Page 99 - Strategies and Applications in Quantum Chemistry From Molecular Astrophysics to Molecular Engineer

84 W. KUTZELNIGG
negligible. In a wide range of r ’flat’ gaussian are more important than ’steep’ ones,
which only matter for small r.
The next step on the way to an expansion of 1/r in a Gaussian basis is to replace
the integral (2.2) by a sum. Before we divide the range between into n
intervals, we apply a variable transformations, such that after this transformation
an equidistant grid can be used.

Let us define the normalized functions (with respect to square integration over r)

Then (2.8b) becomes

Obviously we must choose p(x) such that the domain between and – which
have different orders of magnitude – is covered in a balanced way. One may fur-
ther require that all g(r, x) have about the same weight in the sum. The latter
requirement leads to the condition

Obviously an exponential mapping looks also good in the sense of the first criterion.
One sees easily that f(r) is independent of the choice of such that we may as
well take Of course, this is only a plausiblity argument and we need a
rigorous criterion for the optimum mapping. We come back to this problem in the
conclusions.
We hence have

We now approximate (2.12a) as a sum (with the discretization error).