Page 95 - Strategies and Applications in Quantum Chemistry From Molecular Astrophysics to Molecular Engineer

80 W. KUTZELNIGG

should also mention that there is a fourth type of basis sets (d), namely that of
Gaussian lobes [12,13] i.e. functions of type (1.2) with only but with
centers spread over the molecule, not only at the position of the nuclei. These don't
differ basically from case (c).
It has been shown [14] for both types of basis sets (1.1) and (1.2) that a given set
of dimension n can be regarded as a member of a family of basis sets that in
the limit become complete both in the ordinary sense and with respect to
a norm in the Sobolev space – which is the condition for the eigenvalues and
eigenfunctions of a Hamiltonian to converge to the exact ones. However, as to the
speed of convergence the two basis sets (1.1) and (1.2) differ fundamentally.

In a careful study of basis sets of type (1.1) applied to the ground state of the
hydrogen atom Klahn and Morgan [15] were able to show that the error of the energy
goes as (n being the dimension of the basis) for fixed By optimization
of one can achieve [1.6] that the error goes as . Anyhow this rate of
convergence is as bad as one can imagine and it makes basis set (1.1) absolutely
useless. Convergence as an inverse-power law with a small exponent generally
prevents accurate calculations, as is known from the slow convergence of the partial-
wave expansion for the interelectronic coordinate (equivalently the convergence of
a CI for an atom with the highest angular equantum number l in the basis set
included), where the error goes as Inclusion of a single term with the
right behaviour at the Coulomb singularity (a ’comparison function’ [2]) improves
the rate of convergence, such that the error goes as for the expansion of the
H-atom ground state in basis (1.1) [16] or as for the convergence of a CI
[17].

If one includes functions with n – l even in (1.1) (i.e. one uses set b) the basis is
formally overcomplete. However the error decreases exponentially with the size of
the basis [2,16]. Unfortunately for this type of basis the evaluation of the integrals
is practically as difficult as for Slater type basis functions, such that basis sets of
type (b) have not been used in practice.

The rate of convergence of expansions in the basis (1.2) has received little attention
except for purely numerical studies [3,7,8,9,16] which indicated that the convergence
is at least (unlike for bais set of type) not frustratingly slow. Rather detailed studies
were performed for the even-tempered basis set, i.e. for exponents constructed from
two parameters and (for each l)

In a numerical study of basis sets (a), (b) and (c) for the H atom ground state W.
Klopper and the present author [16] found that for the basis (c) the error goes as

i.e. the convergence is not exponential (which would be ideal, i.e. generally the
case for a basis that describes the singularities correctly) but almost so. This
does not only hold for the energy, but for other properties as well. However there
are properties for which the limit does not yield the correct result, e.g.