The results in equations
5.19, 5.20, 5.22 and
5.23 have been
written in a form which shows that in general each term can be
represented by a real numerical prefactor, integers which are the
powers of
and one further integer
to signify the presence of one of the terms
. When these terms are combined and differentiated by the
, the general term also needs integers to represent powers
of
. Therefore a general term in the
expressions for
and
could be
represented by a data structure consisting of one real variable
and ten integer variables as follows:
(5.45) |
(5.46) |
The cost of calculating the analytic matrix elements increases dramatically as higher angular momentum components are included. In general, a much smaller value of is used than is ``recommended'' by the kinetic energy cut-off. However, these basis functions are being used to describe functions localised in overlapping regions, and in this instance, a degree of ``under-completeness'' is desirable. If the basis functions formed a complete set (up to a given kinetic energy cut-off) in each support region, then a variation which is confined to the overlapping region can be equally described by variations in either region. Symmetric and antisymmetric combinations of these variations can be formed, the antisymmetric variation vanishing and thus leaving the density-matrix invariant. Therefore this superposition results in directions in the parameter space with very small curvature which degrade the efficiency of minimisation algorithms (see section 6.2.3). When working with overlapping support functions, it is therefore better to treat as a convergence parameter along with, rather than derived from, .