J. Phys. Chem. 1996, 100, 6231-6235
6231
Response Function Basis Sets: Application to Density Functional Calculations Gerald Lippert, Ju1 rg Hutter, Pietro Ballone, and Michele Parrinello* Max-Planck-Institut fu¨ r Festko¨ rperforschung, Heisenbergstrasse 1, 70569 Stuttgart, Germany ReceiVed: September 20, 1995; In Final Form: December 5, 1995X
We present a systematic procedure to generate a compact basis for molecular computations made of atomic orbitals and their derivatives with respect to the total electronic charge. All electron density functional computations for small molecules highlight the rapid convergence to the basis set limit. The simplicity of the basis set construction allows a fine tuning with respect to the electronic structure method (Hartree-Fock, configuration interaction, density functional, etc.), resulting in improved performance and in reduced basis set superposition errors. We discuss the connection of our scheme with chemical concepts like Fukui functions or absolute hardness and the relation with the linearized methods (linear augmented plane waves, linear muffin tin orbitals) of solid state physics.
Introduction The simplest picture of molecular bonding starts from atomic levels and involves both an energy shift and the superposition in space of orbitals from different atoms. Most electronic structure computations make use of finite one-particle basis sets, whose accuracy and efficiency depend on their ability to describe these two effects with a small number of degrees of freedom. Molecular calculations in ab-initio quantum chemistry are almost exclusively done with Gaussian-type functions (GTF), which were introduced in 1950 by Boys.1 Since then, many different approaches to construct atomic basis sets from GTF were followed, leaving us with a vast number of choices for molecular calculations. The huge literature and many reviews2-5 on the subject clearly indicate its importance. Only for correlated molecular calculations do consistent basis sets exist6-8 that allow for a controlled improvement of calculations. The recent application of density functional theory (DFT) to chemically relevant problems has further increased the variety of basis functions used in electronic structure computations. Besides GTF,9-11 commonly used basis functions include Slatertype functions (STF)12 and numerical atomic functions.13,14 Moreover, DFT computations may in addition need auxiliary basis sets to represent the charge density and the exchange and correlation potential. To the detriment of simplicity, it has become a common habit to use basis sets optimized for HartreeFock also in DFT calculations. In contrast with the variety of basis sets used in quantum chemistry, the solid state computational community has focused its attention along two major directions. The most widely used is, by far, the plane wave basis set,15 whose completeness can be characterized by a single number, the energy cutoff. Plane waves are almost invariably used together with the valence electron-pseudopotential approximation. The other majority family of basis sets has been originated by the augmented plane wave approach and, in particular, by its linearized versions.16-19 At variance from the plane wave scheme, these methods are mostly used for all electron computations. In the inner region close to each atom the basis reduces to just two functions for each orbital symmetry. The first function φ1 is the atomic orbital at a given reference energy (usually not the atomic level), and the second function φ2 is the derivative of φ1 with respect to the reference energy. Despite being far from complete, this X
Abstract published in AdVance ACS Abstracts, March 15, 1996.
0022-3654/96/20100-6231$12.00/0
basis is very well adapted to describe the changes in the wave function due to the chemical bonding and underlies several of the most impressive and accurate computations in solid state physics. In this paper we introduce a procedure for the construction of atomic-type orbitals for electronic structure calculations based on the successive augmentation of the atomic ground state orbitals by response functions of the valence orbitals to changes in the charge distribution. As discussed below, our scheme is inspired by both chemical concepts like Fukui functions20,21 or absolute hardness22 and the linearized atomic-like basis of solid state. It is also related to the finite charge difference method used by Delley14 and the state averaging technique used by Widmark et al.7 We believe that this method shows some desirable properties. It is applicable to ab-initio quantum chemistry methods (Hartree-Fock and post-Hartree-Fock) and DFT and can make use of any type of primitive functions (GTF, STF, or numerical). It provides a consistent way of improving the basis to reach completeness. Our test computations show that this method produces compact basis sets, since convergence to the basis set limit is achieved with a remarkably small number of functions. In the next two sections we present an outline of our method, and as a first application we generate a general contracted GTF basis for first-row atoms for DFT calculations. In the last section the performance of the new basis is tested in calculations on small molecules. Response Function Basis Sets The generation of the response function basis proceeds in two steps. In a first step we fully optimize the lowest energy configuration for a given atom, within a given method (e.g., DFT or Hartree-Fock), under the constraint of the functional form of the basis (STF, GTF, or numerical grid). These functions build the core of our basis. With the inclusion of the optimized atomic functions in the basis, it is assured that we can describe the atomic ground state to high accuracy within our functional constraint. The second step consists of a systematic procedure to add new functions to the basis. It is a common concept in chemistry to view molecules as assemblies of slightly perturbed atoms, with the effect of the perturbation being described in terms of changes in the formal occupation numbers of the atomic states. A rigorous formulation of this point of view is provided by concepts like the Fukui functions,20,21 the absolute hardness,22 © 1996 American Chemical Society
6232 J. Phys. Chem., Vol. 100, No. 15, 1996
Lippert et al.
etc., relating the atomic response to a generic perturbation to the derivatives of chemical potential, total energy, and electron density with respect to the total number of electrons N. Here we extend this idea to wave functions, by assuming that, in the proximity of an atom in an external potential (as produced by other atoms in a molecule), the orbitals can be expanded in a Taylor series:
∂Ψi(r) ∂2Ψi(r) Ψ(r) ) Ψi(r) + R + ... +β ∂N ∂N2 where N represents the total electron number and, to emphasize the role of occupied orbitals, the derivatives are “left derivatives”; i.e., they are computed as N increases from N - δ to N. The coefficient R, β, etc., are determined by the specific perturbation acting on the atom. We stress that this expansion does not assume that the real charge of the atom is changing in making the chemical bond. It is the purpose of this paper to show that the sequence
{
Ψi(r),
}
∂Ψi(r) ∂2Ψi(r) , ... , ∂N ∂N2
provides a basis with increasing accuracy. Here and in the following the atomic orbitals and their derivatives are assumed to be normalized. Before testing the accuracy of this scheme in molecular computations, we briefly discuss the connection of this basis with the chemical functions mentioned above. The Fukui function for an atom is defined as23
f(r) )
[ ] [ ] δµ δV(r)
)
N
∂F(r) ∂N
V
where µ is the atomic chemical potential, F(r) is the electron density, and V(r) is the external potential. By expressing F(r) in terms of the independent electron orbitals
F(r) ) ∑ ni|Ψi(r)|2 i
(where ni are occupation numbers), we obtain
f(r) ) |ΨN(r)|2 + ∑ niΨi(r) i
∂Ψi(r) ∂N
Together with the unperturbed orbitals, therefore, the functions {∂ψi(r)/∂N} describes the atomic Fukui function and characterize the chemical reactivity of the atom. A simple calculation shows that the first derivatives are sufficient to reproduce also the absolute atomic hardness,23 which is defined as the second derivative of the total energy with respect to N. In general, knowledge of the mth derivative of the orbitals with respect to N allows us to compute the energy response up to order 2m + 1 (the 2m + 1 theorem of perturbation theory). This observation provides a hint on the reasons for the accuracy of the present basis. The connection with the linearized basis of solid state physics is also simple and interesting. A straightforward extension of the scheme used in solid state methods16-19 for the case of molecular computations is not practical, since the derivatives with respect to the energy diverge at infinity. (In the solid state the atomic-like part of the basis is used only within a suitable “atomic sphere”.) Our response functions, instead, are regular
Figure 1. Radial dependence of the valence basis functions (Ψi (full lines) and ∂Ψi/∂N (dash-dotted lines) only) for oxygen. The lowest virtual atomic orbital (dash line) and the natural orbital (dotted line) corresponding to the 3s atomic level are shown by comparison. The natural orbital is from ref 7.
everywhere and contain the same basic information, i.e., how atomic orbitals change under a shift of their energy. As observed in solid state applications, the shape of the derivative functions is reminiscent of the virtual orbitals, each successive derivative having one node more than the previous one (see Figure 1). The similarity is even stronger with the lowest energy natural orbitals, commonly used as a basis for correlated wave function calculations. Both the virtual and the natural orbital, however, are significantly more diffuse than the response functions. Although in the Kohn-Sham or Hartree-Fock method taking derivatives with respect to the occupation of the highest lying atomic orbital is an obvious choice, different choices for the charge derivatives can easily be envisaged. In fact, numerical tests showed that only minor differences occur when the differentiation is performed with respect to different orbital occupations. A further possibility is to differentiate with respect to the nuclear charge; this gives also very similar results. DFT Response Basis Sets for First-Row Atoms As a first application of response function basis sets, we generate small Gaussian basis sets for first-row atoms for DFT calculations. The calculations are done using Becke’s gradientcorrected exchange functional24 and the correlation functional of Lee, Yang, and Parr25 (BLYP) without spin polarization. The derivative functions are calculated from finite differences in the occupation of the highest lying atomic orbital.26 Exponents of the primitive Gaussians were optimized using a quasi-Newton method following the experience gained in Hartree-Fock theory.27,28 In the following we will denote uncontracted sets by brackets and contracted functions by parentheses. Using oxygen as a test, we fully optimized basis sets ranging from seven s-type and three p-type functions [7s3p] up to [13s8p]. In Table 1 we give the energy difference to the exact result calculated with a fully numerical DFT program. We also give results for optimizations under the constraint of equal exponents for s and p functions. Basis sets with shared exponents allow
Response Function Basis Sets
J. Phys. Chem., Vol. 100, No. 15, 1996 6233
TABLE 1: Energy Differences in Hartree for Fully and Constraint Optimized Gaussian Basis Sets for Oxygena
TABLE 3: Errors in Molecular Geometries Introduced by Contracting the [9s5p] Basisa
basis
∆E (fully optimized)
∆E (shared exponents)
7s3p 8s4p 9s5p 10s6p 11s7p 13s8p
0.123 684 0.031 848 0.010 023 0.003 209 0.001 095 0.000 278
0.141 269 0.050 909 0.016 021 0.005 428 0.002 023 0.000 557
(3s2p1d/ 2s1p)
(4s3p1d/ 3s1p)
(5s4p1d/ 4s1p)
[9s5p1d/ 5s1p]
6-311+G(3df,2p)
R ν
0.7473 4408
0.7471 4341
0.7471 4341
0.7471 4341
0.7467 4349
R ν
1.1225 2296
1.1086 2313
1.1077 2314
1.1065 2317
1.1031 2331
R ν
1.2404 1506
1.2369 1488
1.2363 1488
1.2355 1489
1.2283 1501
R ν HF R ν CO R ν CO2 R ν1 ν2 ν3 H2O R R ν1 ν2 ν3 NH3 R R ν1 ν2 ν3 ν4 CH4 R ν1 ν2 ν3 ν4
1.4371 943
1.4348 961
1.4344 960
1.4341 961
1.4311 951
0.9371 3940
0.9341 3937
0.9339 3947
0.9336 3951
0.9329 3939
1.1523 2090
1.1414 2120
1.1400 2116
1.1389 2117
1.1362 2121
1.1856 611 1291 2317
1.1764 621 1306 2324
1.1752 624 1304 2323
1.1745 624 1304 2322
1.1723 641 1309 2325
0.9763 103.9 1604 3660 3767
0.9738 103.9 1591 3643 3748
0.9739 103.9 1589 3650 3755
0.9737 103.9 1590 3652 3755
0.9709 104.6 1596 3680 3782
1.0274 106.6 1018 1636 3375 3501
1.0241 106.7 1017 1639 3346 3466
1.0242 106.7 1025 1640 3349 3465
1.0241 106.7 1021 1639 3347 3468
1.0221 106.7 1021 1631 3371 3482
1.1003 1315 1528 2987 3103
1.0971 1312 1528 2949 3047
1.0969 1312 1527 2949 3046
1.0965 1313 1528 2948 3046
1.0943 1311 1526 2959 3054
H2 N2
a
Calculations with the Becke and Lee, Yang, and Parr functionals. The exact results is -75.023693 hartrees.
O2
TABLE 2: Errors Introduced by Contracting the [9s5p] Basisa
F2
3s2p 4s3p 5s4p
3s2p 4s3p 5s4p
3s2p 4s3p 5s4p
Li
Be
B
C
N
O
F
Ne
190 30 1
0 0 0
29 17 1
55 40 4
115 80 7
89 45 4
35 17 1
0 0 0
Li+
Be+
B+
C+
N+
O+
F+
Ne+
44 8 0
485 101 1
263 2 1
800 40 4
953 45 8
1023 95 13
815 54 6
560 22 2
Li-
Be-
B-
C-
N-
O-
F-
6611 1131 16
85 2 0
309 26 2
271 55 3
448 32 2
408 12 1
288 0 0
a All calculations are done using the Becke and Lee, Yang and Parr functionals in spin polarized form. Energy differences are given in µhartrees.
for considerable savings in CPU time in molecular calculations. This is especially true for calculations of derivative integrals and with general contracted-type functions. As expected, the results for the basis with the constraint exponents are slightly worse then the fully optimized basis but still considerably better than the results with one function less. At this point we decided to make all further tests with a basis of nine s and five p functions with shared s and p exponents. This basis is big enough to be contracted up to five s and four p functions yet small enough to be used in large molecular calculations. In Table 2 are the contraction errrors listed for spin-polarized calculations on first-row atoms and their ions. We see that errors introduced through the contraction to the (5s4p) basis are typically a few microhartrees. With a contraction to 4s and 3p functions the error increases by about 1 order of magnitude. With a (3s2p) contraction the error stays in most cases below 1 mhartree. In general, we can conclude that, for the [9s5p] basis, contraction errors for (3s2p) and (4s3p) contractions are much smaller than the total basis set error and are negligible for (5s4p) contractions. Molecular Test Calculations To test our new basis sets, we performed a series of calculations on small molecules using the program Gaussian94.29 As is the case for other basis set schemes hydrogen causes special problems,30 and we decided to depart from the response function concept. The basis for the hydrogen atoms consists of five primitive Gaussians. We fully optimized the 1s function with respect to the total energy in a spin-unpolarized BLYP calculation. As for Hartree-Fock calculations,30 we scale the exponents by a factor 1.44, corresponding to a scaling of 1.2 for a hydrogenic function. Tests on the H2 molecule revealed that the performance of a basis of Raffenetti type,32 i.e., uncontracting the functions with the most diffuse exponents, gives best results for the 2s and 3s basis sets. In all molecular
a All calculations are done using the Becke and Lee, Yang, and Parr functionals in spin polarized form. Distances are given in angstroms, angles in degrees, and vibrational frequencies in cm-1.
calculations we added polarization functions to our basis. In principle, response function concepts might also be used to generate polarization functions. However, for simplicity, in this paper we considered just one additional Gaussian per symmetry whose exponent has been chosen on the basis of chemical experience. The d exponents for C, N, O, and F were 0.80, 1.00, 1.20, and 1.40, respectively. The p function for the hydrogen basis was chosen to have an exponent of 0.80. In Table 3 results for molecular geometries and harmonic frequencies are given for response function basis sets including the first, second, and third derivatives. Results are compared to calculations with the corresponding uncontracted basis. The mean error in bond length is reduced from 0.0061 Å with one, to 0.0010 Å with two, to 0.0005 Å with three response functions. For the same series contraction errors in harmonic frequencies are reduced from 17, to 2, to 1 cm-1. Finally, results are given for the 6-311+G(3df,2p) basis. This basis was used in earlier studies by Gill et al.33 and should give results close to the basis set limit. This basis is superior to our (5s4p1d) basis mainly through its extended set of polarization functions. Bond lengths typically differ by less than 0.003 Å, with the exception of the oxygen molecule where the difference is 0.007 Å. Harmonic
6234 J. Phys. Chem., Vol. 100, No. 15, 1996
Lippert et al.
TABLE 4: Basis Set Superposition Errors for Different Basis Setsa (3s2p1d/2s1p)
(4s3p1d/3s1p)
6-31G(d,p)
6-311G(d,p)
0.348 0.182 0.386 0.540 0.450 0.700
0.014 0.205 0.442 0.633 0.281 0.775
0.062 1.491 1.425 2.353 0.893 2.201
0.010 0.650 0.834 1.118 0.477 1.568
H2 N2 O2 F2 HF CO
a Values calculated by the counterpoise method. All energies are given in kcal/mol.
frequencies differ for all examples by less than 30 cm-1 with a typical difference of ≈10 cm-1. Basis set superposition errors (BSSE) are one of the biggest problems that occur in the calculation of thermochemical properties with small basis sets. Response function basis sets exhibit only small BSSE, as they are optimized consistently with the functional used in the molecular calculations, and through the use of the general contraction scheme they provide a good description of core states. BSSE as calculated through the counterpoise correction31 are shown in Table 4 where we compare the performance of the density response basis set relative to a more conventional segmented contracted basis which has the same number of contracted functions but a larger number of primitive Gaussians. By increasing the number of contracted functions from (3s2p1d) to (4s3p1d), we increase the flexibility of the basis in the valence region, and therefore, the BSSE slightly increases. Molecules including H do not follow this behavior, as the H basis was constructed differently. BSSE for conventional segmented contracted basis sets are considerably higher. Even the 6-311G(d,p) basis with two more primitives shows BSSE that are approximately twice as large as the ones from response basis sets. Summary The chemical theory of reactivity and the computational experience in solid state DFT applications point to the set of atomic orbitals and their derivatives with respect to occupation as a promising basis set for molecular and condensed-matter ab-initio computations. The scheme provides a systematic way of improving the basis completeness by adding successive derivatives. We have explicitly constructed a basis along these lines for first-row elements within DFT in the BLYP approximation for exchange and correlation, and we have tested its accuracy by molecular computations. Cohesive energy, equilibrium distance, and vibrational frequency display a remarkable fast convergence to the basis set limit. As for other schemes, hydrogen deviates somewhat from the trends observed for the other elements, but this problem can be overcome by the same standard correction used to generate other basis sets. The concept of response function basis sets is independent of the form of the basis functions, which makes it immediately applicable in calculations using any type of primitive function, like GTF, STF, or numerical functions. The simplicity of the basis set construction allows its tuning to the specific approximation used in the electronic structure computation (HF, DFT with different functionals, etc.), thus improving significantly the basis set superposition error in molecular computations. Another field of application of our scheme will be in connection with the use of pseudopotentials. Also in this case it is important to construct the basis consistently with the choice of method and pseudopotentials used in the computation. This
fine tuning results in a very compact basis that provides a crucial advantage for large-scale computations like ab-initio molecular dynamics. In our paper we discuss the connection of our scheme to the Fukui functions, the absolute hardness, and the linearized schemes of solid state physics. We suggest that the accuracy of the basis can be traced to a basic result of perturbation theory (the “2m + 1” theorem). The construction we used (“left” derivatives with respect to the population of the highest occupied atomic orbital) is only one of the possible ways to implement the basic idea. Refinements and further tuning could improve the results presented in the paper. Appendix To allow an easy reproduction of our results, we summarize in this Appendix the procedure we used to generate the response basis set discussed in the present paper. For the first-row elements the starting point is a standard DFT computation for the single atom in the spherical approximation and within the specific exchange-correlation approximation to be used for the molecular computations. In the present case, computations have been performed with the BLYP approximation. The ground state orbitals {Ψi} determined in this way provide the “zero-level” basis for each atom. For the valence states this basis is supplemented by the left derivative of the {Ψi} with respect to the occupation number of the highest occupied atomic state. In principle, this derivative could be computed by perturbation theory. For simplicity, we used instead a finite-difference approximation34 to evaluate the first two derivatives by repeating the atomic computation for few different populations of the highest occupied state spaced by ∆ ) 0.05 electron. The ground state orbital and the first derivative are orthogonal by construction. The second and higher derivatives have to be explicitly orthogonalized (for instance, by a Gram-Schmidt procedure) and normalized to obtain an orthonormal basis set. Since most of the quantum-chemical computer codes are based on Gaussian basis sets, we approximate the response basis set generated above by a Gaussian basis. The exponent optimization procedure and the estimate of the error introduced by this choice of the basis functions are discussed in the section DFT Response Basis Sets for First-Row Atoms. The resulting basis set parameters are also available in ref 35. Finally, polarization functions are added to this basis set in molecular computations. For simplicity, we considered just one additional Gaussian per symmetry whose exponent is listed above for C, N, O, and F. The only exception to the scheme outlined above is H, for which our basis consists of the basis traditionally used in quantum chemistry computations. References and Notes (1) Boys, S. F. Proc. R. Soc. London, A 1950, 200, 542. (2) Dunning, T. H.; Hay, P. J. In Methods of Electronic Structure Theory; Schaefer, H. F., III, Ed.; Plenum Press: New York, 1977; p 1. (3) Ahlrichs, R.; Taylor, P. R. J. Chim. Phys. 1981, 78, 315. (4) Huzinaga, S. Comput. Phys. ReV. 1985, 2, 6. (5) Davidson, E. R.; Feller, D. Chem. ReV. 1986, 86, 681. (6) Almlo¨f, J.; Taylor, P. R. J. Chem. Phys. 1987, 86, 4071. (7) Widmark, P.-O.; Malmqvist, P.-A° .; Roos, B. O. Theor. Chim. Acta 1990, 77, 291. Widmark, P. O.; Persson, B. J.; Roos, B. O. Ibid. 1991, 79, 419. Pierloot, K.; Dunez, B.; Widmark, P.-O.; Roos, B. O. Ibid. 1995, 90, 87. (8) Dunning, T. H., Jr. J. Chem. Phys. 1989, 90, 1007. Kendall, R. A.; Dunning, T. H., Jr.; Harrison, R. J. Ibid. 1992, 96, 6796. Woon, D. E.; Dunning, T. H., Jr. Ibid 1993, 98, 1358.
Response Function Basis Sets (9) Sambe, H.; Felton, R. H. J. Chem. Phys. 1974, 61, 3862. 1975, 62, 1122. (10) Dunlap, B. I.; Connolly, J. W. D.; Sabin, J. R. J. Chem. Phys. 1979, 71, 3396, 4993. (11) Godbout, N.; Salahub, D. R.; Andzelm, J.; Wimmer, E. Can. J. Chem. 1992, 70, 560. (12) Baerends, E. J.; Ellis, D. E.; Ros, P. Chem. Phys. 1973, 2, 41. Rose´n, A.; Ellis, D. E. J. Chem. Phys. 1976, 65, 3629. (13) Averill, F. W.; Ellis, D. E. J. Chem. Phys. 1973, 59, 6412. Delley, B.; Ellis, D. E. Ibid. 1982, 76, 1949. (14) Delley, B. J. Chem. Phys. 1990, 92, 508. (15) Srivastava, G. P.; Weaire, D. AdV. Phys. 1987, 36, 463. (16) Andersen, O. K. Solid State Commun. 1973, 13, 133. (17) Andersen, O. K. Phys. ReV. B 1975, 12, 3060. (18) Andersen, O. K.; Jepsen, O.; Gloetzel, D. In Highlights of Condensed Matter Theory; Proc. Int. School Phys. “Enrico Fermi”, Course LXXXIX; Bassani, F., Fumi, F., Tosi, M. P., Eds.; North-Holland: Amsterdam, 1985. (19) Skriver, H. L. The LMTO Method; Springer: Berlin, 1984. (20) Fukui, K.; Yonezawa, T.; Shingu, H. J. Chem. Phys. 1952, 20, 722. (21) Parr, R. G.; Yang, W. J. Am. Chem. Soc. 1989, 106, 4049. (22) Parr, R. G.; Pearson, R. G. J. Am. Chem. Soc. 1983, 105, 7512. (23) Parr, R. G.; Yang, W. Density-Functional Theory of Atoms and Molecules; Oxford University Press: 1989. (24) Becke, A. D. Phys. ReV. A 1988, 38, 3098. (25) Lee, C.; Yang, W.; Parr, R. G. Phys. ReV. B 1988, 37, 785.
J. Phys. Chem., Vol. 100, No. 15, 1996 6235 (26) It is relatively easy to derive and solve the equations satisfied by the orbital derivatives ∂mΨ/∂Nm. However, simple finite-difference derivatives provided sufficient accuracy for our purposes. (27) Faegri, K., Jr.; Almlo¨f, J. J. Comput. Chem. 1986, 7, 396. (28) Scha¨fer, A.; Horn, H.; Ahlrichs, R. J. Chem. Phys. 1992, 97, 2571. Scha¨fer, A.; Huber, C.; Ahlrichs, R. Ibid. 1994, 100, 5829. (29) Gaussian 94, Revision B.2.; Frisch, M. J.; Trucks, G. W.; Schlegel, H. B.; Gill, P. M. W.; Johnson, B. G.; Robb, M. A.; Cheeseman, J. R.; Keith, T.; Petersson, G. A.; Montgomery, J. A.; Raghavachari, K.; Al-Laham, M. A.; Zakrzewski, V. G.; Ortiz, J. V.; Foresman, J. B.; Peng, C. Y.; Ayala, P. Y.; Chen, W.; Wong, M. W.; Andres, J. L.; Replogle, E. S.; Gomperts, R.; Martin, R. L.; Fox, D. J.; Binkley, J. S.; Defrees, D. J.; Baker, J.; Stewart, J. P.; Head-Gordon, M.; Gonzalez, C.; Pople, J. A. Gaussian, Inc., Pittsburgh, PA, 1995. (30) Tatewaki, H.; Huzinaga, S. J. Comput. Chem. 1980, 3, 205. (31) Boys, S. F.; Bernardi, F. J. Mol. Phys. 1970, 19, 553. (32) Raffenetti, R. C. J. Chem. Phys. 1973, 58, 4452. (33) Gill, P. M. W.; Johnson, B. G.; Pople, J. A. Chem. Phys. Lett. 1992, 197, 499. (34) Press, W. H.; Teukolsky, S. A.; Vetterling, W. T.; Flannery, B. P. Numerical Recipes, 2nd ed.; Cambridge University Press: Cambridge, 1992. (35) Basis functions for the first-row atoms can be retrieved by anonymous ftp to parrix1.mpi-stuttgart.mpg.de (user id: ftp, directory: pub/ outgoing, file: dft-basis).
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