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].