8. Conclusions

We have shown that it is possible to construct a set of basis functions which are solutions of the free-electron Schrödinger equation, subject to being localised in spherical regions. Basis functions within the same region are mutually orthogonal, avoiding the problem of the overlap matrix becoming singular when the size of the basis set is increased. It is also possible to truncate the basis set using the kinetic energy cut-off used to truncate the plane-wave basis.

We have shown in detail how to obtain analytic results for the overlap integral between any two basis functions, and presented these in a form which can be implemented computationally.

The same results can be adapted to obtain matrix elements of the kinetic energy operator, providing an efficient and accurate method of computing the kinetic energy in real space and avoiding the use of finite difference methods.

The projection of basis functions onto ionic core angular momentum states can also be performed analytically so that non-local pseudopotentials can be used.