Page 106 - Strategies and Applications in Quantum Chemistry From Molecular Astrophysics to Molecular Engineer

CONVERGENCE OF EXPANSIONS IN A GAUSSIAN BASIS 91
Only the cosine terms contribute, because the sine terms vanish at
The larger l and the smaller h the more rapidly oscillating is the cosine factor in
(A.9) and the smaller is the contribution to For sufficiently small h usually the
term with l = 1 dominates in the sum.

A very popular method of numerical integration is that of [23]. It has the
advantage that with n points in a integration one gets the same accuracy
as with 2n points on an equidistant grid – provided that the integrand is well
approximated as a polynominal of degree n, or is expandable in an orthogonal basis
like in Laguerre polynomials. For the examples that we study here this condition
is far from beeing satisfied, and therefore the integration is not supposed to
be helpful.

We now study some special examples that are closely related to those that we are
interested in.

B. THE EXPONENTIAL FUNCTION WITH AN EQUIDISTANT GRID

For the example

a closed expression for the truncation error can be obtained

In this case the relative error is the same for all intervals and one gets

We write to indicate that this is a discretization error.

If one expands (B.3) in powers of one gets the same result as from (A.4) namely

noting that