D9 dipole dispersion coefficients

CHEMICAL PHYSICS LETTERS

Volume 171, number 4

10 August 1990

I&, dipole dispersion coeffkients P.W. Fowler Departmentof Chemistry,University ofExeter. Stocker Road, Exeter EX4 4QD. UK Received 2 1 April 1990; in final form 2 May 1990

A formula is derived for 4, the first correction to the usual single-term approximation for the dispersion dipole of a pair of dissimilar atoms. The expression is a Casimir-Polder integral involving dipole and quadrupole polarisabilitia, dipole+bpolequadntpole and dipole-quadrupole-octopole hypetpolarisabilities on each atom. A coupled Hartree-Fock scheme for its ab initio calculation is implemented and static approximations to D9 show it to be of the same sign as D, but larger in magnitude (when both are expressed in atomic units). CHFcalculations on H and He give Dg(H+He-) = - 3390 au, i.e. over 25 ai times D7, and the R -g local dipole in H...H is over 30Rd2 times the better known R-’ moment.

1. Intruduction A pair of well separated atoms has an attractive dispersion energy which is given in the non-relativistic limit by the multipole expansion [ 11

(1) The physical origin of the energy is the correlation between the fluctuations in electron coordinates on the two centres which leads to a polarisation of each charge density into the internuclear region. Each nucleus is attracted to the centroid of its own (polar) electron cloud, giving a force that varies as R -’ [ 2,3]. A concomitant of the R -6 energy is an Rm7 dipole moment on each atom, which leads to an R-’ pair dipole for dissimilar atoms [4]. A fuller treatment shows that the dispersion dipole moment for a pair of spherical atoms is rep resented by the series AB PZ.“dW

=

D7 & $jgjp....

D,,

For non-centrosymmetric molecules with a non-vanishing first hyperpolarisability, the dispersion dipole series begins at R -’ and depends on mutual orientation of the partners [ 5,6]. D7 is relatively well studied [S-l 31, and an exact expression relates it to polarisability and hyperpolarisability properties of the separated atoms [ 51. Whilst Ca and higher dispersion energy coefficients have been calculated for many pairs, D9 does not ap pear to have been discussed. It would seem to be of some fundamental interest to know its sign and to be able to estimate the relative magnitudes of 4 and Dg in specific cases. The undamped energy series ( 1) is characterised by monotonically increasing coefficients C,, and the same could be true of the dispersion dipole series (2). Higher coefftcients may also be large in the dispersion polarisability series; the A8 term (the coefficient of R -s) was found to be significant relative to Aa * for H + H [ 14 ] and may be large in other cases too. The present paper sets out to derive an exact integral expression for D9, to show how it may be calculated ab initio 0009.26 14/90/S 03.50 0 1990 - Elsevier Science Publishers B.V. (North-Holland )

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by a hybrid of coupled Hartree-Fock and finite-field approaches and to provide rules of thumb for its approximation. An illustrative calculation on the HeH pair is reported.

2. Derivation A simple and physically transparent derivation of the dipole coefftcients 0, can be achieved by differentiating the dispersion energy with respect to a uniform and static electric field, F [ 61. By the Hellmann-Feynman theorem (3) and if the energy is expressed in terms of dynamic polarisabilities, the D,, coefficients emerge naturally as functions of hyperpolarisabilities. In constructing the energy expression to be differentiated it is important to include all terms appropriate to interacting anisotropic partners, since the perturbing field induces polarisabilities that would vanish for the spherical atom. At long range, where exchange is negligible, the second-order non-relativistic dispersion energy for interaction of molecules A and B cau be written as a sum over excited states of the composite A +B system [ 31 E vdW

(4)

where the prime on the summation implies pA# mA and PB# m& where m iS the ground State and the St&3 of A+B are products of monomer fUnCtiOn Ip,,pB) e IpA) I&) with energies W,” + WpB. The operator H’ iS the electrostatic interaction between A and B and can be developed in a multipole expansion. For our purposes the relevant part of H’ is [ 31

(R -*)

(5)

.*. ,

where the order in R of each term is noted at the end of the line. The p, 6%52and # symbols denote multipole operators for each subsystem and the T tensors are the Cartesian derivatives T a~...,,= VaV,...Vv IR I - ’3

(6)

where R is the vector from an origin in A to an origin in B. As usual, the Einstein convention of summation over a repeated subscript is implied. All terms in charge qA and qB have been dropped from H’ since they have no off-diagonal elements and so cannot affect EvdW. Expressions for the R -6, R -‘, R -’ and R -9 dispersion energies may be obtained by substituting (5) into (4) and using the Casimir-Polder identity [ 151 ---

XY (X2+W*)(Y2fW2)

dw

(7)

to re-write each sum as an integral over imaginary frequency of two dynamic polarisabilities of A and B. The results for orders R -’ to R-* are well known [ 35161 and are not repeated here. Terms at R-’ arise from 278

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R -3 x R -A and R -4~ R -5 combinations in H’ and the general R -9 second-order dispersion energy for two anisotropic molecules A and B is -

J%v(R-~)= -

&

TaJyw

1[at,(iw)

Zj,&iW) -cr&(iw) Zj&,Jiw)] dw

0

(8) where (Y,C, A, E, H and I are the dynamic dipole and quadrupole polarisabilities, the mixing dipole-quadrupole, dipole-octopole, quadrupole-octopole and dipole-hexadecapole polarisabilities, respectively. All are defined in refs. [ 3,171 or can be obtained by obvious generalisations of the formulas therein. E.,dw(R-9) is of course zero for a pair of atoms (since A = E= H= I= 0 for a spherical system) but the last two terms of (8) give rise to non-zero electric-field derivatives. The third-order perturbation energy contains terms of order R -9 which for three distinct atoms give rise to the triple-dipole three-body dispersion energy [ 11. Its two-body counterpart involves as integral of two hy perpolarisabilities of odd parity, and therefore for a pair of centro-symmetric monomers both C, and its first derivative with respect to electric field are vanishing. There is no third-order contribution to D, for pairs of atoms. Application of a static external field produces linear changes in A and H for an atom through the hyperpolarisability terms Mt,,(iw)=B$d,fi(O, AH&+(iw)

io) Fd+_.. ,

=L &,,(O,

iw, -iw) F,+... ,

(9) (10)

where B is a dipole-dipole-quadnrpole hyperpolarisability [ 5,17 1, and L a dipole-quadrupole-octopole hyperpolarisability, as defined in ref. [ IS]. Note that each has one static and two dynamic indices, as the notation for L is designed to emphasise. The tensor L is perhaps an unfamiliar property, but in the static limit it corresponds to an energy change of AE= -&u,gy,at&&J~~

>

(11)

when a molecule is subjected to a non-uniform field. Some further discussion of L is given in Appendix B of ref. [18]. Thus, for a pair of atoms the dipole moment of order R -9 is

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D

_

CHEMICAL. PHYSICS LETTERS

IO August 1990

WR-‘1

_R9

9-

aFz OD =-

g

T,B,T&

I 0

--~%,~~~(O,ia

[L$,,,(O,

iw, -ia)

didi~)+~~2z,d0,i~)

C~,,,W)

$,+,(io)ldw,

-io)(y~~~iW)-5B~=,~(O,iO)

(12)

where the first two terms of the integrand correspond to a local dipole on A and the latter two to the moment on B. For an atom each of the properties involved is character&d by a single frequency-dependent parameter

[3,5,6,181 cr,,(iw)=cx(iw) &+40,

L&,

io)=tNO,

(13)

io) [3(6,,68,+6,,6~)-26,86,,1

,

C,,,(iw)=~C(io)[3(6,,6~+SasS~)-2~~lrB~yd19

(14) (15)

La,~dO,iw -iw)=iL(O,iw-io)[(6,6,6,+6,,6,6,+6,,6,6,+s,,6,~,+6,,6,6,+~,,6,6,) -~(~,,~,~,,+~,,~,~*+~,,~,~,+~,,~,~,,+~,,~,~*+~,,~,~,+~,~,~,,+~,,~,~,+~,,~,~,,l (16) and by substituting these isotropic operators into (12) and contracting the tensor indices the orientationdependent factor in front of the integral sign reduces to a multiple of TahTaayz. By use of the propagator product 360&

Ta&Taava=-~‘0

(17)

this simplifies to our final expression for the D9 coeffkient, co D9AB =!I

[LA(O, io, -io)

aB(iw)++?*(O,

iw) CB(iW)

0

-LB(O,io, -io)

aA(yB8(0,

io) CA(io)] do,

(18)

which can be compared with the D, coefftcient for two atoms [ 5,6]

D$"-:I [BA(O,iw)aB(iw)-BB(O,ia)aA(iw)]dw.

(19)

0

Since it appears that all the dynamic properties fall in magnitude monotonically to zero as the argument io tends to infinity, several plausible approximation schemes based on assumed frequency dependence can be devised. For example, applying an Unsold-type approximation as used in ref. [ 51

LA(O,io,-iw)

P( 3RZ+W2) 3(L?2+c02)2

zLA(O,O,O)

(20)

and making the approximate factorisation of integrals also as in ref. [ 51 gives AB D9

qc16 AB 6

~(3ACs-CABB),

which corresponds to the approximate formula previously derived for 0, [5,6] 280

(21)

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10 August I990

(22) Many other plausible approximation schemes could be derived. Equations like (2 1) can be useful as they predict the likely sign of Dg - both L and B are negative for ground-state atoms @’and so D9 has the same sign as D,. i.e. both produce the polarity A+B- when A is the more hyperpolarisable atom. Short-range overlap effects usually have the opposite sign and dominate the dispersion dipole at the van der Waals contact distance [ 19-22 1. A rough idea of magnitudes can be gained from (2 1) if we take the known static polarisabilities of the hydrogen atom. For Il we have L=-2025 [18], B=-213/2 [23], a=9/2 [24], C=15/2 [25] and CyH ~6.5 au [ 261 and so the local Rs9 dipole on H in the H..:H pair is expected to be about 30R -* times the R -’ value, with roughly equally important contributions from aL and BC terms. By symmetry, of course, the total pair dipole vanishes in this case. On the basis of the approximate formula we can conclude that D9 and 0, give reinforcing contributions and that & in eaA”may be appreciably larger than 0, in eat for pairs involving for example H or a negative ion; both factors would tend to produce doubt about the ultimate convergence of the dispersion dipole series. To proceed further it is necessary to calculate values of D4 in some particular cases.

3. Calculation The present derivation of D9 suggests a strategy for its computation. The polarisabilities A(iw) and H(iw) are calculated by conventional coupled Hartree-Fock or random-phase methods, but with a finite field added to the Hamiltonian. By numerical differentiation with respect to field strength, the hyperpolarisabilities B and L with one static and two dynamic indices are obtained. This calculation is carried out for a set of 16 frequencies io and the Casimir-Polder integrals evaluated by quadrature. A similar method was used in ref. [ 61 for 0, and other property dispersion coefficients. The calculations use a modified version of the SYSMO RPA program, with finite perturbation and H(ia) modules written in Exeter. Angular momentum considerations limit the choice of example, since for an atom with occupied orbitals of angular momentum 1 the basis should include functions with I’= I? 1, lf2, IA 3 if it is to allow for dipole, quadrupole and octopole excitations. An atom with an occupied s orbital therefore requires an spdf basis; for an occupied p shell the basis would need to include g functions, which is beyond the capabilities of the current program. This extension is in hand, but in the present section we restrict our attention to H and He atoms, calculating the local dipole moment of order R -9 for each partner in the HH, HeH and HeHe pairs, and thus the Dg coefficient for HeH. Variational methods with special basis sets may be used to solve the H+ H problem to any desired accuracy [ 18 1, but a less efficient Gaussian basis set is used here since that is all that will be available for many-electron atoms and molecules. The final uncontracted GTO basis sets are ( lOs8p5d4f) for both H and He. For H the van Duijneveldt [ 27 ] 10s exponents (C) were used to generate the spdf sets in the pattern ( <,-<,o, <3-&o, &,&,, t[&, ) where <, , =0.028 a$ and the 10s set for He from the same source was used to give a pattern (& &,, &&z, &,-& ,, &-&, ) where C I = 2& =0.07 a;*. In both cases the extra exponents were chosen to continue an approximately geometric progression. The spd substrates of these basis sets give an accuracy of 99.9% or better in static values of (Y,B and C for hydrogen, and yield similarly close agreement with Hartree-Fock limits [ 28 ] for these properties of He. The number and range of exponents of the f functions was fixed by trial calculations of the static L polaHI In the crudestclosureapproximation for a one-electron atom, BW -4( r’) /St2 and LS -9( r6>/742, whered isanaverageexcitation energy. The negative sign follows simply from the fact that these properties involve products of 3 multipole operators, each of which is proportional to the charge on the electron, i.e. - 1 in au.

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risability. In the purely spd basis the computed value of L for H is only - 9 10 au; addition of the 3 leastdiffuse f sets changes this to - 1649 au, and a further inner f function produces a smaller change, to - 2004 au. When 4 diffuse f functions are used, L is -2017 au; the exact value of LH(O,0,0) is -2025 au [ 181. For He a similar pattern is found: LHeis -31.25 (spd), -68.15 (spd+3f), -69.23 (spd+4f), -69.24 (spd+5f). Thus the presence of f functions is important for qualitatively correct results, with about half of the final L value arising from (spd) +f virtual transitions. They are of similar importance for the dynamic properties and Casimir-Polder integrals, as numericaI tests on L" (iw, - iw, 0) show. This property governs the dispersion octopole moment of each atom in the H+ H pair. The present ( lOs8p5d4f) basis gives a value for R:,=%ja(iw)L(iw,

-io,O)dw

(23)

0

of - 1090 au, compared with the exact value - 1088.5... au [ 181. The & integrals involve a different sampling of the full L tensor, but should be equally accurate. Using these basis sets, we calculated the dynamic properties a(iw), C(iw), B(0, iw) and L(0,io, -io) for H and He at the quadrature frequencies. Various component integrals and the local dispersion dipole moments are listed in table 1. They show, for example, that the Rp9 dipole moment developed on a hydrogen centre is large, and is indeed about 30Rm2 times the R-' moment as estimated earlier, so that for the H+H pair we have

with expected errors of order 1% in the numerators. The equiIibrium separation in the lowest triplet state of H2 is E 7.85 a, [29 j and so at this distance the undamped contribution of the R -9term to the local dipole is about half that of the leading term. For H+He the pair dipole dispersion coeffkients are D,(H+He-)=118au,

(25)

&(H+He-)x3390au

(26)

( 1 au of D7 is eat, and for Q, an au is cub”) and the ratio of approximately 29 af again bears out the qualitative predictions of the static model. In fact, with the values calculated in the present basis of Cf” =2.72, cxHe= 1.322, BHe= -6.58, CHc= 1.163 and LHe= -69.2 au, the static approximations (22) and (21) yield reasonable estimates of these coeffkients; they are $(H+He) x 102 and &(H+He’ ) x2380 au, respectively. In the absence of other information the latter would be a useful guess for Dg. Finally, it should be recalled that RPA results do not include electron correlation. It is known that correlation in the helium atom causes small increases in magnitude for the static polarisability properties ( 5% in LY,11% in B, 5% in C [ 301) and if we assume a similar effect on L the increase in D9(H+He- ) is rather minor, since this coefficient is dominated by the hyperpolarisabilities of the hydrogen atom. Inclusion of electron correlation Table I Casimir-Folder integrals for hydrogen- and helium-containing pairs. For comparison, all integrals have the same prefactor of 3/x.p, and & arc the local dipole moments of order R-’ and Rw9, respectively, induced on atom B by dispersion interactions within the AB pair. All quantities are in atomic units

H H He He

282

H He H He

-131.6 -11.88 -51.13 - 5.452

-245.7 - 23.52 -46.68 -5.121

-2958 -135.9 - 1232 -65.9

- 12803 -832.8 -4220 -278.2

- 394.9 -35.65 - 153.4 - 16.36

a32 a23 %28 El7

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not alter the main conclusion of this paper, which is that D9/D7is likely to be substantially greater than 1 au for many systems. lf, as found here for H/He, the ratio DJD, is of order 25 ai for other systems, then the Rs9 dipole will equal the R-' dipole in the vicinity of the van der Waals contact distance (where shortrange terms are also important) and will still account for a significant proportion of the total dispersion dipole at larger separation.

would

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