Comparison of models for H2–H2 and H2–He anisotropic intermolecular repulsionT. L. Starr and D. E. Williams

Citation: The Journal of Chemical Physics 66, 2054 (1977); doi: 10.1063/1.434165 View online: http://dx.doi.org/10.1063/1.434165 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/66/5?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Effects of the Anisotropy of the Intermolecular Potential on the Collisioninduced Spectra of H2H, H2He, H2H2,and HDHe AIP Conf. Proc. 645, 216 (2002); 10.1063/1.1525458 Anisotropic intermolecular potentials for HeC2H2, HeC2H4, and HeC2H6, and an effective spherical potential forHeCHF3 from multiproperty fits J. Chem. Phys. 88, 4218 (1988); 10.1063/1.453830 Vibrational relaxation times of F2–He and Ar, H2–He, and D2–He. II J. Chem. Phys. 67, 4730 (1977); 10.1063/1.434643 LASER EMISSION NEAR 8 FROM A H2–C2H2–He MIXTURE Appl. Phys. Lett. 17, 436 (1970); 10.1063/1.1653260 ShortRange Intermolecular Forces. II. H2–H2 and H2–H J. Chem. Phys. 26, 756 (1957); 10.1063/1.1743400

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Comparison of models for H2-H2 and H2-He anisotropic intermolecular repulsion

T. L. Starr and D. E. Williams

Department of Chemistry, University of Louisville, Louisville, Kentucky 40208 (Received 14 September 1976)

Three empirical potential models (dumbbell, Kihara, and ellipsoidal overlap) are fitted to theoretical calculations of H2-H, and H,-He repulsion. The dumbbell model gives a slightly better fit and has better transferability. This model is recommended because of its simplicity and physical meaningfulness.

INTRODUCTION

Theoretical calculations of the potential energy of repulsion between small molecules are available. Con­siderable expenditure of effort may be required to pro­duce these results. Usually the authors of such papers attempt a fit of their results to an empirical model po­tential. Two primary goals may be sought through the use of such empirical models, The first goal may be to obtain some degree of transferability (with accom­panying correlations) between different molecular sys­tems. The second goal may be to obtain some intuitive physical understanding of the repulsion phenomenon, Since molecules are non-spherical, the repulsion poten­tial must be a fUnction of both separation and orienta­tion, particularly for short intermolecular distances. Diatomic molecules are the simplest example and sev­eral repulsive potential energy models have been pro­posed. In this paper we will critically examine three different models for the repulSion potential energy of diatomic molecules. An exponential form for the re­pulsion will be used throughout since this, in general, has shown the best fit to theoretical and experimental results.

INTERMOLECULAR REPULSION MODELS

The dumbbell model considers two repulsive centers located on the molecular axis (Fig. 1). To a first ap­proximation they are located at the nuclei, but in gen­eral, their separation is taken as adjustable parameter d. Especially in the case of the repulSion of hydrogen molecules, d will be less than the internuclear distance. It will be referred to as the foreshortened bond length. The potential is given as

2 2'

V= B L L exp(- CrlJ) , i:l J=I'

where rlJ is the distance between repulsive centers on different molecules. B is referred to as a scale pa­rameter. The anisotropy of the model enters through the foreshortened bond distance d. In the limit d - 0, the potential becomes isotropic.

The modified Kihara core model1,7 considers three

repulsive centers on the molecular axis (Fig. 2): one at the center of the bond and one at each nucleus. The two types of centers have different scale parameters and the potential is given as

3 3'

V = B L L exp[ - C(rlJ - r l - rJ] , 1=1 J=l'

2054 The Journal of Chemical Physics, Vol. 66, No.5, 1 March 1977

where r ij is again the distance between repulsive cen­ters on different molecules, Band C are the overall scale and range parameters, and ri is the scale pa­rameter associated with a particular repulsive centero

The ellipsoidal overlap model2 considers the poten­tial proportional to the overlap of ellipsoids of revolu­tion around each molecular axis (Fig. 3). In general, the overlap model specifies only the angular dependence of the potential. Assuming the radial dependence is exponential, the potential is given as V = B exp(- CR), where R is the distance between molecular centers and B= E:(1 - X cos2e)2, C= p{l- 1/2X[(cOS<Pl + COS<P 2)2/ (1 + X cose) + (cOS<Pl - cos<P2)2/(1 - X cose)]}l/ 2, e = angle between the molecular axes, <PI = angle between molec­ular axis i and the intermolecular vector. In this mod­el both the scale and range parameters Band Care directly modulated by anisotropic functions. The ad­justable anisotropiC parameters are E:, X, and p.

DETERMINATION OF THE MODEL PARAMETERS

The three repulsive potential models are compared using the H2-H2 system for which reliable quantum mechanical calculations are available. For this pur­pose, SCF calculations are preferred to configuration­interaction calculations since the latter include the dis­persion energy.3 Tapia, Bessis, and Bratoz4 give SCF calculations for four orientations in the range 1-3 A. Since quadrupole-quadrupole energies in this region are negligible, 5 these calculations should give pure repulsion energies. Of the 24 values given by TBB, the shortest linear orientation was discarded since one

FIG. 1. Dumbbell model.

Copyright © 1977 American I nstitute of Physics

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T. L. Starr and D. E. Williams: Anisotropic intermolecular repulsion 2055

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intermolecular atom-atom distance was shorter than the intramolecular distance. The longest planar or­thogonal orientation was also discarded since it seemed drastically low and was not fit by any of the models. A least squares fit using relative weighting was per­formed with each model. For the dumbbell model, the three parameters are BHH , CHH, and dH • For the Kihara model, four parameters are specified but only three of these are independent, The scale parameter for the repulsion at the nucleus THl is arbitrarily set to 1 bohr, leaving BHH , CHH, and TH2 (the scalar "param­eter for" the repulsion at the bond center) to be deter­mined. For the overlap model, €HH' PH' and XH are to be found. The resulting parameters are given in Table I and a comparison of the fit is shown in Table II. All models give a good fit to the data with the dumbbell

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FIG. 2. Kihara model.

model slightly superior to the others. The rms rela­tive error of fit ranges from 9.26% for the dumbbell model to 10.57% for the overlap model, with an inter­mediate value of 9. 35% for the Kihara model.

TRANSFERABILITY OF PARAMETERS

A further property desired of a potential model is transferability. The three models are tested for this using the H2-He system and the SCF calculations of Tsapline and Kutzelnigg. 6 For the dumbbell model, the parameter dH is held at the value obtained from the H2-H2 fit the values of B HHe and CHHe are derived. For the Kihara model, THl and TH2 are held constant; the scale parameter for the He repulsion is arbitrarily set at 1 bohr and values for B HHe and CHHe are found. The

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J. Chern. Phys., Vol. 66, No.5, 1 March 1977

FIG. 3. Ellipsoidal overlap model.

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2056 T. L. Starr and D. E. Williams: Anisotropic intermolecular repulsion

TABLE 1. Parameters [or H, -H, potential models (a. u. ).

Dumbbell model

Kihara

Overlap

BHH" 1. 6602 CHH ~ 1. 6588 dH = O. H1226 BHH • 0.005726 CHH = 1. 6564 YHI = 1. 0000 r H1 c. 1. 9004

"HH=6.7495 %=1.6724 XH=0.031029

Atomic units: energy in hartrees and distance in bohrs. 1 hartree=27.198 eV =2624.7 kj/mole. 1 bohr=O. 529167 A = 0.529167 x 1010 m.

ellipsoidal overlap parameters for an atom-molecule interaction have the form B = EHHe,

C = [2p~ P~e/(2p~ + P~e)] (1 - XH cos2¢ )1/2 ,

where ¢ is the angle between the H2 axis and the atom­molecular vector. The values of EH and PH have al­ready been obtained and values for EHHe and PH. are now derived. A least squares fit with relative weighting is again performed using the results in the range 3-6 bohrs. The resulting parameters are given in Table III and the final fit is shown in Table IV. The best fit is again shown by the dumbbell model (5.21%). The Kihara model is only slightly worse (5.31%), but the overlap model suffers from this comparison (8.99%). The overall fit is better than for H2-H2 , perhaps be­cause the theoretical basis data may be more accurate.

DISCUSSION

In a recent paper,7 Raich, Anderson, and England used the Kihara and Gaussian overlap models in fitting

TABLE II. Fit to H2-H2 SCF calculations (a. u.).

Ikhlivl' ,')"l'Ot

Ii V [)umblll'll h.ih;lI':1 (h'l'I'bp .. ~----~----

:--iquan' ., u;) II. 11~() 1 (J. (10:-..;, 0.00\1:; II. 1191i

plana I' ~. ~;l:l (). () :-) \ I ~ I " -0, oiBi - O. 0).,(1, -11.111·1·)

:L :;·t:2 O.O!\I!)il -0. l:.!,.,.., -0, I :IO~ - O. ()91:l

-L ~.)() II. OOnO~1 - (I. IIO!) -(J. 1117 - o. OHI."'" -l. H.-) "I (). 00 I 7~) U. II:),.) - O. ().'''O -O.O;'i4:."\

.J.(i(i!; 0, O()(),j 1 O.O:2"";l n,02:-<.,"\ O.()H7

.l\on-planar ., u:-) II. I Ii 17~ (). (Hi :)~l (t.Oito II. 194::)

orthogonal ~. :--:;;:; O.O:-/j(iO - O. \l,j(j". -0. O .. lJj~ O. (J2r):-\

:{ .. )·!:2 (). () I ~l~() -u. 10IJO -0. J 007 - O. 1),-,0.;7

~. 2.j() O,O():),,",o.; -o,o:..;on -0, (),'"'!!; - O. 0,""j9:..;

..t.!l."io.; 0. (jOlio -O.Oml:! - n. OOU4 - O. O()."i:~

.). (iii Ii D. (JO()·ii O. 111) O. 11:)2 O. lOOG

Linear 2.,'-1:1:1 O. 1 ()(i7~ - O. 1;);)7 -0. l:IB:1 - O. 2-192

:l. :)'I~ O.O;1l."i7 - o. on77 - O. O~H19 - o. U-;24 .J-, ~:lO O,O()!lfiO -O.O,O.;;W ~. O. O~·):I - O. OB09

--1-. ~l:)" O.OO:!'-IO - U. 02!)(j - n. 02,~"0:7 0.02 KH .). (jlili 0, ('OU"'O O.04:;·W O. OSI (i o. 1 ~/'q

Planar 2.12:) O.I,"lS72 o. :!(H14 0.2101 O.mH1S

Orthogonal 2. K:l:I O.OG-l7() O.()!)-!1 0.0925 -O,OU);,

:1. S·l:! U.021-12 P,O:t24 O.O:lO9 -O.O74;~

·L ~5() O.OO!i?O O. O~G9 O.02G2 - O. 0779 .1. ~)S ..... 0.001."17 II. 1422 O. 14:27 O.02H.)

(j" ~((VII- t~/)/V[l) O. ()~):!(; O. O~l;JS 0.1 US,

T ABLE III. Paramele rs [or II, -He poten­tial models (a. u. ).

Dumbbell

Kihara

Overlap

B HHe = 5.2061 CHHe ·= L 9671 dH = O. H1226 B HHe '= Oc 026360 C HHe .1. 9686 YHe 1. 0000 YH1 1. 0000 YH~ 1. 9004 ~HH.·= 11. 483 %e= 1. 13-17 %~1.6724

XH = O. 031029

the anisotropic interactions of H2-H2 but dismiss the dumbbell model as inadequate. Our results, however, show better fit with the dumbbell model, and consider­ing its greater simplicity, suggest it to be the model of choice.

Williams 8 has previously used the dumbbell model to fit the results of SCF calculations for H2-H2 and H2-He, concluding that the repulsive center is shifted into the bond 0.07-0.20 A. Our present calculations using re­cent SCF results indicate a shift of 0.16 A. Stewart, Davidson, and Simpson have shown that the best fit of spherical centers of electron density is significantly inside the nuclear positions. 9 It is antiCipated that for heavier atoms the shift will be considerably smaller and the dumbbell model can be further simplified by setting d equal to the internuclear distance. This situ­ation is predicted because it is expected that only the electrons involved in bonding will shift Significantly into the internuclear region. The remaining electrons will essentially remain centered at the nuclei. Since the repulsion is related to overlap of filled orbitals, the presently found shift of the repulsion center into the bond corresponds to the results of the cited electron density shift study. The electron denSity contours of H2 do not show a local maximum at the center of the bond as would be appropriate for the Kihara or overlap models. Thus, the dumbbell model has a more direct physical interpretation than the Kihara or ellipsoidal overlap models. Extension of the dumbbell model to

TABLE IV. Fit to H 2-He SCF calculations.

Helalive error

Ii V llumbbell Kihara ()v(:'rh~p

Linear 3. 0.035948 0.0591 u.OHl -0.0713

4. 0.005594 - O. 0482 - O. Oti57 - O. 14R3

5. 0.000781 - O. 0465 - O. OG55 - 0.1294

n. 0.000095 O. 0976 0.0741 O. U2:!7

Orthogonal :1. O. 02644:1 O.020H n.0561 O.1301i

4. 0.004020 - O. 04":1 - O. 0211 0.0472

5. 0.000562 - O. 0394 -0.015" (J,0374

Ii. 0.000074 0.0:IC7 O.O:l2H 0.095:1

45 degrees 3. 0.O:lO832 0.0451 0.0:145 O.O;3:lG

4. 0.00477:) - O. 0487 -O.OSH7 -- O. UGIH

5. O. 000673 - 0.0522 -- O. OG:{2 - O. 0649

(:':((Vo - ~c)/Vo)')'" 0.0.")21 o.O;):n (). () "'~)!)

J. Chern. Phys., Vol. 66, No.5, 1 March 1977

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T. L. Starr and D. E. Williams: Anisotropic intermolecular repulsion 2057

polyatomic molecules is simple and immediate in con­trast to the other models which cannot be applied easily to large systems.

The overlap model does have computational advan­tages over the dumbbell and Kihara models which can be of great importance in certain calculations such as molecular dynamics. A trade-off between accuracy and computing speed is necessary for larger systems. Al­though the overlap model has been shown to give quali­tatively correct results for relatively complex sys­tems, 10 quantitative results for Simple molecules such as CO2 and benzene have not been encouraging. 11 Even for CO2, the overlap model cannot give both gas and solid properties for the same set of potential parame­ters. The Kihara and dumbbell models are successful in this case.

CONCLUSION

In a direct comparison of three models for anisotrop­ic intermolecular repulSion, the dumbbell model is shown superior in both fit to SCF data and transferabil­ity of parameters. This, combined with its inherent Simplicity and meaningfulness of its parameters, sug-

gest it to be the method of choice in intermolecular re­pulsive ~otential energy calculations.

tAo Koide and T. Kihara, Chern. Phys. 5, :34 (1974). 2B. J. Berne and P. Pechukas, .J. Chern. Phys. 56, 421:3

(1972). 1R. O. Etters, R. Oanielowicz, and W. England, Phys. Rev.

A 12, 2199 (1975). 40 . Tapia, G. Bessis, S. Bratoz, Int. J. Quanturn Chern.

84, 289 (1971). 5W. England, R. E. Etters, J. Raich, and R. Oanielowicz,

Phys. Rev. Lett. 32, 75H (1974). 6B . Tsapline and W. Kutzelnigg, Chern. Phys. Lett. 23, 17:3

(197:3) . 7J. C. Raich, A. B. Anderson, and W. England, J. Chern.

Phys. 64, 5088 (1976l. BO. E. Williams, J. Chern. Phys. 43. H24 (1965). ~. F. Stewart, E. R. Davidson, and W. T. Simpson, J.

Chern. Phys. 42, 3175 (1965). tOR. Pynn, J. Chern. Phys. 60, 4579 (1974); J. Kustick, B.

J. Berne, J. Chern. Phys. 64, 1:162 (1976). !to. J. Evans, R. O. Watts, Mol. Phys. 29, 777 (1975); T.

B. MacRury, W. A. Steele. B. J. Berne, J. Chern. Phys. 64, 1288 (1976).

12T his investigation was supported by NIH research grant number GH16260 from the Institute of General Medical Sciences.

J. Chern. Phys., Vol. 66, No.5, 1 March 1977

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