Page 102 - Strategies and Applications in Quantum Chemistry From Molecular Astrophysics to Molecular Engineer

CONVERGENCE OF EXPANSIONS IN A GAUSSIAN BASIS 87
The minimum with respect to (fornh fixed – and sufficiently large –) is achieved
if

This means that one should choose roughly and that for
fixed h the error decreases exponentially with n (or for fixed n exponentially with
h).
The estimation of the discretization error is fortunately rather easy, relying on the
results of appendix E (which contains the difficult part of the derivation). In fact
the discretization error given by (2.14) is simply proportional to 1/r. Hence

A derivation of the discretization as

is very lengthy, but leads essentially to the same result, which is not so obvious,
since in appendix E we have done the phase-averaging before integrating over r,
and phase averaging and integration over r need not commute.

We use again the argument that the minimum of appears close to the
value of h for which the arguments of the exponential agree, i.e.

There is one difficulty insofar as (3.8) is only an estimate of the absolute value of
the discretization error. It cannot be excluded that (depending on how the limit
is performed, see appendix and have opposite sign. In
this case the minimum absolute error may vanish, while (3109a) is still valid.
Note that h is related to the of an even-tempered basis (1.3) for the H atom
ground state as

Let the smallest orbital exponent in the Gaussian basis be and the largest
Then for sufficiently large n we have

these results, especially that for are in good agreement with results from a purely
numerical study [21].