J . Phys. Chem. 1989, 93, 7594-7603
7594
Accurate Predlction of Static Dipole Polarizabliities with Moderately Sized Basis Sets Mark A. Spackman Department of Chemistry, University of New England, New South Wales 2351, Australia (Received: February 6, 1989)
A relatively small 6-3 1G(+sd+sp) basis set has been developed for polarizability calculations. Exponents of d functions on first- and second-row atoms and the p function on hydrogen have been optimized with respect to maximum mean polarizability for the first- and second-row AH, hydrides. SCF and MP2 (second-order Mdler-Plesset perturbation theory) dipole and quadrupole moments and polarizabilities are critically compared with experiment for a total of 24 molecules, ranging in size from H F to cyclopropane. SCF polarizabilities are generally between 5% and 15% below experimental static values, except for molecules containing C=C and C=N bonds, where the component parallel to the bond is routinely overestimated, even near the Hartree-Fock limit. For the majority of molecules, MP2 mean polarizabilitiesare typically only 5% below experimental static values, and the polarizability anisotropy is always greater than or equal to experiment. The most exceptional behavior is observed for SO2,and this is attributed to deficiencies in the 6-31G substrate. With scaling, the 6-31G(+sd+sp) basis set should yield polarizability tensors within *2% of experiment for a wide range of polyatomic molecules.
Introduction Molecular dipole polarizabilities are well-known to be of major importance in several areas of current research in chemical physics, including molecular interactions’ and light scattering.2 We have recently become interested in the ab initio calculation of static and dynamic dipole polarizabilities, both for comparison with, and as an aid in the analysis of, experimental determinations, particularly for large m~lecules.~To be worthwhile, such calculations must be capable of distinguishing between two alternative polarizability tensors which are commonly derived from a combination of experimental r e s ~ l t s .Such ~ a task demands extremely high quality calculations, typically an accuracy of 5% in components of the tensor (and hence n). Although it has been adequately demonstrated that this sort of accuracy is more or less routinely attainable for small molec u l e ~ , ~ -almost *~ all such calculations use basis sets that are prohibitively large for application to molecules such as benzene, which will serve as our benchmark “large” molecule. A typical example is the ELP (electrical properties) basis set developed by ( I ) (a) Buckingham, A. D. In Intermolecular Interactions: From Diatomics to Biopolymers; Pullman, B., Ed.;Wiley: New York, 1978; pp 1-67. (b) Maitland, G. C.; Rigby, M.; Smith, E. B.; Wakeham, W. A . Intermolecular Forces-Their Origin and Determination; Clarendon: Oxford, 198 1. (c) Gray, C. G.; Gubbins, K. E. Theory of Molecular Fluids; Clarendon: Oxford, 1984; Vol. I . (2) Bogaard, M. P.; Orr, B. J. In MTP International Review of Science, Physical Chemistry Series 2; Buckingham, A. D., Ed.;Butterworths: London, 1975; VOI. 2, pp 149-194. (3) Coonan, M. H.: Hesling, M. R.; Ritchie, G. L. D.; Spackman, M. A . J . Phys. Chem., in preparation. (4) For example, the work described by MurphfJ on the simulation of the Raman contours of the pure rotational spectrum of SO, results in two sets of numerical values for the components of a, corresponding to positive and negative values of azI- a. ( 5 ) Werner, H.-J.; Meyer, W. Mol. Phys. 1976, 31, 855. (6) Greadv, J. E.: Bacskay, G. B.: Hush, N. S. Chem. Phys. 1977, 23,9. (7) Christiansen, P. A.; McCullough, E. A. Chem. Phys..Lett. 1978, 55, 439. (8) Gready, J. E.; Bacskay, G. B.; Hush, N. S. Chem. Phys. 1978,31,467. (9) Martin, R. L.; Davidson, E. R.; Eggers, D. F. Chem. Phys. 1979,38, 341. (10) Amos, R. D.; Williams, J. H. Chem. Phys. Lett. 1979, 66, 471. (11) Williams, J. H.; Amos, R. D. Chem. Phys. Lett. 1979, 66, 370. (12) Diercksen, G. H. F.; Sadlej, A. J. J . Chem. Phys. 1981, 75, 1253. (13) Amos, R. D. Chem. Phys. Lett. 1982,87, 23. (14) Amos, R. D. Chem. Phys. Lett. 1982, 88, 89. (15) Bacskay, G . B. J . Chem. Phys. 1983, 79, 2090. (16) Amos, R. D.; Handy, N. C.; Knowles, P. J.; Rice, J. E.; Stone, A. J. J . Phys. Chem. 1985,89, 2186. (17) Bishop, D. M.; Maroulis, G. J . Chem. Phys. 1985,82, 2380. (18) Jameson, C. J.; Fowler, P. W. J . Chem. Phys. 1986, 85, 3432. (19) Liu, S.-Y.; Dykstra, C. E. J . Phys. Chem. 1987, 91, 1749. (20) Bacskay, G . B.: Rendell, A. P. L.; Hush, N. S. J . Chem. Phys. 1988, 89, 5721.
0022-3654/89/2093-7594$01.50/0
Liu and Dykstra,Iq which yields 306 basis functions for benzene! We therefore decided to explore the optimization of a much smaller basis set, specifically for polarizability calculations. The choice of angular functions needed for polarizability calculations is straightfor~ard;~*’ d functions are necessary for first- and second-row atoms and p functions for hydrogen. However, the selection of optimum exponents is not so clear, although it is well established that diffuse functions are essential.21 In this paper we report an optimum, relatively small basis set, for calculating polarizabilities of large molecules to an accuracy of 10-15% at the SCF level and near 5% at the MP2 (second-order Maller-Plesset perturbation theory) level. The derivation of this basis set is described in the following section, and results for firstand second-row AH,, molecules are presented and discussed in detail. The basis set is then applied to a large number of polyatomic molecules, both small and relatively large, and the results are critically compared with available experimental data and other theoretical calculations where appropriate. We place particular emphasis on a comparison with experiment, since that is our primary interest. Therefore we devote special care to our choice of experimental data for such a comparison. A variety of experimental methods is used to determine a,including refractive index measurements, dielectric constants, Rayleigh and Raman scattering, the Kerr effect, and the quadratic Stark effect.lCv2 Some of these methods yield static electronic polarizabilities, but most values, particularly of individual tensor components, are observed at optical frequencies, and allowance must be made for this. Still others include a substantial vibrational polarizability component, and we have attempted to allow for this when quoting the experimental results.
Basis Set Optimization There have been several previous attempts to derive optimum exponents for polarizability calculations. Werner and Meyers performed a rough optimization of d function exponents for Ne and the heavy atoms in HF,H 2 0 , NH3, and CH4. This optimization was performed with respect to maximum polarizability, a procedure otherwise described as a second-order energy optimization, since maximizing effectively minimizes the energy of the molecule to second order in the perturbing electric field, F:’a
where Eo is the energy of the molecule in the absence of the field, is the permanent molecular dipole moment, and the suffixes denote Cartesian components, a repeated suffix implying sum-
po
(21) Christiansen, P. A.; McCullough, E. A. Chem. Phys. Lett. 1977,51, 468.
0 1989 American Chemical Society
The Journal of Physical Chemistry, Vol. 93, No. 22, 1989 7595
Prediction of Static Dipole Polarizabilities
TABLE 11: Optimum Values of fa(A) and f,(H) Obtained in Polarizability Calculations with a 6-31C(+d+p) Basis Set for AH, Molecules
:f CH, NH3 H2O
0.042 0.053 0.068 0.090 0.019 0.025 0.029 0.036
HF SiH4 pH3
H2S HCI
cs2
-
--
rc-s E 1.561 A rC+ = 1.099 A; rc* rC-H = 1.087 A; rC-F rC-H = 1.087 A; rc+ LHCH = 112.6' C H F 2 C,,,rp-" = 1.085 A; rr-F 1.329 1.062
D,h H 2 C 0 C, C H l F C, CH2F2 C,,
1.080 1.088 1.080 1.079
= 1.219 A; L H C H = 115.5'
= 1.388 A; LHCH = 109.7' = 1.363 A; LFCF = 109.0'; = 1.342 A; LFCF = 108.5' = 1.217 A = 1.334 A; LHCH
= 1.522 A; LHCH = 1.311 A; LHCH = 1.500 A; LHCH
116.9' 107.7' 117.7' 114.2'
mation over x, y , and z. Similar procedures have been employed in other work,22but to our knowledge no systematic attempt has been made to use such a method to derive optimum exponents for a basis set applicable to a large number of atoms. We note, however, that Christiansen and McCullough' have presented a set of simple rules for choosing Gaussian basis sets for polarizability calculations, which they expected to be especially useful for polyatomic molecule calculations. Their basis set is effectively [5s4p2d/3s2p], which results in 228 contracted basis functions (bf) for benzene, placing it out of consideration for our purposes. A more appropriately sized basis set is the polarized 6-31G** basis of Hariharan and P ~ p l ewhich , ~ ~ is effectively a [3s2pld/ 2slpl basis, resulting in 120 bf for benzene, a more tractable calculation, especially if use is made of molecular symmetry. This basis set has been widely applied, especially in geometry optimizations, and even for quite large molecules.24 It has also been used recently, without modification, to calculate polarizabilities of 1,l-dicyanoethylene and vinylacetylene at the S C F level.25 We chose the 6-31G basis set26as the sp substrate upon which can be built an optimum basis set by addition of diffuse polarization f~nctions.~'We initially added a single d function to firstand second-row atoms, and a single p function to H , resulting in a basis which we label 6-31G(+d+p), and which is identical with the 6-31G** basis except for the choice of exponents of the polarization functions.28 We followed a procedure similar to that (22) (a) Stevens, R. M.; Lipscomb, W. N. J. Chem. Phys. 1964,41, 184. (b) Gutschick, V. P.; McKoy, V. J . Chem. Phys. 1973, 58, 2397. (c) Billingsley, F. P.; Krauss, M. Phys. Rev. A 1972, 6, 855. (d) Fortune, P. J.; Certain, P. R. J . Chem. Phys. 1974,61, 2620. (e) Huiszoon, C. Mol. Phys. 1986, 58, 865. (231 Hariharan. P. C.: PoDle, J. A. Theor. Chim. Acta 1973, 28. 213. (24) An excellent summaiy of applications is given in: Hehre, W. J.; Radom, L.; Schleyer, P. v. R.; Pople, J. A. Ab Initio Molecular Orbital Theorv: Wilev: New York. 1986. (2lo0 bf) overestimate both all and aI. Therefore, it is not surprising that the MP2 corrections to these quantities result in a lowering of both. As noted for HCN, the very good agreement with experiment observed at the S C F level may be fortuitous, as electron correlation appears to reduce all components of a for CzH2,and allfor HCN. The MP2 result for Aa is an overestimate by 16% compared with the static experimental values; this behavior correlates nicely with the low value of n/Aa = 1.9 for acetylene. Both S C F and MP2 quadrupole moments are in good agreement with the experimental value.71 CZH4. For ethylene, polarizability tensor components have been obtained at 514.5 nm from a combination of experimental data.74 The present results, both S C F and MP2, agree well with the available data. Of particular note is the correlation correction to a,,,the component along the C = C bond. This quantity is large and negative (-0.40, or -6.7% of the S C F value) whereas the correlation corrections to both a, and ayyare small and positive. The correlated result is in worse agreement with the experimental value for a,, than is the S C F result. The present S C F value for a=,(5.997) agrees well with other theoretical results of 6.06319 and 6.0010 obtained with quite large basis sets and hence is probably close to the Hartree-Fock limit for a,,. Ahern, Garrell, and Jordan” report SCF, MP2, and CISD polarizabilities for ethylene, using a variety of moderately sized basis sets. Of particular relevance are their calculations with a doubly polarized [5s4p2d/3s2p] basis set. At the S C F level this basis set yields values for (axx,ayy,a,,) of (3.58, 3.92,5.87), in good agreement with the present S C F result. MP2 results obtained by Ahern et al. with the same basis set are (3.59, 4.1 1, 5.61), very close to our present MP2 results; the correlation correction to a,, is again large and negative. As we noted for CzHz and HCN, the apparently good agreement of S C F near Hartree-Fock limit polarizabilities with experimental values is probably fortuitous; inclusion of electron correlation can, and often does, worsen agreement with experiment. The present MP2 value of CE is 3.8% below the experimental static value in Table IV, and the correlated result for Aa appears to be very close to the static experimental value obtained from the dispersion of Rayleigh depolarization data.S3 The quadrupole moment tensor for ethylene is well determined with the 6-31G(+sd+sp) basis: the MP2 results are in excellent agreement with values deduced from collision-induced absorption.73 czH6. SCF values reported in Table IV are virtually identical with those reported by Amos and Williams,lo who used a basis set with 88 bf, substantially more than the present 68 bf. The MP2 a is 5.8% below the static experimental value, and the polarizability anisotropy, Aa = 0.642, is in excellent agreement with the static experimental value quoted in Table IV. 0 is also well determined with the 6-3 lG(+sd+sp) basis set. C3H4.As observed for HCN, C,Hz, and CzH4,the polarizability component along the C=C bond for allene, all in this case, is overestimated by a large amount at the SCF level, and the MP2 correction reduces this to give better agreement with experiment. The MP2 value for a is only 2.3% below experiment, and Aa is apparently just 1.5% too high. Even the S C F prediction for Aa is only 21% above the experimental value; for allene a/Aa = 1.3, and the theoretical results support our earlier conclusion that Aa will be well determined in such a case. C3H6.Amos and Williamse3reported S C F multipole moments and polarizabilities for cyclopropane using various basis sets. Their
-
N
(80) Jaquet, R.; Kutzelnigg, W.; Staemmler, V. Theor. Chim. Acta 1980, 54, 205.
(81) Fowler, P. W. Mol. Phys. 1982, 47, 355. (82) Buckingham, A. D.; Orr, B. J. Trans. Faraday SOC.1969,65, 673.
(83) Amos, R. D.; Williams, J. H. Chem. Phys. Lett. 1981, 84, 104. (84) Buckingham, A. D.; Graham, C.; Williams, J. H. Mol. Phys. 1983, 49, 703.
Prediction of Static Dipole Polarizabilities best results yielded 8 = 8.40 X lo4 C m2, all = 5.1 1, and crl = 5.85 ( 1 14 bf), all very close to the SCF/6-31G(+sd+sp) (84 bf) results given in Table IV. MP2 results in Table IV appear to be the best so far reported for cyclopropane; a is only 3.6% below the experimental value, and Acr is in excellent agreement with experiment, although the experimental value may be spuriously high by -10% due to inclusion of vibrational Raman scattering in the data reported by Bogaard et alas3The MP2 value for 0 appears to be rather high; Amos and Williams discussed the large discrepancy between the SCF and experimental quadrupole moments and explored the possibility that hyperpolarizability effects could be important experimentally. Their results were inconclusive, and the present MP2 results suggest that the problem deserves further attention, both experimentally and theoretically.
Conclusions The use of p and d function exponents optimized for polarizability calculations on first- and second-row hydrides, in routine calculations of polarizabilities for larger molecules, is extremely successful. The 6-3 lG(+sd+sp) basis set is probably the smallest that could be applied to a large class of polyatomic molecules with the present degree of success, and there are notable deficiencies with it, especially for multiply bonded second-row atoms, such as sulfur in SO2. Use of a DZ sp substrate will probably overcome such problems, and we expect the optimum ld(A) and Jb(H) values given in Table I1 to be more or less directly transferable for use with a different sp substrate. This is a direct consequence of the flatness of the a surface as a function of both ldand l as illustrated in Figure l for water. The basis set can be further augmented by addition of d- and p-type polarization functions with energy-optimized exponents, as these functions are essentially orthogonal to the diffuse ones required for polarizability calculations. The 6-3 1G(+sd+sp) basis generally performs exceptionally well when compared to S C F results obtained with much larger basis sets, even those approaching the Hartree-Fock limit. SCF results for 6 are generally between 5% and 15% below experimental static values, with several important exceptions, notably HCN, C2H2, C2H4, and C3H4. In these cases the S C F value is within -2% of experiment, and inclusion of electron correlation worsens this agreement. In all of these cases the component of a along the axis of the molecule, parallel to the G N or C = C bonds, is close to, or an overestimate of, the experimental value, and the correlation correction is negative. These results suggest that Hartree-Fock limit calculations on these molecules, and others like them, will in general be in fortuitous agreement with experiment. The multiply bonded linear molecules C02, N 2 0 , OCS, and CS2are also somewhat anomalous; all(SCF)is typically low by -6%, but a,(SCF) is low by twice this amount. Because of this, S C F calculations yield excellent estimates of Aa, while providing an uneven description of the principal tensor components for these molecules. MP2 corrections to aI are small and positive, while corrections to all are large and positive, leading to an overestimate of Pa more typical of most other molecules studied in the present work. We can offer no explanation for these observations, but it is clear that they are not limited to the moderately sized basis
The Journal of Physical Chemistry, Vol. 93, No. 22, 1989 7603 set used but rather are likely to be generally true for these sorts of molecules. Bearing in mind the exceptions, the majority of molecules we have studied (18 out of 24) display well-behaved trends a t the MP2 level: a is always -5% below experimental static values (the range is from 2.0% for H2S to 8.6% for CH2F2)and Acr is always greater than or equal to experimental values, the magnitude of the discrepancy being closely related to the ratio a/Aa. This behavior should prove useful for predictive purposes. It is also important to observe that MP2/6-3 lG(+sd+sp) values of dipole and quadrupole moments are usually in very good agreement with experiment, and hence the replacement of energy-optimized polarization functions with those of a more diffuse character has not adversely affected the description of these important oneelectron properties, particularly a t the MP2 level. It is clear that a major factor in the success of the present MP2 results is the fact that second-order many-body perturbation theory accounts for the majority of the correlation effects on polarizabilities (and one-electron properties); third- and fourth-order corrections appear to largely cancel. This observation has been made in several earlier publi~ations,'*J~*~~ and it is strongly underlined by the present results for a large range of molecules. We have made no attempt in the present work to account for the effects of vibrational averaging or wavelength dependence of the polarizabilities. Vibrational corrections have been attempted by Werner and M e ~ e rand , ~ they typically increase components of a by varying amounts, and by as much as 3.4%. More detailed work in this direction is clearly necessary, as the effects on Aa will be quite substantial. The calculation of SCF polarizabilities at optical frequencies is now routinely possible with time-dependent Hartree-Fock theory.I6 To date very few applications have been made, but it should be possible to obtain very good estimates of frequency-dependent MP2 polarizabilities by using corrections at nonzero frequencies obtained at the S C F level. For both vibrational-averaging and frequency-dependent studies, the 631G(+sd+sp) basis developed in this work should prove ideal. We have recently used this basis set in calculating the polarizability tensors for dimethyl ether, dimethyl sulfide, ethylene oxide, and ethylene sulfide, and those results will be published in detail el~ewhere.~ We expect the 6-31G(+sd+sp) basis set to be useful in calculating reliable polarizability tensors for still larger polyatomic molecules. In this regard we note that the mass storage requirements for SCF calculations scale approximately as p,with N the number of basis functions. Thus if an SCF calculation of a with the present basis set yields results approaching those obtained with more conventionally constructed basis sets, which are typically 2-3 times as large, it will be quite feasible to obtain MP2 corrections to a using the same amount of mass storage, or even substantially less, that would be required for the SCF calculation with the larger basis set. Acknowledgment. The perspective presented in this paper has benefited greatly from interaction with Prof. G . L. D.Ritchie, Drs. M. P. Bogaard and I. R. Gentle, and Mr. M. R. Hesling. These calculations would not have been possible without generous provision of resources on the Gould NP1 in the Computer Centre of the University of New England.
