Page 103 - Strategies and Applications in Quantum Chemistry From Molecular Astrophysics to Molecular Engineer

88 W. KUTZELNIGG
4. Conclusions

We were able to show analytically – in an unexpectedly tricky way (the mathemat-
ical ingredients of which are in the appendix) – that the error of an expansion of
the function in terms of an even-tempered Gaussian basis of dimension n goes
as provided that the two parameters of the even-tempered basis are
optimized.
We have not shown that this is the optimum convergence, in other words whether
there are other (two- or more-parameter) basis sets for which the convergence is
even faster.

The examples given in the appendix give some indications on the properties which
the mapping function has to satisfy that both the cut-off error and the discretization
error decrease exponentially (or faster) with nh and 1/h respectively and don't
depend too strongly on r. Further studies are necessary to settle this problem.

For quantum chemistry the expansion of in a Gaussian basis is, of course,
much more important than that of The formalism is a little more lengthy than
for 1/r, but the essential steps of the derivation are the same. For an even-tempered
basis one has a cut-off error and a discretization error
such that results of the type (2.15) and (2.16) result. Of course, is not well
represented for r very small and r very large. This is even more so for 1/r, but this
wrong behaviour has practically no effect on the rate of convergence of a matrix
representation of the Hamiltonian. This is very different for basis set of type (1.1).
Details will be published elsewhere.

At this point one can conjecture that the relatively rapid convergence of Gaussian
geminals [22]

to describe the correlation cusp, has a somewhat similar origin as the example
studied here, and goes probably also as exp with n the dimension of the
geminal basis.

Acknowledgement
The author thanks Stefan Vogtner for numerical studies of expansions of 1/r in a
Gaussian basis which have challenged the present analytic investigation. Discus-
sions with Christoph van Wüllen and Wim Klopper on this subject have been very
helpful.
This paper is dedicated to Gaston Berthier, from whom I have learned a lot. Al-
though Berthier's publications have mostly dealt with applications of quantum
mechanical methods to chemical problems, he never liked black boxes or unjustified
approximations even if they appeared to work. The question why the quantum
chemical machinery does so well although it often lies on rather weak grounds has
concerned him very much. I am therefore convinced that he will appreciate this
excursion to applied mathematics.