 (3)] 1/2 _ ilél___ . £1 - £1 £1 ‘ 8n [3 5 68z 3 6Bz] 3 682 R 115 _ (15)”2 . 16H ‘ 2 = —§"—1/—2 R = a(2,1) c(211;oo) 3(15) 1 8n => a(2,1) =——,—o————R (c.6) C(211,00) 3(151/2 Is there a contribution from A = 2, L = O in Eq. (C.2)? No, since C(201;m,-m) = 0. There may be a contribution from A = 1, L = 1. c * Integrating with [Yj’fifl Y?(R)] , the left hand side of Eq. (C.2) becomes A A O A 0 A * f er f an r8 r2 RB [Y1(r) Y1(R)] 3 A A fl; f er f dQR rB rZ RB coser 0030 3 A A A A A W I er PB P2 {‘2 f dQR RB RZ R . R But the first integral vanishes no matter how B is chosen. Thus there is no contribution from A = 1, L = 1. Similarly, the A = 1, L = O instance makes no contribution. What about A = O, L = 1? . . o A o A * . Integrating With [Yo(r) Y1(R)] gives - A 0. 0‘ * fdnrfanr r R [Y0(r) Y1(R)] 8 z B (3)“2 . - = ——E;—— f er f dQR PB rz RB COSGR 1/2 . A _ (3) A A - ”n f er PB r'z f dQR RB R2 R ALL/3.111, .in, R _ Mn 3 82 3 82 = JET/7 R = a(O,1) C(O11;OO) 3(3) -> (O1)~——-1——--—lil——R (C7) ‘ a ' ~c(o11;00) 1/2 ' ‘ 3(3) il'-H . (.:,? ‘- 1 new) noiaudt1$nnfi ,¢_‘ a; ' ‘.. - «'1 par zfia'TT J! ' ('c“; I - '. ..'; _l'.‘ ..'.l-I; ... - . - (8.. - J51 artisslaxll “ns'.* ' 1 ..'v,l.”' .- _- utmoosd ll6 Finally, there is no contribution from A = O, L = 0 since C(001;m,-m) = 0. Using Eqs. (C.6) and (C.7) in Eq. (C.2) gives 1 - n n R = z a(2,1) YEW») Y1m(R) C(211;m,-m) m=-1 o- 0‘ + a(0,1) Y0(r) Y1(R) C(011;OO) 1 1 8n m A -m A 2 —_—— - — R Y (r) Y (R) C(211;m,-m) =_1 C(211,00) 3(1,5)1/2 2 1 1 Mn 0 A O A + m ' W R Yo(f‘) Y1(R) C(011;00) o (C.8) Using this method, expansions of all the Cartesian terms were carried out. These were then inserted into the expressions for VDWz and INDZ, which were used in Eqs. (3.22). (3.36), (3.N9), and (3.72) to yield the symmetry-adapted dipole coefficients D and D . AL A1A2AL 117 Appendix D. Clebsch-Gordan Coefficients Clebsch-Gordan coefficients are used in combining states of different angular momentum. Suppose that we want to add two commuting angular momentag1 and £2. The product ket |J1sz1m2> = IJ1m1>132m2> . (D.1) where |j1m1> and |j2m2> are eigenvectors of if and g: respectively, constitutes a basis in the product space. From this basis, we can construct a new basis which comprises the eigenvectors of Jz and i?, where g_is the total angular momentum of the combined system. The transformation equation is |J1J2JM> = Z |J1J2m1m2> <3132m1m2 JM) , (D.2) where the summation is carried out over m and m for fixed values 1 2’ of j1 and 32. The ket [j1j2JM> is the new basis, and the transformation coefficients are the Clebsch—Gordan 2m1m2 coefficients. Several different notations are used for these coefficients; the one used in the dissertation is C(j1j2J;m1m A 2). general formula for calculating Clebsch-Gordan coefficients is [101] C(j1j2J;m1m2) = [(J+j1-jz)!(J-j1+j2)!(j1+j2-J)!(J+m1+m2)!(J-m1-m2)!]1/2 . . . . —1/2 x [(J+J1+j2+1)1(J1—m1)!(J1+m1)!(J2'm2)!(.]2+m2)!] -...-. . ' it" ' ”*"-.-' . ”w..- " 1' '_ _. .. ':'I:a " .' 'u .1 ‘ To adieu-minim mud-.64». .. 01.1.1 bbs 0.1 new aw Jan: ecoflfl":5a .. ..‘ij '15.3;;:, .'--.:-i -. iiuhrqu firi'!‘ ._l: ans 1‘33 IM'MU‘, . w.!.. 1._ . ..i . - _ #1. "“1. 1 Er :fltfi'l - Ign'rflstrtl -_ .. mu. < 1;“ “19ml 'aud 1 .--.--.1u:ti.‘r.mm --1 -. ._'v'1'e.nu:: b sonnw ‘rl: '-.p 118 k+' J +m x zk(—1) 2 2 (2.1+1)”2 (J+12+m1—k)!(j1-m1+k)! -1 .' ' _ I — — 1 1 ' _' — — l x [(J J1+J2 k).(J+m1m2 k).k.(k+\]132 m1 [112).] , (13.3) where the summation is over all k for which the factorials in the denominator are nonnegative. This equation may be simplified for the case m = m = 0. Let 2g = j1 + j2 + J. Then [101] 1 2 C(j132J;OO) = 0 if 2g is odd, and C(j132J;OO) = (—1)8+J (2.1+1)”2 A(j132J)g![(g-j1)!(g-j2)!(g-J)!]-1 (0.1) if 2g is even, where A(i J J) = [(J +3 -J)'(J +J-i )'(J +J-j )'/(i +1 +J+1)']‘/2 1 2 1 2 ° 1 2 ' 2 1 ' 1 2 ' ' (0.5) Table D.1 lists values for several Clebsch-Gordan coefficients. 119 Table D.1 Values of Clebsch-Gordan coefficients 31 32 J m1 m2 C(3.132‘1‘1'Hm2) o 1 1 o o 1 2 1 1 o 0 (2)102 91/2 2 3 1 o o (35) 111/2 11 3 1 o o (3;) u 5 1 o o (5)”2 181/2 6 5 1 o o (TIE 2 o 2 o o 1 2 2 o o o (1§)1/2 2 2 2 o o - ($1M 181/2 2 2 4 0 O ng) 1) 0 L1 0 o 1 11 2 2 o 0 ($41” 201/2 U 2 4 O 0 (7?) 11 2 6 o o (5)”2 #:N 3 1 (1‘69 ’8 1119’ 11/2 - ‘3’. 3 1/2. 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