Page 105 - Strategies and Applications in Quantum Chemistry From Molecular Astrophysics to Molecular Engineer

90 W. KUTZELNIGG
the exact by This also holds if f is only (2k – 1) times differentiable,
such that one has has to truncate the expansion anyway.

The discretization studied here is related to that of the Euler-McLaurin method
well-known in numerical mathematics (see e.g. [23]). The difference is that in this
method one approximates the mean value of f ( x ) in the interval by the average of
the values at the boundaries of the interval, while we approximate it by its value
at the center of the interval. This choice is more closely related to the expansion of
a function in a basis.

For the Euler-McLaurin discretization an error formula similar to (A.4) holds,
namely without the factor which corresponds to the expansion co-
efficients of coth(x/2).
An equidistant integration grid may not be the best choice. Let us therefore consider
that we perform a variable transformation in the integral before we discretize.

To define the error by (A.1) and to apply the error formula (A.4) we must replace
and and respectively

We are mainly interested in the transformation

Eqn. (A.4) or its counterpart with h replaced by and allows
us to estimate for small h it is less convenient for so large that
the Taylor series within an interval converges slowly or diverges.
There is an alternative – and for our purposes more powerful – way to estimate
the discretization error, namely in terms of the Fourier expansion of a periodic
function. We write see (A.1), as [24]