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L. .m.w...§.r .¢»fifl$§$§% . . . . .. . .11 ...: . .r i. ..c i. . .. .. . . . ...:... 2. Magmmwa my..v..§mu._i..¥.v.m.% x v\~.t . . THESIS ZCOl LIBRARY ‘ Michigan State Unlverslty This is to certify that the dissertation entitled COLLISION-INDUCED DIPOLES AND POLARIZABILITIES presented by MARK HEBERT CHAMPAGNE has been accepted towards fulfillment of the requirements for Pb . D . degree in W KW; cw Major professor 9- Date I A/00 MS U is an Affirmative Action/Equal Opportunity Institution 0- 12771 PLACE IN RETURN BOX to remove this checkout from your record. TO AVOID FINES return on or before date due. MAY BE RECALLED with earlier due date if requested. DATE DUE DATE DUE DATE DUE 11m WpGS-p.“ COLLISION-INDUCED DIPOLES AND POLARIZABILITIES By Mark Hebert Champagne A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Chemistry 2000 repel able 1 a the: col 1 is [Gill's COLLISION-INDUCED DIPOLES AND POLARIZABILITIES By Mark Hebert Champagne During a molecular collision the electron clouds of the molecules repel and distort one another, which leads to experimentally observ- able changes in the properties of the molecules. This thesis presents a theory for types of these collision-induced effects: for two-body collisions, the collision-induced dipole (CID), and for three-body collisions, the nonadditive three-body polarizability. Compressed gases and liquids consisting of molecules of tetrahe- dral and centrosymmetric linear symmetry absorb far-infrared radia— tion, due to transient dipole moments induced during molecular colli- sions. In earlier theoretical work on far-infrared absorption by CH4/N2 mixtures, good agreement was obtained between calculated and experimental spectra at low frquencies, but, at higher frequencies, (from 250 to 650 cmrl) calculated absorption intensities fell signifi- cantly below the experimental values. In this work, we focus on an accurate determination of the long-range, collision-induced dipoles of Td--4Lm pairs, including two polarization mechanisms not treated in the earlier line shape analysis that are consequences of the disper- sion and nonuniformity in the local field gradient acting on the l} moiecu cients only I The ca pute 1 slight Co hydroq l.ntera this I CUies based (013) 1Mm molecule. Numerical values are given for the long—range dipole coeffi— cients, and constant—ratio approximations are developed that require only the static susceptibilities and C5 van der Waals coefficients. The calculated dipole coefficients for CH4 and N; are then used to com- pute theoretical collision—induced absorption line shapes that show slightly higher absorption in the high frequency range. Collision—induced light scattering spectra of the inert gases and hydrogen at high densities provide evidence of nonadditive three-body interaction effects, for which a quantitative theory is needed. In this work, the three-body polarizabilities Amfl3) for interacting mole- cules with negligible overlap are derived and evaluated. The results, based on nonlocal response theory, account for dipole-induced-dipole (DID) interactions, quadrupolar induction, dispersion, and concerted induction-dispersion effects. SUDC DEDICATION This work is dedicated to my wife for her constant encouragement and support throughout my time in graduate school. iv ing m ageme woulc tions lecu ing supI Cilf (Chan mom fu] (Rial to . 1:r ACKNOWLEDGEMENTS I am grateful to my research advisor, Dr. Katharine Hunt, for giv- ing me the opportunity to work for her. Without her guidance, encour- agement and support, I would not have succeeded in this endeavor. I would also like to thank Dr. Robert Cukier for his help and sugges- tions while this dissertation was being prepared. Much of what I learned about spherical tensor analysis and intermo- lecular forces was taught to me by Dr. Xiaoping Li. Thank you Xiaop- ing for all your patient and informative explanations. During my time as a graduate student I had many, many helpful dis- cussions with and support from Dr. Merrick Dewitt, Dr. Jannavi Srini- vasan, Dr. Edmund Tisko, and Peter Krouskoup. Thank you all for your help and for making graduate school more enjoyable. I would like to thank the members of my family for their love and support. To my brother, Dr. Matthew Champagne, my sisters Jennifer Gilmore and Sarah Nadrowski and their respective counterparts, Dana Champagne, Donald Gilmore and Kevin Nadrowski, thank you. To my mother, Donna Champagne, this dissertation is a true sign of the wonder- ful job you did raising me - thank you. And to my father, Vincent Champagne, who died just over a year ago - I'm sorry I took too long to finish, but I finished because of the determination I inherited from you - thank you. To my father-in-law and mother-in-law, Jack and Marlene Crawford, thank come caree for h the gr thank you for your words of support and for always making me feel wel— come and especially for feeding me all through my college and graduate careers. And finally I would like to thank and praise my exceptional wife for her love, her advice, her strength and her belief in me. You're the greatest. vi LIST C LIST C CHAPTE CHAPTE CHAPTEF 3.1 3.2 3.3 CHAPTER 4.1 TABLE OF CONTENTS LIST OF TABLES LIST OF FIGURES CHAPTER 1. INTRODUCTION CHAPTER 2. COLLISION-INDUCED DIPOLES FOR Td AND Don MOLECULAR PAIRS 2.1 Introduction 2.2 Induction Dipole 2.3 Dispersion Dipole CHAPTER 3. NUMERICAL RESULTS FOR THE COLLISION-INDUCED DIPOLES OF Td AND Dab MOLECULAR PAIRS 3.1 Introduction 3.2 Numerical Results 3.3 Summary and Conclusions CHAPTER 4. THEORY OF LINE SHAPE FOR COLLISION-INDUCED ABSORPTION 4.1 Introduction 4.2 Line Shape Theory 4.3 Line Shape Theory for Two—Component Molecular Mixtures 4.4, Dipole Components CHAPTER 5. NUMERICAL RESULTS FOR COLLISION-INDUCED ABSORPTION LINE SHAPES 5.1 Introduction 5.2 Spectral Moments vii ix 16 25 25 25 36 4O 4O 42 4s 48 55 SS 55 CHAR CHA Pl ‘4 Appenc 5.3 Spectral Contributions 57 5.4 Summary and Conclusions 60 CHAPTER 6. DERIVATION OF THE IRREDUCIBLE THREE-BODY POLARIZABILITY 89 6.1 Introduction 89 6.2 Derivation of the Irreducible Three-Body Polarizability 91 CHAPTER 7. NUMERICAL RESULTS FOR THE IRREDUCIBLE THREE-BODY POLARIZABILITIES 105 7.1 Numerical Results 105 7.2 Summary and Conclusions 124 Appendix 131 viii 2.1 3.1 3.2 3.3 4.1 4.2 5.1 7.1 7,2 7.3 an: DOT Int 64- am? In, 64s; aDDr Cons Imag. LIST OF TABLES Coefficients a, from Eq.(35), b, from Eq.(36), c, from Eq.(44), and d, from Eq.(46). Molecular properties used in computing pair dipoles Van der Waals coefficients C33" used to estimate dispersion dipoles Collision—induced dipole coefficients Induced dipole components used to compute the far infrared spectrum involving collisions between a diatomic or symmetrical linear molecule (1) and a tetrahedral molecule (2) for single molecule transitions. Induced dipole components used to compute the far infrared spectrum for the double transitions of a diatomic or symmetrical linear molecule (1) and a tetrahedral molecule Values of the moments Mo (10‘61 erg cm5), M1 (10“9 erg cm5) and M2 (10'35 erg cm6 5'2) Molecular properties used in computing three—body polarizabilities Integrals of the form Ljf‘Ciw)a3(iw)aC(iw)dw, evaluated by 64-point Gauss-Legendre quadrature or by constant-ratio approximations. Integrals of the form J“f‘(iw)a3(iw)dw, evaluated by 0 64-point Gauss-Legendre quadrature or by constant-ratio approximations. Constant-ratio approximations from Eqs. (50) and (51) for imaginary-frequency integrals in A0853) and Aaf'” . ix 22 26 27 28 51 51 62 106 108 110 113 3.1 3.2 3.3 3.4 5.1. 5.2. 5.3. 5.4. 5.5. 5.6. A! At Ab Ab Ab LIST OF FIOJRES Polarization terms in the CH4---N2 dipole as a function of intermolecular separation R, for configuration 1, shown at \l on .10. .11. .12. .13. .14. upper right. CH4---N2 dipole, for configuration 2 CH4---N2 dipole, for configuration 3 CH4---N2 dipole, for configuration 4 . Absorption coefficient contribution , 9N2 acu . Absorption coefficient contribution, enzacu4o at 297 K . Absorption coefficient contribution, enzacm: at 195 K 4, at 162 K . Absorption coefficient contribution, @NZaCI-M, at 297 K . Absorption coefficient contribution, §"2ac”4’ at 195 K . Absorption coefficient contribution, énzacnp at 162 K Absorption coefficient contribution, (1,590.4, at 297 K Absorption coefficient contribution , (1N2 QC" 4, at 195 K Absorption coefficient contribution , “"2 (20,4 , at 162 K Absorption coefficient Absorption coefficient Absorption coefficient Absorption coefficient at 297 K Absorption coefficient at 195 K contribution, aN2§cn4v at 297 K contribution, aNzém4, at 195 K contribution, anz§m4s at 162 K contribution, YNZQCH4 - GuzAcn4. contribution, YNZQCH4 - ®"2AC“4’ 30 31 32 33 63 64 65 66 67 68 69 7O 71 72 73 74 75 76 5.1! 5.16 7-2 as (w 7 .3 do EqL 7 .4 Ac- 7.5 7.6 .15. .16. .17. .18. .19. .20. .21. .22. .23. .24. Absorption coefficient contribution, YNZQCH4 - GNZACH4’ at 162 K Absorption coefficient contribution , 7N2 @014 + 9N2 E014 . at 297 K Absorption coefficient contribution, YNZ @014 + 8N2 E014 . at 195 K Absorption coefficient contribution, YN2§CH4 + GNZECH4’ at 162 K Absorption coefficient contribution, Dd(0001;0), at 297 K. Absorption coefficient contribution, Dd(0001;0), at 195 K. Absorption coefficient contribution, Dd(0001;0), at 162 K. Total absorption coefficient at 297 K. Total absorption coefficient at 195 K. Total absorption coefficient at 162 K. .mn3> for H---H---H in an equilateral triangle of side R (with R and Ad in a.u.). Aafi” for Kr---Kr---Kr in an equilateral triangle of side R (with R and Ag in a.u.). mm for H2- - -H2--.H2 with molecular centers arranged in an equilateral triangle of side R (R and Ad in a.u.). Ad“” for H---H---H in a linear, centrosymmetric array. Ad“” for Kr---Kr- -Kr in a linear, centrosymmetric array. aaf” for Hz-wa---H2 in a linear, centrosymmetric array. xi 77 78 79 80 84 85 86 116 117 118 119 120 121 Th dipoIe colliS' lecular is slig bIe. In I are dete inductlo and the the dISp. Gar Waal; Static e] Sion‘indu DOIEntja]; into the . t0 the Com MerriCa] 5; SYStemS’ ar CHAPTER 1 INTRODUCTION This thesis presents new work on the theory of collision-induced dipoles and polarizabilities. It encompasses two-body and three-body collisional effects at long range, where it is assumed that the intermo— lecular separation is sufficiently large that electron cloud overlap is slight and electron exchange between colliding molecules is negligi- ble. In Chapter 2, the collision—induced dipoles of Ta and Day, molecules are determined, complete to order Rr5 in the intermolecular separation R. The analysis includes quadrupolar and higher-multipolar induction;L4 it accounts for the nonuniformity of the local fieldi” and the local field gradient57 acting on each molecule. In addition, the dispersion dipole #dfipnll4 which determines the change in the van der Waals interaction energy due to the application of a uniform, static electric field Fe, is evaluated. The derivation of the colli- sion-induced dipole begins with the presentation of the interaction potential15 in Cartesian tensor form, followed by its incorporation into the first-order correction to the pair wavefunction and finally to the conversion, using angular momentum algebra,”18 to the more sym metrical spherical tensor form for easier line shape analysis. In Chapter 3, the collision-induced dipole is found for specific systems, and numerical values are given for the dipole coefficients for Cl tion ‘ these tion t range ( nation physice Voyager N2, CH, ZOO-600 crepanc- greates1 possible depender (bound-t 0f the ‘ the dIS( IQrmS f whim] tial an- Work Cor form'ty and Buec of (0111 for CH4 or CF4 interacting with H2, N2, C02, or C52, for direct applica— tion in calculating the collision-induced absorption spectra. Of these pairs, CH4---N2 holds particular interest because far-IR absorp- tion by CH4/N2 mixtures has been investigated experimentally over a range of temperatures -(162—297 K)— and, in addition to providing infor- mation on intermolecular interactions, the results have potential astro- physical applications: The atmospheric opacity on Titan (observed by Voyager) is attributed principally to collision-induced absorption by N2, CH4, and fb,19¢4 with absorption by CH4---Nz predominating in the ZOO-600 cmil region.21 Yet this is the frequency range where the dis- crepancies between the observed and calculated absorption spectra are greatest. Birnbaum, Borysow, and Buechelé25 have analyzed a number of possible explanations for the discrepancies—including rotational-state- dependence of the multipole moments of CH4, effects of dimer formation (bound-bound, bound-free, and free—free transitions), and deviations of the quantum line shape from the model form. They concluded that the discrepancies for w > 250 cmrl might be due to the omission of terms from the calculated pair dipole,25 i.e., to terms beyond the multipolar induction, field-gradient effects,*7 and single exponen- tial anisotropic overlap terms included in their work. The current work contains numerical values for dipoles due to dispersion and nonuni- formity of the local field gradient not treated by Birnbaum, Borysow, and Buechele. These dipoles will be incorporated into the calculation of collision-induced absorption (CIA) spectra in Chapter 5. traI lar Ott fied the calcz spect induc funct tiona' rules coeffi In Chapter 4, the theory for constructing far-IR rotational- translational collision-induced absorption spectra for Td---Dgh molecu- lar pairs is presented, according to the treatment first used by Mary- ott and Bi rnbaumz“. The Bi rnbaum-Cohen (BC) model” which uses a modi- fied Bessel function and a set of time factors is utilized to model the translational contribution to the spectrum. The time factors are calculated from the zeroth, first, and second moments of the spectrum,23v29 which in turn are calculated from the contributing induced dipole coefficients and the intermolecular potential function.30 The rotational component is then calculated using the rota- tional state and spin state populations of CH4 and N2 and the selection 3132 contained in the Clebsch-Gordan rules for rotational transitions coefficients. The two components are then combined to form a spectrum of translationally broadened rotational lines.”42 In Chapter 5, numerical values for CH4 and N; from Chapter 3 are combined with the theory for collision—induced absorption spectra devel- oped in Chapter 4 and the values of the induced dipole components (including the new terms for dispersion and nonuniformity of the local field gradien7.and additional overlap terms) are tabulated. These new terms are added to the induced dipole components of Birnbaum, Borysow and Buechele already used for spectral computation, bringing the com- puted spectrum into better agreement with experiment in the ZOO-600 cm-1 range . In Chapter 6, the nonadditive three-body polarizabilities of coll indu and a no: body nonlo densi‘ where hYDerp the po the int “Tar di In “Hafiz; IIEIdS’ sion Int moIQCUleg orientafi are of Co ITO” Of t] colliding molecules are determined. The theory accounts for dipole- induced—dipole (DID) interactions, quadrupolar induction, dispersion, and concerted induction-dispersion effects. The approach is based on a nonlocal response theory developed for three-body energies and three- body dipoles.33-34 This theory gives the interaction energy in terms of nonlocal polarizability densities a(r,r';m) and hyperpolarizability densities BCr,r',r";-wo;w,w') and 7(r,r',r",r"';-wa;w,w',w"), where we denotes the sum of the remaining frequency arguments in the hyperpolarizability. The polarizability density aCr,r';w)35-37 gives the polarization P(r,w) induced by an applied field Fe(r';w) acting at r . Thus it characterizes the distribution of polarizability within the interacting molecules; similarly B and 7 account for the intramolec- ular distribution of the non-linear response to an applied field.33»39 In Chapter 7, numerical results for the nonadditive three-body polarizabilities including quadrupole polarization, static reaction- fields, third-body fields, dispersion and concerted induction-disper- sion interactions are found for real systems of inert gases and the molecules H2, N2, C02, and CH4 for linear and equilateral triangular orientations. The three-body contributions are significant as they are of comparable magnitude with the two-body terms, due to a cancella- tion of the first—order, two—body DID contributions to AG. REFE lPhe 23.1 ‘K. L 5M. Mo, 5hr. Bye 100 P ‘‘L. Ca]E REFERENCES 1Phenomena Induced by Intermolecular Interactions, edited by G. 2]. 3L. 4K. 5A. 6M. 7J. 3W. 9K. 100. 11L 123 13p 14X. 15A. 16A. Birnbaum, NATO ASI Ser. B 127 (Plenum, New York, 1985). L. Hunt and 3.0. Poll, Mol. Phys. 59, 163 (1986). Frommhold, collision-Induced Absorption in Gases (Cambridge University Press, Cambridge, 1993). L. C. Hunt and X. Li, in Cbllision- and Interaction-Induced Spectroscopy, Vol. 452 of NATO ASI Ser., Ser. C, edited by G. C. Tabisz and M. N. Neuman (Kluwer, Dordrecht, 1995). D. Buckingham and A. J. C. Ladd, Can. J. Phys. 54, 611 (1976). Moon and D. N. Oxtoby, J. Chem. Phys. 75, 2674 (1981); 84, 3830 (1986). E. Bohr and K. L. C. Hunt, J. Chem. Phys. 87, 3821 (1987). Byers Brown and D. M. Whisnant, Mol. Phys. 25, 1385 (1973); D. M. Whisnant and w. Byers Brown, ibid. 26, 1105 (1973). L. C. Hunt, Chem. Phys. Lett. 70, 336 (1980). P. Craig and T. Thirunamachandran, Chem. Phys. Lett. 80, 14 (1981). . Galatry and T. Gharbi, Chem. Phys. Lett. 75, 427 (1980); L. Galatry and A. Hardisson, J. Chem. Phys. 79, 1758 (1983). . E. Bohr and K. L. C. Hunt, J. Chem. Phys. 86, 5441 (1987). w. Fowler, Chem. Phys. 143, 447 (1990). Li and K. L. C. Hunt, J. Chem. Phys. 100, 9276 (1994). D. Buckingham, Adv. Chem. Phys. 12, 107 (1967). J. Stone, M01. Phys. 29, 1461 (1975). 150. 19R. 23:2. BR. 8]. 24A 25C. 2"c. 23M. ‘9]. Birn 3234 Blrnt 17R. 180. 19R. 20R. 21R_ 22R. 23] 24A. 256. 26A, 27c_ 28M. 29] 30“. 31A. 32”. 33X. 34X. 35W. N. Zare, Angular Momentum (Wiley Interscience, New York, 1988). A. Varshalovich, A. N. Moskalev, and V. K. Khersonskii, Quantum Theory of Angular Momentum (World Scientific Publishing Co., New Jersey, 1988). E. Samuelson, N. R. Nath, and A. Borysow, Planet. Space Sci. 45, 959 (1997). Courtin and D. Gautier, Icarus 114, 144 (1995). D. Lorenz, C. P. McKay, and J. I. Lunine, Science 275, 642 (1997). Courtin, Icarus 51, 466 (1982); 75, 245 (1988); C. P. McKay, J. B. Pollack, and R. Courtin, ibid. 80, 23 (1989). . L. Hunt, J. 0. Poll, 0. Goorvitch, and R. Tipping, Icarus 55, 63 (1983). Borysow and C. M. Tang, Icarus 105, 175 (1993). Birnbaum, A. Borysow, and A. Buechele, J. Chem. Phys. 99, 3234 (1993). A. Maryott, and G. Birnbaum, J. Chem. Phys. 36, 2026 (1962). Birnbaum and E. R. Cohen, Can. J. Phys. 54, 593 (1976). Moraldi, A. Borysow, and L. Frommhold, Chem. Phys. 86, 339 (1984). . D. Poll and J. van Kranendonk, Can. J. Phys. 39, 189 (1961). J. M. Hanley and M. Klein, J. Phys. Chem. 76, 1743 (1972). Rosenberg and J. Susskind, Can. J. Phys. 57, 1081 (1979). W. Galbraith, J. Chem. Phys. 68, 1677 (1978). Li and K. L. C. Hunt, J. Chem. Phys. 105, 4076 (1996). Li and K. L. C. Hunt, J. Chem. Phys. 107, 4133 (1997). J. A. Maaskant and L. J. Oosterhoff, Mol. Phys. 8, 319 (1964). 3PT. 373. ‘- ‘UH a. O 36T. 37] 38K. 39H. Keyes and B. M. Ladanyi, Mol. Phys. 33, 1271 (1977). . E. Sipe and J. van Kranendonk, Mol. Phys. 35, 1579 (1978). L. C. Hunt, J. Chem. Phys. 80, 393 (1984). Ishihara and K. Cho, Phys. Rev. B 48, 7960 (1993). 2.1 Int Com radiatic molecule CH, - H35 racy. absorpti. transit;< GIG-molec 7'rlrermoye In th dIDOIeS O' AQCUAar Se have been line Shape pled “Ith . dipoie, as Md "0190.”! A .4” 8 § CHAPTER 2 COLLISION-INDUCED DIPOLES FOR Td AND Dog. IDLECULER PAIRS 2.1 Introduction Compressed gases and liquids composed of nonpolar molecules absorb radiation in the far infrared, due to transient dipoles induced when molecules collide.1-4 Absorption intensities due to bimolecular CH4-H2s and CH4-N26'7 collisions have been measured with high accu- racy. The fhr~IR spectra of (}u/N2 mixtures consist of continuous absorption bands arising 'from ‘translationally-broadened rotational transitions of one or both speciesm7. Since such transitions are sin- gle-molecule forbidden, the line shapes contain information about the intermolecular dynamics. In this chapter, results are given for the collision-induced dipoles of Td-Dmh molecular pairs, complete to order R“6 in the intermo- lecular separation R. Using spherical tensor methods&42, the results have been cast into the symmetry-adapted form needed for spectroscopic line shape analyses.‘7 The properties of molecules A and B are cou- pled with functions of the vector R (from A to B) to obtain the pair dipole, as a rank-one spherical tensor. For molecule A of Ta symmetry and molecule B of Dmh symmetry, the pair dipole has the form [Jan-B = flJ—g‘)‘ Z (2a+1)1/2 Dcanbvlefln') flamh'CQA) Y:"CQB) a,b,A, L,m,m' , (1) "T..,q xY.,L_q(Qa) (abmrn' ' lAq> (ALqM-qllm where dipoIE and L: In that d1 Iecular tion of cal har tor R, Cient. In d8riyed the con ter‘ms. 2.2 If“ where ,ufi'” denotes the spherical-tensor component M of the pair dipole, ”Sue = “9mg (2) and uh"; = 1(1/2)”2 (uQ"‘B.tl°u9"'B). (3) In Equation (1), D(a,b,A,L;m') is a dipole expansion coefficient that depends on the length, but not on the orientation of the intermo- lecular vector R, 1);.(QA) is a Wigner rotation matrix taken as a func- tion of the Euler angles of molecule A, K$.(QB) and YfiqCQR) are spheri- cal harmonics dependent on the orientations of molecule B and the vec- tor R, respectively; and (a b c d | e f) is a Clebsch-Gordan coeffi— cient. In the following two sections the induction and dispersion dipoles derived using spherical tensor methods are given for Tg-Dmh pairs and the constant—ratio approximations are developed for the dispersion terms . 2.2 Induction Dipole Through order Rf; the induction dipole of a T¢4Lm molecular pair results from the direct (first-order) polarization of each molecule by the field and field gradients from the permanent moments of its colli— sion partner. The induction dipole is computed starting with the inter- action potential V in the Hamiltonian represented in multipolar form.13 V=_“A,T<2>,“B _ %—uA-T<3)298 _ %uA.T<4); QB (4) In E pole Synm gradi the c In Eq Sian 1 At “Ere 4 it, s X'C' +%9A:T‘3) -u3 + %9A:T‘4’ :88 + 315—6":T‘5) 5 93+... -%¢VWMfi-JmNF”wL.H +fi§Ax T‘s) -u3. .. In Equation (4), u. 9, Q, and 6 denote the dipole, quadrupole, octo- pole, and hexadecapole operators, respectively. Molecule A has Ta symetry and B has Dob symetry. Tm) denotes the propagator with n gradient operators: T“” = (vv---v)| R r4, where R is the distance from the center of A to the center of B. The vector R is given by RB-R“. In Equation (4) and below, the symbols -, :, 3, and 3 denote Carte- sian tensor contractions. At first order, the dipole moment induced in molecule A by B is u?“ = (1 + C)<€Po|u‘|@1). (5) where C denotes the complex conjugate of the expression that follows it, leg) is the unperturbed wavefunction for the A --B pair, and lel) is the first order correction to the pair wavefunction, IQ1)= - G V I‘l'o)- (5) G is the reduced resolvent for the A---B pair, 5 = (1*|@0)(@o|) (H-Eo)*1 (l-l90>(@o|)- (7) To cast paw into the form used in line shape analyses, we need to convert between the Cartesian tensor contractions in Equation (4) for V and the rank-zero components of the corresponding spherical tensor products: ”AoTa) :e8 = -\/—7_[u"® (T‘3)® 983(1)]‘0’ . (8) 10 ) b L and C‘- Equatic contrib. The Symtl tion (5) A I ”... in. In Equat‘ tule A, and 95 ar 9019 of n NeXt’ [A6, “A .115) :§8 = _./11[“A® (Tm 3 @8)(1)](0) (9) 9A:T(4) :93 = 3[9“® (Two GEN-“1(0) . (10) and 9A 2 T‘s) :eB = -\/fi[QA® (T‘s) ® 93)‘3’]‘°’. (11) Equations (8)-(11) give the lowest order A -~B interaction terms that contribute to the polarization of A, since u = Q = 0 fbr molecule 8. The symbol 3 denotes a tensor product. With Equations (8)-(11), Equa- tion (5) transforms to give #4,... = (1 + C)(— l?) (@8l{u‘®[6‘u‘l93)® (7% eg)<1>]},g1> VT— +<1 + C)(— ,0; )<@Gl{u‘®[c‘u‘l%>® 0% e3)‘1>l‘°>}.‘.1’ (12) +(1 + C)(- le )<@3|{u‘®[G‘e‘|@8)® (Two GSJ‘Z’J‘WH.” +<1 + C)(- %)® new eg><3>1<°>}.<,1>. In Equation (12), log) denotes the unperturbed wavefunction for mole— cule A, GA is the reduced resolvent for molecule A (when isolated), and 93 and Q3 denote the permanent quadrupole and the permanent hexadeca- pole of molecule B, respectively. Next, a spherical tensor coupling relation14'is used, (a) (b) (d) (h <' = __ ,b,d,- a b c .. .. 23w mun x{[A(a)® BM] (c)® D(d)}(j) (13) where A, B, and D are arbitrary spherical tensors of ranks a, b, and c (respectively), the quantity in brackets is a 6j symbol, and Habmz = (2a + 1)1/2(2b + 1)1/2- . -(22 + um, (14) With Equation (13), taking A = “A; B = pA, 9A, 99; and D = (T"”® 83) or (T””® o3), the operators are recoupled in Equation (12). With the Operators for molecule A coupled first, the resulting equation 11 contains quantities of the form (1 + C)(@3|[A e GAB B]“”|e3 ), which are susceptibilities of molecule A. Specifically, the transformed equa- tion contains the polarizability a, the dipole-quadrupole polarizabil— ity A, and the dipole-octopole polarizability E, for molecule A. The spherical tensor components of these susceptibilities are one = (1 + cmsl [we c‘um‘) we» <15) A3“) = (1 + Owel [we we‘ll“) I93). (16) and £3“) = (1 + C) (QISI[uA® ammo I113). (17) Since molecule A has Td symmetry, a,“” is nonzero only if c= 0; AQ“) ¢ 0 only if c = 3, and E$“” ¢ 0 only if c = 4. With these obser- vations, Eqs. (13)—(17), and the relation [A(3) 3 309)];0 ____, (_1)a+b—c[B(b) ® A(a)])((C)’ (18) Eq. (12) yields 113.... = (- .197.”me 681L$D — (lg—mew o3] <4>®T<5>},<,1> 131-)2 {i i §}H{[AA‘3’®981®T‘4)},§1’ (19) C F‘ 4 2 C . 4%)?{5 1 3}H.{EE“‘>®eSJ“Ian‘nlfi’. By the definition of the spherical tensor products, (mew 93] ®r<4)},<,1> = Mqu ”933.?- 1.112, <32mm' lcq>. <20) and similarly for the quantities {[ (wow 931(2) e T<3> 1,31). {[ aA(0)® ogjm e T}’<'1>, and {[ 5A(4)® 83]“) 3 PM}? In Equation (19), the molecular moments, response tensors, and propagators are given in the space—fixed frame. For an arbitrary spher- ical tensor P of rank k, the components P(k,p) in the space-fixed 12 frame P(l Since 1* permane' 6%: 95(2 and ;§(4 frame are related to the components P(k,q) in the molecule-fixed frame by14 P(k,p) = Z 1333(9) P(k,q). (21) q If the only non—zero component P(k,q) in the molecule-fixed frame has q =‘0, then in the space-fixed frame P(k.p) = [4n/(2k + 1))”2 mm) P(k.q=0). (22) Since molecule B has Dmh symmetry, the only nonzero components of the permanent quadrupole and hexadecapole in the molecule—fixed frame are 83(2,0) and §3(4,0). These are given by (23) 930,0) = x/E/z 93,22 713/2 93 24 and o3(4,0) = \/70/4 63.2222 5 V70/4 Q8. ( ) In the pair-fixed frame with the z axis defined by the vector R, the nonzero components of the propagators appearing in Equation (19) are T(3,0) = -3m R“, (25) 1(4,0) = (ax/‘76 R-S, (26) and T(5,0) = -9o\/1_4 M. (27) The nonzero components of the susceptibilities of molecule A (in its own frame) are aA(o,0) = -f (um-mun) —<—‘—‘§’1)\/I5—Z weak-6 1985Y.:(QB)Y.§_,CQR)(040mm) x(4Sn‘M-mllM) —(%)(4n) 2. ”(2c + 1)1/2 c,m,m ,m 3 2 c {4 :l 2 xv}. ((28) 11:4... (QR) (32mm' lcm' ' ) (c4m' 'M-m' ' llM) fig; A*(3.q)eg 15223.36.) +(gm/3.5 (471) 2 (2c + 1)1/1’{4 2 C}Z l-I“(4.CI)98R‘6 (32) c,m,m',u" S 1 3 q XZZ:(QA)K§(QB)Kimn(QR)(42mm'Icm")(c5m"M-m"I1M). Then from Equations (29)-(32), is obtained the following contributions to the dipole coefficients: D(0223;0) = -fiaAegR-4, (33) 0(0445;0) = 450.431“, (34) 019(3224;¢2) = : 1a,,AAegR-5, (35) and Dge(4225;0) = (%)\/760£e(4225;:4) = bAEAegR—G’ (36) with coefficients a, and b, listed in Table 2.1. The dipole moment induced in molecule B at first order in the interaction with molecule A is derived similarly, starting from the analogue of Equation (5). For molecule A, the nonzero moments of lowest order are 9 and o. The isotropic polarizability a9(0,0) of molecule B is obtained as in Equa— tion (28); but molecule B also has an anisotropic polarizability, given by 0180.0) = LC‘i-(azz — a.u.). (37) in the molecular frame. The dipole—quadrupole polarizability A8 van— ishes. The dipole—octopole polarizability EB is non-zero, but it does 14 not C ishes Then ' The DEI 14( w and ¢A( From Eq the dip, 0(3( DQJC 0(40 and De, (4 The facu Table 2.1 (“9019 an We) Thr not contribute to the induced dipole in B to order Rffi, since 96 van- ishes. To find “am, we use the analog of Equation (9) and #3 °T(4) :QA = 3[u8 ® (T(4}® 989(1)](0) . (38) Then for the dipole induced in molecule B, is obtained 14: ..., = (1141“...) Z (-1>1+C(2c + 1)1/2{9 3 C}s23<3.q)a8 ,9 D(d) 1(9) ,3 [am e E(e)](h)}(k) where the quantity in brackets is a 9j symbol, and Hcfgh is defined as in Equation (14). Also used are the defining relations for the imagi- nary-frequency polarizability and hyperpolarizability, 018(c)(l'w) = 2 2'Lu8n ® #301“) Ano (Ago + hzwz)-1 (56) n and22 BA(0;iw.-iw|9.a; ma) = 2' [Cumodufi‘f )er—lA)rO A;(1,(Aro + 171004 r,s,ll,q q q + (uQ)os(u2°)sr(u3_q)ro Ag?) (Aro ’ 172004 + (us-q)o.(u3°)sr(ua r0 4.2%(450 + ”WV (57) + (#8)05(H39q)sr(lla) r0 AF%(ASO ’ 171404 + (144,30s(ufi°)sr(u3)ro(Aso + lhw)’1(Aro + lhw)‘1 + (ué)os(ufi°)sr(u$-q)ro(Aso + ihw)‘1(A.z-o - 172004] x<1lq(m-q)IQM)<19MmIama> In Equations (56) and (57), the primes on the summations indicate that the ground state is omitted from the sum over states r and s of mole- cule A and n of molecule B; the indices m, q, and M label the spheri- 17 cal turb 3‘(O: cies which frequ compo that after SYNmetr Vanishj BYXZ = E‘ (< FrOm EQL cal tensor components of the dipole operators; Amo denotes the unper- turbed energy—level difference (En-Eo). In a3“3(iw), the two dipole operators are coupled to give a tensor of rank c; in [3"(0; iw, —iw |g, a; m.) the dipole operators associated with the frequen- cies iw and -iw are coupled to give a resultant tensor of rank g, which is then coupled with the third dipole operator (associated with frequency 0) to give a tensor of rank a overall. The Cartesian components of the transition dipoles are all assumed to be real, and that BOBYCO; ‘iw, —iw) = 30,3,(0; -iw, it»). From Equations (SD—(57), after algebraic manipulation the following is obtained a b c: a c A uh = ”a” (WWW-1w“L 95L {3 c: 1:} {L 1 g} (“/70 (58) x ram; [ BAco; iw,-iwlg,a) o a3(‘)(1'w)](") ® ”(3) ® T(2)](L) .21). Symmetry analysis shows that for a Td molecule there is a single non- vanishing component of the B hyperpolarizability, Bxyz ([3,,yz = 3,2, = Byxz = Byzx= Bzxy = Bzyx). and BA(0;iw,-iwl2,3,:2) = i'i fiaxyzmnwpiw) (59) From Equations (58) and (59), D,(3032;¢2) = :(3ih f1? R-6/1on) = fa... B§,,(o;1w,-iw) aBC‘iw) (5°) and 0, (32mm) = :f,L(ihR’5/n) Id... agyzco;iw,-iw)(aEZCiw)-a2,(iw)). (61) In Equation (61), the coefficients fM_ have the values ffio==‘¢15]/SO, f12 = 6/175, 132 = 47/35, f32 = aria/175, f34 = 3J1—0/175, f4, = —3\/€/35, and f54 = 6V11/35. All other coefficients fAL van- 18 ish. for orderh E(O;ic polari cients The hy heavier ability estimat 7'3th a ish. Equations (60) and (61) give the dispersion dipole coefficients for molecule A in a quantum mechanically rigorous form, to leading order. For a direct evaluation of the integrals in these equations, BCO;‘iw,-'iw) and a(iw) are required. Since the imaginary-frequency polarizability determines van der Waals interaction energy coeffi- cients, a(iw) has been computed for a number of atoms and molecules. The hyperpolarizability 3(0;iw,-iw) is not known for methane or heavier molecules of Td symmetry; however, the static B hyperpolariz- ability has been determined in ab initio work, and it can be used to estimate the integrals in Equations (60) and (61) within a constant ratio approximation, developed next. [0 dw BQn(O;iw;-iw) a8 (m) = (62) 51 Ida) aACiw) aeciw)[B§,z(o; o; 0)/ (M0)] andrdw 54,,(o;iw;-iw)(a§z(iw)-a§,(-iw)) = o (63) 52 fat, wciwxagzcm-afixcmn[Béyzcm o; 0)/ axon are first given. Then the aA-a3 integrals are obtained from the isotropic and anisotropic C6 van der Waals coefficients, and ab initio values are used for a(0) and Bnq(0;0,0) on the right hand sides in Equations (62) and (63). The $1 and S; (but no other quantities) are determined within the Unsold approximation. This gives fawmflconm-m) a8 (1m) = «1.2/3 mu.»a)-1[B§nco;o,0)/a*‘(0)] X It C200/3h (64) and fat.» B§n(0;iw,-iw) (agzcw)-a§,(iw)) =-(1+2/3 A)(1+A)'1 X[B§yz(0;0.0)/a‘(0)]n C22°/3h (65) 19 quadrupo Fm. ‘ Utes t0 1 where A is the ratio of the excitation energies for molecules 8 and A, in the Unsold approximation23, i.e., A = QB/QA, Cgm and C220 -(3n/n)j: dw wow) aBow) <66) -(n/n) j: dw wow) (chow-ago») (67) Equations (60), (61), and (64)—(67) provide the basis for our numeri- cal estimates of dispersion dipoles in the next section. The leading term in the dispersion dipole of molecule B is of higher order. It stems from the combined effects of fluctuating u-u and u-e interac- tions and varies as R42 (The dipole of A contains a corresponding term.) After isotropic averaging over the orientations of A and B, only one component of the net dispersion dipole is nonzero; it yields the coefficient o,(0001;0) = (gm-Umj: an BS(O;iw,-iw)(iw)aB-88(O;iw,-iw)aA(iw)]dw(68 where BCO;iw;-iw) denotes the dipole-dipole-quadrupole hyperpolarizabil- ity; 33(o;iw,—1‘w) = (2/5)[ 3:2,,(0;iw,-iw) + 2 Bszz(0;iw,-iw)] (59) for the Td molecule and 88(0;iw,-iw) = (2/15)[483,,,(o;iw,-iw) + B§x22(0;iw,-iw) +2 Bgzxz(0;iw,—iw) + 232,2,(0;iw,-1'w) + BEZZZCO;1w,-1w)] (70) for the Dmh molecule. Bah6(0;iw,-iw) has dipole indices a and B, and quadrupole indices 7 and 6; the frequency (-iw) is associated with the quadrupole indices. For completeness, it is noted that classical induction also contrib- utes to the pair dipole at order R-7; but there are moments and suscep- 20 tibi~ induc iSOtr tion f lar f7 izes E tibilities in the R57 terms that are not yet known accurately. Direct induction effects, which contain a single propagator l“5>, vanish on isotropic averaging over the orientations of A and B. One back—induc- tion term is nonvanishing after orientational averaging: the quadrupo- lar field of B polarizes A, and the reaction field from A then polar- izes B anisotropically. 21 Table from from Table 2.1. Coefficients a, from Equation (35), b, from Equation (36), c; from Equation (44), and d; from Equation (46). A a, bx CA d1 3 T93 0 m3 o 4 {-37 E/2 «E/s -— L 55 5 4%: 7R? 0 4% 22 REFER 1Pheno 3L. Frc Uni ‘K. L. Spe Tab SC. Birr Rad: 6'I. R_ [3 Read 76. Birnl (199. REFERENCES 1Phenomena Induced by Intermolecular Interactions, edited by G. 2J. 3L. 4K. SC. 6I. 7G. 8X. 9T. 10R 11A. 12A. 13T. 14R 15p. Birnbaum, NATO ASI Ser. B 127 (Plenum, New York, 1985). L. Hunt and J. D. Poll, Mol. Phys. 59, 163 (1986). Frommhold, Cbllision-Indbced Absorption in 63595 (Cambridge University Press, Cambridge, 1993). L. C. Hunt and X. Li, in collision- and Interaction-Induced Spectroscopy, Vol. 452 of NATO ASI Ser., Ser. C, edited by G.C. Tabisz and M.N. Neuman (Kluwer, Dordrecht, 1995). Birnbaum, A. Borysow, and H.G. Sutter, J. Quant. Spectrosc. Radiat. Transf. 38, 189 (1987). R. Dagg, A. Anderson, S. Yan, w. Smith, C. G. Joslin, and L. A. A. Read, Can. J. Phys. 64, 1467 (1986). Birnbaum, A. Borysow, and A. Buechele, J. Chem. Phys. 99, 3234 (1993). Li and K. L. C. Hunt, J. Chem. Phys. 79, 1758 (1983). G. A. Heijmen, R. Moszynski, P. E. S. Wormer, and A. van der Avoird, Mol. Phys. 89, 81 (1996). . Samson and A. Ben-reuven, J. Chem. Phys. 100, 9276 (1994). J. Stone, Mol. Phys. 29, 1461 (1975). D. Buckingham, Adv. Chem. Phys. 12, 107 (1967). Bancewicz, Mol. Phys. 50, 173 (1983). . N. Zare, Angular Mbmentum (Wiley Interscience, New York, 1988). Isnard, D. Robert, and L. Galatry, Mol. Phys. 31, 1789 (1976). 23 15K. L. C. Hunt, J. Chem. Phys. 92, 1180 (1990). 17w. Byers Brown and D. M. Whisnant, M01. Phys. 25, 1385 (1973). 180. P. Craig and T. Thirunamachandran, Chem. Phys. Lett. 80, 14 (1981). 19B. Linder and R. A. Kromhout, J. Chem. Phys. 84, 2753 (1986). 208. J. Orr and J. F. Ward, Mol. Phys. 20, 513 (1971). 21X. Li, M. H. Champagne, and K. L. C. Hunt, J. Chem. Phys. 109, 8416 (1998) 24 3.1 Th evalua abilit' lated a The throUQh tions 0 ChaDIEr linear t Stant~ra 2 for tr Static Iisted i1 3'2' The 1at0r Str ratio of 3‘2 ~“mer- The nu TabTe 3.3 CHAPTER 3 NUMERICAL RESULTS FOR THE COLLISION-INDUCED DIPOLES OF Td AND 00., CDLLISIONAL PAIRS 3.1 Introduction The dipole coefficients derived in the last chapter will now be evaluated for specific systems. The values for the multipoles, polariz- abilities and hyperpolarizabilities of each of the molecules are tabu- lated and used in the collision-induced dipoles for each system. The properties needed to evaluate the A --B induction dipole through order Rfi‘are known from experiments or from ab initio calcula— tions on H2, N2, C02, CS2, CH4, and CF4. Numerical values used in this chapter are listed in Table 3.1: 90, oo, o, and [azz-cnu] for the linear molecules; Qo, do, a, A, and E for the T3 molecules. The con- stant-ratio approximation of Equations (35), (36), and (41) in Chapter 2 for the R“6 and R'7 dispersion dipoles are used; this requires the static susceptibilities BA, 8", BB (ReferenceslandZ), a“, and as listed in Table 3.1, and the van der Waals coefficients given in Table 3.2. The C; values have been derived from pseudospectral dipole oscil- lator strength distributions.}5 For simplicity, A = 1 is used for the ratio of the excitation energies of A and B. 3.2 Numerical Results The numerical results for the dipole coefficients are listed in Table 3.3. With Equation (1), these coefficients determine the induc- 25 «hwy NZ u‘.‘ LK u.l I r .‘I lllll I l . llll. ,Ill .fl-W.-Ji-“ CHWLMJPMM>V vaPOEMWIIUluU'wrvalimU—LIMMMumfflshowlit.fi unvw‘di Vnwim.ULmWQCLQ Lfivh‘drvnwhoz . N .M, aw~h-lv5 .em mucocwmwmc .Nm .mwm cmrm mom "Hm oucwcwmwmo .mm monocommme .Om oucwcowomF .NN oucocmwwmx .mm muchmwmmw .HN wu:w...wu_.w~_.F .wm oucmcowwm; .o~ mucoemmoma .nm oucwcmmwmC .mH oucoewwwmu .oN oucmcwmomu .wH monocowomu .mm mucwcwwwmn .NH wucocmmwmm Emma- 0.2;. cm .mfi- 22:- L2- $2- 8.92 35 52m ...: 03 co: . mm ..m . 8 ES 8 . o n 2 Sam 28:- EN - 037 :24 . mm - 03. um: - 8 . one 5.5 53:. 3.8». - 0:7 393. .22- $2. 8661mm. 0 o o o mod 32;. 5.055 .583 12.2: Lama: 33.4 .92- 03:- Ed o o o o ”4.2 .085 :3. ..mm .3 ENE at... USS .H o o .30 i as ...:, . mm 18% . D .13 . S 58% . m u: .2 .13 . D a 52.0: ..E .H- 13;..- 33% .o .35 .229. on o o o o 18.4- €$9~ 39m .32.». - 5w: 4. 18:36 o o 8 5 NB Nz f 5 .6 282°: a.u.m cw mwapm>v mmpoawu ewmn mcvuaneou cw tom: mowucoQOLa cmpzuopoz .H.m ornmh 26 CH;, CF. {Ex (H4 CF, ‘Refere DCOMDUI given 1 CRefere Cflcs 9Q“(A' Table 3.2. Van der Waals coefficients C33” used to estimate dispersion dipoles (values in a.u.). H2 N2 co2 cs2 C200 CH4 _39.57a -96.94a —142.6b —331.4c CF4 -44.7d —110d ~161d -375“ C220 CH4 .4.84e -12.7° .38.56 —1139 cu:4 -5.47e -14.4°- -43. 5° -1279 aReference 1. ”Computed from pseudospectral dipole oscillator strength distributions given in Ref. 1 for CH4, and Ref. 2 for CO2. CReference 3. ”C200 (CF4-X) estimated as [a (CF4)/a (CH4) ] C200 (CH4-X) . “-‘Cg20 (A—X) estimated as -§- [oxzz (X) -axx (X) ] C200 (A-X) /a (X) . 27 85 5: oh: 8: 55 555 55:- So. 5.55:. 55:55:55 55H- 55: 5.: H55- 53 5: 5.55 55- 55:55.55 5.5m 5.55- 5.2- 3.: 5: 5m.m 55 55.5 55:55.5 55.5- 5.5 35 55.5- :5 m5 5.: 5.: 55:55:“. 555- 55 5.5 H55- 55 55 o: 5.3- 51555555 5.55 5.55- 555- 5.: 55 55.5 55.5 :5: 555555 5.3- 5.3 5:5 55.5- N: 5.5m 5.3 55.5 555555 5.3: .255: 5:5: 55.: 5.5: .55.: .555: .55.: 5115:5535 .55: 5:: 5.2: 5.5: 5.5: .555: .555: 5:5: 55:553.: .55.: .555: .55.: .55.: .555: .555: .554: .555: 5515:5593 .52: 5.5: .555: .555: 5.8: .55.: .555: 55.5: 35:55:15 .555: 55.5: .55.: .555: .555: .55.: .55.: 5:: 5515:5533 .55: .55.: .55.: .55.: 5.5: 5.2: .555: .535: 35:55.35 .555: .555: .555: .585: .555: .55.: .55.: .55.: . 55:552. 5m: 55.: .55.: .55.: .554: .555: .555: .555: 5515:5555 .555: .555: .555: .555: .55.: 5:: .58.: .53.: 5515:5555 .555: .555: .555: .555: 55.5: .55.: 5.: .55.: 5515:5535 5H 5.5 55 5.: :55- 5.53- 5.5:- 55.55- 55:52.5 .5 5.: 5.5 m5 5:- 5.55m. 5.55- 555- 55535 .555: .555: .52: .555: .555: 55.5: 5.5: .555: 2515835 .5: .55: .55: .55: .555: .55: .55: .55.: 2.51552 :5- 5.5 55 55.57 855- 5.5 55 5.5:- 5555555 5.55. 5.5: 55.5 5.5:- 5.2- 5.55 55.5 55.3- 555555 NWU...¢H_U ~0U...vn_U ~2...vn_U 3*...va ~mU...vIU ~0U...£U NZ...vIU ~I...¢IU A55 :3 3.5a 3.3205 590...? Low mucwwuriooo £02.“. uwuaucwucowmwppou .m.m £92. 28 tion and dispersion dipoles of CH4-- H2, CH4- -N2, CH4 --C02, CH4- -CS2, CF4---H2, CF4---N2, CF4---CO2, and CF4---CS2, complete to order R'6 in the molecular interaction. These pairs show quite different patterns in the D coefficients, due to differences in the molecular properties that generate the polarization: For example, 90 and 50 are both posi- tive for H2 and CS2, and negative for N2 and C02; 90 is positive for CH4 and negative for CF4, and the signs of 50 are reversed for these two molecules; also 50 [calculated at the self-consistent field (SCF)level] is very small for CF4. The polarizability anisotropy is ~37%~39% of the isotropically averaged polarizability a for H2 and N2, but ~83% for CO2, and for CS2, ozu — an is actually larger than a. The static B hyperpolarizability is negative for CH4 and positive for CF4. Because of the experimental importance of the CH4- -N2 dipole, the effects of each polarization mechanism were examined separately for this pair. Figures 3.1—3.4 show the relative significance of the M = 0 component of the various contributions to the dipole for CH4 and N2 separated along the z axis, with 2(N2) > 2(CH4). These figures pro- vide "snapshot" illustrations of the magnitude, sign, and dependence of each mechanism on the orientations of CH4 and N2: In Figures 3.1 and 3.2, the N2 molecule is aligned along 2. The Euler angles for CH4 iri Figure 3.1 are (a,B,7 ) = (O,—arcsin[(2/3) U2],-n/4); thus one CH bond points up along 2 toward the N2 molecule. For this configura- tion, the dipole p2 is independent of the angle a. In Figure 3.2, (a.£%i’) = (n/4,0,0) and the coordinates of the H nuclei are {0,V5?,1}, 29 0.010 \ I 0.008 (2 v 0.006 \ <0 “0 (a-U-) \A 0.004 \E: 0.002 K 43a + Bo: -0.002 7 7.5 8 8.5 9 9.5 10 Fl(a.u.) Figure 3.1. Polarization terms in the CH4--.N2 dipole as a function of intermolecular separation R, for configuration 1, shown at upper right. 0.0125 0.0100 06) . 0.0075 >< H0 (a.u.) 0.0050 \ ad) 0.0025 0 — 0.0025 — 0.0050 7 7.5 8 8.5 9 9.5 10 R (a.u.) Figure 3.2. CH4---N2 dipole, for configuration 2 (upper right). 31 0.008 0.006 520, L10 (a.u.) 0.004 . A: \ 0.002 \ /I3 —0.004 -0.006 / R (a.u.) Figure 3.3 CH4---N2 dipole, for configuration 3 (upper right). 32 0.002 \aq, l1 (a-U- \ >< 0 ) \EC-)\ ...... AG): /7 /Ba+Ba -0.002 —0.004 / —0.006 » (ha —0.008 » (19 / 7 7.5 8 8.5 9 9.5 10 R (a.u.) Figure 3.4 CH4---N2 dipole, for configuration 4 (upper right). 33 {—NCI,0,—l}, {0,-:2,1}, and {~0?,0,-1}. In Figures 3.3 and 3.4, the N2 molecule is aligned along the x axis. For Figures 3.3, the CH4 Euler angles are ((2.5,7) = (0,—arcsin[(2/3)1"2],—n/4) while for Figure 3.4, (a,B,Y) = (n/4,0,0); thus for Figure 3.4, the two nitrogen nuclei are staggered with respect to the two nearest H nuclei of the CH4 mole— cule. The terms are grouped in Eq. (1) from Chapter 2 for the M = 0 pair dipole (i.e., the products of the numerical prefactors, D coeffi- cients, Wigner rotation matrices, spherical harmonics, and Clebsch-Gor- dan coefficients) according to the moments and response tensors involved. In Figures 3.1—3.4, the label a8 represents dipole induc- tion in CH4 due to the permanent quadrupole of N2; a§ represents induc- tion in CH4 due to the hexadecapole of N2; Qa induction in N2 due to the octopole of CH4; A9, response of CH4 to the quadrupolar field gradi- ent of N2; E9, response of CH4 to the gradient of the field gradient; and Ba+Ba, the total dispersion dipole. The dispersion dipole’"18 is the sum of an R‘6 dipole induced in CH4 [designated the Ba term, and obtained from Equations (35) and (36) in Chapter 2] and the net R“7 dipole [designated the Ba term, and obtained from Eq. (41) in Chapter 2]. In all of the configurations studied, the largest contribution to the pair dipole comes from the as term, i.e., dipole induction in CH4 due to the N2 quadrupole. At R = 7 auu. the value of the a9 term ranges from 51% to 264% of the total pair dipole in the configurations 34 studied (high values reflect near cancellation of several contribu- tions to p). In all cases the Qa term — dipole induction in N2 by the CH4 octopole—is second largest in absolute value, ranging from -158% to 22% of the total dipole. Here and below, all results are quoted for R = 7 a.u.; for comparison, the value of o in the Hanley-Klein potential for N2 --CH4 is 6.903 a.u.6 For configurations 1, 2, and 4, dipole induction in CH4 by the N2 hexadecapole (a5) is the next largest term, accounting for 11% —24% of the total dipole. For configuration 3, the a5 term is —41% of the total, but the effects of hexadecapolar induction by CH4 (5a) and of the nonuniformity of the N2 quadrupolar field (A9) are both larger, at -S3% and 46% of the total, respec- tively. For all four configurations, the 5a terms range from -53% to +25% of the total; the A9 terms contribute 4% to 46% of the total; the E9 terms, —8.6% to 13% of the total; and dispersion terms (the sum of Ba and Ba components), between —S.6% and 29% of the total dipole. For CH4- -N2 at R = 7 a.u., the Ba term in the dispersion dipole [from Dd(0001;0)] is roughly an order of magnitude larger than the Ba tenn. In Figures 3.1—3.4, Equation (41) of Chapter 2 is used for Dd(0001;0); the result is —9S7.5 R'7. Equation (40) in Chapter 2 yields a coefficient of larger absolute value, Dd(0001;0) = —1197 R”; hence the plots are based on a conservative estimate of the dispersion effects. Intermolecular correlation gives rise to greater distortions in the CH4 electronic charge distribution than in the N2 charge distribu- tion: the ratio Bo/a is about 44% larger for CH4 than for N2. Gener- 35 ally, dispersion dipoles have greater relative importance for lighter species. For CH4---H2, however, the 80/0: values for CH4 and H2 nearly cancel, yielding Dd(0001;0) = -100 R77, based on the estimate in Equa- tion (41) in Chapter 2. 3.3 Summary and Conclusions The collision-induced dipoles of Tg---Dmh molecule pairs with weak or negligible charge overlap have been determined, in the symmetry- adapted form needed for line shape analyses of far-IR absorption by mixtures. For CH4 or CF4 interacting with H2, N2, C02, or CS2, the numer— ical results for the D coefficients are listed in Table 3.4. The analy- sis includes two polarization mechanisms that have not been evaluated in earlier work on Ta---Dmh pairs: (1) dispersion and (2) effects due to nonuniformity in the quadrupolar field gradient (E tensor terms). The exact equations for the dispersion dipole coefficients depend upon products of the hyperpolarizabilities [3(0; iw,—iw) or BOCO; iw,-iw) with the polarizability a(iw), integrated over imaginary frequencies. To obtain numerical results for the dispersion dipoles, a constant— ratio approximation has been developed that requires only the static polarizabilities, static hyperpolarizabilities, and the C5 van der Waals coefficients. For CH4-- N2 at R = 7 a.u., the dispersion contribu- tions range up to ~29% of the total pair dipole, and the E-tensor terms range up to ~13% of the total, for the configurations studied. The R“6 dispersion term gives rise to transitions with AJ = 0 or :3 on the Td molecule and A] = :2 on the Chm molecule; but for CH4 --N2 at R 36 = 7 a.u., the dispersion contribution that varies as RS7 is actually larger than the R‘6 term. One component of the R'7 dispersion dipole remains nonzero after isotropic averaging over the orientations of the two interacting molecules (unlike the dipoles due to the other polariza— tion mechanisms treated in this chapter). This component produces a very broad translational absorption band in the far-IR spectrum. The E-tensor terms give rise to double transitions with AJ = :4 on the Td molecule and AJ = :2 on the Dmh molecule (for an isotropic intermolecu- lar potential). Thus, inclusion of these terms in the CH4-2-N2 dipole should increase the calculated absorption in the high—frequency wings of the far-IR spectrum, bringing the calculations into closer agree- ment with experiment. 37 REFERENCES 10. 2B. 3A. 4]. 5W. 6K. 7D. 8L. 9]. 10p 11X. 12K. 13K. 143 15R. 166. 17R. 18G. 196. 200. J. Margoliash and W. J. Meath, J. Chem. Phys. 68, 1426 (1978). L. Jhanwar and W. J. Meath, Chem. Phys. 67, 185 (1982). Kumar and W. J. Meath, Chem. Phys. 91, 411 (1984). E. Bohr and K. L. C. Hunt, J. Chem. Phys. 87, 3821 (1987). Byers Brown and D. M. Whisnant, Mol. Phys. 25, 1385 (1973); D. M. Whisnant and W. Byers Brown, ibid. 26, 1105 (1973). L. C. Hunt, Chem. Phys. Lett. 70, 336 (1980). P. Craig and T. Thirunamachandran, Chem. Phys. Lett. 80, 14 (1981). Galatry and T. Gharbi, Chem. Phys. Lett. 75, 427 (1980); L. Galatry and A. Hardisson, J. Chem. Phys. 79, 1758 (1983). E. Bohr and K. L. C. Hunt, J. Chem. Phys. 86, S441 (1987). . W. Fowler, Chem. Phys. 143, 447 (1990). Li and K. L. C. Hunt, 3. Chem. Phys. 100, 9276 (1994). L. C. Hunt and J. E. Bohr, J. Chem. Phys. 83, 5198 (1985). L. C. Hunt, J. Chem. Phys. 92, 1180 (1990). . Linder and R. A. Kromhout, J. Chem. Phys. 84, 2753 (1986). P. Feynman, Phys. Rev. 56, 340 (1939). Birnbaum, A. Borysow, and H. G. Sutter, J. Quant. Spectrosc. Radiat. Transf. 38, 189 (1987). D. Amos, Mol. Phys. 38, 33 (1979). Maroulis, J. Chem. Phys. 105, 8467 (1996). Maroulis, Chem. Phys. Lett. 259, 654 (1996). M. Bishop and J. Pipin, Int. J. Quantum Chem. 45, 349 (1993). 38 216. 226. 23C. 24D. 25U. 266. 27] 280. 29” 30M. 31c_ 32C. MarouJis and A. J. Thakkar, J. Chem. Phys. 88, 7623 (1988). MarouJis and A. J. Thakkar, J. Chem. Phys. 93, 4164 (1990). MarouJis, Chem. Phys. Lett. 199, 250 (1992). M. Bishop and J. S. Pipin, J. Chem. Phys. 98, 4003 (1993). Hohm and K. KerJ, M01. Phys. 69, 803 (1990). MarouJis, Chem. Phys. Lett. 226, 420 (1994). . Komasa and A. J. Thakkar, Mo]. Phys. 78, 1039 (1993). B. Lawson and J. F. Harrison, J. Phys. Chem. A 101, 4781 (1997). . J. Bridge and A. D. Buckingham, Proc. R. Soc. London, Ser. A 295, 334 (1966). R. BattagJia, A. D. Buckingham, D. Neumark, R. K. Pierens, and J. H. WiJJiams, M01. Phys. 43, 1015 (1981). J. Jameson and P. W. FowJer, J. Chem. Phys. 85, 3432 (1986). E. Dykstra, J. Chem. Phys. 82, 4120 (1985). 39 CHAPTER 4 THEORY OF LINE SHAPE FOR COLLISION-INDUCED ABSORPTION 4.1 Introduction In this Chapter a derivation of the absorption coefficient used to modeJ coJTision-induced absorption (CIA) Jine shapes wiJJ be pre- sented according to the work of George Birnbaum and Richard Cohenl. However, this fieJd of study is broad and has many contributors and their work shoqu be detaiJed. The first pressure-induced absorption spectrum was identified by Crawford, Werh and Locke2 in compressed oxygen in 1949 and the coJJision-induced rotationaJ spectrum of hydro- gen was first observed by KeteJaar, Cona and Hooge3 in 1955. It was subsequentJy studied by Cona and KeteJaar4 in 1958 and Kiss, Cush, and WeJShS in 1959. At the same time that the experiments on CIA were being performed, the theoretica] aspects of this phenomena were being eprored. Van Kranendonk and Bird6 worked out the rotationaJ transi- tion matrix eJements and absorption coefficients for a two-moJeque system using a sphericaJ harmonic expansion for the CID dipoJe, but did not take into account the transJationaJ broadening term. Bosomworth and Gush7 presented CIA theory on rare-gas mixtures in 1965, utiJizing a basic exponentiaJ function to describe the transJa- tional broadening. Over the next few years different theories were presented, some for the pure transJationaJ terms of rare-gas mixtures, for exampTe the theory of McQuarrie and Bernstein3, and some for the 40 combined rotationa] and transJationaJ terms, such as, the theories of Trafton9 and Tanimotouh. However, these theories were generaJJy com- pJex and computationaT time was iengthy. In 1967 Levine and Birnbaum11 constructed a ciassicai theory of CIA in rare-gas mixtures that incorpo— rated a transJationaJ broadening term in the form of a Besse] function that greatJy reduced the compJexity of the equations and the time for computation. Then in 1976 Birnbaum and Cohen1 refined the transJa- tionaJ term, creating what is now known in the Titerature as the Birn- baum—Cohen (BC) Mode]. Armed with this new convention a Targe number of researchers began expiorations into the fieJd of CIA. These incJude the study of rare-gas mixtures [Ref. 12, 13] the study of moie- que and rare-gas mixtures, Refs. [14-16], and for two-moiecule sys- tems, Refs. [17—26]. In the mid 1980's Borysow and Frommhon wrote a series of papers focusing on the systems of Hz-He”, Hz-szs, Hz—N229, N2— N230, Hz-CH431 and CH4-CH432. These papers performed an in—depth anaJy- sis of the different types of translationaJ broadening modeis and poten- tiaJs that cou1d be used in CIA, concJuding that the (BC) function and a new mode], the extended Birnbaum-Cohen (EBC) function, that adds an extra exponentiaJ fitting term are the most effective of the fitting options. These papers proved the vaJidity of the equations used by Bi rnbaum, Borysow, and Bueche'le33 and used in this chapter. This Tine shape theory mnder the transJationaJ and rotationaJ bands of the spectrum for Td--JL¢ moJequar coJJisions. It is assumed at the Tow energies of incident radiation (ZOO-600 cmrl) that the moJe- 41 cuies remain in their vibrationaT and eiectronic ground states during the coJTision. This is not a theory derived from first principJes because it does not begin with the equations of motion. (This first principJes approach is shown in detaiJ Ref. 11). The approach in this chapter is a derivation using a ciassicaJ theory of Debye absorption3 that utiJizes a correJation function, greatJy siminfying the equa- tions for the Tine shape. 4.2 Basic Line Shape Theory The derivation begins with the equation?”-35 50.)) = 32,173 (1 - e“Bh“’)fme’i“’t(M(0)-M(t)) dt (1) which is the absorption coefficient per unit path Jength for a v01ume V of a mixture. (3:: g, and (M(0)-M(t)) is the dipoJe moment correJa- tion function. M is the totaJ dipoJe moment and the brackets denote a thermaJ average for the system. The next step is to reduce the totaJ corre'lation function for the system to a moiequar corre'lation func- tion. (MCOJ-MCtD = N (i.dil uC0)-|f.df>av i f (5) where i and f are the initai and finaJ states and d,- and df are the degeneracies of the states and Di=€fiVZhi€Mi 1' (6) The frequency of excitation term uqf has been factored out of the corre- Jation function, where (...)av is the average over a1] of the dynamica] paths. It is assumed in these equations that the aborption bands are on1y the sum of the absorptions due to each individuaJ Jine, i.e., the cross reTaxing coHisions that coupJe spectra] “line amintudes are nngigibJe. In Birnbaum and Cohen's work1 a reduced corre1ation function C(t) is defined as (1%: (igd-iI HCO)-|f,df)(‘i, dil “(O)|f’df)av = luiflz Cifct) (7) where lumz = .1241, l(i,d,~lu(0) I f, df)|2 (s) is a dipoJe matrix eJement. The corre1ation function is rewritten as Mt) = 1;: ml Hiflz eiwift Cir“). (9) Substituting this term into Eq.(4) gives 4:0») = Z o'lu- :2 (220-1 e-Ww-w tc- (t) dt. 1f 1 if I: if (10) A condition of detaiied baJance is necessary for this type of system. The detaiJed baJance condition is written as oC-t) = §(t +iBh). (11) To a110w this to be true, the t in C”(t) is replaced by the compJex time 43 = (t2 - maul/'2. (12) This approximation was first used by Egelstaffflfl This requires (;f(y) to be a real even function of y so the real part of Cfi{y) is even in t and the imaginary part is odd in t. A function (thY) that satisfies Eq.(12) is CifCY) = 6Xp{ti1[t2 - (tfi + y2)LQ]} (13) where t1 and t2 depend in general on the initial and final states. Though other model correlation functions can be used, Eq.(13) models the translational band shape well. Using the expansion of the model correlation function, c,,(y) = 1 + 1' t 25ft; 25122 (1 + Bz’féffm) + (14) the spectral function obtained from Eq. (10) and (13) is (Mm) = {:1 Oil uiflz Tidal-) (15) where Piwa-)= (2n) 1 [file “’1‘ ”) tC ”(t) dt. (16) Solving the integral gives _ _ / h /2 2 K Inwa-) etz n 93 w ‘1‘??? (17) where w, = w iIOHf and z, = [1 + wfiti 11"2 [ :5 + (Bh/Z)2]LQ/t1 . K1(z) is the modified Bessel function of the second kind. It has the following properties: zK1(z) —> 1 as 2 —> 0 (18) and K1(z) —>(7r/22)1/2 e"? if lzl )) 1. (19) If only absorptions are allowed in Eq. (15), that is, i < f, then W») = :1 Diluii'z I‘iij) + g; 'Hif|2[pirif(w) + DfTiwa.)] (20) 44 The first term denotes nonresonant (iai) contributions and the second term denotes the resonant (Taf) contributions. This equation substituted into Eq.(3) gives 5(a)) = 43”;:V~(1 - 8’31“”); Diluiilz Tiij) + g; luifIZLDiTiwa) + pfrwcwa] (21) which is the starting point for the line shapes computed in this work. 4.3 Line Shape Theory for Two—Component Molecular Mixtures This derivation was originally used for rare—gas mixtures and for one- component molecular systems. In Birnbaum, Borysow, and Buechele?3 colli- sion-induced absorption for two-component mixtures was developed and a new form of Eq. (21) was utilized. In Birnbaum, Borysow, and Buechele the absorption coefficient due to collisions between molecules 1 and 2 is divided into three contributions 5(a)) = 812(0)) + 521(0)) 4» 51%)(10) (22) The first term represents induction in molecule 1, N2, by the multi- pole field of molecule 2, CH4, and conversely for the second term. eu2Cw) and £Q1Cw) contribute through rotational transitions in mole- cule 2 and 1 respectively. 530») contributes through all the double transitions in which both molecules are rotationally excited. The explicit forms of the absorption coefficient contributions are given below. 512(0)) = C F1200). (23) _ (A2) (232+1) g2 5+1) _ . where F12Cw)- g1 293235 2124.1 PM“) (”3232) (24) 45 —Bhw) ’ (25) ' 3hc and for the second contribution, 521(0)) = C F21Cw): (26) where F210») = (:3 323.433.0314) 601413300) r (w) w dw = Z—gfljggckmzda (34) 2 r . ‘- x((ODZ(c) (R) /6R) + #:3714030 (RD and the second moment M?” is given by20 2 a) ”$0 = 5%zh—jo QCR) [-D(C) (R) %- 640(0 (R) /C3R4 (35) -(aD(c) /aR) 40330.0 (“’) n 141400)): {exp[ti0 +tow] z K1[z 1}. (42) 4.4 Dipole Cbmponents The induced dipole components, Dflnab, used in the calculations are listed in Tables 4.1 and 4.2 (These terms used by Birnbaum et al. have a different form than those used earlier in this work in Chapters 2 and 3. This is due to a summing procedure used by Birnbaum et al to simplify the final forms of the dipole coefficients. However, they contain exactly the same information as those used in Chapters 2 and 3. In the appendix, the relationship between the two forms of dipole coefficients is given). Four of the terms are the same as those given by Bi rnbaum, Borysow and Buechele. They are flea R“, which is the dipole induction in CH. due to the permanent quadrupole of Nz;~/§§a it“, which is the the dipole induction in CH. due to the permanent hexa- decapole of N2; \[%§.cnzl?5, the induction in N2 due to the octopole of CH. and Vfigfaé R4fi the induction in N2 due to the hexadecapole of CH4. Each of these last two coefficients have exponential terms that are added to take into account the anisotropic overlap of the colliding molecules. Also, the final forms of the last two terms are the simpli- fied sums of several similiar terms. The simplification can be found in the appendix. The next term, (fgg7(279 — \fiZfeA) Rfs, is a combina- tion of two double transition terms, the octopole induction of the anisotropic polarizability of N2 and the response of CH. to the quadrupo- 49 lar field gradient of N2. These terms must be summed because of inter- ference. The coefficient, 79 used by Birnbaum, et al., was found to be incorrect and needed to be multiplied by a factor of VE?. The final term from Birnbaum et al., the hexadecapole induction of the anisotropic polarizability of N2, was found to have a interference term that must be added: the response of CH. to the gradient of the field gradient. Bi rnbaum et al. used two other terms, V70: (05); R‘8 and (8/3NEEI)71(Q5)2, that were not found to be significant. Birnbaum et al. did not use dispersion terms in their calcula- tions. The dispersion terms added to the calculation of the lineshape are listed in Table 3.1. They are they octopole-induced-dipole, the octopole—induced-quadrupole, and the 0(0001;0) term that comes about from the combined effects of fluctuating u-u and “-9 and varies as R42 The derivation and final forms of all of these terms are shown in Chapter 2. SO Table 4.1. Induced dipole components used to compute the far infrared spectrum involving collisions between a diatomic or symmetrical linear molecule (1) and a tetrahedral molecule (2) for single molecule transitions. DA(Al,A2,L;R) AlszL Selection rules fielazR-4 2023 A31 = 0,42 \fs‘alazR-é 4045 2:]. = 0,42,:4 x/iSS-aleR's + )L438’(R ' ONO 0334 A32 = 0,il,...,i3 ./—6,£a142R-6 + Ag4e‘(R - CW 0445 a]; o,:1,...,¢4 Table 4.2. Induced dipole components used to compute the far infrared spectrum for the double transitions of a diatomic or symmetrical linear molecule (1) and a tetrahedral molecule (2). 0(A1,A2,L;R) AIAZL Selection rules ./T85— (fins); — V2161A2)R'5 234 A31 = 0,12; AJ;=0,¢1,12,¢3 %%Y1 QZR‘6 245 A31 = 0,:t2; A32 = 0,i1,...i4 51 REFERENCES 16. Birnbaum and E. R. Cohen, Can. J. Phys. 54, 593 (1970). 2M. F. Crawford, H. L. Welsh, and J. L. Locke, Phys. Rev. 75, 1607 (1949). 3J. A. A. Ketelaar, J. P. Colpa, and F. N. Hooge, J. Chem. Phys. 23, 413 (1955). 4 LA . P. Colpa and J. A. A. Ketelaar, Mol. Phys. 1, 14 (1958). 5Z. J. Kiss, H. P. Gush, and H. L. Welsh, Can. J. Phys. 37, 362 (1959). 53. van Kranendonk and R. Byron Bird, Physica 17, 953 (1951). 7D. R. Bosomworth and H. P. Cush, Can. J. Phys. 43, 751 (1965). 8D. A. McQuarrie and R. B. Bernstein, J. Chem. Phys. 49, 1958 (1968). 9L. M. Trafton, Astro. J. 146, 558 (1966). 100. Tanimoto, Progr. Theoret. Phys. (Kyoto) 33, 585 (1965). 11H. B. Levine and G. Birnbaum, Phys. Rev. 154, 86 (1967). 12M. Moraldi, Chem. Phys. 78, 243 (1983). 133. L. Hunt and J. 0. Poll, Can. J. Phys. 56, 950 (1978). 146. Birnbaum, S. I. Chiu, A. Dalgarno, L. Frommhold, E. L. Wright, Phys. Rev. A 29, 595(1984) 15J. Van Kranendonk and D. M. Cass, Can. J. Phys. 51, 2428 (1973). 16J. 0. Poll and J. L. Hunt, Can. J. Phys. 54, 461 (1976). 17M. Gruszka and A. Borysow, Mol. Phys. 88, 1173 (1996). 18R. H. Taylor, A. Borysow and L. Frommhold, J. Mol. Spec. 129, 45 (1988). 52 19G. Birnbaum, G. Bachet and L. Frommhold, Phys. Rev. A 36, 3729 (1987). 20M. Moraldi, A. Borysow and L. Frommhold, Chem. Phys. 86, 339 (1984). 21Y. Fujita and S. I. Ikawa, J. Chem. Phys. 103, 9580 (1995). 22F. Strehle, T. Dorfmuller and J. Samios, Mol. Phys. 72, 993 (1991). 23P. Dore, M. Moraldi, J. 0. Poll and G. Birnbaum, Mol. Phys. 66, 355 (1989). 24K. Fox and I. Ozier, Astro. J. 166, L95 (1971). 25C. G. Gray, J. Chem. Phys. 55, 459 (1971). 25]. K. 6. Watson, J. Mol. Spec. 40, 536 (1971). 27A. Borysow, M. Moraldi and L. Frommhold, J. Quant. Spec. Rad. Trans. 31, 235 (1984). 23]. Borysow, L. Trafton and L. Frommhold, Astro. J. 296, 644 (1985). 29A. Borysow and L. Frommhold, Astro. J. 303, 495 (1986). 30A. Borysow and L. Frommhold, Astro. J. 311, 1043 (1986). 31A. Borysow and L. Frommhold, Astro. J. 304, 849 (1986). 32A. Borysow and L. Frommhold, Astro. J. 318, 940 (1987). 33G. Birnbaum, A. Borysow and A. Buechele, J. Chem. Phys. 99, 3234 (1993). 34R. Kubo, Lectures in Theoretical Physics, vol. 1, ed by W. E. Britten and L. C. Dunham (Interscience Publisheers, Inc., New York), p. 151.(1959). 35]. H. Van Vleck and V. F. Weisskopf, Rev. Mod. Phys. 17, 227 (1945). 36nL = 6.022 1023/22413.6 cm3. 37A. Rosenberg and J. Suskind, Can. J. Phys. 57, 1081 (1979). 53 38H. J. M. Hanley and M. Klein, J. Phys. Chem. 76, 1743 (1972). 39Egelstaff, P. A. 1960. Proceedings Symposium on Inelastic Neutron Scattering, IAEA, Vienna, 25. 54 CHAPTER 5 NUMERICAL RESULTS FOR COLLISION-INDUCED ABSORPTION LINE SHAPES 5.1 Introduction In this chapter, numerical results will be given for the spectral moments and these values will be compared to those calculated by Birn- baum, Borysow and Buechele.1 The moments are then used with the dipole coefficients found in Tables 4.1 and 4.2 to compute the various contributions to the absorption coefficient spectra for the methane- nitrogen system. There are several discrepancies between the values of the Birnbaum group and those of the Hunt group. These discrepancies are due to dispersion terms added in this work, calculation cutoffs by the Birn- baum group and some incorrect data found in the tables of Birnbaum, Borysow and Buechele. These discrepancies are studied and corrected where necessary. The final spectra of the Hunt group show a slightly better fit to experiment compared to the Birnbaum group's spectra. This only slight improvement is commented upon and future work is detailed. 5.2 Spectral Moments The zeroth, first, and second moments, denoted M0, M1, and M2 respec- tively, are listed in Table 5.1 for each contribution to the absorp— tion coefficient. Discrepancies were found between the values from 55 Birnbaum et al. and those calculated in this work though the same equa- tions were used. The discrepancy is due to cutoffs in the numerical integrations in the calculations of Birnbaum et al. In this work the integrals were either calculated exactly with use of Mathematica, or if numerical integration was necessary, the number of iterations was increased until no further changes in the result occurred. The new values of M0, M1, and M2 for each of the contributions to the absorp- tion coefficient spectrum are found in Table 5.1. These contributions include the nitrogen quadrupole-induced-dipole in methane, the nitro- gen hexadecapole-induced-dipole in methane, the methane octopole- induced—dipole in nitrogen, the methane hexadecapole-induced-dipole in nitrogen, the response of the anisotropic polarizability of nitrogen to the octopole of methane, the response of the dipole-quadrupole polar- izability of methane to the quadrupole field-gradient from nitrogen, the response of the anisotropic polarizability of nitrogen to the hexa- decapole of methane, the response of the dipole—octopole polarizabil- ity of methane to the nonuniformity in the quadrupolar field-gradient from the quadrupole of nitrogen and the leading term in the dispersion dipole of methane. Using the values in Table 5.1 and Equations (40)-(42) of Chapter 4, the theoretical spectra for collision-induced absorption in the far— infrared range are found for the (}u---N2 system at three different temperatures, 297 K, 195 K and 162 K. The spectra were calculated from 0 to 700 cm-1 for 297 K and 195 K and from 0 to 600 cm“1 for 162 56 K. The shorter range is given for the 162 K system, since the interac- tion effects fall away faster at this lower temperature. 5.3 The Spectral Contributions. Figures 5.1-5.3 represent the quadrupole-induced—dipole, GNzaCHu contribution to the absorption coefficient 5 for the three tempera- tures, 297 K, 195 K, and 162 K, respectively. In this contribution the quadrupole field of the nitrogen polarizes the methane molecule, inducing a dipole. There were no dispersion effects large enough to merit calculation and addition into this term. As noted in the cap- tion, the computed spectrum of Birnbaum et al. is shown in dashes and the spectrum computed in this work is shown in dots. There is a slight difference between the two plots (at 100 cur1 the separation is 3%). The main reason for the slightly higher peak given in this work is that a substantially different value was used for sz. Figures 5.4-5.6 represent the hexadecapole—induced-dipole, §N20{H4, for the three temperatures given above. In the calculations performed in this work, the value for the hexadecapole of nitrogen was the same as that used by Bi rnbaum et al. There are no dispersion terms added to this contribution and thus, the plots overlap nearly exactly. Figures 5.7-5.9 represent the methane octopole-induced-dipole, QNZQIH4. for the three temperatures given above. It is seen in Table 4.1 that there is an extra exponential term, 2.; exp(-(R-o)/p), added 57 into this contribution, which takes into account the overlap of the two molecules during collision. o is the potential parameter and p is the range of overlap. The amplitude of overlap, A43, is an approxima- tion based on ab initio work for inert gas pairs. The term is a summa- tion of squares of the terms found in Table 3.3: D(3034;:2). The mathe- matical method used to sum these terms is found in the Appendix.2 The dispersion terms of Dp(3032;:2) from Table 3.3 must be added also. These two terms interfere with the that of Birnbaum et al. and were added into the calculation using the mathematical method shown in the Appendix. However the dispersion effect increased the spectral inten- sity by less than one percent and the difference cannot be seen in the spectra. Figures 5.10—5.12 represent the methane hexadecapole-induced- dipole of nitrogen, QNZQCH‘. There is an overlap term given as 154. The terms summed to form the term of Birnbaum et al. are D(4045;0) and D(404S;:4) from Table 3.3. Dispersion terms were not found to be large enough to tabulate. Figures 5.13-5.15 represent two different types of induction, the response of the anisotropic polarizability of nitrogen to the octopole of methane, YNZQCH... and the response of the dipole-quadrupole polariz- ability of methane to the quadrupolar field gradient from nitrogen, ) 74 x nmm um . vIU<~z® .uczz .e:mn::wm .1. 1 £00 42» . :o .545 .558 5:0 .5 .5653 :0 596.22 $58 5:032... .mH.m mmDUHm 00h 000 00m 00¢ 00m 00m 00H 0 I 4 J 41 4 4 4 ' ”'OH I v I f not I all t v 000/ 4". m I ‘0 0 at so 1 ... M. coll \ o coil 44. filo." u on I so. ole/I \o ‘0 OOH/I \\00 m 00” \‘O. m .0 II \ 0 a cell! \\00 I. ....IOI.’ IIIII ‘\H.\00 “ Cocoonololololooooo ‘0 . 0-0." .m, T- e w an A mic." w... 4... ( 75 . Hen: .sawn:cwm 1.. v. 34 um .4:u<~z@ 14504255545558 50.5.5500 55:54.94 .36 $5an 005 $58 3:03.42“. 000 00m 00¢ 00m 00m 00H 0 4 ”C, 4 4 1 4 1 F1 ”IOH ole/I 99 ..I a V t .u, . m .1 I 0 a, 40 J 00’ s. d I ‘ 00’ ‘0 m- .I 4. u 010/ so. 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Woe M r 86 .03 Tea $0 u3~> « a 9.3m; w>.._:u 593.2 2.? nuts .03 to.“ 3 m3; cw mcwmmmgucw w>gau gwsuw; >Fw>wmmuuuzm sumo spwz .o u « mucmmwgawg m>g=u umwzor as» .« we marm> pcwgwmmwu m mucmmmgawg w>g=u cumm .ucwaw$$wou commgwamwu K-x wsu ow nQ\no-mvuvaxw 4 .Eguu ampgm>o cm acmuum mo uuwwmw msh .m~.m wgsmwu :Euv 35:3: DOB 08 gm 00¢ 8m CON OCH 0 . - . . . 93 HV 0. S m w. m. WOH 0 «J 0 m. m N. W 4 0:0." \m w...- p. w B # muOH am. (.... 87 REFERENCES 16. Birnbaum, A. Borysow and A. BuecheJe, J. Chem. Phys. 99, 3234 (1993). 2The materiaJ in the appendix is taken from the appendix of reference 1 and from P. Dore, M. MoraJdi, J. D. P011 and G. Birnbaum, M01. Phys. 66, 355 (1989). 3X. Li, M. H. Champagne and K. L. C. Hunt, J. Chem. Phys. 109, 8416 (1998). 88 CHAPTER 6 DERIVATION OF THE IRREDUCIBLE THREE-BODY POLARIZABILITY 6.1 Introduction CoJJision-induced Tight scattering (CILS) spectra of the inert gases and hydrogen at high densities show nonadditive three-body inter- action effects}5 The three-body contributions to CILS spectra'l moments have been determined experimentaJJy, but they have not yet been eprained quantitativeJy. This chapter focuses on one source of these effects in CILS: the nonadditive three-body poJarizabiJity Aa‘” of interacting moJecu1es with nngigibJe eJectron overJap. For a set of isotropic systems with centers at r,, rj, rk,..., the totaJ poJarizabiJity a can be expressed as a cJuster sum"*8 (1 = Z1; a<1>(r,-) + 12;; Aa<2)(r,-,rj) + 1.§::k13cyx<3>(r,-,rj,rk) + -~-, (1) where a““(r{) is the poJarizabiJity of the moJeque Jocated at r3, when iso'lated; Aa<2>(r,-,rj) is the interaction-induced poJarizabiJity’47 of the pair of moieques at r, and rj; and Aa<3>(r,-,rj,rk) is the nonadditive three-body po'larizabi'litym'22 of the moJeques at r,, rj, and rk. The terms in the c1uster expansion of the intermoJequar potentia] e have been more extensiveJy studied; e = 1'24. ¢(2)(r,-,rj) + gin. ¢(3>(ri,rj,rk) + (2) Here ¢(2)(r,-,rj) is the two-body interaction energy, and ¢<3>Cr1,rj,rk) is the nonadditive three-body contribution to the potentia1.2132 89 The nth moment Wk of the CILS spectrum of a compressed gas is deter- mined by the t = O va1ue of the nth time derivative of the poJarizabiJ- ity autocorreJation function C;,,B(t),5“8 cam) = mam acaco». (3) with a3 = 22 for poJarized (VV) Tight scattering, and a5 = XZ for depo- Jarized (VH) scattering, where the axes are specified in the 'labora- _ —M1" tory frame. Wk has the viriaT expansion‘i“8 1an+2Mn 02+3an3+'°'. (4) 3;, where p is the number density, and 2M" and 3H,. are the pair and trip- Jet terms, respectiveJy, in the nth moment. From Eq. (3) and the expan— sions for a and e, there are nonvanishing tripJet contributions to the spectra] moments that appear as a requt of three-body dynamicaJ corre'lations,7'33‘39 even in the pairwise-additive (PA) approximation, in which the series in Eqs. (1) and (2) are truncated at mm and ¢<2). SpecificaJJy, the tripJet term in the nth spectraJ moment (3Mh) con— tains both the PA contribution 3M,‘,2’ and an i rreducibJe three-body effect 3Nu3),4° 3Mn = 3an) + 3M?) - (S) The derivation of the irreducibJe three-body poJarizabiJity is based on a nonJocaJ response theory deveioped for three-body energies and three-body dipoJes.41'42 This theory gives the interaction energy in terms of nonJocaJ poJarizabiJity densities a(r,r';w) and hyperpoJariz- abiJity densities BCr,r',r";—wo;w,w') and yCr,r',r",r ;—wo;w,w',w"), where wo denotes the sum of the remain— 90 ing frequency arguments in the hyperpoJarizabiJity. The poJarizabiT- ity density a(l",r";w) 43“" gives the poJarization P(r,w) induced by an apined fieJd F'Cr'uu) acting at r'. Thus it characterizes the distri- bution of poTarizabiJity within the interacting moJecu1es; simiJarJy B and 7 account for the intramo'lecuTar distribution of the nonJinear response to an appTied fieJd.47-49 '. |A-‘m 5'- Starting from recent requts‘u’42 for the three-body energy AE‘3’, Aa<3> is evaJuated for an A---B---C duster by taking the functionaT ‘-+ derivative of the three-body energy AE‘” with respect to an applied eJectric fieJd F‘Cr), Aa<3> = Jdr dr' [-62AE(3)/6F‘(r)6F‘(r')]. (5) 6.2 Derivation of the Irreducible Three-Body PoTarizabi'lity IrreducibJe three-body terms appear in the interaction energy at second and higher order in the A—B—C interaction.‘1-42'5°v51 In this chapter, the notation E(”'") is used to represent an n-body contribution to the mth order interaction energy; thus (19(3) in Eq.(Z) is obtained by suming Em” over m. The nonadditive second—order energy E‘Z'” requts from induction in each moJeque due to the fiest from the other two.43v49'5°'51 For moJeque A ,41'42 E3” = — Jdr- . -dr"'aACr,r'):[T(r,r")-P8(r")] (7) x[T(r' ’rl t ')'P8(r" I 9)]. 91 Here, T(r,r') denotes the dipoTe propagator, TCr,r') = vv |r-r' vi. The poTarization of moiecuTe B in the absence of A and C is given by PSCr). This property is reJated to the charge density p3(r) of B by v P8(r) = —p8(r); simiJarJy for P3, the perma- nent poTarization of moJeque C. The response of moJecuie A to the fiest from its neighbors is determined by its static, nonlocaJ poJarizabiJity density aCr,r') a a(r,r',w = 0),4}47 (3) aaBCr.r';w) = (OIPaCP) C(w) PBCP')|0)+<0|Pb(r') GC-w) PLCPJIO). (9) In Eq. (8), C(w) denotes the reduced reskoent, 60.)) = (1 — Do)(H - Eo - fiw)'1 (1 - 00). where H is the unperturbed moJequar Hamthonian, E0 is the unper- turbed ground-state energy, and 90 is the ground-state projection opera— tor, p0 = |0) were used to determine the three-body poJarizabiJity at second order, AafiL3>. The functionaJ derivatives of the poJarization P3Cr') and the poJarizabiJity density aACr,r') satisfy 5P3,,(r)/6F§(r') = OQBCPJ'), (10) 52P3a(r)/6F§(r')6F:(r") = 5035C". P')/5F:(l‘") = Bémcrvr'anul) and 62@B(r,r')/6F$(r")6F§(r") = 725,5(I‘.r',r'-,r"'). (12) In Eq. (11). th(r,r',r") is the static, nonJocaJ hyperpoJarizabiJity of moJeque A; that is, BQBYCrm'm”) = 5“ (r,r',r";0,0,0). The 037 frequency-dependent B tensor satisfiessz'S3 92 —_ - -a-)li,.h.un 5.; _. F‘ _ BQBYCPJ'J'”; -wo; an. (dz) = 512[(0|Pa(r) 6000) P30") 6001) PBCP'NO) + <0IPYCr-") CC-wz) P20") cc—wo)P.(r)I0> + (13) (OIPYCP'U C(-w2) P309 C(w1)Pp(P')l0)]- where the operator $12 denotes the sum of terms obtained by permuting the frequencies ml and w; in the expression foJJowing it, and simtha- neousJy permuting the operators P,3(r') and P,(r'). In Eq. (13), we = wl + 0);, and P3,,(r) a Pa(r) — (OlPa(r)|0). The second hyperpoJarizabiJ- ity density 735,6(r,r',r",r"'; -wo; ml, mg, mg) is obtained by extend— ing the perturbation analysis to the next higher order;52-53 735,6(r,r',r",r"') in Eq. (12) is its static Jimit. From Eqs. (7)—(12), is derived the genera] requt —5ZE,§2-3>/5Fg(r)5rg(r') = far". . -dr" 735,6(r,r',r",r"') xTYECI‘". r“) PBGCr‘V) TMCr'”. r") P3,,(r') + SaBSBCJdr" . - ~dr" 333,6(r,r”,r"') T,€(r", r“) x agflcrivm') T5¢(r"', r") P3,.(rV) (14) + Sscfdr". . -dr" a¢6(l"",l""') T..—Cr", riv) x 320,3 (r‘V,r,r') T6¢(r"', r") P3¢(rV) + Sagjdr"- - -dr-v a¢6(r",r"') T,e(r", r“) x a2a( r",r) T6¢(r"', r") agB(rV, r'). In Eq. (14), 50.3 denotes the sum of terms obtained by permuting the subscripts a and B in the expression fo‘lJowing it, and SEC denotes the sum of terms obtained by permuting the moJecu1e Jaber B and C. 93 For neutraT atoms in S states, Po(r) is nonzero, but its integraJ over aTT space vanishes, as do the moment integraTs that yieJd that quadrupoJe 90, the octopoJe 90, or the higher-order charge moments. The same requts hon for moJecu1es after isotropic averaging over aTJ of the moJequar orientations. Thus in the Tong range Jimit, onTy the Jast term in Eq. (14) and the corresponding terms from EQL3’ and EXL3) contribute to the second-order, three-body poJarizabiJity AaUL3> for isotropic systems A, B, and C. By TayJor expansion of the Jast term in Eq. (14) and use of Eq. (6), for isotropic species we obtain5 Aaéfi'” = aAaBaCSABCTMCRA,RB)T35(RA,Rc)+(1/3)SABCCAa3aCTam5(RA,RC) (15) xTB,5(R*,RC) + where the species centers are Jocated at R“, R3, and RC, respectively; .Smc denotes the sum over a1] permutations of the TabeTs A, B, and C in the expression that foTTows it; and the propagators Tag...,,(r, r') of arbitrary tensor rank are given by TaB...,,(r, r') = vavfln-vulr-r'I-l. In Eq. (15), a is the poTarizabiJity of the isotropic system, Chg = a 605. The C tensor determines the quadrupoTe induced by a uniform fieJd gradient, within Jinear response.6 For isotropic systems C takes the form6 Came = C [(5.17 686+ 5:16 5137) " (1/3) 6(136761- (16) Equation (16) defines the scaJar C'A appearing in Eq. (15) above. Equation (15) is equivaJent to Aagé'3)= or‘aBaCSAgc(9rQBr},CcoseA —3 rgerfis—B rgcr’gC-apéaBMfiRgé +(1/3) SABCCAaBaC[r§3r‘}3C(225 c0529A — 9) (17) 94 -9O coseA(r33r’}§3+ rQCrABC) + 18 coseA6a3]R;§Rgé +- - - , where 9A, 93, and 9c are the interior angJes of the ABC triangTe at the vertices A, B, and C, respectiveTy; rfiB denotes the a component of a unit vector pointing from A to B; and RM; is the distance from A to B. The terms omitted from Eq. (17) are of order (1‘10 or higher in a repre- sentative distance d between the molecu1es in a cJuster. At third order in the A—B—C interaction, three distinct mechanisms contribute to the nonadditive three-body energy AE9”: cTassicaJ three- a. body induction, dispersion (van der WanS effects), and induction- dispersion interactions.‘1'42 The c1assica1 induction energy AEéza” in turn is a sum of three componentsz7»8 the static reaction fieJd energy ang) the thi rd-body-fie'ld energy [15:39), and the hyperpo'larization energy AS1353). The static reaction-fier terms Acgfi3> account for the energy shift due to c10$ed poTarization Joops, e.g., the poJarization P3(r) of mole- que A sets up a fier that poJarizes B; the fier from the induced poJarization of B in turn poTarizes C, thereby setting up a reaction fier that affects the energy of A. Accounting for both possibTe poJar- ization routes (A—aB-aCaA and A—>C->B—>A) gives for mo'lecu'le A ,7-3 mag-3g}, = - far. . .dr" P3(r)-T(r,r')-a3(r',r")- (18) T(r",r"')-ac(r"',r"V)-T(r"V,rV)-P3 (rV). The net static reaction—fieJd energy is the sum of AESQA, AEéizr-fi, , and AEéifé . From Eqs. (6),(10)-(12), and (18), we obtain the static reac- tion-fieJd effect on the three-body poTarizabiJity. In the Tong-range 95 limit, Aa;2;3> = 5m aAaBacaA Ta,(RA,RB) T,6(R°,RC) T6,,(RC,RA) .03 = SABC aAaBaCaACZ7 rfiBng‘cos 93 cos ec + 9 rQBr'fBCcos eg (19) + 9 rQBrgAcos 9A + 9r3crcg‘cos 6c + 3 rgar‘gh- 3 r393; _ -3 -3 + 3 '39"; ‘5a33 RAgRscRAc- The third-body-field energy A5439) is similar to AE(3'3), but the srf Tl polarization routes that contribute to mag,” begin and end on differ- i ent centers; 7v3for example, the term in AEég'f” associated with the 4.- route A—)B—>C—>B 'iS AEéiflsc = - I dr- ~dr" PSCr)-T(r.r')-accr',r")- (20 TC!‘”,I’"”)-aB(I""',riv)'T(rivorv)'P8(rV)o ) The total thi rd-body—field energy neg,” equals smaeggggmc; then from Eqs.(6),(10)—(12), and (20), the third-body-field contribution to the polarizability satisfies Aaggg’gw = SABC SOB a8 ozC a3 a:A (9 rgc rgA cos 93 —3 rgc rgc (21) + 3 If"? ' 50KB) RAgRég- The third classical induction effect results from the hyperpolariza- tion of each molecule by the fields from the other two, giving the (3.3) yp For molecule A ,7-3 energy change A Aegis}, = ~(1/2)(1 +9“) I dr' - °drVBA('v"v"') ‘ (22) T(r,r"')-P3(r"')] [T(r.PiV)-P3(r'iv)][T(l"'.f"’)-P(C)(f"')] where Q“ permutes the labels B and C. For interacting atoms in S 3.3) states , AEéyp’A is nonzero only at short range; there it reflects the direct electrostatic overlap effects that lead to hyperpolarization. 96 Similarly, Aaéiflm is nonvanishing at short range, but in the long- range limit, (23) In addition to the static reaction-field effects treated in Eq. (18), there are dynamic effects due to the polarization fluctua- tions that generate the irreducible three-body dispersion energy AEQ”. For example, a spontaneous, quantum mechanical fluctuation in the charge density of molecule A generates a field that polarizes B; the polarization of B induces a polarization in C, which gives rise to a dynamic reaction field at A. The average energy shift due to the reac- tion field depends on correlations of the fluctuating polarization of A - which are determined by the imaginary part of the polarizability density of A, acccording to the fluctuation-dissipation theorem.9 From earlier work,17 allowing for all permutations of the labels A, B, and C, was derived the nonadditive three-body dispersion energy at third order,1°'11 AEé3'3) = (-n/n)fdwfdrm dr" Tr[T(rV,r‘°V).aC(r‘°V,r"';iw)~ (24) T(r"',r")-a3(r",r';iw)-T(r',r)-aA(r,rV;iw)]. From AEQL3), Eqs. (6),(11), and (12), and Taylor-expansion of the coor- dinates in the propagators about the molecular centers, we obtain the leading, long-range dispersion effect Aafi in the three-body polarizabil- ity of atom A, mfg”: (h/Ir) f dw T,5(RA,RC) agefiw) T€¢(RC,RB) 0 (25) 0:3) (‘iwyfin (RB ,RA)meB(-1'w; 'iw,0,0) . 97 For isotropic systems, yfimé(—iw;iw,0,0) = fiC—iw;iw,0,0)6a,36,5 + fiC-iw;iw,0,0) (26) XC5G7655+605537) . From Eqs. (25) and (26), and the dispersion polarizabilities of B and C, we obtain 11018512) = (h/Zn)SA3cfaB(RA,RB,Rc)fdw fiC—‘iw;‘iw,0,0) aBC‘iw) aCCT'w) 0 +(n/2n)s.3cg,B(RA,RBAbra.) ygc-iw; ‘iw,0,0) (1360)) 0636.007) 0 where 1:03 (RA 1R8 9RC) T76 (RAQRB) T66 (Rcvks) T67 (RB,RA)6aB = -3(1 + 3 coseA cos 9.; cos ec)6aBR;§ Rag R33 (28) and 9,5an ,RB ,RC) [Ta,(RB,R‘) T,5(R‘.R°)T65(R‘.RC) + TaYCRA.RC)T}5(RC.RB)TBBCRC.RA)] = “3(27r33rgA c0593 cosec + gram-gt c0593 (29) + 9r‘,,‘,'3rgA cos 6A + 9rf,‘,crf,A cos 9c 4» Bryrgfl + arm - madam-gage. The perturbation of the two-body dispersion interactions by the field of the third molecule yields the final third-order, three-body energy: the induction-dispersion term £15353) . The pair dispersion energy of A and B is altered by the presence of C via two mechanisms: (1) Both A and B are hyperpolarized by the concerted action of the fluctuating field from the partner molecule (8 or A) and the field from the polarization of C; and (2) the field from C alters the intrin- sic, quantum mechanical fluctuation correlations that determine the A-B dispersion energy.12v13 Earlier workl‘“15 treated the first effect 98 by use of a field—dependent nonlocal polarizability density to describe the response of each molecule to the fluctuating field of its neighbors and treated the second by use of a field-dependent imaginary part of a(r,r'; w) in the ffiuctuation-dissipation relation for the polarization flucutations. The resulting change in the A-B dispersion energy due to C is16 Aegiz?13),_C = - (1 + pBC)(h/2n)j:dwfdr dr" agmo'm'unium) T}5(r",r"') aSECr'",er;iw)Tg5(rfiV,r') Tg¢(r,r”) P3¢(rV). (30) From Eqs. (6), (10)-(12), (30), and the analogous equations for the (A---C)+B and (B---C)eA energies, we obtain the three-body induction- dispersion polarizability in the long-range limit, Aafiafifi = 5m (1 + pawn/2n) j: dw ye...<-iw;iw.o.0) 013(in aC(0)T6,(RA,RB)T,€(RA,R8)T,,,(RA,RC) (31) Then from Eq. (26), 401333.13 = 5m: (smog dw fiC-iw;‘iw,0,0) 018(1'w) aC(0)(3rgCrgC - (Sap) R38 R32 (32) + 5mm + papa/m]: dw yac-iw ;iw,0,0) aBC‘iw) aC(0) x(9r{,§3r‘,§,C cos 6;), - 3r33rg3 + BrQCrQC - 60,3)Rggkgé When summed, Eqs. (17), (19), (21), (27), and (32) give the individual tensor components of the irreducible three-body polarizability, to order d‘9. The three-body effects on the scalar polarizability were derived by averaging isotropically over the orientations of the A-- B --C clus- ter to give [10(2'3) a (l/3)Aa,§3). At second order, the classical induc- 99 tion terms yield Ad‘2'3) s SABC [aAaBaCCB COSZGA - DRAgRAE (33) + SCAaBaCCS cos3eA - 3 coseA)R,-,g RAE]; while at third order, induction yields A 013;” = _zaAaBaC(aA + a8 + ac) (1 + 3 coseA C0593 cosec)R;\3RAER63 and (34) A0439 = ZSABcaBaCaBaAO coszea.;,-1)R,;u33 Rig. (35) Dispersion effects on the scalar three—body polarizability are given by Ads“) = —SA3c (h/n)R;gR;gR§g(1 + 3 coseA c0593 cosec) “(a/2)]: dw yfi-imiw ,o,0) aBCiw) aCCiw) (35) +J: d... y; (4... ;iw,0,0) aBC'iw) atom] and induction-dispersion effects by mfg,” = 2 sABC (II/7r) J: d.) y; (-iw;iw,0,0) (180'...) aC(0) (37) x(3 coszeA - 1)R;§R;g. (L3) The 7? term in Ath+d vanishes upon contraction, when the isotrOpic average is computed. 100 . '. 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Mukamel, J. Chem. Phys. 98, 7046 (1993). 50D. E. Stogryn, Phys. Rev. Lett. 24, 971 (1970). 51F. Piecuch, Chem. Phys. Lett. 110, 496 (1984). 528. J. Orr and J. F. Ward, Mol. Phys. 20, 513 (1971). 53D. M. Bishop, J. Chem. Phys. 100, 6535 (1994). 104 CHAPTER 7 NUMERICAL RESULTS FOR THE IRREDUCIBLE THREE-BODY POLARIZABILITIES 7.1 Introduction The induction contributions to Ad“” and to the individual tensor components Aug? depend on the static dipole polarizability a a a(0) and the static quadrupole polarizability C of molecules A, B, and C. Values are used for a and C from ab initio calculations or from experi- mentalmeasurementsfl"13 listed in Table 7.1. For the molecular spe- cies, response tensors obtained by isotropic averaging over all orienta- tions of the molecular axes are used, as indicated by the notation (X) for molecule X. The averaging reduces the C tensor to the form in Eq. (16) of Chapter 6, with the scalar C values given in Table 7.1. For [Lm molecules, the scalar C is related to the components in the mole- cule-fixed frame by C = (4/5) Cxx'xx + (1/10)sz,zz + (4/S)CXZ,XZ (1) where z is the symmetry axis. For Ta MO19CU195: C = (3/5) sz,zz + (6/5) sz,xz - (2) The dispersion and induction-dispersion contributions to Add?” depend on polarizabilities and hyperpolarizabilities at imaginary fre- quencies. The single-molecule polarizability a(iw) has been computed for a number of species, since a(iw) yields van der Waals interaction 14—21 energy coefficients and photoionization cross sections.”*24 At 105 Table 7.1. Molecular properties used in computing three-body polariz- abilities. Static dipole polarizability, a, quadrupole polarizability C, and hyperpolarizability 7 for isotropic systems; the notation (X) denotes isotropic averaging over all possible orientations of molecule X. Values in atomic units (a.u.) Species a Ca 72 H 4.59 7.5“»9 444.375f He 1.383199 1.22269 14.3689 Ne 2.3766h 3.2102h 22.86h Ar 10.757h 25.096h 319.6h Kr 16.791 49.931 748.3h Xe 27.76j 108.25 1947h (H2) 5.3966k 8.3571k 221.23‘ (N2) 11.675' 40.37- 276.8' 16.39n 60.45n 770.7n 17.626° 77.757o 399° aFor S-state atoms, C = (3/2) C22,;z ”71 = 71(0;0,0,0). See Eqs.(3)-(S). “Reference 2. dReference 3. 9Reference 4. fReference 5. 9Reference 1. "SCF values from Ref. 6. 1'Semiempirical 0050 result from Ref. 7. jSDQ-MP4 calculations from Ref. 41; C = 1/2 a2, from this reference. “Reference 9, values at R = 1.4 a.u. 7Reference 10. ”Reference 11. "CCSD(T) values obtained at R =2.052 a.u. with an (115 7p 4d 2 f/65 2p 1d)[6s 4p 4d 2 f/4s 2p 1d] basis set, Ref. 12. °SDQ-MP4 results from Ref. 13. 106 present, the hyperpolarizabilities y‘l‘(—iw;iw,0,0) and 73(—iw;iw,0,0) are known with high accuracy only for H, He, and H2; Bishop and Pipin have calculated these properties using explicitly correlated wave functions.10 For S-state atoms, 71(-iw;iw,0,0) = yxxzz(-iw;iw,0,0) (3) and YZC-Tw;'iw,0,0) = [Yzzzz(-1'w;‘iw.0.0) " yxszC-iw;'iw,0,0)]/2. (4) For th molecules , VIC—1w;'iw,0,0) (1/15)yzzzz(-‘iw;1'w,0,0) + (4/15)yzzxx(-1'w;‘iw,0,0) +(4/15)Yxxzz(-1'w;iw.0.0) - (4/1537xzsz-iw;iw,o,0) (5) +(1/3)xxxyy(—iw;iw.0.0) + (1/15)7xxxx(-1'w;iw.0.0). and 72C-1'w:1'w.0.0) = (1/15)Yzzzz(-1'w;1'w.0.0) - (1/15)Yzzxx(-‘iw;iw,0,0) -(1/15)Yxxzz(-1'w;1'w.0.0) + (2/15)szxz(-iw;iw,0,0) - (1/6)Yxxyy(-1'w;1'w.0.0) + (7/30) YxxxxC-iw;iw,0,0)(.6) The susceptibility 705,5(-1'w;1'w, 0, 0) is symmetric under interchanges of a and B or 7 and 6, but it lacks full permutation symetry (with respect to the indices alone) unless w = O. This accounts for the forms of Eqs. (3)—(6). The integrals in Eq. (27) of Chapter 6 for the three-body disper- sion term Aaé3'3’ are evaluated for clusters containing H, He, or H; by use of 64-point Gauss-Legendre quadrature, with the values given by Bishop and Pipin for the imaginary-frequency susceptibilities.10 The numerical results are listed in Table 7.2. The integrals in Eq. (32) 107 owma m.nmma OOmH n.¢m¢H wwmm.0m A~Iv A~Iv ANIV wmm Hm.mwm Hmv oo.wa¢ vomo.ma A~Iv wI ANIV wHH m~.mOH NNH mm.mHH mnoon.¢ wI mI ANIV oooa m.¢oaa CmHH H.¢~HH me~.wm ANIV I ANIV Hem wv.mom mmm Hm.m~m mv~.HH w: I ANIV mmw vo.mnw can mm.m~m mumm.m~ I I A~Iv «Ha mm.o~H MNH m¢.n~H vono.mH ANIV ANIV wI m.¢m mnw.vm ~.nm wmm.mm mn00n.¢ ANIV m: w: m.mw mm.~m .mm ~w.mm mv~.HH A~IV I mI omwm .oawm owmm H.wnm~ Hom~.wm ANIV ANIV I «mm mw.mmm mmm wm.wmn m¢~.HH A~IV w: I ommH m.m~mH owON m.wHHN mumm.m~ A~Iv I I w.oa wmm.0H N.HH w~.HH mMme.H wI m: w: m.m~ mom.m~ v.n~ HoH~.w~ Hm-¢.m wI I wI v.mo mmH.~n w.on wmm.mn mo¢w¢.w I I 0I HON nm.mna mam o.oo~ Hmmmv.m wI wI I va mm.o~m mum mm.owm mmvwv.w mI I I OOmH m.mmmH omwa o.mme ammo.- I I I ~9 Uwumzrm>w .Sbmawvudmaowvmsmsewvfisaq ELOL. wzu $0 mPMmeHCH .N.N wPQMH 108 of Chapter 6 for the induction-dispersion term 864:3) contain the prod- ucts y(-iw;iw,0,0) a(iw) for each pair. These same ya integrals deter- mine the effects of two-body dispersion interactions on polarizabilities;”*27 hence for H, He, and Hz, the ya integrals have already been computed with high accuracy.1°'26 In Table 7.3 are listed 13) numerical values for the coefficients in Aaéfl, , derived from the two- body integrals given by Bishop and Pipin10 or found by direct 64-point sf memaJI: { Gauss-Legendre quadrature. For species beyond H, He, and H2, mg”) and 2101,3353) must be approxi- mated, since the hyperpolarizabilities yl(-iw;iw,0,0) and 72(—iw;iw,0,0) are not yet known. A form of the constant—ratio app roxi mati 0:128'29 is used which relates the yaa integrals to static a and 7 values and the three-body van der Waals energy coefficient C9. In its simplest version, this approximation yields identical values for the Yiaa and ygaa integrals, because 71(0;0,0,0) = 72(0;0,0,0) for all isotropic systems. The integrals are not equal, however: the integrand drops more rapidly from its zero—frequency value in the yzaa integral than in the 71aa integral. From Table 7.2 for triplets con— taining H, He, and Hz, the ratios of the Yzaa integral to the ylaa inte- gral are 0.923 1 0.024. For this reason, for all of the heavier spe- cies we have adopted a modified constant-ratio approximation (CRA) of the form Fdw r} (4...; iw,0,0) aBC‘iw) aCCiw) = (gjyg/aA)13(A,8,C) 0 7 )ofadw aACTw) aBCTw) aCCTw) ( ) 0 109 cam mm.mm~ -m em.o~m Hk~6.~a ANIV ANIV mam Hm.m- oe~ e~.we~ mmme.m I ANIV m.e~ m~m.a~ e.m~ Amo.oM eo~o~.e ANIV II Nae mm.ame mmm HI.HIm mmwmfl.m ANIV I m.w RHI.I we.m Refle.m kkm~m.fi II 6I .ma ~mu.o~ H.H~ mwm.- meemm.~ I 8I ”4H mm.mmH 46H ~o.mmH meemm.~ «I I wee mm.mmH 46H ~o.mmH meemm.~ 8I I cum em.mwm ”He mw.wm¢ memow.e I I 25 86.33.72» :5 86.33.15» 83., I < .=.m cw mupzmwm -xucwacweu .OH .mmm eccw mucwon wczumenmza wsu um mowuwanwuamumam uconcwamu .n~<¢u new Hn nonmapm>w .3umewve8msvvsmh% Ecov mnu mo mpmcmwucH .m.n ornmh 110 with :2 = 0.9235. The symbol y’i denotes fi(0;0,0,0) , and the func- tion Ig(A,B,C) is defined below. For the 710a and aaa integrals for H-~H---H, He---He-~He, and H2---H2---H2, the mean value of g1 is 0.726. cl = 0.726 and :2 =0.670 are taken as fixed values in the CRA for all of the ma integrals. (For comparison, a direct calculation within the Unséld approximation gives :1 = 19/24.) For S-state atoms, 71(0;0.0.0) = (1/3) Yzzzz(0;0.0.0); (8) for Dmh molecules, Y1(0;09090) = (1/5) Yzzzz(0;ovooo) + (4/15) Yxxzz(o;osoao) + (8/45) Yxxxx(0:0.0.0); (9) and for T... molecules, Y1C0:0.0.0) = (1/5) YzzzzC0;0.0.0) + (2/5) Yxxzz(0:0.0.0)- Static values of 71 are listed in Table 7.1. (10) The function 13(A,B,C) is defined by I3(A,B,C) = [Idem/‘018 aC/fdwaAaBac] [de y‘a‘cfi/{dwa‘a‘aA ]’1 (11) with all integrals evaluated in the Unséld approximation. This is the only stage of the calculation in which this approximation is used. In Eq. (11), fdwf‘aa a:C is used to denote the integral of Y1(0;0.0-0) (23(iw) aC(iw) over all frequencies from zero to infinity, and deaAaB ozC to denote the integral of QAC'iw) aBCiw) aCC‘iw) over the same frequency range. 13(A,B,C) depends on the ratios of the excita- tion frequencies QA, QB, and QC in the Unsdld approximation; these can be estimated by taking the ratios of the ionization potentials for A, 111 B, and C. Explicit equations for 13(A,B,C) are given in the Appendix. 1. If A, B and C are identical, 13(A,B,C) Since the Amilrod-Teller three-body dispersion energy coefficient CSBC sati sfies30 c539C = (3h/7r) J: d... (190'...) aBCiw) acciw), (12) the constant-ratio approximation then becomes Fdw y; (4...; 16.0.0) a8(‘iw) aCCT'w) = ngjygI3cA,8,C)c9ABC/(3na9). (13) 0 As above :1 = 0.726 and g; = 0.670. In Table 7.3, results from the CRA [Eq. (13)] are compared with the quadrature results. CRA1 approxi- mates the 71 aa integrals and CRA; approximates the 72 aa integrals. The root-mean—square (r.m.s.) error in these 36 integrals is 4.65%, which suggests that the CRA will be useful for heavier species. In Table 7.4, results are given from the constant-ratio approximations of Eq. (13) of Chapter 6 applied to the species Ne, Ar, Kr, Xe, N2, C02, and (JD; the values of C9 have been taken from Refs. 30-34. To approximate the ya integrals in the induction-dispersion polariz- ability, we set J: dw Y‘j‘C-‘iw; iw,0,0) aBC‘iw) = 7r gjfiI2(A,B)CQB/(3haA) (14) where Iz(A,B) = [J 6... 7‘08 / Id... (1" a8 1U d... 99 (1A / [6... (1A (1“ 1-1, (15) with the integrals in Eq. (15) evaluated in the Unsbld approximation. This gives I2(A,B) = (4/17) (6 + 8 A3 + 3 A§)/(1 + A3)2 (15) 112 .Hm .mwm eoLm mu com mow .om new em .mmm 50;» omoo onamma Now now; uwuzaeoue .wm wucweomwmt .Hm .mwm sot; mu cow nov.cw and em .mmm scew omoo ouzoma eIu saw; omuaasoue .mm mocwcmmwmu .Nm wucmcmmmmu .Hm mucwcmmwme o 32338865233; % o mmuchn so 6 fix% xrcmpwswm "suaswvcfisvvcmo.o.3w"avuvhm9a Facmwucw wcu mechu so 8 c NIEI 2: SS 8...: Seem .22 1.3 28V . . . 28v . . . $83 0me 2mm 8»? 88m .83 .052 2671.67 .AvIuv 83 em: 82: SN: 1.8.36 .2 .m tzv . . . 22v . . . A ~23 com: 802 822 82mm .33 am . an ex. . .mx. . .ex com 8% 83.. 8mm 9:: 3832 3.. . 5.. . .5. cm: 8.: SIS 82H ..m 83 gm .8 .2. . .._<. . .2 .3 .9. 8.8 4.3 33.: Imwmd 82:82:82 3U 6C4. 3t as; an Gaga a3b 685% mu C .836 pm a T..... .:.m cw mupammm .Imawcq new Immwca cw mrmcmwucv xucwacmcmnxcmcwmmew cow Away new Amav .mum scum meowumEIxOLaam avowenucmumcou .v.n wreak 113 with A3 2923/9", and I; (A, A) = 1. In Eq. (51), cgoo denotes the isotro- pic van der Waals energy coefficient, defined as a positive quantity: .. (17) c6 = (3h/n) Jo d... GAC‘iw) GBC‘iw). Setting 61 equal to the mean value obtained for H --H, He --He, and H2---H2 integrals gives 51 = 0.622. (The value of £1 obtained from the Unsold approximation is 17/24.) The average of the ratios of the )Qt: to no: integrals is used to set «52 = (0.900§1) = 0.560. Table 7.3 lists the numerical values of the ya integrals derived from the con- stant-ratio approximation of Eq.(14), for comparison with the accurate results from 64-point Gauss-Legendre quadrature, for A and B = H, He, or (isotropically averaged over orientations). The r.m.s. error in these 18 integrals is 4.99%. This improves on our previous constant— ratio approximation to integrals of the ya type,29 and provides a” use- ful basis for estimating the integrals in 2101,5333) [from Eq. (32) of Chap— ter 6] for heavier species. Numerical values for A---B-~-C clusters containing Ne, Ar, Kr, Xe, N2, C02, and CH4 are given in Table 7.4. In each case, the molecular response tensors have been isotropically aver- aged over molecular orientations prior to calculating the AGE” coeffi- cients. For a cluster of identical molecules A-- A --A arranged in an equi— lateral triangle, each of the three-body effects reduces the scalar polarizability 861(3) . Through order R'9 in the separation between cen- ters, for the equilateral triangular configuration Aa<2-3> = -(3/2)013R‘6 -(105/4)a2CR‘8, (13) 114 A6313) = —(33/4)a4 R‘9 (19) A 643-,” = — 3 a4 It", (20) A6533"! = -(33h/47r) R-9 [(3/2) F d... nc-iw; 1'... 0,0) 0.20“...) 0 + J: d... 7201...; 96.0.0) (1208)] (21) and 16,5333) = -(3h/n)R”9a(0)f d... y;(-‘iw;'iw,0,0) aCiw). (22) In Fig. 7.1, are compared the magnitudes of different three-body contri- butions to ac: for H---H---H arranged in an equilateral triangle. Fig- ures 7.2 and 7.3 show the three-body contributions to Ad! for Kr---- Kr- -Kr and H2---H2---H2 (respectively), also in equilateral triangular arrays. Susceptibilities for each H2 molecule have been averaged iso— tropically over the orientations of the internuclear axis. In Figs. 7.1-7.3, the second-order, three-body term A¢H23> has been separated into its dipole-induced-dipole (DID) component, which gives the MR-6 term in Eq. (18), and a quadrupole polarization effect, which gives the aZCR-8 term in Eq. (18). The remaining curves show the effects of static reaction fields, third-body fields, dispersion, and concerted induction-dispersion interactions. For a collinear triplet Ao--A---A, all of the three-body effects except for quadrupole polarization increase the scalar polarizability. For the linear, centrosymmetric configuration, with each of the termi— nal molecules separated by distance R from the central molecule, ecul-3) = 50138-6 - (35/2) (120,... (23) 10.3,” = (3/2)a4R-9; (24) 115 A643,,” = (73/8)a4R-9; (25) Aaé3’3) = (36/2n)R-9[(3/2)f° d... yl(-‘iw;'iw,0,0) (120'...) m 0 (26) + [a d... new; 98.0.0) 8.268)] and (3 3» -9 . . . (27) 3a,; = (73h/87r)R a(0)J: d... y;(-1w;1w,0,0) (10...). Figure 7.4 shows each of the three-body terms in Ad for linear, centrosymmetric H---H---H. As in Fig. 7.1, Ada-3’ has been separated into the DID component [the a3 R-6 term in Eq. (23)] and the quadrupole polarization effect [the aZC R-8 in Eq. (23)]. Effects due to static reaction fields, third—body fields, dispersion, and concerted induc- tion-dispersion interactions are also plotted. Figures 5 and 6 show the three-body terms in A0: for linear, centrosymetric Kr---Kr---Kr and H2--- H2--- H2 respectively; again, susceptibilities for each H2 molecule have been averaged isotropically over the orientations of the internuclear axis. (3.3) 3.3) 2'3), 3643,23), Add , and Ad‘s“, are compared for Ad‘2'3), Aas‘f the equilateral triangular and linear, centrosymmetric A---A- -A config- urations, for each of the species in this work, at R values approxi- mately 1 a.u. outside the minima in the pair potentials. In all cases, the largest term is A0193). For the triangular configuration, this term accounts for ~83%-93% of the total three-body Adm value. The dipole-induced-dipole effects in Aa‘2'3) range from 44% of the total Aa<3> for C0; to 68% for He; quadrupolar polarization effects (the azC R‘8 term) in Aa‘2'3) are quite large, ranging from 24% of the 116 R 0 I4 tbf/ fd -2 . srf 104AE -4* dis 1 -6~ DID -fg. Figure 7.1. Aafi” fOr H---H~--H in an equilateral triangle of side R (with R and A0 in a.u.). The plot shows the effects of second-order dipole-induced interactions (DID), quadrupole polarization (a3C), static reaction fields (srf), third-body fields (tbf), dispersion (dis), and concerted induction-disper- sion interactions (i+d). The equilibrium internuclear separa- tion in H2 in its lowest triplet state is Rm = 7.85 a.u. (Ref. 35) 117 R O . /7r85=="'9 10 ’H-d ‘1 tbf 102A— “ dis -2. srf aZC —3 DID l -4: T KT , Kr Kr ..5 . [ Figure 7.2. Ad“” for Kr-- Kr --Kr in an equilateral triangle of side R (with R and Ad in a.u.). The equilibrium internu— clear separation in Krz is Rn = 7.85 a.u. (Ref. 36). Curves labeled as in Fig. 7.1. 118 -8- DID H2 -10: HZAH2 -12 . Figure 7.3. [101(3) for Hz-qu-qu with molecular centers arranged in an equilateral triangle of side R (R and Ad in a.u.). Individual H2 molecule susceptibilities have been aver- aged isotropically over molecular orientations. Curves labeled as in Fig. 7.1. 119 6. 104AE . ‘ DID 4 i+d tbf 2\, :dls ‘er G 2 8 Fl 9 10""— -2. ’ aZC H—H—H -4. / Figure 7.4. Add” for H---H---H in a linear, centrosymetric array. Curves labeled as in Fig. 7.1. (R and Ad in a.u.). 120 102AE DID 2 tbf 2 ..aa 9 10"”— Kr—Kr—Kr Figure 7.5. A683) for Kr---Kr---Kr in a linear, centrosymmet- ric array. Curves labeled as in Fig. 7.1. (R and Aa in a.u.). 121 8 ' DID Figure 7.6. zwfl3) for H2---H2---H2 in a linear, centrosymmet- ric array. Individual H2 molecule susceptibilities have been averaged isotropically over molecular orientations. Curves labeled as in Fig. 7.1. (R and Ad in a.u.). 122 total Aa<3> for H or He to 45% for C02. In the triangular configura- tion, the largest corrections to A119-9 stem from the static reaction field and dispersion terms for all clusters (except C02---C02---C02, for which the third-body-field terms exceed the dispersion terms). The static reaction field terms accout for ~2% of the total and”) for He and 9% for Xe; other cases are intermediate. Dispersion terms range from ~2% of the total Aux”) for C0; to 7% for H atoms (6% for H2). The dispersion effects are more important for the lighter spe— cies (H, He, Ne, and H2), while the static reaction field effects are larger for the heavier species (Ar, Kr, Xe, N2,<3h, and C02). In the linear, centrosymmetric configuration, 21a9L3’ accounts for 90%~98% of 1160”. 'Third-body-field effects and concerted induction— dispersion terms typically produce the largest corrections to Aczav” . (Again C02---C02---C02 is the exception, for which the static reaction-field terms exceed the induction-dispersion terms.) The third-body-field terms range from ~1% of the total Acfl3) for He to 7% for Xe, while induction-dispersion effects range from ~1% of the total A41”) for C02 to 2% for H. The third-body-field effects exceed the induction-dispersion effects for all species except H and He. For all of the species, the relative importance of static reaction field effects and the third-body-field effects changes significantly when the configuration shifts from triangular to linear. For equilateral triangular A---A --A, 123 aagg;3>/ Aa‘3'3) = 0.364, (28) srf while for linear, centrosymmetric A- -A --A, Aaég%3)/ A03?) = 6.083. (29) The purely geometrical factors relating Adj-(3&3) to Aaff’” are the same as those that relate 1643:” to Aagi'f” for the two configurations. How- ever, the calculated ratios of Ace-(333) to Aaé3'3’ are species- dependent[unlike Eqs. (28) and (29)], because the frequency integrals in Eqs. (21) and (26) differ from those in Eqs. (22) and (27). 7.2 Summary and Conclusions The irreducible three-body effects on the polarizability of an A --B --C cluster at long range have been derived within the nonlocal response theory in Chapter 6. At second order, DID interactions and quadrupole polarization contribute to Ad”); Eq. (17) of Chapter 6 gives the second-order, three-body term Aaég'”. At third order, DID interactions produce the static reaction field effects in Eq. (19) of Chapter 6 for Aaéifla and thi rd-body—field effects in Eq. (21) of Chap- ter 6 for AaéfiflB. Both are proportional to a“ and vary as d_9 in the representative intermolecular distance d. Third—order, three-body hyperpolarization effects on Aa<3> vanish at long range. Dispersion effects yield A6123? in Eq. (27) of Chapter 6, and concerted induction- dispersion effects give A (23315 in Eq. (32) of Chapter 6. The contribu- tion of each polarization mechanism to the isotropic cluster polariz- 124 ability Ad‘” is listed in Eqs. (33)-(37) of Chapter 6 for use in molecu- lar dynamics simulations of light scattering or dielectric properties. Equations (18)-(22) specialize the results to equilateral triangular arrays, and Eqs. (23)-(27) give the results for linear systems. Tables 7.1-7.4 provide the quantities needed to evaluate all of the coefficients in these equations for clusters containing H, He, Ne, Ar, Kr, Xe, H2, N2, C02, and CH4. Ordinarily, the magnitude of the three-body terms in Ad, relative to the two—body terms, is governed by the parameter dd’3; but this does not hold for the isotropic interaction—induced polarizability Ad of a cluster A-- B-- C with negligible electronic overlap. The two-body and three-body terms in Ad are comparable in significance. Both vary as ch in the distance between molecular centers and both scale as d3, to leading order. For two S-state atoms or other isotropic systems separated along the z axis, DID interactions increase 0:; for the pair by 40:2R‘3 at first order, while the DID effects decrease an and dW by ZaZR'3. Hence the isotropic, two-body terms in Ad vanish at first order, enhancing the relative importance of three-body interactions. The classical induction contributions to the two-body polarizabil— ity Ad““ of interacting isotropic systems are10 86153,,” = 4 a3 R-6 + 20 a2 c R-8 + (30) at second order and10 (39) AC1]: nd =4a4 R79+... (31) at third order. In addition, dispersion affects the pair polarizabil— 125 ity; the dispersion term satisfies 8632'” = (471/71) R-6 I d... [ 3/2 71 (-iw;iw,0,0) + y;(-iw;iw,0,0)] (1618) + ABA R‘8 + (32) I. A6.d R“6 + A8“, R‘8 + The coefficient A&d depends on integrals over imaginary frequencies, as given by Eqs. (3) and (4) of Ref. 14. The integrals contain dCiw), C(iw), Y1 (-iw; iw, 0, 0), and the higher-multipole hyperpolarizabili— g. .".‘..mW3 "I! ties B(-iw; iw,0), P(-iw; 0,0,im), and Q(-iw; 0,0,iw). The coeffi- cients Amd are known with high accuracy only for H and He atoms, from 0 a variation-perturbation analysis for H atom pairs1 .and from ab initio calculations for He pairs.1‘4 From Eqs. (30)-(32), for three isotropically polarizable molecules arranged in an equilateral triangle, the sum of the two-body terms in the polarizability Ad satisfies Aa<292> + Aa(3'2) = 12 a3R'6 + (12h/n)R'6J: d... [3/2 n(-iw;iw,0,0) + new; 18.06)] a('iw) + 60a2CR'8 (33) + 3A5,dR‘8 + 12a4R-9 + This should be compared with the sum of the three-body terms from Eqs. (18)-(22). For a linear, centrosymmetric configuration, the sum of the two-body terms is Aa(2'2) + Aa(3'2) = (129/16)a3R'5 + (129h/16n)R’6 xj: dw[3/2 nC-iw;iw,0,0) + Y2C-‘iw;‘iw,0,0)] (34) x aCiw) + (2565/64) 0802-8 + (513/256)A,,,,,,R-8 + (1025/128) a4R-9 + 126 and this should be compared with the sum of the three-body terms from Eqs. (23)-(27). For H---H---H and He --He---He, all of the coefficients in Eqs. (33) and (34) for the two-body polarizabilities and in Eqs. (18)-(27) for the three-body effects are known with high accuracy, from Ref. 14, Ref. 75, and the current work. For H---H---H with R = 8.85 a.u. (approximately 1 a.u. outside the minimum in the pair potential), the three-body terms equal —S.5% of the two body-terms for an equilateral triangular array, and 12% of the two-body terms for a linear, centrosym- metric system. FOr He-- He --He with R = 6.6 anu., the three-body terms equal -S.4% of the two body terms for the equilateral triangular array, and 13% of the two-body terms for a linear, centrosymmetric system. The ratios of the three-body to two-body terms are signifi- cantly larger than the order parameters d/R3 ( 0.0048 for He --He-- He at R = 6.6 a.u.). The coefficients Amfi are not yet known for the heavier species, but approximate values for the integrals in Eqs. (33) and (34) are given in Table 7.4. With these, the three-body effects as a percent of the two-body effects were determined, both truncated at order R4i In the equilateral triangular arrays, the three-body terms equal -4% of the two-body terms for Ne, -7% for Ar, -8% for Kr, —8.5% for Xe, -5% for H2, -8% for N2, -7% for CH4, and -9% for C02. In the linear, centrosymmetric arrangement, the three-body terms equal 19% of the two- body terms for Ne, 35% for Ar, 39% for Kr, 42% for Xe, 24% for H2, 40% 127 for N2, 37% for CH4, and 47% for C02. Thus three-body interactions contribute significantly to Adfi”, and their relative magnitude gener- ally increases as the molecular size increases. 128 REFERENCES 1D. 21. 3C. 4A. 5C. 6D. 7A. 86. 9D. 100 llc _ 126. BC. 14A. 15R. 16L 17w . 183. 19c_ 20A. M. Bishop and J. Pipin, Chem. Phys. Lett. 236, 15 (1995). Waller, Z. Phys. 38, 635 (1926). A. Coulson, Proc. R. Soc. Edinburgh, Sect. A: Math. Phys. Sci. 61, 20 (1941). D. Buckingham, C. A. Coulson, and J. T. Lewis, Proc. Phys. 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A. van Gisbergen, J. G. Snijders, and E. J. Baerends, J. Chem. Phys. 106, 5091 (1997). 226. H. F. Diercksen, W. P. Kraemer, T. N. Rescigno, C. F. Bender, B. V. McKoy, S. R. Langhoff, and P. W. Langhoff, J. Chem. Phys. 76, 1043 (1982). 23L. Veseth, Phys. Rev. A 44, 358 (1991). 24N. Stein, C. Hattig, and B. A. Hess, Chem. Phys. 225, 309 (1997) 25K. L. C. Hunt, 3. A. Zilles, and J. E. Bohr, J. Chem. Phys. 75, 3079 (1981). 25F. W. Fowler, K. L. C. Hunt, H. M. Kelly, and A. J. Sadlej, J. Chem. Phys. 100, 2932 (1994). 27K. L. C. Hunt and J. E. Bohr, J. Chem. Phys. 84, 6141 (1986). 28W. Byers Brown and D. M. Whisnant, Mol. Phys. 25, 1385 (1973). 29]. E. Bohr and K. L. C. Hunt, J. Chem. Phys. 86, 5441 (1987). 300. J. Margoliash, T. R. Proctor, G. D. Zeiss, and W. J. Meath, Mol. Phys. 35, 747 (1978). 31A. Kumar and W. J. Meath, Mol. Phys. 54, 823 (1985). 320. J. Margoliash and W. J. Meath, J. Chem. Phys. 68, 1426 (1978). 33S. A. C. McDowell and W. J. Meath, Chem. Phys. 67, 185 (1982). 34B. L. Jhanwar and W. J. Meath, Chem. Phys. 67, 185 (1982). 35W. Kolos and L. Wolniewicz, J. Chem. Phys. 43, 2429 (1965). 36A. K. Dham, A. R. Allnatt, W. J. Meath, and R. A. Aziz, Mol. Phys. 67, 1291 (1989). 130 Appendix: Relationships of Dipole Coefficients Birnbaum et al. (Reference 1 of Chapter 5) use a dipole coeffi- cient with the symbolism D(AlAzA;R) or D“3(R) where C represents the terms, A1, A2, and A, while Li, Champagne and Hunt (Reference 3 of Chap- ter 5) use the symbolism D(a,b,A,L;m'). It is the purpose of this appendix to show that each set of terms contains the same information and to explain the mathematical method of simplifying (summing) like terms. The relationship between coefficients is as follows: A1 = b; A; = a; A = L; R = R and A can have the values la-bl s A s |a+b|. A is not part of the dipole coefficient used by Birnbaum et al. and neither is m . This is due to simplification made by them using the symmetry properties of the dipole coefficients as shown in the following exam- ple. From Table 4.1 the dipole coefficient D(044S), without the overlap term is \/§;13Ni§CH‘. The equivalent terms used by the Hunt group are given in Table 3.3 as D(4045;-4), D(4045;0) and D(4045;+4). To sim- plify, the terms are squared and summed over A and m' and the square root of the sum is taken. In this example A = 4 only and the summa- tion is over m' alone: 131 90(4. 0. 4. 5: —4>2 + W. 0. 4. s; 0)2 + D(4. o. 4. s; 4)2 = WWW—0) 9» 99f + (9999. 9cm)2+ ((593799) om. 960% \/ £579 0992901. When two or more different induction and dispersion terms inter- fere, they are summed, the square is taken of the sum and that value is added to the total contribution. 132