Page 111 - Strategies and Applications in Quantum Chemistry From Molecular Astrophysics to Molecular Engineer
P. 111
96 W. KUTZELNIGG
The essential message is that the error goes as and the optimum h as
This means very fast convergence with the number n of intervals, very
different from the example of appendix B where the error only decreased as
In this appendix we have argued that (C.5b) is valid for ’sufficiently
small’ h. That meant that h should not be significantly larger than which
is not very restrictive. However (C.2) only holds for h satisfying (C.6), which limits
its validity to extremely small h, in the limit (C.2) becomes even invalid.
The two references to ‘small’ h must be clearly distinguished.
D. THE EXPONENTIAL FUNCTION WITH A LOGARITHMICALLY
EQUIDISTANT GRID
We consider again (B.1), but with the transformation
The lower integration limit is now changed from 0 to If we want to discretize,
we must also introduce a lower cut-off. I.e. rather than (D.2) we must consider
The integrand in (D.3) falls off rapidly for but more slowly for
Therefore the ‘lower’ cut-off is more critical than the ‘upper’ cut-off We have
We divide the domain into n intervals, hence
We minimize the error with respect to for hn fixed, ignoring terms of
