2272
J . Phys. Chem. 1991, 95. 2272-2218
and calculated properties a t the optimized structure of these crystals. As can be seen from Table XII, there is good agreement between our elastic constants and those of SLB. The elastic constants of PE have also been obtained theoretically by Odajima and Ma& (OM),29Wobser and Blasenbrey (WB),37 and Tashiro et al. (TKT),38 as listed in Table XII. The values from WB are in reasonable agreement with our values and those of SLB. However, the elastic constants obtained by OM and by TKT do not agree well with other calculated elastic constants, especially for components perpendicular to the chain directions ( C I I C22, , CI2,C6&. This is probably because they used an unreasonably small cutoff distance (4 A) for the intermolecular interactions. The vdW parameters of SLB are slightly modified from those of Williams, while the vdW parameters of WB were adjusted to (37) Wobscr, G.; Blascnbrey, S . Kolloid 2.Z . Polym. 1970, 241, 985. (38) Tashiro, K.; Kobayashi, M.; Tadokoro, H. Macromolecules 1978, 11, 914. (39) Olf, H. G.; Fanconi, B. J. Chem. Phys. 1973,59,534. (40) Vergoten, G.; Fleury, G.; Tasumi, M.; Shimanouchi, T. Chem. Phys. Lerr. 1973, is, 191. (41) Wu, C-K.; Nicol, M. J . Chem. Phys. 1973, 58, 5150. (42) Mizushima, S.;Shimanouchi, T. J . Am. Chem. Soc. 1948.71, 1320.
fit the energy and cell parameters of PE. The setting angles of these calculations are 1.2' and 1.8O larger than ours, which is in turn 0.9O larger than experiment.
IX. Summary Force field parameters adequate for molecular mechanics calculations for PE crystal are developed and tested. Valence force field parameters are obtained by using a theoretical Hessian as well as experimental data for n-butane. Nonbond parameters are determined empirically from graphite and PE crystals. The cell and atomic coordinate optimizations were carried out on PE crystal, and elastic constants and phonon bands were calculated by using analytical second derivatives. To examine the behavior of the crystal under large strain in the directions perpendicular to the chain, stress and energy are plotted as functions of deformation distances. Yield stresses and surface energies are calculated from these relations. Acknowledgment. This work was partially supported by grants from the Air Force Office of Scientific Research (No. AFOSR88-005 l), the Caltech Consortium in Chemistry and Chemical Engineering, and Imperial Chemical Industries, Wilton Materials Research Centre, Cleveland, England. Registry No. Polyethylene (homopolymer), 9002-88-4.
Basls Set Effects on the Intermolecular Interaction Energies of Methane Dimers Obtained by the Merller-Plesset Perturbation Theory Calculation Seiji Tsuzuki* and Kazutoshi Tanabe National Chemical Laboratory for Industry, Tsukuba, Ibaraki 305, Japan (Received: February 28, 1990) Intermolecular interaction energies of methane dimer were calculated by using several basis sets up to 6-3 11G(3d,4p) with electron correlation energy correction by the Maller-Plesset perturbation method and basis set superposition error (BSSE) correction by the counterpoise method to evaluate the basis set effect. The calculated interaction energies depended on the basis set considerably. Whereas the interaction energies of repulsive component calculated at H F level were not affected by the change of basis set, the dispersion energy component depended greatly on the basis set used. The dispersion energies calculated with the Maller-Plesset second- and third-order perturbation by using 6-3 1 1G(2d,2p) basis set were 0-101 and 4 4 % smaller than those obtained with the fourth-order (MP4(SDTQ)) perturbation, respectively. The BSSEs calculated by the counterpoise method were still about 30% of the calculated intermolecular interaction energies for the conformers of energy minima even at the MP4(SDTQ)/6-311G(2d,2p) level. The calculated interaction potentials of dimers at the MP4(SDTQ)/6-3 11G(2d,2p) level were considerably shallower than those obtained by MM2 force fields but were close to the potentials given by the Williams potential and by the recently reported MM3 force field.
Introduction
The nonbonded interactions of carbon and hydrogen atoms are important to understand the intramolecular and intermolecular interactions of organic molecules. These interactions frequently play a crucial role in the determination of the conformation of molecule and crystal structure.',* These interactions have attracted much attention from chemists and have been studied by various methods. The intermolecular interaction potential of methane has been estimated from the measurement of compressibility of gaseous methane.- The intermolecular interactions (1) Burkcrt, U.;Allinger, N. L.Molecular Mechanics;American Chemical Society: Washington, DC, 1982. (2) Dale, J. Srereochemisrry and Conformational Analysis; Verlag Chemie; New York, 1978. (3) Schamp, Jr.. H. W.; Mason, E. A.; Richardson, A. C. B.; Altman, A.
Phys. Fluids 1958, I , 329. (4) Dymond, J. H.; Rigby, M.; Smith, E. B. J . Chem. Phys. 1965, 42. 2801.
( 5 ) Snook, 1. K.; Spurling, T. H.J . Chem. Soc., Faraday Trans. 2 1972, 68, 1359. (6) Hanky. H. J. M.; Klein, M. J . Phys. Chem. 1972, 76. 1743. (7) Pope, G. A.; Chappelear, P. S.;Kobayashi, R. J. Chem. Phys. 1973, 59, 423. ( 8 ) Matthews, G. P.;Smith, E. B. Mol. Phys. 1976, 32, 1719.
0022-365419 1/2095-2272302.50/0
of organic molecules have also been investigated from the analysis of crystal structure and lattice energy.lWl5 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. The atom-atom interaction energy term can be described as E(total) = E(repu1sion) E(dispersion) E(Cou1ombic) (1) A lot of equations have been proposed to describe the atom-atom interaction energy term.I6I8 The parameters of the equations,
+
+
(9) Sherwood, A. E.; Prausnitz, J. M. J . Chem. Phys. 1964, 41, 429. (10) Williams, D. E.;Starr, T. L. Compur. Chem. 1977. 1, 173. (11) Bates, J. B.; Busing, W. R. J . Chem. Phys. 1974, 60, 2414. (12) Lii, J.-H.; Allinger, N. L. J . Am. Chem. Sa. 1989, 111, 8576. (13) Warshel, A.; Lifson, S.J . Chem. Phys. 1970, 53, 582. (14) Sprague, J. T.; Allinger, N. L. J. Compur. Chem. 1980, I , 257. (15) Allinger, N. L.;Miller, M. A.; Vancatledge, F.A.; Hirsch, J. A. J. Am. Chem. Soc. 1967,89, 4345. (16) Lennard-Jones, J. E.Proc. R. Soc. London, Ser. A 1924, 106,463. (17) Buckingham, A. D.; Utting, B. D. Annu. Reu. Phys. Chem. 1970,21, 287.
0 1991 American Chemical Society
Interaction Energies of Methane Dimers
The Journal of Physical Chemistry, Vol. 95, No.6, 1991 2273
which represent the atom-atom interaction, were fitted to reproduce experimental measurements.I0 Equation 2 is the exp(6-1) type nonbonded interaction potential, which is frequently used to describe this interaction energy term." (2) E/] = Bij exP(-Cijr/j) - A/jrij4 + 9iqfij-l One of the difficulties in determining the atom-atom interaction potential from experimental measurement is that experimental measurement gives only limited information of the potential. Whereas the measurement of the compressibility of methane gives the interaction energy potential of methane, the anisotropy of the methane-methane interaction is not revealed from this measurement. From the measurement of crystal structure, the shape of the intermolecular interaction potential near a minimum can be revealed, because molecules locate on the equilibrium position where the intermolecular interaction energy becomes minimum. Thus the other region of the potential is not covered from the measurement of crystal structure. If the intermolecular interaction energy can be estimated precisely by theoretical calculations, this difficulty can be eliminated. The intermolecular interaction energy of any orientation of the supermolecule can be calculated. Ab initio molecular orbital calculation has been applied to the study of intermolecular interaction. For example, the interaction of noble gas,I9v2Ohydrogen b ~ n d i n g , ~ and I - ~ the ~ interaction of aromatic c l ~ s t e r have ~ ~ - been ~ ~ studied. The dominant components of the intermolecular interaction energy of molecules, which have polar functional groups, are the repulsive and Coulombic energy components. The dispersion energy component is relatively small. Thus the large part of the interaction energy can be obtained by the Hartree-Fock level a b initio molecular orbital calculation, which cannot evaluate the dispersion energy.22 On the other hand, the Coulombic energy component of the interaction energy of a nonpolar molecule, like a noble gas, is small. The major components of the interaction energy of this type of molecules are the repulsive and dispersion energy components. Thus a careful evaluation of the dispersion energy is necessary to obtain the intermolecular interaction energy potential of a noble gas.20 Since the polarity of alkanes is also small, the major components of the intermolecular interaction energy of alkane are also the repulsive and dispersion energy components. The repulsive energy component of alkanes has been investigated from the ab initio SCF calculation of the methane dimer.28*29 Unfortunately the dispersion energy, which is the other important energy component, was not calculated with the a b initio method. Thus we turn our attention to this attractive energy component. Since the dispersion energy has its origin in electron correlation, the calculation of electron correlation energy is necessary to estimate this energy component .29 The calculated electron correlation energy often depends on the size of the basis set and the level of the c o r r e c t i ~ n . This ~~ dependence presents a difficulty in evaluating the calculated potential. In this paper we evaluate the effect of basis set and level of the correction of electron correlation on the calculated intermolecular interaction energy of methane dimer using various basis sets with several levels of electron correlation correction. (18) Hill, T . L. J . Chem. Phys. 1948, 16, 399. (19) Szalewicz, K.; Jeziorski, B. Mol. Phys. 1979, 38, 191. (20) Tatewaki, H.; Tanaka, K.; Ohno, Y.; Nakamura, T. Mol. Phys. 1984, 53, 233. (21) Matsuoka, 0.;Clementi, E.; Yoshimine, M. J . Chem. Phys. 1976.64, 1351. (22) Clementi, E.; Habitz, P. J . Phys. Chem. 1983, 87, 2815. (23) Wojcik, M. J.; Hirakawa, A. Y.; Tsuboi, M.; Kato, S.;Morokuma, K. Chem. Phys. Lett. 1983, 100, 523. (24) Kumpf, R. A.; Damewood, Jr., J. R. J . Phys. Chem. 1989,93,4478. (25) Bredas, J. L.; Street, G. B. J . Chem. Phys. 1989, 90,7291. (26) Carsky, P.; Selzlc, H. L.; Schlag, E. W. Chem. Phys. 1988,125, 165. (27) Karlstrinn, G.; Linse, P.; Wallqvist, A.; J o n d n , B. J. Am. Chem. Soc. 1983, 105. 3777. (28) Kola, W.; Ranghino, G.; Clementi, E.; Novaro, 0.Int. J . Quantum Chem. 1980, 17,429. (29) Williams, D. E.; Craycroft, D. J. J . Phys. Chem. 1987, 91, 6365. (30) Clark, T. A Handbook of Computational Chemistry; John Wiley: New York, 1985.
H
Figure 1. Two orientations of the methane molecules considered in the present work.
The intermolecular interaction energy is obtained from the subtraction of the monomer energy from the supermolecule energy as shown in eq 3. If the energies of supermolecule and monomers E(inter) = E(supermolecu1e)
- E(monomers)
(3)
are not calculated in a basis set consistent manner, basis set superposition error (BSSE) may appear. To obtain correct intermolecular interaction energy, BSSE correction is ne~essary.~' The BSSE was corrected by counterpoise methods.32 The effect of the BSSE correction is also discussed. Several empirical intermolecular interaction potentials for hydrocarbons have been proposed and are used for crystal packing analysis, molecular mechanics, and other s i m u l a t i ~ n s . ' ~ J ~ ~ ~ ~ * ~ ~ These empirical potentials are not optimized to methane, but to various hydrocarbons. However, the comparison of the calculated potentials with these empirical potentials would be interesting. Williams et al. have reported the atomatom interaction potentials of carbon and hydrogen from the analysis of crystal ~ t r u c t u r e . ~ Allinger et al. have determined the potentials based on experimental data for the conformational a n a l y ~ i s . ' , ' ~ , ' ~ The . ' ~ , ~dif~ ference between the calculated interaction energy potentials of methane dimers by the ab initio method and those from empirical potentials will be discussed.
Method The GAUSSIAN 86 program35was used for the molecular orbital calculations. The 6-31G*36and the 6-31 lG*37 basis sets have a set of d functions on carbon atoms (ad(C) = 0.8 and 0.626, respectively). The 6-31 1G** has a set of d functions on carbon atoms and a set of p functions on hydrogen atoms ((Yd(c) = 0.626 and a,(H) = 0.75). The 6-3 11G(2d,p) has two sets of d functions on carbon atoms and a set of p functions on hydrogen atoms ((Yd(c) = 1.252 and 0.313).38 The 6-311G(2d,2p) has two sets of d functions on carbon atoms and two sets of p functions on hydrogen atoms ( a (H) = 1.5 and 0.375). The 6-31 1G(3d,2p) has three sets of d functions on carbon atoms and two sets of p functions on hydrogen atoms (ad(c) = 2.504,0.626, and 0.1565). The 6-31 1G(4d,2p) has four sets of d functions on carbon atoms and two sets of p functions on hydrogen atoms ( q ( C ) = 5.008, 1.252,0.313, and 0.07825). The 6-311G(2d,3p) has two sets of d functions on carbon atoms and three sets of p functions on (31) Ransil, B. J.; J . Chem. Phys. 1961,34, 2109. (32) Boys, S. F.; Bernardi, F. Mol. Phys. 1970, 19, 553. (33) Allinger, N. L. J . Am. Chem. SOC.1977, 99, 8127. (34) Jorgensen, W. L.; Madura, J. D.; Swenson, C. J. J . Am. Chem. Soc. 1984, 106, 6638. (35) Frisch, M. J.; Binkley, J. S.;Schlegel, H. B.; Raghavachari, K.;
Melius, 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. GAUSSIANB~; Carnegie-Mellon Quantum Chemistry Publishing Unit: Pittsburgh, PA, 1986. (36) Hariharan, P. C.; Pople, J. A. Chem. Phys. Letr. 1972, 16, 217. (37) Krishnan, R.; Binkley, J. S.;Seeger, R.; Pople, J. A. J. Chem. Phys. 1980, 72, 650. (38) Frisch, M. J.; Pople, J. A.; Binkley, J. S.J . Chem. Phys. 1984, 80, 3265.
2214 The Journal of Physical Chemistry, Vol. 95, No. 6, I991 TABLE I: Cilculnted Dispersion Energiee by the Maller-Plesset Perturbation Metbod" level of calculation dist, A MP2 MP3 MP4(SDQ) MP4(SDTQ) Dimer A -1.821 3.8 -1.735 (0.953) -1.740 (0.956) -1.599 (0.878) -1.269 4.0 -1.188 (0.936) -1.213 (0.955) -1.117 (0.880) 4.2 -0.818 (0.923) -0.846 (0.954) -0.781 (0.882) -0.886 4.4 -0.567 (0.915) -0.591 (0.953) -0.548 (0.883) -0.620 -0.437 4.6 -0.397 (0.910) -0.416 (0.952) -0.386 (0.885) 4.8 -0.282 (0.907) -0,296 (0.950) -0.276 (0.886) -0.311 -0.226 5.0 -0.205 (0.906) -0.215 (0.949) -0.200 (0.887) 5.2 -0.151 (0.907) -0.158 (0.949) -0.148 (0.887) -0.167 5.6 -0.087 (0.912) -0.090 (0.948) -0.085 (0.888) -0.095 -0.036 6.4 -0.033 (0.927) -0.034 (0.950) -0.032 (0.888) -0.008 8.0 -0.007 (0.949) -0,007 (0.953) -0.007 (0.886) 3.0 3.2 3.4 3.6 3.8 4.0 4.2 4.4 4.8 5.6 7.2
-2.545 (0.983) -1.785 (0.979) -1.256 (0.976) -0.889 (0.976) -0.636 (0.977) -0.459 (0.979) -0.336 (0.982) -0.248 (0.986) -0.141 (0.994) -0.052 (1.006) -0.011 (1.016)
Dimer B -2.479 (0.958) -2.320 -1.748 (0.959j -1.632 -1.233 (0.959) -1.149 -0.873 (0.959) -0.813 -0,623 (0.958) -0.580 -0,449 (0.958) -0.418 -0.327 (0.959) -0.304 -0.242 (0.959) -0.224 -0.136 (0.960) -0.126 -0.050 (0.962) -0.046 -0.010 (0.964) -0.010
10.896) (0.895j (0.894) (0.892) (0.892) (0.891) (0.890) (0.890) (0.890) (0.889) (0.886)
-2.588 -1.824 -1.286 -0.91 1 -0.650 -0.469 -0,342 -0,252 -0.142 -0.052 -0.01 1
'Energies are in kcal/mol. The 6-31G(2d,2p) basis set was used for the calculation. BSSE's were corrected by the counterpoise method. Ratios with the MP4(SDTQ) level dispersion energy are in parentheses.
hydrogen atoms (ap(H) = 3.0,0.75, and 0.1875). The 6-31 1G(2d,4p) has two sets of d functions on carbon atoms and four sets of p functions on hydrogen atoms (ap(H) = 6.0, 1.5, 0.375, and 0.09375). The 6-31 1G(3d,3p) has three sets of d functions on carbon atoms and three sets of p functions on hydrogen atoms. The 6-31 IG(4d,3p) has four sets of d functions on carbon atoms and three sets of p functions on hydrogen atoms. The 6-31 1G(3d,4p) has three sets of d functions on carbon atoms and four sets of p functions on hydrogen atoms. The 6-31 lG(2df,2p) has two sets of d and a set o f f functions on carbon atoms and two sets of p functions on hydrogen atoms ( a f ( C )= 0.8). The 631 lG(2d,2pd) has two sets of d functions on carbon atoms and two sets of p and a set of d functions on hydrogen atoms (ad(H) = 1.0). The 6-31 l+G has a set of diffuse sp functions on carbon atoms (a,,(C) = 0.0438). The 6-31 1++G has a set of diffuse sp functions on carbon atoms and a set of diffuse p functions on hydrogen atoms (a,(H) = 0.036).39 The geometry of a single methane molecule was optimized at the HF/6-311G** level. This optimized geometry, in which C-H bond distance is 1.084 12 A, was used for the calculations of the interaction energy of the dimers.
Results and Discussion Electron Correlation Energy. The electron correlation energies of the two orientations of methane dimers, which are shown in Figure 1, were calculated by the second-, third-, and fourth-order M~rller-Plesset methodqw3 using the 6-31 1G(2d,2p) basis set to evaluate the effect of the order of the perturbation. The calculated electron correlation energies are summarized in Table I. The calculated electron correlation energies are largest at the MP4(SDTQ) level. The dispersion energies are underestimated at other levels. The dispersion energies calculated at MP2, MP3, and MP4(SDQ) are 0-9%, 4-5%, and 10-12% smaller than those calculated at MP4(SDTQ) level. The MP3 dispersion energies are only a few percent smaller than that obtained at MP4(SDTQ). (39) Clark, T.;Chandrasekhar, J.; Spitznagel, G. W.; Schleyer, P. v. R. J. Comput. Chem. 1983.4, 294. (40) Maller, C.; Plesset, M. S.Phys. Rev. 1934, 46, 618. (41) Pople, J. A.; Seeger, R.; Krishnan, R. Int. J. Quant. Chem. Symp. 1977, I I , 149. (42) Krishnan, R.; Pople, J. A. Int. J . Quuntum Chem. 1978, 14, 91. (43) Krishnan, R.; Frisch, M. J.; Pople, J. A. J. Chem. Phys. 1980, 72, 4244.
Tsuzuki and Tanabe TABLE II: Calculated Intermolecular Interaction Energies Corrected Perturbation Method" with Several Levels of the MaUer-PI-t dist, level of calculation A HF MP2 MP3 MP46DO) MP4(SDTO) Dimer A 4.388 2.653 2.648 3.8 2.789 2.567 1.121 1.096 1.192 2.309 4.0 1.040 0.433 0.396 0.369 0.328 1.214 4.2 0.092 0.073 0.049 0.020 0.640 4.4 -0.046 -0.057 0.340 -0.076 4.6 -0.097 -0.093 -0.099 -0.128 0.183 -0.113 4.8 -0.I25 -0.099 -0.104 0.101 -0.113 5.0 0.057 -0.101 -0.110 -0.094 -0.091 5.2 0.020 -0.070 -0.075 -0.066 -0.064 5.6 -0.028 0.005 -0.029 -0.03 1 -0.027 6.4 0.001 -0.007 -0.007 8.0 -0.007 -0.006 3.0 3.2 3.4 3.6 3.8 4.0 4.2 4.4 4.8 5.6 7.2
5.699 2.930 1.494 0.758 0.384 0.195 0.100 0.053 0.016 0.003 0.000
3.153 1.145 0.239 -0.131 -0.252 -0.264 -0.235 -0.195 -0.125 -0.050 -0.01 1
Dimer B 3.219 1.181 0.261 -0.1 16 -0.240 -0.254 -0.227 -0.189 -0.120 -0.047 -0.010
3.379 1.298 0.345 -0.055 -0.196 -0.223 -0.204 -0.171 -0.1 IO -0.044 -0.009
3.1 IO 1.106 0.208 -0.153 -0.267 -0.274 -0.241 -0.199 -0.126 -0.049 -0.010
OEnergies are in kcal/mol. The 6-311G(2d,2p) basis set was used for the calculation. BSSE's were corrected by the counterpoise method.
The same tendency is observed for the dispersion energy calculations using other basis sets. Hence the MP3 method was frequently used to reduce computational time for further calculations using large basis sets. The calculated interaction energies using the 6-3 11G(2d,2p) basis set are summarized in Table 11. Due to the underestimation of the dispersion energies at the MP2, MP3, and MP4(SDQ) levels, the calculated potential depths at these levels are shallower than those obtained at the MP4(SDTQ) level. The interaction energies of the dimer A, at the carbon-carbon distance of 4.8 A, calculated at MP2, MP3, and MP4(SDQ) levels are 23, 12, and 27% smaller than that at MP4(SDTQ) level, respectively. Those of the dimer B at 4.0 A are 4, 7, and 19% smaller, respectively. The configuration interaction (CI) energy calculation is another method to estimate electron correlation energy.41 To compare the performance of this method with the M~rller-Plesset perturbation calculation, the CI energies of the dimers were calculated. The intermolecular interaction potentials corrected with the CI energies of single and double excitation by using the 6-31 1G** basis set are compared with those obtained by the MP4(SDTQ)/6-311G** level calculation in Table 111. The depths of the potentials obtained by the M~rller-Plessetmethod are deeper than that obtained by the CI method. Basis Set Superposition Error. The intermolecular interaction energies and the BSSEs of the dimers A and B calculated by the counterpoise method32using several basis sets are shown in Table IV. The BSSEs are considerably large if a crude 6-31G* basis set is used. The use of the improved basis sets decreases the BSSE. However, the BSSE's given by our calculations are still large and cannot be ignored. The BSSEs of the dimers A and B were calculated by changing the carbon-carbon distance to evaluate the dependence of the BSSE on the distance between the methane molecules. The calculated BSSE's shown in Table V indicate that the BSSE rapidly increases if the distance becomes short. The intermolecular interaction energy of the dimer A and that of the dimer B given by the MP4(SDTQ)/6-311G(2d,2p) level calculation reach minima if the carbon-carbon distances are 4.8 and 4.0 A, respectively. The calculated interaction energies of these dimers are -0.178 and -0.364 kcal/mol, respectively. The BSSEs of these dimers given by the counterpoise method are 0.050
The Journal of Physical Chemistry, Vol. 95, No. 6, 1991 2215
Interaction Energies of Methane Dimers TABLE III: Calculated Intermolecular Interaction Energies Corrected with the Electron Correlation Energy Obtained by the M0lle~Pleswtand by the Configuration Interaction Energy Calculations with All Single and Double Substitutions' dist, A MP2 MP3 MP46DTOI CISD Dimer A 4.4 0.123 0.022 -0.002 -0.03 1 4.6 -0.046 -0.120 -0.141 -0.164 4.8 -0.108 -0.164 -0,180 -0,198 5.0 -0.191 -0. I65 -0.121 -0.176 -0.166 -0.113 -0.147 5.2 -0.155 -0.110 -0.078 -0.099 5.6 -0.103 -0.037 -0.034 6.4 -0.035 -0.027 -0.006 -0.006 -0.004 -0.005 8.0 .I
3.6 3.8 4.0 4.2 4.4 4.8 5.6 7.2
-0. I29 -0.239 -0.248 -0.221 -0.185 -0.120 -0.046 -0.008
Dimer B -0.135 -0.241 -0.248 -0.219 -0.182 -0.117 -0.044 -0.008
-0.173 -0.268 -0.267 -0.233 -0.192 -0.123 -0.046 -0.008
-0.05 1 -0.1 IO -0.153 -0.150
-0.131 -0.088 -0.034 -0.006
OEnergies are in kcal/mol. The 6-31 1G** basis set was used for the calculation. To avoid the size consistency problem of the CI calculation, the interaction energies were obtained by the subtraction of the energy of the reference configuration, in which the two methane molecules are separated 20 A, from the calculated energies of the dimers. and 0.090 kcal/mol, respectively. These BSSEs are 28 and 25% of the calculated interaction energies, respectively. Effects of Basis Sets. The intermolecular interaction energies of the dimers A and B were calculated by using several basis sets with electron correlation energy correction by the Mdler-Plesset perturbation methodm3 and BSSE correction by the counterpoise method3* to evaluate the effect of the basis set. The calculated interaction energies are summarized in Tables VI and VII. The choice of the basis set slightly changes the carbon-carbon distance of the dimers of energy mimima. On the other hand, the depths of the potentials depend considerably on the basis set used. The calculated depth of the potential of the dimer A is -0.128 kcal/mol a t the MP4(SDTQ)/6-31 IG(2d,2p) level. The depth of the potential is underestimated if a cruder basis set is used. The calculated interaction energies of the dimer A, at the carbon-carbon distance of 4.8 A, using 6-31G*, 6-31 1G*, 6-31 IG**, and 6-31 1G(2d,p) basis sets with the same level of the electron correlation energy correction, are 50, 70, 43 and 30% smaller than that a t the MP4(SDTQ)/6-3 1 1G(2d,2p) level, respectively. The same tendency is observed in the calculation of the dimer B. The calculated depth of the potential of the dimer B is -0.274 kcal/mol a t the MP4(SDTQ)/6-311G(2d,2p) level. The calculated interaction energies of the dimer B,a t the carbon-carbon distance of 4.0 A, using 6-31G*, 6-31 1G*, 6-31 lG**, and 6-31 1G(2d,p) basis sets with the same level of electron correlation energy correction, are 89, 70, 45 and 25% smaller than that a t the MP4(SDTQ)/6-31 IG(2d,2p) level, respectively. The calculated interaction energies with the 6-31 1G(d,2p) basis set are close to those obtained by using the 6-3 11G(2d,2p) basis set. The interaction energies are also calculated by using variously augmented 6-3 1 1G(2d,2p) basis set. The calculated interaction energies are shown in Table VII. The use of three sets of d orbitals on carbon atoms or the use of three sets of p orbitals on hydrogen atoms increases the interaction energy as much as 11-3596, while the use of four sets of d orbitals on carbon atoms or four sets of p orbitals on hydrogen atoms does not further increase the interaction energy. The augmentation of a set o f f orbitals on carbon atoms or a set of d orbitals on hydrogen atoms changes the calculated interaction energies slightly. The addition of diffuse functions increases the interaction energy as much as 3-7%. In order to evaluate the details of the basis set effect, the interaction energy was separated to the dispersion energy component and the repulsive energy component. Table VI11 shows
TABLE I V Calculated Intermolecular Interaction Energies and BSSE's" interacn energy basis set interacn energy BSSE + BSSE corrn Dimer A 6-31G* -0.280 0.216 -0.064 6-31 1G* -0.226 0.188 -0.038 6-31 lG** -0.198 0.125 -0.073 -0.215 0.125 6-31 IG(2d,p) -0.090 -0.231 0.102 6-311G(d,2p) -0.129 6-31 IG(2d,2p) -0.178 0.050 -0.128 6-31 IG(3d,2p)* -0.189 0.064 -0.125 6-31 lG(4d,2p)* -0,190 0.062 -0.128 -0.310 0.163 6-31 lG(2d,3p)* -0.147 -0.329 0.185 6-31 IG(2d,4p)* -0,144 -0.325 0.173 -0.152 6-31 IG(3d,3p)* 6-31 lG(4d,3p)* -0.320 0.170 -0.150 6-31 IG(3d,4p)' -0.314 0.184 -0.130 6-3 1 lG(2df,2p)* -0.113 -0.158 0.045 -0.160 0.047 6-31 lG(2d,2pd)* -0.113 6-31 l+G(2d,2p)* -0.177 0.060 -0.1 I7 6-31 l++G(2d,2p)* -0.199 0.08 1 -0.118 6-31G* 6-311G* 6-31 1G** 6-31 IG(2d,p) 6-31 1G(d,2p) 6-3 1 lG(2d,2p) 6-31IG(3d,2p)* 6-31 1 G(4d,2p)* 6-31IG(2d,3p)* 6-31 IG(2d,4p)* 6-31IG(3d,3p)* 6-31 lG(4d,3p)* 6-31 IG(3d,4p)' 6-31 IG(2df,2p)* 6-31 lG(2d,2pd)* 6-31l+G(2d,2p)* 6-31 I++G(2d,2p)*
Dimer B -0.215 -0.209 -0.267 -0.343 -0.331 -0.364 -0.409 -0.394 -0.445 -0.449 -0.487 -0.483 -0.471 -0.326 -0.334 -0.346 -0.372
0.185 0.127 0.1 I6 0.137 0.073 0.090 0.1 12 0.107 0.133 0.141 0.151 0.163 0.143 0.069 0.072 0.077 0.101
-0.030 -0.082 -0.151 -0.206 -0.258 -0.274 -0.297 -0.287 -0.3 12 -0.308 -0,336 -0.320 -0.328 -0.257 -0.262 -0.269 -0.271
Energies are in kcal/mol. Electron correlation energies were corrected by the fourth-order Maller-Plesset perturbation method (MP4(SDTQ)). BSSE's were corrected by the counter ise method. The distance between carbon atoms of dimer A is 4.8 r a n d that of dimer B is 4.0 A. *Electron correlation energies were corrected by the third-order Maller-Plesset perturbation method (MP3). Electron correlation energies were corrected by the second-order Maller-Plesset perturbation method (MP2). the basis set effect on the calculated interaction energy at HF level, which corresponds to the repulsive energy component. The choice of basis set little affects this energy component except for the crudest 6-31G* basis set. The calculated electron correlation energies by the M d l e r Plesset perturbation method, which correspond to the dispersion energy component, are summarized in Tables IX and X. In contrast to the repulsive energy component, the calculated dispersion energy changes greatly by the choice of the basis set. The dispersion energies calculated by using 6-3 1 1G(2d,2p) basis set are considerably larger than those calculated by using cruder basis sets. The calculated dispersion energies of the dimer A, a t the carbon-carbon distance of 4.8 A, using 6-31G*, 6-31 1G*, 631 lG**, and 6-31 1G(2d,p) basis sets a t the MP4(SDTQ) level of electron correlation energy correction, are 30, 30, 20, and 14% smaller than that a t the MP4(SDTQ)/6-311G(2d,2p) level. The calculated dispersion energies of the dimer B, a t the carbon-carbon distance of 4.0 A, using these basis sets with the same level of electron correlation energy correction are 51, 41, 26, and 14% smaller than that a t the MP4(SDTQ)/6-311G(2d,2p) level. The calculated dispersion energies with the 6-3 11G(d,2p) basis set are close to those obtained by using 6-31 1G(2d,2p) basis set. The further improvement of the basis set increases the calculated dispersion energy slightly. The calculated dispersion energies are increased by the use of three or four sets of d orbitals on carbon atoms or by the use of three or four sets of p orbitals on hydrogen
2276 The Journal of Physical Chemistry, Vol. 95, No. 6, 1991 TABLE V: Calculated Intermolecular Interaction Energies and BSSEW interacn energy + dist, A interacn energy BSSE BSSE Dimer A 3.8 2.220 0.347 2.567 4.0 0.8 17 0.223 1.040 4.2 0.186 0.142 0.328 4.4 -0.073 0.093 0.020 4.6 -0.163 0.066 -0.097 -0.I78 0.050 -0.128 4.8 5.0 -0.I64 0.039 -0.125 5.2 -0.140 0.030 - 0 . 1 IO 5.6 -0.093 0.018 -0.075 6.4 -0.036 0.005 -0.03 1 8.0 -0.007 0.000 -0.007 Dimer B 3.O 3.2 3.4 3.6 3.8 4.0 4.2 4.4 4.8 5.6 7.2
2.709 0.81 1 -0.01 3 -0.318 -0.389 -0.364 -0.308 -0.248 -0.152 -0.056 -0.010
0.401 0.295 0.221 0.165 0.122 0.090 0.067 0.049 0.026 0.007 0.000
3.1 IO 1.106 0.208 -0.153 -0.267 -0,274 -0.241 -0.199 -0.126 -0.049 -0.0 IO
" Energies are in kcal/mol. BSSE's were estimated by the counterpoise method. Energies were calculated at the MP4(SDTQ)/6-31 IC(2d,2p) level. TABLE VI: Calculated Intermolecular Interaction Energies Using Various Basis Sets" dist, basis setb A A B C D E F Dimer A 3.8 2.567 4.0 1.610 1.563 1.329 1.040 4.2 0.663 0.673 0.527 0.328 4.4 0.212 0.243 0.152 0.119 0.024 0.020 4.6 0.012 0.045 -0.011 -0.035 -0.097 -0.097 4.8 -0.064 -0.038 -0.073 -0.090 -0.129 -0.128 5.0 -0.084 -0.067 -0.089 -0.102 -0.126 -0.125 5.2 -0.081 -0.070 -0.084 -0.095 -0.111 -0.110 5.6 -0.057 -0,054 -0.063 -0.075 6.4 -0.023 -0.024 -0.027 -0.03 1 8.0 -0.005 -0.005 -0.006 -0.007 Dimer B 3.0 3.2 3.4 3.6 3.8 4.0 4.2 4.4 4.8 5.6 7.2
2.142 0.917 0.340 0.079 -0.030 -0.069 -0.077 -0.062 -0.028 -0.006
1.983 0.795 0.245 0.008 -0.082 -0.105 -0.102 -0.074 -0.032 -0.007
1.609 0.559 0.093 -0.093 -0.151 -0.153 -0.136 -0.092 -0.038 -0.008
-0.016 -0.171 -0.206 -0.194 -0.165
-0.112 -0.241 -0.258 -0.231 -0.192
3.110 1.106 0.208 -0.153 -0.267 -0.274 -0.241 -0.199 -0.126 -0.049 -0.010
"Energies are in kcal/mol. Energies were calculated by the Maller-Plesset fourth-order perturbation method (MP4(SDTQ)). BSSE's were corrected by the counterpoise method. bBasis set A is 6-31G*, B is 6-311G*, C is 6-311G**, D is 6-311G(2d,p), E is 6311G(d,2p), and F is 6-311G(2d,2p).
atoms as much as 3-19%. The augmentation of a set off orbitals on carbon atoms or a set of d functions on hydrogen atoms or the augmentation of diffuse functions increases the dispersion energy less than 4%. Polarizability. The origin of dispersion energy is the interaction between induced dipoles of molecules. Thus the evaluation of the calculated polarizability by ab initio method would be interesting.
Tsuzuki and Tanabe TABLE VII: Calculated Intermolecular Interaction Energies Using Variow Basis Setsa dimer A dimer B basis set MP2 MP3 MP2 MP3 6-31 I G ( 2 d , 2 ~ ) ~ -0.099 -0.113 -0.264 -0.254 6-31 IG(3d,2p) -0.112 -0.125 -0.303 -0,297 6-31 1G(4d,2p) -0.114 -0.128 -0.295 -0.287 6-31 IG(2d,3p) -0.129 -0.147 -0.316 -0.312 6-31 IG(2d,4p) -0.127 -0.144 -0.313 -0.308 6-31 1G(3d,3p) -0.134 -0.152 -0.337 -0.336 6-311G(4d,3p) -0.131 -0.150 -0.323 -0.320 6-31 1G(3d,4p) -0.130 -0.328 6-311G(2df,2p) -0.100 -0.113 -0.267 -0.257 -0.1 13 -0.273 -0.262 6-31 IG(2d,2pd) -0.099 6-311+G(2d,2p) -0.102 -0,117 -0.278 -0.269 -0.118 -0.280 -0.271 6-311++G(2d,2p) -0.103 'Distance between carbon atoms of dimer A is 4.8 A and that of dimer B is 4.0 A. Energies are in kcal/mol. BSSE's were corrected by the counterpoise method. bCalculated interaction energies of dimers A and B corrected with MP4(SDTQ) are -0.128 and -0.274 kcal/mol, respectively.
TABLE VIII: Calculated Intermolecular Interaction Energies of Rewlsive Enerev Comwnent Usine Various Basis Sets' basis setb dist, A A B C D E F Dimer A 3.8 4.388 4.0 2.239 2.288 2.299 2.309 4.2 1.152 1.199 1.192 1.214 4.4 0.589 0.631 0.621 0.627 0.634 0.640 4.6 0.301 0.334 0.328 0.331 0.335 0.340 4.8' 0.155 0.179 0.177 0.179 0.180 0.183 0.081 0.098 0.097 0.098 0.099 0.101 5.0 5.2 0.044 0.055 0.055 0.056 0.056 0.057 5.6 0.015 0.020 0.020 0.021 6.4 0.004 0.005 0.005 0.005 8.0 0.001 0.001 0.001 0.001 Dimer B 3.0 3.2 3.4 3.6 3.8 4.od 4.2 4.4 4.8 5.6 7.2
2.866 1.458 0.745 0.383 0.198 0.103 0.054 0.015 0.002 0.000
2.893 1.464 0.740 0.375 0.193 0.101 0.054 0.017 0.003 0.000
2.918 1.482 0.751 0.382 0.195 0.102 0.054 0.017 0.003 0.000
0.751 0.381 0.195 0.101 0.054
0.758 0.384 0.196 0.101 0.053
5.699 2.930 1.494 0.758 0.384 0.195 0.100 0.053 0.016 0.003 0.000
'Energies are in kcal/mol. Energies were calculated by the Hartree-Fock method. BSSE's were corrected by the counterpoise method. "asis set A is 6-31Gb, B is 6-311G*, C is 6-311G**, D is 631 IG(2d,p), E is 6-311G(d,2p), and F is 6-311G(2d,2p). CEnergies calculated at HF/6-31 IG(3d,2p), HF/6-311G(4d,2p), HF/6-31 IC(2d,3p), HF/6-31 1G(2d94p),HF/6-31 IG(3d,3p), HF/6-31 IG(4d,3p), HF/6*311G(3d,4p), HF/6-31 IG(2df,2p), HF/6-31 lG(2d,2pd), HF/ 6-31 1+G(2d,2p), and HF/6-31 l++G(2d,2p) levels are 0.180, 0.183, 0.187, 0.183, 0.186, 0.186, 0.181, 0.184, 0.185, 0.185, and 0.186 kcal/mol, respectively. Energies calculated at the levels shown in footnote c a r e 0.197, 0.196, 0.195, 0.197, 0.197, 0.196, 0.198, 0.195. 0.196, 0.195, and 0.196 kcal/mol, respectively.
The polarizabilities of methane were calculated at the HF and MP2 levels by using the basis sets employed for the interaction energy calculations. They are summarized in Table XI. The calculated polarizability increases as the basis set used becomes large. The calculated polarizabilities at the MP2 level are larger than the corresponding HF level polarizabilities except for the crudest 6-31G* basis set. The MP2/6-311G(3d,4P) level calculation gave the largest polarizability of 16.35 au. This value is still smaller than the experimental value of 17.28 au.44 Werner (44) Werner, H.-J.; Meyer, W. Mol. Phys. 1976, 31, 855.
The Journal of Physical Chemistry, Vol. 95, No. 6, 1991 2277
Interaction Energies of Methane Dimers TABLE I X Calculated Intermolecular Interaction Energies of Dispersion Component Using Various Basis Sets' basis setb dist, A A B C D E F Dimer A 3.8 -1.821 4.0 -0.629 -0.726 -0.970 -1.269 4.2 -0,489 -0.526 -0.665 -0.886 4.4 -0.377 -0.387 -0.469 -0.508 -0.610 -0.620 4.6 -0.289 -0.289 -0.340 -0.366 -0.432 -0.437 4.8 -0.219 -0.217 -0.250 -0.269 -0.309 -0.311 5.0 -0.166 -0.164 -0.187 -0.200 -0.225 -0.226 5.2 -0.125 -0.125 -0.138 -0.150 -0.167 -0.167 5.6 -0.071 -0.074 -0.082 -0.095 6.4 -0.026 -0.028 -0.031 -0.036 8.0 -0.006 -0.006 -0.007 -0,008 Dimer B 3.0 3.2 3.4 3.6 3.8 4.0 4.2 4.4 4.8 5.6 7.2
-0,725 -0.541 -0.405 -0.304 -0.229 -0.173 -0.131 -0.077 -0.030 -0.006
-0.910 -0.669 -0.495 -0.368 -0.275 -0.206 -0.156 -0.091 -0.035 -0.007
-1.309 -0.923 -0.658 -0.475 -0.346 -0.255 -0.190 -0.109 -0.041 -0.009
-0.767 -0.552 -0.401 -0.295 -0.219
-0.871 -0.625 -0.453 -0.332 -0.246
-2.588 -1.824 -1.286 -0.911 -0.650 -0.469 -0.342 -0.252 -0.142 -0.052 -0.01 1
Energies are in kcal/mol. Dispersion energies were calculated by the fourth-order Maller-Plesset perturbation method (MP4(SDTQ)). BSSEs were corrected by the counterpoise method. bBasis set A is 6-31G*, B is 6-311G*, C is 6-311G**, D is 6-311G(2d,p), E is 631 IG(d,2p), and F is 6-31 IG(2d,2p).
TABLE X Calculated Intermolecular Interaction Energies of the Dispcnion Component Usinn Various Basis Sets' dimer A dimer B basis set MP2 MP3 MP2 MP3 6-31 lG(2d,2p)' -0.282 -0.296 -0.459 -0.449 6-31 lG(3d,2p) -0.292 -0.305 -0.500 -0.493 6-31 IG(4d,2p) -0.297 -0.311 -0.491 -0.483 6-31 1G(2d,3p) -0.316 -0.334 -0.511 -0,507 6-31 1G(2d,4p) -0.310 -0.326 -0.510 -0.504 6-31 IG(3d,3p) -0.320 -0.338 -0.535 -0.533 6-31 lG(4d.3~) -0.318 -0.336 -0.519 -0.516 -0.526 6-31 1G(3d,4p) -0.31 1 6-31 IG(2df,2p) -0.283 -0.297 -0.462 -0.452 6-31 IG(2d,2pd) -0.285 -0.298 -0.468 -0.458 6-31 I+G(2d,2p) -0.287 -0.303 -0.473 -0.464 6-31 I++G(2d,2p) -0.289 -0.304 -0.476 -0.467 'Distance between carbon atoms of dimer A is 4.8 A and that of dimer B is 4.0 A. Energies are in kcal/mol. BSSE is corrected by the counterpoise method. bCalculated dispersion energies of dimers A and B corrected with MP4(SDTQ) are -0.311 and -0.469 kcal/mol, respectively.
and Mayer have reported that the vibrational correction increases the calculated polarizability of methane as much as 4%.44 Comparison with Experimental Potential. Spherically averaged potential for methane has been reported from several experimental - ~ carbon-carbon distance measurements in the gas p h a ~ e . ~The of the potential minimum scatters in the range of 3.84-4.27 A. The depth of the potential scatters in the range of 0.33-0.46 kcal/mol. Our potentials cannot be compared directly with these spherically averaged potentials. However, the calculated potentials with large basis sets are not widely different from these experimental potentials. Comparison with Empirical Force Fields. We selected three empirical potentials to compare with the calculated potentials. Williams et al. reported three types of atom-atom interaction potentials for hydrocarbons in 1977 and have concluded that the set I1 potential gives significantly better fit to the crystal structures of 18 hydrocarbons and lattice vibrational frequencies than the
TABLE XI: A Comparison between the Calculated and Experimental Polarizabilities' basis set RHF MP2 6-31G* 12.16 12.02 12.75 6-311G* 12.85 13.19 6-31 IG** 13.23 13.81 13.90 6-31 1G(2d,p) 14.44 14.31 6-31 1G(d,2p) 14.29 6-31 1G(2d,2p) 14.46 15.21 6-31 lG(3d,2p) 15.53 15.60 6-31 1G(4d,2p) 16.05 15.67 15.35 6-31 lG(2d,3p) 15.80 6-31 IG(2d,4p) 16.29 15.82 15.43 6-31 1G(3d,3p) 15.78 6-31 1G(4d,3p) 16.26 15.84 6-31 1G(3d,4p) 16.35 14.30 6-31 lG(2df,2p) 14.45 14.32 14.48 6-31 IG(2d,2pd) 14.61 6-31 I+G(2d,2p) 14.88 14.75 6-31 l++G(2d,2p) 15.08 17.28 exptb
I, In au. Reference 44. TABLE XII: A Comparison between the Calculated Intermolecular Interaction Energies by ab Initio and by Empirical Force Field Methods" dist, Williamsb MM2' MM3d ab initioeJ ab initids 8, (1977) (1977) (1989) A B Dimer A 3.8 2.905 4.506 2.889 2.567 4.0 1.089 1.439 1.115 1.040 4.2 0.292 0.237 0.320 0.328 4.4 -0.032 -0.184 -0.010 0.020 4.6 -0.145 -0.293 -0.130 -0.097 -0.128 -0.152 4.8 -0,168 -0.287 -0.157 -0.125 -0.246 -0.149 5.0 -0.156 -0.110 5.2 -0.134 -0.200 -0.128 5.6 -0.090 -0,126 -0.086 -0.075 6.4 -0.039 -0.052 -0.037 -0.031 8.0 0.009 -0.012 -0.009 -0.007 3.0 3.2 3.4 3.6 3.8 4.0 4.2 4.4 4.8 5.6 7.2
4.087 1.381 0.202 -0.258 -0.396 -0.398 -0.351 -0.292 -0.190 -0.078 -0.017
2.295 0.141 -0.627 -0.806 -0.759 -0.642 -0.519 -0.410 -0.251 -0.099 -0.021
Dimer B 2.603 0.815 0.021 -0.289 -0.374 -0.364 -0.318 -0.265 -0.172 -0.071 -0,015
3.110 1.106 0.208 -0.153 -0.267 -0.274 -0.241 -0.199 -0.126 -0.049 -0.010
-0.336
Energies are in kcal/mol. Reference IO. Reference 33. dReferences 12 and 45. CInteractionenergies calculated at the MP4(SDTQ)/6-311G(2d,2p) level. fBSSEs were corrected by the counterpoise method. shteraction energies calculated at the MP3/631 lG(3d,3p) level.
other two sets.I0 This potential was used for comparison. In addition to this potential, the nonbonded interaction potentials used in two molecular mechanics force fields were taken. The nonbonded interaction potential used in a commonly used force field MM233and that used in a recently reported revised force field MM312945346 were compared. The intermolecular interaction energy potentials of the dimers were calculated by using these three empirical potentials. The comparison is shown in Table XII. The carbon-carbon distances of the dimers A and B a t the energy minima obtained at the MP4(SDTQ)/6-311G(2d,2p) level with BSSE correction are 4.8 and 4.0 A, respectively. Whereas the minima of the interaction (45) Allinger, N. L.; Yuh, Y. H.; Lii, J.-H. J. Am. Chem. Soc. 1989,111, 8551. (46) Lii, J.-H.; Allinger, N. L. J. Am. Chem. Soc. 1989, 111, 8566.
2278
J. Phys. Chem. 1991, 95, 2278-2282
energy potentials given by Williams’ potential and by MM3 force field are close to those given by the a b initio method, the carbon-carbon distances of the potential minima given by MM2 force field are shorter. The depths of the potentials of the dimers A and B obtained at the MP4(SDTQ)/6-31 lG(2d,2p) level are -0.128 and -0.274 kcal/mol, respectively. The calculated interaction energies at MP3/6-311G(3d,3p) for the same geometries are -0.152 and -0.336 kcal/mol, respectively. The depths of the potential of the dimers A and of B obtained by Williams’ potential are -0.168 and -0.398 kcal/mol, respectively. Those obtained by MM3 force field are -0.1 57 and -0.374 kcal/mol, respectively. These values are 3-1 1% larger than the values obtained at the MP3/6-311G(3d,3p) level calculation. The cause of the difference between the ab initio potentials and those obtained by Williams’ potential and by MM3 is not certain. There are a lot of possible reasons for this difference. The empirical potentials are optimized to various hydrocarbon molecules. Thus they can be different from the interaction potential of methane dimer. The crystal-derived potential could include many-body effect. The molecular mechanics potentials depend on the degree of molecular strain. The potential depths obtained by using MP3 level electron correlation correction might be 7-12’31 shallower than that obtained by using the MP4(SDTQ) correction. The BSSE given by the counterpoise method could be slightly larger than the real BSSE.4749 The intermolecular interaction potentials calculated by the MM2 force field have significantly deeper minima than those given by the former three types of calculations. The depths of the potentials of the dimers A and B given by MM2 are -0.293 and -0.806 kcal/mol, respectively. The depths of these potentials are (47) Schwenke, D. W.; Truhlar, D. G.J . Chem. Phys. 1985, 82, 2418. (48) Gutowski, M.;Lenthe, J. H. v.; Verbeek, J.; Duijneveldt, F. B. v.; Chalasinski, G. Chem. Phys. L t r . 1986, 124, 370. (49) Frisch, M.J.; Del Bene, J. E.;Binkley, J. S.;Shaefer 111, H.F. J . Chem. Phys. 1986,84, 2279.
larger than those obtained at the MP3/6-31 lG(3d.3~) level calculation as much as 93 and 140%. Recently, a lot of defects in the MM2 force field were found and the improved force field MM3 was p r o p o ~ e d Probably . ~ ~ ~ ~ the ~ ~overestimation ~~ of the attractive interaction energy of MM2 force field was a cause of the defects of this force field. The shape of the potential of the repulsive region of the dimer A given by MM2 force field is steeper than that obtained by MP4(SDTQ)/6-311G(2d,2p) level calculation. The potentials of this region given by Williams’ potential and by MM3 are close to the a b initio potential. The potential of the repulsive region of the dimer B given by MM3 is close to that from the same level ab initio calculation. The calculated potential of this repulsive region by Williams’ potential is steeper and that by MM2 force field is shallower than the a b initio potential. Conclusion The calculations of the intermolecular interaction energies of two orientations of methane dimers with electron correlation and BSSE correction using various levels basis sets showed that the calculated dispersions energies depended greatly on the basis set used. The calculations using a crude basis set underestimated the dispersion energies. The calculated intermolecular interaction potentials of the methane dimers obtained by the second- and third-order Maller-Plesset perturbation calculations were slightly shallower than those by the fourth-order (MP4(SDTQ)) calculation. The BSSE’s of the methane dimers estimated by the counterpoise method were about 30% of the calculated interaction energies even at the MP4(SDTQ)/6-311G(2d,2p) level for the dimers of the potential minima. The comparison of the MP4(SDTQ)/6-3 1IG(2d,2p) level intermolecular interaction potentials with those derived from empirical nonbonded potentials showed that the potentials from the ab initio calculations were much shallower than those derived from the MM2 force field but were close to those derived from Williams’ potential and from the recently reported MM3 force field.
Theoretical Study of the Bonding of Sc, Y, and La Singly Charged and Dipositive Ions to C2H2,C2H4,and CBHs Charles W. Bauschlicher, Jr.,* and Stephen R. Langhoff NASA Ames Research Center, Moffett Field, California 94035 (Received: March 19, 1990; In Final Form: July 10, 1990)
The interaction of the Sc and Y singly charged and dipositive ions with C2H2,C2H4, and C3Hs is studied using electronic structure calculations that include high levels of electron correlation. These results are compared with comparable calculations performed previously for La+ and La2+. For C2H2and CzH4, all three metal ions insert into the C-C r bond, making a three-membered ring. The optimal structures for the MC3H6+ions all involve rearrangement to make a four-membered ring. The strength of the metal-ligand bond for the singly charged ions follows the order La > Sc = Y. In contrast, the bonds involving the dipositive ions are electrostatic, so that the binding energy increases as the size of the ion decreases, leading to the trend Sc > Y > La.
Introduction The interaction of ligands with metal ions has become a very active area of research, in part due to the numerous experimental methods that have been developed to study ions in the gas phase.I-2 The determination of accurate experimental dissociation energies is leading to new insight into the nature of metal-ligand bonding. (1) (2)
Armentrout, P. B.; Georgiadis, R. Polyhedron 1988, 7, 1573. Buckner, S.W.; Freiser, B. S.Polyhedron 1988, 7 , 1583.
The singly charged ions of the transition metals on the left side of the row are interesting, not only because of the availability of experimental data but also because of the diverse reactions they undergo with hydrocarbon^.^.^ Since the second ionization potential (IP) of Sc, Y, and La is lower than the first IP of many (3) Sunderlin, L. S.; Aristov, N.; Armentrout, P. B. J . Am. Chem. Soc. 1987, 109, 78. (4) Sunderlin, L. S.; Armentrout, P. B. J . Am. Chem. SOC.1989, 1 1 1 , 3845.
This article not subject to US.Copyright. Published 1991 by the American Chemical Society
