Page 98 - Strategies and Applications in Quantum Chemistry From Molecular Astrophysics to Molecular Engineer

CONVERGENCE OF EXPANSIONS IN A GAUSSIAN BASIS 83

In doing so we make two ’cut-off ’ errors

The error function erfx has a power series expansion for small x and an asymptotic
expansion for large x

and the following inequalities hold

which allow us to estimate and in two alternative ways.

We have indicated the order of errors of these estimates after the semicolons. We
see that (2.6a) is a close estimate for if while (2.6c) is a close estimate
for . On the other hand the relative error approaches 1, i.e. 100%
for and for . Note that the cut-off error never exceeds 100%.
The range of r-value for which f(r) is a good approximation to 1/r is

In this range the total cut-off error is determined by the
’lower-cut-off’ error (2.6a), with respect to which the ’upper-cut-off error’ (2.6c) is