Page 101 - Strategies and Applications in Quantum Chemistry From Molecular Astrophysics to Molecular Engineer

86 W. KUTZELNIGG

The analytic expression for this integral is

The limit of (3.3) is not obvious. To get it we must expand the
first line of (3.3) in powers of and insert the asymptotic expansion of erfc in the
second line before we collect powers of . We get for the first and second lines of
(3.3) respectively

Of course, and as defined by (3.4) are the ’cut-off’ errors due to limitation
of the integration domain to to

We next approximate the integral (3.3) by a numerical integration after performing
the variable transformation (2.11) with . This means we first replace (3.3) by

Before we study the ’discretization errors’ let us look on how the ’cut-off errors’
and depend on the number of points chosen in (3.5c). In view of (3.5a), (3.4)
and (2.13b) we have