Page 94 - Strategies and Applications in Quantum Chemistry From Molecular Astrophysics to Molecular Engineer

Convergence of Expansions in a Gaussian Basis

W. KUTZELNIGG
Lehrstuhl für Theoretische Chemie, Ruhr- Universität Bochum,
Universitätsstr. 150, D-4630 Bochum, Germany

1. Introduction
Few papers have had as much impact on the progress of ab-initio quantum chemistry
as that of Boys [1] where he proposed to use Gaussians (GTOs) as basis sets. The
great breakthrough of ab-initio theory would never have been possible without the
invention of Gaussians. Nevertheless, even nowadays it is difficult to explain to
a beginner why one should rely on Gaussians, which have the wrong behaviour
both near the nuclei and very far from them. The ease; with which two-electron
integrals over GTOs can be computed is certainly an argument. However, if one
has thought a little bit on the importance of choosing basis sets with the right
behaviour at the singularities of the Hamiltonian [2], one cannot but be deeply
surprised that expansions in GTOs converge decently well in spite of their failure
at the singularities of the Hamiltonian.
To appreciate this point somewhat better it is useful to compare three types of
Gaussian basis sets, (a) a set of Gaussians with common orbital exponents (for one
l) but a sequence of principle quantum-numbers

(We consider here only the case of a single center), (b) the same set (1.1) but with
n – l = 1,2,3,4, ..., (c) a set of Gaussians with the lowest possible n for each l, but
with a sequence of orbital exponents

Sets of orbital exponents have been proposed mainly by Huzinaga [3], van
Duijneveldt [4], Pople et al. [5]. A systematic construction of basis sets of arbitrary
dimension is possible in terms of the ’even tempered’ concept of Ruedenberg et
al. [6,7 ], or of some more sophisticated generalizations [8,9,10]. For a recent
comprehensive review on basis sets see Feller and Davidson [11].

It does not make a significant difference that in practice one uses ’cartesian Gaus-
sians’ rather than Gaussians with explicit inclusion of spherical harmonics. One
79
Y. Ellinger and M. Defranceschi (eds.), Strategies and Applications in Quantum Chemistry, 79–101.
© 1996 Kluwer Academic Publishers. Printed in the Netherlands.