Page 108 - Strategies and Applications in Quantum Chemistry From Molecular Astrophysics to Molecular Engineer

CONVERGENCE OF EXPANSIONS IN A GAUSSIAN BASIS 93

and

which implies

from which one is immediately led to the equivalence of (B.8) and (B.9)

If one limits the sum (B.9) to the term with l = 1 and expands in powers of h, the
coefficient of the leading term in instead of the correct value
(see B.10). Convergence with l for small h is pretty (though not extremely)
fast.
We want to make the overall error minimal for fixed n. We express the total error
in terms of h and n

We want to minimize ε as function of h for fixed n. Since the discretization error
only depends on h, it is obvious that one should make h as small as possible, in
order to minimize it. We can therefore assume that h is so small that

Asymptotically for large n the solution of this transcendental equation is

Since lnn is a slowly varying function of n, the error goes essentially as This is
the typical behaviour of a discretization error for a numerical integration [23], but
is atypical for the examples that we want to study.

C. THE GAUSSIAN WITH AN EQUIDISTANT GRID

Our next example is