Page 97 - Strategies and Applications in Quantum Chemistry From Molecular Astrophysics to Molecular Engineer

82 W. KUTZELNIGG

2. We replace the integral from s 1 to s 2 by a sum over a regular grid. We do
this by applying first a variable transformation (to be specified by some criteria)
such that after this transformation an equidistant grid can be used. An estimate
of the discretization error is possible by means of tricky and non-trivial application
of analysis. Details on this are given in the appendix, which is a rather important
part of this paper.
The integral (1.7), which is the starting point for the expansion of a hydrogen-like
1s function in a Gaussian basis, is rather complicated. There is a much simpler
counterpart of (1.7) which is relevant for the expansion of the Coulomb potential
1/r in a Gaussian basis, namely

It has, in fact, been found in a numerical study [21] that this type of expansion has a
very similar convergence behaviours as that of , i.e. that the error also goes as
. The origin of this behaviour is essentially the same for the expansion
of the two functions. Since (1.8) is formally much simpler, it is recommended to
study the expansion of 1/r first.

In fact only the expansion of 1/r will be treated here in detail, while a full study
of the expansion of will be published elsewhere.

The key feature is – both for the expansion of 1/r or in terms of ’even-
tempered' Gaussians – that, for large n, the cut-off error goes as
with h the step size and that the discretization errors goes as with
a and b constants. While – for fixed n – a small h is good for the discretization
error, it is bad for the cut-off error and vice versa. The best compromise is that
which implies that the overall error goes as
The similarity between 1/r and , as far as the expansion in a Gaussian basis
is concerned, leads to another interesting aspect. In many-electron quantum me-
chanics we have in principle to solve both Schrödinger and Poisson equations. We
don't realize this usually because the Poisson equations are first solved in closed
form – which is not possible for the Schrödinger equation. This procedure destroys
the equivalence between the matter field and the electromagnetic field and one may
want to consider an approach in which one solves the Poisson equations numeri-
cally in a basis of Gaussians rather than solving it exactly. Work on these lines is
in progress [21].

2. Expansion of 1/r in a Gaussian basis
We proceed in two steps. Starting point is the identity (1.8) or equivalently