Page 96 - Strategies and Applications in Quantum Chemistry From Molecular Astrophysics to Molecular Engineer

CONVERGENCE OF EXPANSIONS IN A GAUSSIAN BASIS 81

which is for the exact H ground state wave function, but which
vanishes for the expansion in (1.2) for all finite n.

Similarly is equal to while this second derivative is negative for
any finite expansion with an apparent divergency to for Some prop-
erties like the density at the nucleus and the variance of the energy converge very
slowly to the exact values. These are, nevertheless, relatively minor defects.
Again by adding to the basis at least one function that has the correct behaviour
at r = 0, e.g.

the convergence can be speeded up – and the last-mentioned defects can be removed
[10,16]. However, the improvement is much less spectacular than for basis (1.1) –
unless one is interested in the density at the nucleus or the variance of the energy.
There are hints [9,10,18] that the rate of convergence for basis sets of type (1.2) is
even better than (1.4), if one uses better optimized basis sets than those of even
tempered type (1.3),

and that the same convergence pattern is found for the expansion of as for

There is no doubt that the convergence behaviour of standard Gaussians is much
better than one should have expected in view of their failure at
What is the fundamental difference of basis sets of type (1.1) and (1.2)? Without
claiming to give a definite answer we can say that the expansion in the basis (1.1)
is closely related to the expansion in terms of Laguerre functions, i.e. in a typical
orthogonal basis and that a theory much like that for Fourier series applies. There it
generally holds that the singularities of the function to be expanded determine the
rate of convergence [19]. An expansion in the basis (1.2) can hardly be traced back
to something like a Fourier series. It must rather be viewed as a discretization of
the integral representation of an exponential (or another exponential-like) function.

and entirely different features determine the error. (As to a direct application of a
numerical discretization of the integral transformation (1.7) see ref. 20).

To get analytic results for the convergence behaviour of an expansion in a Gaussian
basis we shall proceed in two steps.
1. We replace the integral (1.7) by an integral from s 1 to s 2 rather than from 0 to
The errors due to this restriction of the integration domain – the cut-off errors
– can easily be estimated.