Page 112 - Strategies and Applications in Quantum Chemistry From Molecular Astrophysics to Molecular Engineer

CONVERGENCE OF EXPANSIONS IN A GAUSSIAN BASIS 97

and we get for the discretization error

We want to take the limit and in order to obtain There
is the difficulty that in this limit becomes an indefinite phase.
Pictorially it is clear what this means.
The intervals near the maximum of the integrand F(z)give the largest contributions.
If one changes both integration limits, the intervals close to the maximum are not
only changed in length, but also their positions with respect to the maximum are
shifted. It makes, especially for large h, a lot of a difference if the ’innermost’ inter-
val has its center or a border at the maximum. The limit for the integration from
depends somewhat on the position of the innermost interval, especially
for large h.
Since the limit and is not unique, we can either choose a procedure
to make it unique, e.g. fix that there is always a border of an interval at z = 0, or
– what is more realistic – we accept the non-uniqueness and hence an incomplete
information and average over the indefinite phase in some consistent way. Leaving
the phase unspecified we get

Since [25,26]