## Basis Sets Used in Molecular Orbital Calculations

1. Introduction

Each molecular orbital (one electron function) yi is expressed as a linear combination of n basis functions Fm. By convention, molecular orbitals are abbreviated by the greek letter y (psi), while the basis functions are symbolized by the greek letter F (phi).

The coefficients cmi are called molecular orbital expansion coefficients or simply MO coefficients. Usually these basis functions are located at the center of atoms and are therefore often called atomic basis functions. This is, however, not strictly true and in order to avoid confusion, we will in the following use the term basis function.

The basis functions used in molecular orbital calculations are usually described through an abbreviation or acronym such as "6-31G(d)". The complete description of the theoretical model used in MO calculations is given in the format "theoretical method/basis set". A calculation using Hartree-Fock theory in combination with the   6-31G(d) basis set would then be referred to as "HF/6-31G(d)". Often the molecular geometries are optimized at one level of theory (method 1) and relative energies are then recomputed using these geometries, but a different theoretical method (method 2). The conventional description of this level of theory is "method 2//method 1". If, for example, geometry optimization has been performed at the HF/6-31G(d) level of theory and relative energies have then been calculated at the MP2/6-31G(d) level, this approach would be described as "MP2/6-31G(d)//HF/6-31G(d)".

In principle, any mathematical function can be used to construct molecular orbitals. There are, however, only a few solutions, which have found widespread application.

1.1 Slater type orbitals (STOs)

STOs are constructed from a radial part describing the radial extend of the orbital and an angular part describing the shape of the orbital (equ. 3).

The radial part rn-1 exp(-zr) depends on the distance r from the origin of the basis function (usually the location of the nucleus), the orbital exponent z (greek: zeta), and the principal quantum number n. The spherical part Ylm depends on the angular quantum number l and the magnetic quantum number m. The normalization constant N is chosen such that the integral over the square of the basis function yields unity. Taking a 1s orbital as an example, the spherical part as well as the rn-1-factor equate to unity and after evaluating the normalization constant, we get:

Slater type orbitals are not very useful for electronic structure calculations, because two electron integrals over STOs are difficult to calculate.

1.2 Gaussian type orbitals (GTOs)

GTOs are also constructed from a radial and a spherical part, but the radial part now has a different dependence on r (equ. 5):

The radial extend is proportional to exp(-ar2), a being the exponent. The normalization constant N serves a similar purpose as for STOs. The spherical part is now expressed through the cartesian coordinates x,y, and z in powers of a, b, and c, respectively. The exponents a, b, and c, should not be confused with quantum numbers. The sum of these exponents L=a+b+c is used to define the angular momentum of  the basis functions: s-type (L=0), p-type (L=1), d-type (L=2), f-type (L=3), g-type (L=4) . . .. For a simple Gaussian function representing an 1s-type orbital, we get:

Gaussian type orbitals are very useful for electronic structure calculations, because two electron integrals over GTOs are NOT difficult to calculate. GTOs have the advantage that all the intergrals needed for the LCAO calculations can be done by using simple formulae. (S. F. Boys, Proceedings of the Royal Society of London Series A, 1950, 200, 542-554.)

Finally, a comparison between STOs and GTOs: There are two major differences (i) the behaviour for r = 0 and for (ii) r = ∞. The STO displays a cusp at r = 0 while the GTO has an incorrect behaviour at the nucleus. Moreover GTOs fall off too quickly as the electron gets further from the nucleus.

last changes: 01.04.2008, AS
questions & comments to: axel.schulz@uni-rostock.de