Page 109 - Strategies and Applications in Quantum Chemistry From Molecular Astrophysics to Molecular Engineer

94 W. KUTZELNIGG
At first glance this looks similar to (B.1). However, there are two differences be-
tween (B.1) and (C.1) that have spectacular consequences.

1. While the function f(x) in (B.1) is convex for all x, the f(x) in (C.1) is concave
from x= 0 to the inflection point and convex from to This
means that the discretization error is negative for intervals between 0 and and
positive between and such that a partial cancellation of the error is possible.
2. While for f ( x ) in (B.1) all derivatives at x= 0 are non-zero, the odd-order
derivatives of the f ( x ) in (C.1) vanish at x = 0. Since these enter the error
formula (A.4) there is no contribution of the boundary at x = 0 to the given by
(A.4), whereas for (appendix B) the derivatives at x = 0 determine
the error.
Prom (A.4) we conclude that for sufficiently small h

Not only is this error negative, meaning that we overestimate the integral (C.1),
but it also appears that the error decreases very rapidly with y, such that one is
tempted to conclude that in the limit (and hence vanishes,
independently of h.

In fact for the odd-order derivatives of f ( x ) vanish at either boundary such
that (A.4) gives the result zero. Of course (A.4) only holds for h smaller than
the radius of convergence of the series. There is no reason why should be
independent of y, and we shall, in fact see that This makes the
estimate (C.2) rather useless because its range of validity is too limited (unlike for
the example of appendix B).
The explicit expression for the discretization error is

Unlike for the example of appendix B a closed summation is not possible. However,
(C.3) allows us to discuss the behaviour of for large h, where the sum is dominated
by the first term

For large h one cannot reduce the error significantly by increasing n. There is
obviously a limiting function for which for large h is given by
(C.4). For small h (C.3) is not convenient because it is slowly convergent.
Fortunately the Fourier expansion method helps us for small and intermediate h
but large n. We get in the limit