J. Phys. Chem. 1988, 92, 3029-3033 Gaussian Basis Sets for High-Quality ab Initio Calculations 3029 Jan Almlof,* Trygve Helgaker, Supercomputer Institute, University of Minnesota, Minneapolis, Minnesota 55455 and Peter R. Taylor ELORET Institute, Sunnyvale, California 94087 (Received: August 7, 1987) Different types of Gaussian basis sets for accurate LCAO calculations are discussed. For calculations designed to recover a substantial portion of the correlation energy, we suggest the use of basis sets comprising natural orbitals from correlated calculations on the atoms. These basis sets have proven to be very efficient in accounting for large fractions of the molecular correlation energy. For cases in which an SCF or MCSCF treatment is adequate the use of a floating basis provides a rapid convergence to the large-basis limit. Introduction In ab initio calculations, deviations from the exact nonrelativistic limit are caused by truncations of the one- and N-particle basis sets. It is desirable to improve these approximations in a systematic way toward a well-defined limit. Due to recent full CI benchmark calculations' the truncation effects on the N-particle basis (the configuration expansion) are now rather well understood. In contrast, the construction of basis sets for LCAO calculations is an area where development has been slow. In Schaefer's words,2 the selection of an appropriate basis set is still "more art than science". In correlated calculations, the size of the final contracted basis set often constitutes a bottleneck, since this determines the amount of work required to optimize the wave function. In contrast, the integral time is of minor concern. Clearly there is a need for compact basis sets even if it means a more expensive evaluation of integrals. A configuration basis can be gradually extended, at least in principle, from one configuration to the full CI limit. Extensions can be made by adding successively less important configurations (or classes of configurations) to the basis, and truncation effects can be explored in a systematic way in a C I calculation. Similarly, a perturbation treatment of correlation energy provides a well- defined path toward the exact solution. In contrast, most studies of one-particle basis set effects have employed rather arbitrarily chosen basis sets of different size. Variations in atomic and molecular properties are erratic and difficult to interpret, since a smaller basis set does not normally span a subspace of the larger one. The basis sets used for calculations on polyatomic molecules are almost invariably Gaussians centered on the positions of the atomic nuclei: +(l,m,n,a) = (x - x,)'a(y - y,)ma(z - za)nne-aa(r-ra)2 (1) Variational optimization of Gaussian basis sets is normally restricted to orbital exponents. Recent studies3 have shown that although many basis sets in present use are incompletely optimized, those deficiencies have a minimal effect on computed atomic and molecular properties. Attempts to improve basis sets further must therefore focus on other parameters of the basis, Le., contraction schemes and/or basis set centers. Basis sets are usually optimized for the free atom (see, however, the recent work by Schlege14on exponent optimization in mole- cules). The conventional approachS is to carry out an SCF cal- culation with an uncontracted basis of fixed size, optimizing the orbital exponents either with a numerical grid-based procedure or by using analytic derivative techniques.) The coefficients for the expansion of the self-consistent AOs are then used as con- traction coefficients. Krishnan et aL6 optimized the exponents and contraction coefficients of their 6-31 1G** sets in MP2 cal- culations. These basis sets were designed specifically for correlated calculations, which is an obvious advantage, even though a var- iational treatment of MP2 energies raises some formal objections. Ideally, procedures such as those outlined above would lead to basis sets made up by the optimum atomic orbitals. An SCF calculation on the atom with this contracted basis would thus reproduce the results for the (usually much larger) primitive basis. However, this is seldom the case in practice. For reasons of convenience and tradition, segmented and nodeless basis sets are often used. In a segmented basis, a primitive Gaussian can contribute to only one contracted function. Clearly, for all atoms beyond He the nodal structure of the atomic orbitals is compro- mised in such a basis. (The popular small STO-nG basis sets are also segmented, but they do incorporate an approximate nodal structure in basis functions with n > 1 + 1. As a result, these basis sets sometimes produce molecular results which are competitive with those from energy-optimized, nodeless basis sets.) In general, the errors introduced by compromising the nodal structure are several orders of magnitude larger than those due to inadequacies in the primitive basis. As a remedy for the node problem, Raffenetti' suggested the use of a general contraction scheme, Le., one where each primitive function can contribute to all the contracted ones. This, of course, would make it possible to use the "true" atomic SCF orbitals as a basis, which would in turn automatically provide the correct nodal structure. There are some problems with this scheme, however. One is that only a minimal basis set is defined. have been made to circumvent this problem by augmenting the basis set with virtual orbitals, using either an N o r an N - 1 electron potential, but the results were not always encouraging. Another objection is, of course, that the entire procedure is biased toward SCF calculations on atoms. Polarization Functions A basis set that provides a good description of the atoms is a necessary, though by no means sufficient, condition for balanced calculations on molecules. In addition to describing the atoms well, the basis must have the necessary flexibility to account for the deformation that the atoms undergo when a molecule is (1) Bauschlicher, C. W., Taylor, P. R. J. Chem. Phys. 1986,85,2779, and (2) Rothenberg, S.; Schaefer, H. F., 111 J. Chem. Phys. 1971, 54, 2764. (3) Almlof, J.; Faegri, K., Jr. J. Comput. Chem. 1986, 7, 396. (4) Tonachini, G.; Schlegel, H. B. J. Chem. Phys. 1987, 87, 514. references cited therein. (5) Huzinaga, S. J. Chem. Phys. 1965, 42, 1293. (6) Krishnan, R.; Binkley, J. S.; Seeger, R.; Pople, J. A. J. Chem. Phys. (7) Raffenetti, R. C. J. Chem. Phys. 1973, 58, 4452. (8) Feller, D. F.; Ruedenberg, K. Theor. Chim. Acta 1979, 52, 231. 1980, 72, 650. 0022-3654/88/2092-3029$01.50/0 0 1988 American Chemical Society