SIMPLIFIED INTRODUCTION TO AB INITIO BASIS SETS. TERMS AND NOTATION

Jan K. Labanowski, Ohio Supercomputer Center,
1224 Kinnear Rd., Columbus, OH 43212-1163,


E-mail: jkl@ccl.net, JKL@OHSTPY.BITNET

INTRODUCTION

Some straightforward reviews on basis sets are available: (Ahlrich & Taylor, 1981), (Andzelm et al., 1984), (Dunning & Hay, 1977), (Feller & Davidson, 1986), (Feller & Davidson, 1990), (Poirier et al., 1985).

Historically, the quantum calculations for molecules were performed as LCAO MO, i.e. Linear Combination of Atomic Orbitals - Molecular Orbitals. This means that molecular orbitals are formed as a linear combination of atomic orbitals:

where  is the i-th molecular orbital,  are the coefficients of linear combination,  is the  -th atomic orbital, and n is the number of atomic orbitals.

Strictly speaking, Atomic Orbitals (AO) are solutions of the Hartree-Fock equations for the atom, i.e. a wave functions for a single electron in the atom. Anything else is not really an atomic orbital. Some things are similar though, and there is a lot of confusion in the terminology used. Later on, the term atomic orbital was replaced by "basis function" or "contraction," when appropriate. Early, the Slater Type Orbitals (STO's) were used as basis functions due to their similarity to atomic orbitals of the hydrogen atom. They are described by the function depending on spherical coordinates:

where N is a normalization constant,  is called "exponent". The r,  , and  are spherical coordinates, and  is the angular momentum part (function describing "shape").The n, l, and m are quantum numbers: principal, angular momentum, and magnetic; respectively.

Unfortunately, functions of this kind are not suitable for fast calculations of necessary two-electron integrals. That is why, the Gaussian Type Orbitals (GTOs) were introduced. You can approximate the shape of the STO function by summing up a number of GTOs with different exponents and coefficients. Even if you use 4 or 5 GTO's to represent STO, you will still calculate your integrals much faster than if original STOs are used. The GTO (called also cartesian gaussian) is expressed as:

where N is a normalization constant,  is called "exponent". The x, y, and z are cartesian coordinates. The l, m, and n ARE NOT QUANTUM NUMBERS but simply integral exponents at cartesian coordinates.  .

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http://www.ccl.net/cca/documents/basis-sets/basis.html