Basis Sets for Molecules

For a given electronic Hamiltonian (that is, a proper description of the electron-nucleus and electron-electron interactions), the LCAO ansatz may deliver the molecular orbitals ipi and the many-electron wave function Y provided that there is a set of useful basis functions (p, for example, atomic orbitals. At this point, it is probably the right time to review briefly what type of atomic orbitals are mostly used within molecular quantum chemistry. By 

In general, an atomic orbital - a solution of Schrodinger s equation for a one-electron atom of a given atomic number Z - may be formulated as a product of a radius- (r) and an angle-dependent (0, f) component such as in 

It is possible, of course, to simply take the numerical solution of the radial Schrodinger equation for the particular atom under question and use it as the radial part of the atomic orbital. It is also possible, and mathematically advantageous, to approximate the exponential decay of the radial part of the atomic orbital Rn (r) in the outer region. One particularly successful approach goes back to Slater s recipe [13], 

As the reader may have noticed, this is just the same expression that determines the sizes of Pauling s ionic radii (see Section 1.1) small orbitals (and thus atoms) will be characterized by large exponents. Sets of Slater exponents have been tabulated for various atoms, and there are several different recipes for determining values at the outset. We mention, however, that any orbital exponent needs modification whenever an atom s effective potential within a molecule deviates substantially from the original atomic potential, because then the orbital exponent must reflect the new potentials. This is a general (and also serious) problem with fixed basis sets. 

Nonetheless, Slater-type orbitals (STOs, in short) are a very reasonable approximation for the true atomic orbitals each atomic orbital is replaced by 

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A minimum basis set for molecules containing C, H, O, and N would consist of 2s, 2p, 2py, and 2p oibitals for each C, N, and O and a 1 j orbital for each hydrogen. The basis sets are mathematical expressions describing the properties of the atomic orbitals. 

S. Huzinaga, J. Andzehn, M. Klobukowski, E. Radzio-Andzelm, Y. Sakai and H. Tatewaki, Gaussian Basis Sets for Moleculations, Elsevier, New York, 1984. 

This method of representing the molecular orbital wave function in terms of combinations of atomic orbital wave functions is known as the linear combination of atomic orbitals-molecular orbital (LCAO-MO) approximation. The combination of atomic orbitals chosen is called the basis set. A minimal basis set for molecules containing C, H, O, and N would consist of Is-, 2s-, Ip -, 2py-, and 2p -orbitals for C, O, and N, and a Is-orbital for hydrogen, inclusion of additional orbitals in the basis set leads to an extended basis set. Economy dictates use of minimal basis sets whenever possible. 

L. Adamowicz and R. J. Bartlett, Chem. Phys. Lett., 105, 167 (1984). Extended Floating Spherical Gaussian Basis Sets for Molecules. Generation Procedure and Results for HjO. 

The criteria for choice between gaussian and exponential basis sets for molecules do not seem obvious at present. In fact, it appears to be constructive to regard them as being complementary, depending on the specific physical property required from molecular electronic structure calculations. 

Flehre W J, Ditchfieid R and Popie J A 1972 Self-consistent molecular-orbital methods XII. Further extension of Gaussian-type basis sets for use in molecular orbital studies of organic molecules J. Chem. Phys. 56 2257-61 Flariharan P C and Popie J A 1973 The influence of polarization functions on molecular orbital hydrogenation energies Theoret. Chim. Acta. 28 213-22... 

A very important difference between H2 and molecular orbital calculations is electron correlation. Election correlation is the term used to describe interactions between elections in the same molecule. In the hydrogen molecule ion, there is only one election, so there can be no election correlation. The designators given to the calculations in Table 10-1 indicate first an electron correlation method and second a basis set, for example, MP2/6-31 G(d,p) designates a Moeller-Plesset electron coiTclation extension beyond the Hartiee-Fock limit canied out with a 6-31G(d,p) basis set. 

The smallest basis sets are called minimal basis sets. The most popular minimal basis set is the STO—3G set. This notation indicates that the basis set approximates the shape of a STO orbital by using a single contraction of three GTO orbitals. One such contraction would then be used for each orbital, which is the dehnition of a minimal basis. Minimal basis sets are used for very large molecules, qualitative results, and in certain cases quantitative results. There are STO—nG basis sets for n — 2—6. Another popular minimal basis set is the MINI set described below. 

J Chem. Phys., 52, 431 (1970)] is a relatively inexpensive one and can be used for calculations on quite large molecules. It is minimal in the sense of having the smallest number of functions per atom required to describe the occupied atomic orbitals of that atom. This is not exactly true, since one usually considers Is, 2s, and 2p, i.e., five functions, to construct a minimal basis set for Li and Be, for example, even though the 2p orbital is not occupied in these atoms. The 2sp (2s and 2p), 3sp, 4sp, 3d,. .., etc. orbitals are always lumped together as a shell , however. The minimal basis set thus consists of 1 function for H and He, 5 functions for Li to Ne, 9 functions for Na to Ar, 13 functions for Kand Ca, 18 functions for Sc to Kr,. .., etc. Because the minimal basis set is so small, it generally can not lead to quantitatively accurate results. It does, however, contain the essentials of chemical bonding and many useful qualitative results can be obtained. 

The Veillard basis set [23] (1 ls,9p) has been used for A1 and Si, and the (1 ls,6p) basis of the same author has been retained for Mg. However, three p orbitals have been added to this last basis set, their exponents beeing calculated by downward extrapolation. The basis sets for Al, Si and Mg have been contracted in a triple-zeta type. For the hydrogen atom, the Dunning [24] triple-zeta basis set has been used. We have extended these basis sets by mean of a s-type bond function. We have optimized the exponents a and locations d of these eccentric polarization functions, and the internuclear distance R of each of the studied molecules. These optimized parameters are given in Table 3. 

Hehre, W. J., Ditchfield, R., Pople, J. A., 1972, Self-Consistent Molecular Orbital Methods. XII. Further Extensions of Gaussian-Type Basis Sets for Use in Molecular Orbital Studies of Organic Molecules J. Chem. Phys., 56, 2257. 

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