J . Phys. Chem. 1994,98, 1830-1833
1830
Refinement of Nonbonding Interaction Potential Parameters for Methane on the Basis of the Pair Potential Obtained by MP3/6-31 lG(3d,3p)-Level ab Initio Molecular Orbital Calculations: The Anisotropy of H/H Interaction Seiji Tsuzuki,’ Tadafumi Uchimaru, and Kazutoshi Tanabe National Institute of Materials and Chemical Research, Tsukuba, Ibaraki 305, Japan Satoru Kuwajima CRC Research Institute, Inc., Engineering-System Division, 1-3- Dl 7 Nakase, Mihama- ku, Chiba-shi, Chiba 261 -01, Japan Received: August 2, 1993’
The intermolecular interaction energies of 132 geometrical configurations of methane dimers were calculated by an MP3/6-3 1lG(3d,3p)-level a b initio method. Nonbonding interaction parameters were fitted using three models (isotropic hydrogen, anisotropic hydrogen, and anisotropic hydrogen with electrostatic interaction) to reproduce the ab initio results. The incorporation of the anisotropy of the nonbonding interaction of hydrogen decreased the RMS error drastically. On the other hand, the incorporation of the electrostatic interaction improved the fitting little. The density, heat of evaporation, and self-diffusion coefficient of liquid methane were calculated by molecular dynamics simulations with the second- and the third-model parameters. The calculated density of 0.38 g ~ m - the ~ , heat of evaporation of 1.7 kcal/mol, and the self-diffusion coefficient of 6.0 X 10-9 m2/s from the second-model parameters and 0.38, 1.7, and 6.5 from the third-model parameters were close to the experimental values of 0.42, 2.0, and 5.4, respectively.
Introduction The nonbonding interactionsof organic molecules are important to understand the structures and properties of organic molecules in the condensed phase.1-3 Recently developing force field simulations of organic molecules in the condensed phase also require accurate intermolecular interaction potentials.2-5 Intermolecular interaction potentials of organic molecules were investigated from the measurement of the compressibility of gases,6-12 the measurement of the properties of liquids,” and the analysis of crystal structures and lattice energies.l”19 Unfortunately, however, these experimental measurements give only limited information of the intermolecular interaction potential. Since the measurements in the gas and in the liquid phase give the information on the spherically averaged intermolecular interaction potential, the anisotropy of the interaction is not revealed from these measurements. From the measurement of the crystal structures, several points on the potential curve can be measured. However, the high-energy region of the potential is not covered from the measurement of the crystal structure. Recently, ab initio molecular orbital calculations have often been applied to estimate the intermolecular interaction potentials of several mole~ules.4~5.2~2~ Ab initio molecular orbital methods can calculate the interaction energy of any orientation of the supermolecule. Force field parameters are refined from the intermolecular interaction potentials obtained from ab initio calc~lations.~~5 A simple empirical representation of the intermolecular interaction energy is a sum of pairwise additive atom-atom interaction energy terms, with each term being the sum of several energy components. This representation is also used for force field~alculations.1~~25~26 The atom-atom interaction energy term can be described as E(tota1) = E(repu1sion)
+ E(dispersion) + E(Cou1ombic)
(1) The dominant components of the intermolecular interaction energy of polar molecules are the repulsive and the Coulombic e
Abstract published in Aduance ACS Abstracts, January 1 , 1994.
0022-3654/94/2098- 1830$04.50/0
energy components. The dispersion energy componentis relatively small. Thus the large part of the intermolecularinteraction energy of polar moleculescan beevaluated by the Hartree-Fock method, which cannot evaluate the dispersion energy.$ On the other hand, the Coulombic energy component of the interaction energy of a nonpolar molecule, like a saturated hydrocarbon molecule, is small. The major components of the intermolecular interaction energy of this type of molecule are the repulsiveand dispersion energy components. Thus the evaluation of the dispersion energy is necessary to obtain the correct intermolecular interaction energy potential of these molec u l e ~ . Since ~ ~ .the ~ dispersion ~ ~ ~ ~ energy has its origin in electron correlation, the calculation of the electron correlation energy is necessary to estimate this energy component.27J9 The difficulty of the evaluation of the dispersion energy by ab initio molecular orbital methods is the basis set dependence of the calculated dispersion energ~.~O.~l-” The calculated dispersion energy is greatly underestimated, if a small basis set is used. A large basis set including multiple polarized functions is necessary to evaluate the dispersion energy correctly. Nonbonding interaction potentials for force field calculations of polar molecules have been refined on the basis of ab initio molecular orbital cal~ulations.4~~ On the other hand, only little work has been done to refine the nonbondinginteraction potentials of saturated hydrocarbon molecules, including the dispersion energy component, on the basis of ab initio methods.2’ The nonbonding interaction of hydrogen atoms of hydrocarbon molecules has large ani~otropy.l~J5-~~J4 One of the ways to incorporate this anisotropy in force field simulations is to relocate the nonbonding interaction center of hydrogen to the direction of the connected carbon atom.14,25-27,34 Several force fields including MM2 and MM3 use this method to incorporate the anisotropy.14925J6 Unfortunately, however, the anisotropy of the nonbonding interaction of hydrogen was not incorporated in the previously reported parameter fitting on the basis of the ab initio pair potential of methane.21 In this paper we refine the parameters of nonbondinginteraction of methane, including the dispersion energy component, on the 0 1994 American Chemical Society
The Anisotropy of H/H Interaction basis of the pair potential obtained by an MP3/6-3 11G(3d,3p)level ab initio calculation. Three models are employed for the parameter fitting to evaluate the effects of the anisotropy of the nonbonding interaction of hydrogen and the effect of electrostatic interaction. Molecular dynamics simulation of liquid methane is carried out using the newly refined nonbonding parameters to evaluate the accuracy of the pair potential of methane obtained by an MP3/6-3 1lG(3d,3p)-level ab initiocalculation. Molecular dynamics simulation is also carried out using the parameters of saturated hydrocarbon molecules in MM225and MM326force fields.
The Journal of Physical Chemistry, Vol. 98, No. 7, I994 1831 H
H
Refinement of Nonbonding Potential of Methane The intermolecular interaction energies of the 12 orientations of methane (Figure 1) with different intermolecular distances were calculated by the ab initio method. The total number of configurations of methane dimer considered in this work is 132. The calculated energies are shown in Table 1. The exp-6-1-type pair potential, which is called the Buckingham
= bij exp(-c..r..) - a..r..d + qq.r..-’ 1J 1J 1J 1J 1 J U was used for the fitting of the nonbonding interaction potential. The nonbonding parameters for H / H interaction were taken from
H-C
H
B \H
B
A
H
H %-H
/
H
Computational Technique The GAUSSIAN 86 program35 was used for the molecular orbital calculations. The 6-31 1G(3d,3p) basis set36J7was used for thecalculations. The electron correlationenergywas corrected by the third-order Merller-Plesset perturbation method.3841The basis set superposition error (BSSE)42was corrected by the counterpoise method.43 The geometry of a single methane molecule was optimized at the MP2/6-3 11G(2d,2p) level, in which the C-H bond distance is 1.082 491 A. This geometry was used for the calculation of methane dimer. Recently we reported the calculationsof the dispersion energies of small molecules with multiple polarized basis sets reported by Pople et al?6J7 and electron correlationcorrectionwith the MerllerPlesset perturbation meth0d3~~1 to evaluate the effects of basis set and electron correlation correction.31-33 The calculations of methane dimers show that the dispersion energy, calculated as the electron correlation energy, is greatly changed with the basis set used.” The dispersion energy is underestimated when a small basis set is used. The calculated dispersion energy is increased by the improvement of the basis set up to the 6-3 11G(3d,3p) basis set. The further improvement of the basis set affects the calculated dispersion energy little. The argumentation of sets of diffuse functions on carbon and hydrogen atoms, a set off functions on carbon atoms, and a set of d functions on hydrogen atoms to the6-311G(2d,2p) basisset gaveonlylittleeffecton thecalculated dispersion energies of methane dimers.” The dispersion energy of methane dimer calculated at the MP3 level is close to that at the MP4(SDTQ) le~el.3~ Molecular dynamic simulationsof 200 rigid methane molecules were carried out in the NPT ensemble. The equations of motion of rigid bodies were integrated by the leapfrog algorithmu with a time step of 2 fs. Periodic boundary conditions with cubic unit cells were employed. Constant-temperature and -pressure conditions were maintained by using the method of N0se~5.4and the method of Ander~en.~’The nonbonding forces were truncated at 12 or 10 A, depending on whether (12 A) or not (10 A) Coulombicinteractions were involved. The truncation corrections were included in both the pressure and potential energy. The systems were equilibrated initially by 10-ps simulation. The density, nonbonding potential energy, and self-diffusioncoefficient were obtained from the following 50-ps simulation. The heat of evaporation AHvwas calculated from the nonbonding potential per molecule, Ei,by AHv -Ei + RT.2
\
\
C
D
E
F
H
\
H-C
H-C
b
G
H
I
J
K
L
Figure 1. Twelve orientations of the methane dimers considered in this
work.
a recent work.33-4es*The parameters for C/C interaction were optimized on the basis of the intermolecular interaction energies obtained by an MP3/6-31 lG(3d,3p)-level ab initio calculation. The nonbonding parameters for C/H interaction were obtained by the geometrical mean of the corresponding parameters for C/C and H / H interactions. Several studies have shown that the nonbonding interaction of hydrogen atoms is highly ani~otropic.l~,25,26s~.52-~~ In order to evaluate the importance of this anisotropy, an isotropic hydrogen model (Figure 2, model I) and an anisotropic hydrogen model (Figure 2, model 11) were employed for the refinement of the nonbonding interaction parameters. In order to incorporate the anisotropy of the nonbonding interaction of hydrogen, the interaction center of hydrogen was slightly relocated toward the connected carbon atom in model 11, as shown in Figure 2.” The effect of the electrostatic interaction for the improvement of the fitting was also considered by using model 111,in which +6 charges were put on hydrogen atoms and a -46 charge was put on the carbon atom. The optimized carbon nonbonding parameters and RMS errors obtained by using the three models are shown in Table 2. The RMS error of the isotropic model I is 0.3 19 kcal/mol. This large error is almost equivalent to the depth of the intermolecular interaction potential of methane dimer. The RMS error is decreased by the use of the anisotropicmodel 11. This improvement shows that the incorporation of the anisotropyof the nonbonding interaction of hydrogen is important for the fitting of the potential. The interaction centers of hydrogen atoms are relocated toward the connected carbon with several ratios. The smallest RMS error of 0.074 kcal/mol is obtained when the interaction centers of hydrogens are relocated toward the connected carbon as much as 13%of the C-H bond distance.
1832 The Journal of Physical Chemistry, Vol. 98, No. 7, 1994
TABLE 1: Intermolecular Interaction Energies of Methane Mmer Obtained by ab Initio Calculationsa orientationC distanceb A B C D E F 3.4 3.6 3.8 4.0 4.2 4.4 4.6 4.8 5.0 5.2 5.4 5.8 6.0 6.2 6.6 7.6 7.8 8.2
1.042 0.314 -0.002 -0.122 -0,151 -0.143 -0.123 -0.100 -0.063
1.550 0.387 -0.113 1.042 -0.291 0.314 -0.323 -0.003 -0,295 -0.122 -0,248 -0.151 -0.200 -0.144 4.123 -0,125 -0.100 -0.063 -0.048
1.546 0.386 -0.113 -0.291 -0.323 -0,295 -0,248 -0.201
1.297 0.335 -0.082 -0.234 -0.263 -0.241 -0.204 -0.165
-0.125
Model 1
H
Model II
-0.103 -0.103 H I
-0.049 -0.041
-0.026
1.297 0.335 -0.082 -0.234 -0.263 -0.241 -0.204 -0.165
Tsuzuki et al.
-0.041
-0.010
I
I
).
-0.026
r*
-0.010
r I
-0.008 -0.008 -0.006
-0.006
H‘
orientationC distanceb
G 2.270 0.624 -0.068 -0,313 -0.361 -0.332 -0.278 -0.223
3.0 3.2 3.4 3.6 3.8 4.0 4.2 4.4 4.6 4.8 5.0 5.2 5.6 5.8 6.0 7.2 7.4 7.6
H 1.912 0.449 -0,157 -0,361 -0.389 -0.348 -0.289 -0.230
I
1.408 0.298 -0.160 -0.312 -0,329 -0.293 -0.243 -0,195
J
1.407 0.298 -0.160 -0.312 -0.329 -0.293 -0.243 -0.195
-0.137 -0.141
K 1.433 0.391 -0.054 -0.214 -0.248 -0.229 -0.194 -0.157
-0,121 -0.121
n
n
L 2.280 0.753 0.055 -0.225 -0.306 -0.300 -0.261 -0.215
r* = r x RF RF: reduction factor
Model 111
I
-0.136 -0.099
-0.052
-0.054 -0.047 -0.047
-0.054 -0.039
-0.010
-0.011 -0.010
-0.010
-0.01 1 -0.008
a Energies in kcal/mol. Intermolecular interaction energies were obtainedby an MP3/6-31lG(3d,3p)-levelabinitiocalculation withBSSE correction by the counterpoise method. b Intermolecular distance in angstroms. Orientations of methane dimers are shown in Figure 1.
Figure 2. Three models used for the optimization of nonbonding interaction parameters of methane in this work.
TABLE 2 Optimized Carbon Parameters of the Buckingham Equation*
f
The fitting is improved little by the incorporation of the electrostatic interaction. The interaction centers were relocated with several ratios for the case of model I1 as several amounts of charge were put on the hydrogen and carbon atoms. The smallest RMS error of 0.059 kcal/mol is obtained by using model I11 when the interaction center is relocated 12%of the C-H bond distance and 0.14 e are put on hydrogen atoms and -0.56 e is put on carbon atom. The RMS error of model I11 is close to that of model 11.
Molecular Dynamics Simulation of Liquid Methane In order to evaluate the accuracy of the newly refined nonbonding interaction parameters, molecular dynamics simulations of liquid methane were carried out using the newly refined parameters. Model I1parameters were implemented in a 5-center methane model in which the H centers were relocated toward the C atom. The unit atomic mass of the H atom was given to the relocated H centers. Model I11 was represented by a 9-center methane model which contains both charge and van der Waals centers. The unit atomic mass was given to the charge centers. The calculated density, heat of evaporation, and self-diffusion coefficient are compared with the experimental values, as shown in Table 3. The calculated density and heat of evaporation with model I1 nonbonding parameters are 0.378 g cm-3 and 1.67 kcal/ mol, respectively. These values are only 11 and 15% smaller
A ( k d
B C RMSerror modelb RFb chargeb (e) A6/mol) (kcal/mol) (A-l) (kcal/mol) I 1.0 0.0 483.95 439 2.457 0.319 I1 0.87 0.0 449.53 1894 2.693 0.074 I11 0.88 0.14 389.34 5196 3.084 0.059 a Nonbonding parameters for hydrogen (A = 51.71,B = 2105,and C = 3.361) from ref 33 were used for the optimization of the carbon parameters. See Figure 2 and text.
H
than the experimental values, respectively.58 The calculated density and heat of evaporation with model I11 parameters are close to those from model I1 parameters. To compare our potential with the nonbonding potential used in conventional force field, the simulations of liquid methane were carried out using the nonbonding parameters of saturated hydrocarbon molecules used in MM225and MM326 force fields. The density and the heat of evaporation with MM2 parameters are much larger than the experimental values. Especially, the heat of evaporation from MM2 parameters is 92% larger than the experimental value. This large heat of evaporation shows that the MM2 potential overestimates the attractive interaction. The agreement of the calculated density and heat of evaporation with the experimental values is improved when MM3 parameters are used. The calculated self-diffusion coefficient of liquid methane with model I1 parameters is 6.0 X le9m2/s. This value is close to theexperimentalvalueof 5.4 X 10” m2/s.59Model I11parameters
The Anisotropy of H/H Interaction
The Journal of Physical Chemistry, Vol. 98, No. 7, 1994 1833
TABLE 3. Density, Heat of Evaporation, and Self-Diffusion Coefficient of Liquid Methane Obtained by Molecular Dynamics Simulations. param
densityb
heat of evapb
self-diffusion coeffb
model IIc model IIIc MM2d MM3' exP
0.378 (-11%) 0.380 (-10%) 0.539 (+27%) 0.363 (-14%) 0.424f
1.66 (-15%) 1.68 (-14%) 3.76 (+92%) 1.52 (-22%) 1.96f
6.0 (+11%) 6.5 (+20%) 1.1 (-80%) 8.5 (+57%) 5.4
The average temperature and pressure in the simulation are 11 1.66
K and 1 atm, respectively. Detailed conditions of the simulations are shown in the text. Density in gem-', heat of evaporation in kcal/mol, and self-diffusion coefficient in 10-9-m2-s-1. The differences from the experimental values are shown in parentheses. Parameters shown in Table 2 were used. d M M 2 parameters for saturated hydrocarbon molecules were used. The interaction center of hydrogen was relocated toward the connected carbon atom as much as 8.5% of the C-H distance. MM2-optimized geometry of a single methane molecule was used for the ~ i m u l a t i o n . ~M~M 3 parameters for saturated hydrocarbon molecules were used. The interaction center of hydrogen was relocated as much as 7.7%. MM3-optimized geometry was used.26JReference 58.8 The value of 111.6 K and 1 atm is estimated under the assumption that the self-diffusion coefficient is proportional to the square of density and the square root of temperature.59
give a close self-diffusioncoefficient. The calculated self-diffusion coefficients with MM2 and MM3 parameters are far from the experimental value.
Conclusion The isotropic hydrogen model is not suitable to describe the nonbonding interaction potential of methane. The incorporation of the anisotropy by relocation of the hydrogen atom toward the carbon atom improves the agreement of the fitted potential with the ab initio potential. The incorporation of the electrostatic interaction improves the agreement little. The accuracy of the pair potential of methane obtained by an MP3/6-31 lG(3d,3p)-level ab initio method is satisfactory. The density,heat of evaporation,and self-diffusioncoefficient of liquid methane obtained by molecular dynamics simulations with the newly refined nonbonding parameters are close to the experimental values. In addition, the nonbondinginteraction parameters in the MM2 force field overestimate the attractive interaction of methane. The calculated density and heat of evaporation using MM2 parameters are greatly larger than the experimental values. The agreement is better when the MM3 parameters are used. References and Notes (1) Wright, J. D. Molecular Crystals; Cambridge University Press: Cambridge, U.K., 1987. (2) Jorgensen, W. L. J . Am. Chem. Soc. 1981,103, 335. (3) Jorgensen, W. L. J . Am. Chem. Soc. 1981,103, 345. (4) Matsuoka, 0.;Clementi, E.; Yoshimine, M. J . Chem. Phys. 1976, 64, 1351. ( 5 ) Clementi, E.; Habitz, P. J. Phys. Chem. 1983, 87, 2815. (6) Schamp, H. W., Jr.; Mason, E. A.; Richardson, A. C. B.; Altman, A. Phys. Fluids 1958, I , 329. (7) Dymond, J. H.; Rigby, M.; Smith, E. B. J . Chem. Phys. 1965,42, 2801. ( 8 ) Snook, I. K.; Spurling, T. H. J . Chem. Soc., Faraduy Trans. 2 1972, 68, 1359. (9) Hanley, H. J. M.; Klein, M. J . Phys. Chem. 1972, 76, 1743. (10) Pope, G. A,; Chappelear, P. S.;Kobayashi, R. J. Chem. Phys. 1973, 59, 423. (11) Matthews, G. P.; Smith, E. B. Mol. Phys. 1976,32, 1719. (12) Sherwood, A. E.; Prausnitz, J. M. J . Chem. Phys. 1964, 41, 429. (13) Greiner-Schmid, A.; Wappmann, S.;Has, M.; Ludemann, H.-D. J. Chem. Phys. 1991, 94, 5643. (14) Williams, D. E.; Starr, T. L. Comput. Chem. 1977, I , 173. (15) Bates, J. B.; Busing, W. R. J . Chem. Phys. 1974, 60, 2414. (16) Lii, J.-H.; Allinger, N. L. J . Am. Chem. Soc. 1989, I l l , 8576. (17) Warshel, A.; Lifson, S.J . Chem. Phys. 1970, 53, 582. (18) Sprague, J. T.; Allinger, N. L. J . Comput. Chem. 1980, I , 257. (19) Allinger, N. L.; Miller, M. A,; Vancatledge, F. A.; Hirsch, J. A. J. Am. Chem. Soc. 1967,89,4345.
(20) Szczesniak, M. M.; Chalasinski, G.; Cybulski, S.M.; Scheiner, S.J . Chem. Phys. 1990, 93,4243. (21) Gay, D. H.; Dai, H.; Beck, D. R. J. Chem. Phvs. 1991. 95.9106. (22) Albcrts, I. L.; Rowlands, T. W.; Handy, N. C. J.-Chem. ihy;. 1988, 88, 3811. (23) Hobza, P.; Selzle, H. L.; Schlag, E. W. J . Chem. Phys. 1990, 93, 5893. (24) Dinur, U.; Hagler, A. T. J. Am. Chem. Soc. 1989, I l l , 5149. (25) Burkert, U.; Allinger, N. L. Molecular Mechanics; American Chemical Society: Washington, DC, 1982. (26) Allinger, N. L.; Yuh, H. Y.; Lii, J.-H. J. Am. Chem. Soc. 1989,111, 855 1. (27) Williams, D. E.; Craycroft, D. J. J . Phys. Chem. 1987, 91, 6365. (28) Novoa, J. J.; Whangbo,M.-H.; Williams, J. M.J. Chem. Phys. 1991, 94, 4835. (29) A Hartree-Fock calculation can evaluate only part of the electron correlation between the same spin electrons. See ref 30. (30) Hehre, W. J.; Radom, L.; Schleyer, P. v. R.; Pople, J. A. Ab Initio Molecular Orbital Theory; John Wiley: New York, 1986. (31) Tsuzuki, S.;Tanabe, K. J . Phys. Chem. 1991,95, 2272. (32) Tsuzuki, S.;Tanabe, K. J. Phys. Chem. 1992,96, 10804. (33) Tsuzuki, S.;Uchimaru, T.; Tanabe, K. J. Mol. Struct. (Theochem) 1993, 280, 273. (34) Williams, D. E. J. Chem. Phys. 1965, 43, 4424. (35) Frisch, M. J.; Binkley, J. S.;Schlegel, H. B.; Raghavachari, K.; Meliu, C. F.; Martin, R. L.; Stewart, J. J. P.; Bobrowicz, F. W.; Rohlfing, C. M.; Kahn, L. R.; Defrees, D. J.; Seeger, R.; Whiteside, R. A.; Fox, D. J.; Fleuder, E. M.; Pople, J. A. Gaussian 86; Carnegie-Mellon Quantum Chemistry Publishing Unit: Pittsburgh, PA, 1986. (36) Krishnan, R.; Binkley, J. S.;Seeger, R.; Pople, J. A. J . Chem. Phys. 1980, 72, 650. (37) Frisch, M. J.; Pople, J. A.; Binkley, J. S.J . Chem. Phys. 1984, 80, 3265. (38) Maller, C.; Plesset, M. S . Phys. Reu. 1934, 46, 618. (39) Pople, J. A.; Seeger, R.; Krishnan, R. In?.J . Quantum Chem. Symp. 1977, 11, 149. (40) Krishnan, R.; Pople, J. A. In?. J. Quantum Chem. 1978, 14, 91. (41) Krishnan, R.; Frisch, M. J.; Pople, J. A. J . Chem. Phys. 1980, 72, 4244. (42) Ransil, B. J.; J . Chem. Phys. 1961,34,2109. (43) Boys, S.F.; Bernardi, F. Mol. Phys. 1970, 19, 553. (44) Hockney, R. W. Methods Comput. Phys. 1970, 9, 135. (45) Nose, S.Mol. Phys. 1984,52,255. (46) Nose, S.J. Chem. Phys. 1984,81, 511. (47) Andersen, H. C. J . Chem. Phys. 1980, 72, 2384. (48) Buckingham, A. D.; Utting, B. D. Annu. Rev. Phys. Chem. 1970,21, 281. (49) It is possible todetermine both H / H and C/C interaction parameters from the calculated intermolecular interaction energies of methane dimers, if a larger number of dimer orientations are considered in the parameter refinement. However, the calculation of the interaction energies of such a large number of orientations requires a huge amount of CPU time and is not practical. Thus we used the H / H nonbonding interaction parameters determined on the basis of the intermolecular interaction energies of hydrogen molecule dimers obtained by an MP4(SDTQ)/6-31 IG(3p)-level ab initio calculation. (50) The interaction parametersshould be transfmedat thesameaccuracy level. However, the definition of the same accuracy level is a difficult issue in the comparison of ab initio molecular orbital calculations. The accuracy For example, methane level is not the same even if the same basis act is d. and neon molecules have the same number of electrons. However, the 6-3 1G* calculation uses 23 base functions for the methane molecule but only 15 base functions for the neon molecule. The accuracy levels of the H / H interaction energy calculations with the 6-31 IG(3p) basis set and those of the methane dimercalculationswiththe6-31lG(3d.3~)basissetarenotthesame. However, theevaluationofthebasisset effectson thecalculated intermolecularinteraction energies of hydrogen dimer and methane dimer shows that the further improvement of these basis sets gives only a slight effect on the calculated intermolecular interaction energies. The calculated interaction energia at the MP3 level are close to those at the MP4(SDTQ) level. Thus it is reasonable to think that the errors due to the differences in accuracy levels are small in this case. (51) The isotropic H / H parameters reported in ref 31 were also used for the fitting of the C/C parameters with the (isotropic) model I. However, the obtained RMS error of the (isotopic) model I was still larger than that of the (anisotropic) model 11. (52) Kochanski, E. J. Chem. Phys. 1973, 58, 5823. (53) Price, S. L.; Stone, A. J. Mol. Phys. 1980, 40, 805. (54) Burton, P. G. Chem. Phys. Lett. 1983, 100, 51. (55) Wiberg, K. B.; Murcko, M. A. J. Comput. Chem. 1987, 8, 1124. (56) Starr, T. L.; Williams, D. E. J. Chem. Phys. 1977, 66, 2054. (57) Widely used molecular mechania force fields MM2 and MM3 use this model to incorporate the anisotropy of the nonbonding interaction of hydrogen. (58) Rossini, F. D.; Pitzer, K. S.;Arnett, R. L.; Braun, R. M.; Pimentel, G. C. Selected Values of Physical and Thermodynamic Properties of Hydrocarbons and Related Compounds; Carnegie Press: Pittsburgh, PA, 1953. (59) Hams, K. R.; Trappeniers, N. J. Physica 1980, 104A, 262.
