Page 104 - Strategies and Applications in Quantum Chemistry From Molecular Astrophysics to Molecular Engineer

CONVERGENCE OF EXPANSIONS IN A GAUSSIAN BASIS 89
Appendix
Estimation of the discretization error

A. GENERAL CONSIDERATIONS

We want to approximate the integral by dividing the integration domain
into n intervals of the same length h and by approximating f(x) in each interval
by its value at the center of the interval. The discretization error is then

To estimate (in a more traditional way) we make a Taylor expansion of f(x)
around in the k-th interval. We write (assuming that f ( x ) is differentiable
an infinite number of times, which is the case for the functions that we study here)

We express

and proceed similarly with in a next step and so on such that
finally

The are Bernoulli numbers.
The expansion coefficients in (A.4) are essentially those of cosech(x/2).

The equality sign in (A.4) only holds if the series converges. Otherwise the series
is at least asymptotic in the sense that the sum truncated at some k differs from