Quadrupole - Quadrupole and Dipole - Octupole Dispersion Forces Between Axially Symmetrie Molecules *

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B A N D a ZEITSCHRIFT FUR NATURFORSCHUNG HEFT Quadrupole - Quadrupole Dipole - Octupole Dispersion Forces Between Axially Symmetrie Molecules * ALWYN.

1 B A N D a ZEITSCHRIFT FUR NATURFORSCHUNG HEFT Quadrupole - Quadrupole Dipole - Octupole Dispersion Forces Between Axially Symmetrie Molecules * ALWYN. VAN DER MERWE Department of Physics, University of Denver, Denver, Colorado ( Z. N a t u r l o r s c h g. a, S 1 «9 7 [ ] ; r e c e i v e d 17 a n u a r y 1967) Formulas are derived that express the contributions of the electric quadrupole-quadrupole dipole-octupole forces, respectively, to the dispersion energy of two unexcited molecules possesg axial symmetry. The formulas are valid in situations where the optical dispersion curve of each molecule is adequately characterized by a pair of resonance frequencies. 1. Introduction Let us consider a microscopic system composed of two equal but opposite electric charges, such that the positive charge is located at a fixed origin, while the negative charge e, say, is displaced f r o m it by the vector T. Suppose a similar system displacement r " has its origin at a point whose position vector is R relative to the first origin. The potential energy pertaining to the COULOMB interaction between the two systems is then V = e(r~1 + \R-r' + r" _1 - j R - r ' _1 - R + r". (1) If both T R" are taken to be less than R, this function can be exped in a power series for the reciprocal distance R~l, which commences third-order terms. One may write this series as V=V1 + V+V3 + V, +..., () where Vx, V, etc. are identical the potential energies of respectively two dipoles, a dipole a quadrupole, two quadrupoles, so on for electric multipoles of increag order. Not every term of ( ) we should remark is of a different order; f o r instance, while it is true that V\ is of the order three V of the order four in R~x, both V3 F 4 are fifth-order terms. The expansion ( ) has been made the point of departure by M A R G E N A U 1 o t h e r s ' 3 f o r quantum-mechanical second-order perturbation calculations of the dispersion forces between molecules in spherically symmetric ground states. These calculations are in general based on a model according to which a molecule consists effectively of a certain * Work supported by the National Aeronautics Space Administration, under Grant NsG H. M A R G E N A U,. Chem. Phys. 6, 896 [1938]. R. H E L L E R,. Chem. Phys [ ]. 3 P. R. F O N T A N A, Phys. Rev. 13, 186 [1961]. number, integral or fractional, of electrons behaving as isotropic simple harmonic oscillators of the same frequency. Now, such a model can obviously not be expected to furnish useful results for molecules, such as the diatomic ones, whose electronic wave functions are markedly lacking in spherical symmetry. At least it should be realized, as L O N D O N 4 pointed out, that the stiffness constant of an electronic oscillator along a bond direction would be different from the constant perpendicular to it. This view has given rise to the so-called anisotropic oscillator model of L O N D O N, which, on application to unexcited axially symmetric molecules, leads in the dipole-dipole approximation, V V1, to the well-known R~e formula of L O N D O N f o r the dispersion energy 4 < its characteristic dependence upon the orientations of the molecule axes. In a recent study 6, the better approximation V = V1 + V has been employed: its effect is to augment the L O N D O N formula by a certain orientation-dependent dipole-quadrupole energy term, which varies as In the present paper we shall carry the generalization of LONDON'S moreover the definition of formula a step further by including fifth-order terms V x in the V. Because of the mere asymptotic validity of the dispersion energy series 7, this is probably as far as one should go for immediate physical purposes. W e shall not bother, in what follows, to define all our symbols, the method of calculation will likewise be given short shrift, leaving us mainly a summary of results. For enlightenment on the misg details, references 6 should be consulted. Phys. Chem. 4 6, 3 0 [ ]. Physik 196, 1 [ ]. M E R W E, Z. Physik 196, 3 3 [1966].. T. L E W I S. Proc. Phys. Soc. London 4 F. LONDON,. A.. VAN DER M E R W E, Z. 8 A.. VAN DER 7 A. DALGARNO 7 A [196]. 69,

2 . Classical Potential Energy On performing the expansion of V, we TAYLOR find that V3 F4 in Eq. ( ) are given, R = R/R, by f73= [/*r"* + (r'-r'v -0(R0-r')(R0-r") r' (R0-r") (r r " ( R 0 - r ' ) -r") + 3 ( R 0 V ) (R0-r")] (3) F4 = [1r'(R0-r')(R0-r") + 1(R0-r') (r'-r") +1 r"(r0-r') (R0-r") A resolution of Y v " into their components located at the centers of the first second mole- (i j k, m) = 1 djj Q k Qm 0 cules, respectively, allows us to rewrite the foregoing i j Is I x'i x appropriately (i j k, m), Qk Qm (8) terchanging gle double primes in the function x' k following it. In order to express the foregoing 4 3 x'm + (m, i j k) x'i' x'j x'k' x'm], here "transpose" denotes the result obtained on in- [ (i j K m) i j Ic m L Qj = + transpose of (m, i j k) ; () x\ z'i z'k' xz m Vi = + 1 ; c o s ö j s i n ö ^ s i n Q m c o s L I I 0' j, k m) n - 3 dij Qjc Q m <I>km equations as 4 (4) + 1 ( R 0 - r " ) ( r ' - r " ) - 3 ( R 0 r ) ( R 0 - r " ) 3-3 ( R 0 - r " ' ) ( R 0 - r ' ) 3 ]. relative to two rectangular Cartesian sets of axes I - 3 r'(r'-r") - 3 r"(r'-r") defined coefficients (6) (ij,km), (m, i j k). coefficients in as few variables as possible, it is furthermore necessary to notice that Q[ = ' + 90, Q'i = Q" + 9 0, If one's purpose is to study isotropic molecules, it will clearly be proper advantageous to choose the gly doubly primed sets of axes to be mutually parallel, fixed in space, such that one of the coordinate directions coincides R ; the coefficients in ( ) ( 6 ) will then simply be some integers. But, when we are dealing anisotropic 0=90, Qs=Q'; Q'z = 9 0, Q'i =0"; ^ n = ^13 = ^ = ^31 = ^33 = ()? = 03 = (P + 9 0, 0 1 = <I>.3 = 0 - (9) 90. In this table Q' Q" signify the angles made by the molecule axes the vector R, 0 the angle included by the projections of these axes onto a plane perpendicular to R. molecules, the choice of axes is no longer solely a matter of preference, as each set of axes must be 3. Quadrupole-Quadrupole Perturbation Energy attached rigidly to the molecule it spans; relative to such axes, rotating the molecules, the coefficients are rather complicated functions of the rotation angles. For the case under consideration, where the molecules are posited to have axial symmetry, the axes are defined as in ref., these functions turn out to be (i j, km) 7a dij Ok 0m + i 18 we assume next that the wave function of a system of two widely separated molecules is the product of six oscillator wave functions, as described in ref.. Our problem is then to find the perturbation energy associated the (not too close) approach of two molecules in their ground states, when the = di j dkm (9t- Qj Gk Qm + In keeping writh the anisotropic oscillator model, Q j Qj Q j Qk dkm Q k Q j Qm Qk Qm m = ± transpose of ( k m, i j ) (Pik interaction V=V1 7a 0jm (Pjm (7) + V + VS + VA is used per- turbation terms of third higher order are negcapital Greek letters replace here the corresponding small letters in the notation of ref. 6. Although the use of ir- reducible tensors would have made our computations more compact, we have chosen to express the outcome in terms of simple trigonometric functions, which permit a straightforward interpretation of our final results.

3 lected. Inspection reveals the fortunate fact that any of the matrix elements of V here required, receives a contribution from at most one of the four terms Vj into which V has been analyzed. Consequently, one is entitled to treat each Vj separately from the others; the corresponding energies furnish on addition the total perturbation energy. In this section we confine ourselves to V3, assume that the interacting molecules contain respectively f f" oscillating electrons. The first-order perturbation energy is non-zero found to be ^ [l-( 6>' + 6>") (1Q) - 1 & 6" + (4 &' Q" Q' O" ], where Q" is similarly defined. E^ is recognized as being the classical COULOUMB interaction energy of two rigid quadrupoles, of moments Q' Q", whose axes coincide those of the molecules 8. The appearance of this term was, of course, inevitable in view of the non-vanishing quadrupole moments of non-spherical molecules. The second-order perturbation energy corresponding to V3 provides the quadrupole-quadrupole dispersion energy; we obtain - AF&*FR = (I/ßl-1/ß;;) (T, -) + [I6(VI + v ' ) ß ' ± ß i ] ~ 1 ( V ß l - I / / 0 ( T A, - ) +(I6V,; /3; L )-I D / ß ' : - I / ß ' ; ) ( A, - ) + [64(vl +vl)(ß'±ß'i)*]- l a*,t*) +[8(vl +v';)(ß'±ß'i;) ]-I(T,A ) (1) + [16(v,; +V\;) (ß' ß\L) V (A, A ) + [16( vl +v" W ) ß'ßLFÄV 1 (T,TA) + [3(vl+*; +v'l+v\l)ß'±ß\\ßlß'^- i (TA.TA) + [8(v'x + / + vx) ß'X ß\ ß'L ]' 1 (TA, T ) + [16(v,; + rx) (ß\\ ßl) ]' 1 (A, T ) + [8(v' +v'l +v\[) ß\\ ßl (A, TA) + transpose; herein the coefficients (T, ), (TA, ),etc. (T = transverse, A = axial) are the following rather complicated functions of the orientation angles: (T, -) = [(1,33) + (1, 33)] + (11, 33) + (, 33), (TA, -) =[(31,33) +(13,33)] + [ (3, 33) + (3, 33) ], (A,-) = (33, 33), (T, T ) =4(11, ll) + 8(, ll) +4(, ) + [(1,1) + (1,1) + (1,1) + (1, 1) ] + 4 [(1,11) + (1,11) ] + 4[ (, 1) + (, 1) ], (T, A ) = (11, 33) + (, 33), (A, A ) = (33, 33), (T, TA) =[(1,31)+ (1,31) + (1, 13) + (1,13) ] + [ (1, 3) + (1, 3) + (1, 3) + (1, 3) ] + [(, 31) + (,13) ] +-[(, 3) +(,3)], (13) (TA, TA) =[(31,31)+ (13,31) + (31,13) + (13,13)] + [ (3, 31) + (3,31) + (3,13) + (3,13) ] + [ (3, 3) + (3, 3) + (3, 3) + (3, 3) ], (TA,T ) = [(31,11) + (13, ll)] + [(3,11) + (3, ll)], (A, T ) = [(33,1) + (33,1) ], (A, TA) = [ (33, 31) + (33,13) ] + [ (33, 3) + (33, 3) ]. In the special case of spherical molecules, it is evident from Eq. (11) that Q' = Q" = 0 thus E 1^ = 0; calculation reveals furthermore that the formula (1) reduces to ( ) 31 e 4 f f" 1 = _ 31 h 3 (A'A") S/ ' 1 ( U l 16h(ß r ß") (v' + v") R em'/«(/' a")'/«+ (/" a')v«rio» all of which agrees the results of HELLER FONTANA 3. 8 W. H. KEESOM, Physik. Z., 19 [191].

4 4. Dipole-Octupole Dispersion Energy As was to be expected, no contribution to the energy arises from F4 in first-order perturbation theory. The second-order energy is the dipole-octupole dispersion energy turns out to be given by h F' ), /,, /, / / / / / / - a j w r r + *l)ß±ß±t l (T,T) + [I6(V + v±)ßllß±t 1 (A,T) + [ 16 (V+ V;,') ß ß\i] - 1 (A, A) + [8 ( rl + ri + v") ß? ß[ ßl] (T A, T) + [8( vi +vn)ß, jßiß, ;r 1 (T a,a) + [so vi +/;) ß'jß'ir 1 (T 3,D + [8(v'i +vi +v'l)ß'_ßl*ßlr 1 (T A, T) + [8(3 v'± + v'l) ß'±* PIT 1 (T 3, A) (1) + [8(v'±+v't -")ß'^ßiß'iT l (TA,A) + [8 (3 v'\\ +v±) ß j 3 ß±](A 3, T) +- [8(3 r,' +v'l) ß'^ßuT 1 (A 3, A) + transpose; the angle-dependent coefficients appearing in this formula are defined as follows: (T,T) = (l/ß'±-l/ß'll) {[3(111,1) + (1, 1) ] + (33, l) + (33, ) } + (l/ßl - l/ß'^d/ßl-l/ßu) {[3(111,1) + (1,1)] [3(1,111) + (1,1)] + (l/ß" - l/ß ') [3 (, 111) + (, 1) ], (A,T) =9(1//?1 l/ß'll) [ (333, l) + (333, ) ] -(33, 1) [3(, 111) + (,1)] + (33,) (,33)} + 6(l/ß'± -l/ßdd/ßl - I//?!,') {(333,) (3,33) -(333,1) [3(3,111) + (3, 1) ] } + (l/ß" - l/ß\;) {[3(3, 111) + (3,1)] + (3, 33) }, (A, A) = 9(l/ß'± ^/ß\\) (333, 3) +3(1//?1 -l/ßl)(l/ßl - 1/ß'i) (333, 3) (3, 333), (T A, T) = [(113,1) + (131,1) + (311, l)] + [(113,) + (131,) + (311, ) ] + (3,1) + (3, ) +[(31, l) +(31,) ], (16) (T A, A) =[(113,3)+ (131,3) + (311, 3)] + (3, 3) +(31, 3), (T 3,T) = (11, 1) + (11, ) + (1, 1) + (1, ) + 3[(111, l) + (111, ) + (, l) +(,) ], (TA, T) =[(133, 1) +(313, 1) +(331,1)] + [(133,)+ (313,) +(331,)] + (33,1) + (33, ), (T 3, A) = (11, 3) + (1, 3) +3[(111,3) + (, 3) ], (TA, A) =[(133,3)+ (313,3) + (331, 3)] +(33, 3), (A 3, T) =3[ (333, l) + (333, ) ], (A 3,A) =3(333,3). For spherical molecules, Eq. (1) reduces to () 10 e 4 f f" hß'ß" R 10 " (v' + 3v") ß' (v"+3v') again as required by the work of other authors ' h 3 a a" 4 e TO'/. RI o (/'_(/vn a") V» + 3 (/" a) ^, (/" (/"//') a') ^ + 1/s 3 (/' a") Vt ' (17). Formulas for Some Special Orientations In view of the unwieldiness of the general formulas (1) (1), it will be useful, for calculations other purposes, to display here their simplified forms for the special orientations (a) to (d) defined in ref.. We have, for similar molecules, a) = - (3/4 R ) (fc 3 /16em 3/ä / 1/ ) (18) X [(17ai / +9af) (a] 1 ' - a l l~) ai / + 04(ai a;i) s/s (ai / + a 1 / )+ 9 ap],

5 b) <f> = -(3/4tf ) (h*/8em'<<f' 1 ) X [ (4 a f + 16 a f ) ( a f - a f ) + 1 a f (19) + 14 (ai at n) * /a ( a f + a f ) + 8 a f a. a ( a f + 3a' )-'], c) < q q> = - (3/4 R ) (fc 3 /16 e m' /s / Vs ) X [(ctf + a f ) ( a f - a f ) + 97 a f (0) + 108(ai at11) 3/s ( a f + a f ) a f + 1a a ( a f + 3 a f r d) E ( > = - (3/4 R ) (h 3 / e m s/s /,/ä ) 1 ], x [ 8 ( a f + 8 a f ) ( a f _ a f ) + 3 a f + 48(ai a ) s/s ( a f + ctf) + 8 a f ] ; (1) a) ^ = - (1/ R ) (h 3 /16em u -f>) () X [(13 cd' + 4 ctf) ( a f - a f ) + 61 (ctj. an)''' ( a f + a f ) - i a f + 7 ctf], b) ( f> = - (1/ fl ) (h 3 /3 e m^- f«) X [81 a f (ctf - c t f ) + 346(ctia, )^ ( a f + c t f ( a f - a f ) + 376a _a ( a f + 3 c t f ) (a ± ( a f + a f (3) + 08 a a (3 a f + a f ) + 07 a f a f ], c) E^ 0 (1/ /? ) (3 fc 3 /e m / > /,/s ) X [6 a f ( a f - a f ) + 6 aj. a ( a f + a f ) ( a f - a f ) (4) - 7 a i a ( a f + 3 a f ) + 7 a f + 6 (a ± a, ) * ( a f + a f ) + 3 a ±a (3 a f + a f ), d) Ef0 = - (1/ R ) (h 3 / e m 3/ ' / 1/ ) X [4(3 a f + 4 a f ) ( a f - a f ) + 144(ai a,,)'^ ( a f + a f + 9 a f + 4 a f ]. () The author wishes to thank Professor H. MARGENAU for his helpful suggestions his interest in this work.

CHM Physical Chemistry II Chapter 12 - Supplementary Material. 1. Einstein A and B coefficients

CHM Physical Chemistry II Chapter 12 - Supplementary Material. 1. Einstein A and B coefficients CHM 3411 - Physical Chemistry II Chapter 12 - Supplementary Material 1. Einstein A and B coefficients Consider two singly degenerate states in an atom, molecule, or ion, with wavefunctions 1 (for the lower