an intermolecular potential for ch4ch4 calculated within the electron gas approximation

Volunte 52, number 1 CII~MICAL 1’IIYSICS LLXTERS 15 Novumber 1977

AN INTERMOLECULAR POTENTIAL FOR CH,-CH, CALCULATES WITHIN THE ELECTRON GAS APPROXIMATION

G.C. TABISZ * Dcptzrttne~lt of 7lmxetioal Cllernistry, Utuknity Chemical Laboratory. Gmzbridge CR.2 IEW. UK

Kecc1ved 3 August 1977

An irttcrmotc&rr potental for CI14--CH4 ic calculuted wrtbin the electron g&s approknn:ttion. The bcrt rer;ult is obtain- ed when no c~rrc~tions are apphcd to the ~xldividu~l ~o~~tr~~utso~~~ to titc total electron pax cs~er~y. ‘fltc potential param- ctcrs and ~bape of the high energy repuIswc wall arc in good agrcctncnt with e\pcrimentd1 determinations.

1. Introduction

The appfication Of the electron gas approximation to the calculation Of pair potentials has been a subject of recent interest, particularly since the work of Gordon and Kim [l--5]. Atom-atom [l---7] and atom- molecule [EL 111 interactions have received the most attention. To the author’s knowlcdgc, there arc only two examples in the literature of the extension of the method to the molecule-molecule case, HF-IIF [12] and UF6 -UF6 1131. Severrll studies have been con- cerned with corrections to the calculated potential [5,7,9,14,15].

The object of the present paper is threefold: (I) t0 present an addit~onat example, Cl!,-CH,, of the use of the electron gas approximation in calculating mole- cule-molecule potentials; (ii) to test, in the molecular case, the corrections found necessary and successful for interatomic potentials; and (iii) to dctermme ac- curately the repulsive wall for the interaction of two nearly spherica! molecules. impetus was provided by the recent publication by Matthews and Smith [lG] of an empirical potential for CM,-CIl, which gives a sound basis for compa~son between ~a~culatlon and experiment.

The last objective is of interest to the author be- cause of his study of collision-induced light scattering

* Permanent addrcu: Department of Physics, University of Manitoba, Winnipeg, Manitoba, Canada, R3T 2N2.

[ 171. The intensity and frequency distribution of the scattered ii&t are sencitivc 10 rfctails of the ~nterlllo- lecular potential, and particularly of the repulsive wdl.

Scattering data exists for molecular flu& comprised of such molcculcs as CH,, CF, and SF, [ 18 J; and rch- able iOterpretatiOn in terms of molecular dyna[ni~s re- quires a good knowledge of the pair potent~al.‘Ihc elcc- tron gas approximation is expected to give an acCUratC

description of the interaction in the overlap region.

2. The electron gas approximation

The basic method will not bc elaborated here; de-

tails may be found in the papers of Gordon and Kun [l-51. The essential ideas are that the molccutcs arc described by their charge dcnslties which are nssumed to bc additive in the overlap re@on. The coulombic in- teraction energy is calculated directly; the kinetic (V,), exchange (V#:) and correlation (V,) energies are ob- tained by thetheory of the uniform, infinite electron gas - wherein the energy is r&ted to the electron density.

The electron gas energ does not contam a contri- bution from IOng-range dispersion energy; it must be included a posteriori. The difficulty lies in making the transition from the region, where the dispersion series alone is accurate, to short ranges, where the electron gas approximation is valid. Several approaches have been tried; the method adopted here is that originally

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due to Pack and his co-workers [7,19] and modified by Clugston [20]. it gives the most consistent results for the inert gas interactions. The dispersion series is approximated by its leading term, -C6/R6; the others are deliberately neglected. It is found that the ratio of the dispersion energy V, to the electron gas correla- tion energy V, goes through a minimum at a distance r,., which lies about 1 bohr inside the potential mini- mum at r,,. lt is assumed that, inside rc, VC gives the correct correlation energy and that, outside rc, the true correlation energy is given by V, plus a fraction of V,, chosen to provide continuity at rc. At long dis- tances, the correlation energy becomes essentiaIly V,.

Multiplicative correction factors may also be ap- plied to the electron gas energy terms [ 14, IS]. These arise because the error in the individual energy terms is greater than the error in the total energy. Specifical- ly, for the inert gases, if no correction is made, VK is underestimated and VE is overestimated.

3. Computational procedure

A main feature Of the electron gas approximation is that it is computationally rapid, especially for atoms where. only several seconds are required to calculate the potential at a given interatomic separation. The di- rect extension of the atom-atom procedure to the molecule-molecule case !eads to six-dimensional in- tegrals and a subsequent increase in computational tirnc; modifications are required to preserve the com- putational advantage over a5 inilio methods [ $21. In the present case, interest is in the spherically aver- aged potential. Therefore, the following simplifica- tions were introduced. The electron densities were cal- cuIated with the one-centre ilartree--Fock wavcfunc- tions of Moccia [21 j. The densities of the individual molecules were then spherically averaged analytically and these averaged densities were used in all calcula- tions. The e!ectron gas energy terms then reduced to two-dimensional integrals as in the atomic case [ 11. The Coulomb interaction was calculated at a number of specific orientations of one molecule with respect to the other and an average taken. Despite the use of spherically averaged electron densities, the potential determined is in good agreement with experiment (as shown in section 4).

The numerical integration required in the calcula-

tion of the electron gas energy was performed by a method due to Handy and Boys [22]. The procedure consists in making a change of variable, for example in the integral J,bf(x) dx, such that x = x(q) with the property that d”x/dq” is zero at x = a and x = 6; II = 0, 1,2, ._. N, where N is an integer to be chosen. The re- sulting integral is then suitable for numerical integra- tion using equally-spaced points and equal weights in q-space. FOI multi-dimensional integrals such a trans- formation is made for each dimension separately.

The integrals for VK, VE and V, reduced to the form V = 11 dr,dr,,f(r,, I-,.,) where I-~ and rb are dis- tances from the centres of molecules a and b, respec- tively. A first transformation was made to spheroidal Coordinates and a second made to set the limits of in- tegration between 0 and 1. Then

V= j-j.n(x,v) dxdy. 00

(1)

The substitutions,

dx = 30 ST (S1 - 1)’ dS,

and

dy = 30Sz(S2- 1)2 dS2,

(3)

with the intcgrand and its first derivative zero at 0 and 1. It was found that V converged satisfactorily for 20 X 20 points of numerical integrition.

In the determination of the CH,-CH, potential, Matthews and Smith [16] used a value for c6 of 150.3 au as calculated by Dalgarno [23]. This value was also adopted in the present work to describe the dispersion contribution; rc was found to be 6.5 au. Corrections to VK and V, were made by multiplying by the Waldman- Gordon (WG) factors [ 151; those appropriate to Ne, i.e. a ten-electron particle, were used: 1.075 for VK and 0.821 for VE.

The potential parameters are summarized in table 1. The potential, calculated without the WG factors, is shown in fig. 1, together with the empirical potential of Matthews and Smith.

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Table 1 Summary of potential parameters

--. ---- 0 a) ‘m

a) E/k b,

@J) (au) WI _-_-_-------------

prcscnt (WIthout WG factors)

present (with WG factors)

6.79 7.62 237

7.56 8.25 122

Matthews and Smith [ 161 6.725 7.307 217

Lcnnard-Jones [ 25 1 7.336 8.234 137 ____--- - - -- -_

a) 1 au = 1 bohr = 0.5292 X 10W8 cm.

b, 1 au = I hartree = 3.1578 x IO’K.

-1c I I I I I I 1 1 I

6 8 10 12 I.5

r (LI.UJ

kig. 1. Intcrrnolecular potential for CHq-CH+ calculated without tilt: USC of the WC. correction factors. The open cir-

cles represent the cmpirkal potential of Matthews and Smith. The atomic unit of energy is the hartrce and the atomic unit of distance is the bohr.

Uest agreement is obtained with the Matthews and Smith potentlai when the WC factors are omitted in the calculation; the values of u. rn, and ~/lk are greater than the empirical ones by I%, 4% and 9%, respective- ly. This conclusion has been reached in other elect ran gas calculations of mtermolecular potentials. I’arker ct al. [8] obtained their best results for IIF--1lF without the Raz correction factor [I41 for VL; they did not at-

tenipt a correction to vK. Similarly Schneider et al.

[ 131 used no correction factor and achieved good agreement with experiment for UF,-UF6.

Leonas [24] has measured the short-range potcntlal for Cli,-CII, by high-energy scattermg techniques. He found that V(r) 0: exp (- Xr) where h is 8.22 (au)-‘. Below 6.5 au, the present potential varies as exp (- -8.56 r). The shape of the repulsive wall at lilgh energy therefore agrees with cxpctiment within 4<%.

Tile magnitude of the present potential in thn rcglon is, ttowcver, about five times greater than mcasure- ment. The potential of Matthews and Smith does not extend into this region, but its magnitude too appears to be larger than suggested by the scattering data.

The previous comments illustrate the success of the electron gas approximation. It is important how- ever, to point out unsatisfactory aspects of the meth- od. The long-standing Lennard-Jones potential [25] differs markedly from the potential of Matthews and Smith: 9% in u, 13% in rn, and 37% in e/k. When the WC correction factors are applied to VK and V,, there is a good agreement between calculation and the Lennard-Jones form (table 1). This result demonstrates that the tin:11 potential particularly in the attractive re- glen depends critically on how corrections to the orig- inal Gordon-Kim method [l] are made. These corrcc- tions do not seriously affect the shape of the repul- sive wall.

In summary, a potential for Cl-I, -CM4 has been calculated, within the clcctron gas approximation, which agrees with experimental data. As for simdar calculations of molecule-molecule potentials, best rc- sults arc obtained when no correction is applied to the individual electron gas energy terms; the physical reason for this requirement is not clear.

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Volume 52, number 1 CiIEhIICAL PIi YSICS LETTERS 15 November 1977

Acknowledgement

It is a pleasure to acknowledge rnttny informative

discussions with Dr. 1h4.J. Clugston concerning the

basis of the electron gas approximation Gratitude is

also expressed to Dr. N.C. Handy for his valuable

advice on the computational aspects of the calculation.

References

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