Page 113 - Strategies and Applications in Quantum Chemistry From Molecular Astrophysics to Molecular Engineer

98 W. KUTZELNIGG
the term with l = 1 dominates for sufficiently small h. If we take only this term in
(D.8) and form the mean square average over the phase we get

This estimate is independent of a as is the estimate (D.6c) of the cut-off error.
Note that is a monotonically increasing function of h, while both
Re and Im oscillate between and

The discretization error for finite integration limits and contains in ad-
dition to (D.8) two extra terms (under the sum) that contain incomplete Gamma
functions. We don't need their explicit form for the estimation of the dominating
part of the overall error. Of course, expanding these extra terms in powers of h
would lead to the error estimation (A.4), that holds for extremely small h (and
sufficiently small l) which is rather irrelevant in the present context.
Somewhat similar to appendix C we have a discretization error that goes as
and a cut-off error The minimum as function of h is
achieved (for large n) if

If and happen to have opposite sign, the optimum error vanishes, while close
to its zero ε (h) has an inflection point.

The optimum interval length goes as and the error as exp This is
certainly a much faster convergence than for the choice of an equidistant grid for
the exponential function as studied in appendix B.
We have not considered the next term in an 1/n expansion of which
would be needed to get the prefactor of

E. A GAUSSIAN WITH A LOGARITHMICALLY EQUIDISTANT GRID
We consider now (C.1) but with the transformation

Everything is similar to appendix D.

Now (D.3) is replaced by