We present a set of localised functions which are related to the plane-wave basis set and share some of its attractive features. A significant problem associated with localised basis functions is that they are not in general orthogonal, so that as the size of the basis is increased, the overlap matrix becomes singular. We demonstrate that the basis functions introduced here are orthogonal, by construction, to others centred on the same site, and that the overlap matrix elements for functions centred on different sites can be calculated analytically, and hence evaluated efficiently and accurately when implemented computationally.
Another disadvantage of using basis functions localised in real-space arises in the calculation of the action of the kinetic energy operator. To take advantage of the localisation it is necessary to focus on real-space and calculate all quantities in that representation. However, since the kinetic energy operator is diagonal in reciprocal-space, the kinetic energy matrix elements are most naturally calculated in reciprocal-space. Methods to evaluate the kinetic energy using finite-difference schemes can be inaccurate when used with localised functions. It is particularly difficult to obtain accurate values for second derivatives in the vicinity of the support region boundaries so that this error is of the order of the surface area to volume ratio. For the one-centre integrals this is not significant, but for the two-centre integrals, the intersection of two spheres may have a large surface area to volume ratio and the error may therefore be large. Indeed, investigations show that the estimates of such integrals obtained by finite differences may often be of the wrong sign! With the new choice of basis, the matrix-elements of the kinetic energy operator between any two functions can also be calculated analytically, thereby overcoming this problem.
One final advantage arises in the inclusion of non-local pseudopotentials which traditionally required significant computational effort. We present two methods of obtaining the matrix-elements of the non-local pseudopotential operator by performing the projection of the basis function onto a core angular momentum state analytically.