5. Localised basis-set

One elegant and popular choice of basis in traditional calculations has been the plane-wave basis. However, because of the extended nature of these basis functions they cannot be used in linear-scaling calculations, and a different choice has to be made, in which the basis functions are localised in real-space. Gaussians [144] are a popular choice since many quantities can be calculated analytically [145], an advantage shared by the basis-set proposed here. Other choices include truncated atomic orbitals [146], -splines or ``blip'' functions [147], wavelets [148] and real-space grids [149].

In this chapter we consider a localised spherical-wave basis set suitable for
linear-scaling
total-energy pseudopotential calculations. The basis-set is
conveniently truncated using a single parameter, the kinetic energy
cut-off used with the plane-wave basis. We present analytic results
for the overlap integrals between any two basis functions centred on
different sites, as well as for the kinetic energy matrix-elements
which can, therefore, be evaluated accurately in real-space. Two methods
for analytically performing the projection of the basis states onto
angular momentum states required for the use of non-local
pseudopotentials are also presented. This work has been published in
[150].

- 5.1 Introduction
- 5.2 Origin of the basis functions
- 5.3 Fourier transform of the basis functions
- 5.4 Overlap matrix elements
- 5.5 Kinetic energy matrix elements
- 5.6 Non-local pseudopotential
- 5.7 Computational implementation