Page 110 - Strategies and Applications in Quantum Chemistry From Molecular Astrophysics to Molecular Engineer

CONVERGENCE OF EXPANSIONS IN A GAUSSIAN BASIS 95
This is (at variance with C.3) a rapidly converging series for

For h suficiently small the first term with l = 1 is a good approximation to the sum
(C.5a).

If the upper integration limit in (C.5a) is y = nh rather than i.e. for finite n,
a simple closed expression is not obtained. However, one can estimate the leading
term in an expansion in powers of such that

The asymptotic expansion of (C.5b) in powers of h agrees with (C.2). In fact the
first term neglected in (C.5b) starts with In the limit of course, all
terms of an expansion in powers of h vanish. has an essential singularity at
h = 0.

From this asymptotic expansion in powers of no conclusions on the radius
of convergence of are possible, but there are some hints that the radius of
convergence is that of cosech i.e.the series (A.4) probably converges for

This conjecture is consistent with the result that for the radius of conver-
gence reduces to 0.
At the arguments of the exponential functions in (C.5a) and (C.5b) agree,
which implies that near goes through zero. Between h = 0 and
is slightly negative and rather well approximated by (C.2), while for
increases rapidly and soon approaches 1.
Near the cut-off error

and the discretization error have the same order of magnitude, hence the minimum
of is also close to The minimum error therefore goes as

The prefactor of the exponential in (C.7) is less easily obtained. To get it one
has to solve the transcendental equation for h and insert this into
Numerically one obtains that this factor is close to 1/2.