Page 100 - Strategies and Applications in Quantum Chemistry From Molecular Astrophysics to Molecular Engineer

CONVERGENCE OF EXPANSIONS IN A GAUSSIAN BASIS 85

Estimates for the discretization error are derived in the appendix. Unlike the esti-
mates (2.6) these are not obtained as strict inequalities, but rather as leading terms
of asymptotic expansions. For the integral (2.12a) with the integration limits
to the discretization error is (for large n and sufficiently small h, see appendix
E)

To arrive from (E.2) and (E.7b) at (2.14) one must identify a of appendix E with
and realize that (E.2) or equivalently f ( x ) in (C.1) is normalized to 1. To
establish the relation to (2.1) one must multiply (E.2) by The relative
discretization error happens to be independent of r (at least as far as its dominant
term is concerned). Using the arguments of the appendix one finds for the optimum
interval length as function of dimension n of the basis

and for the overall error (for that range of r values for which and are suffiently
small).

3. Estimation of the error of an expectation value of 1/r

In practice one will – in fact – not be interested in the accuracy of f(r) as a
function of r, but rather in the error of matrix elements like that over a hydrogenlike
1s function

as

To estimate this error we insert (2.2) into (3.1b) and integrate first over r such that