Calculation of electron density of periodic systems using non-orthogonal localised orbitals

Methods for calculating an electron density of a periodic crystal constructed using non-orthogonal localised orbitals are discussed. We demonstrate that an existing method based on the matrix expansion of the inverse of the overlap matrix into a power series can only be used when the orbitals are highly localised (e.g. ionic systems). In other cases including covalent crystals or those with an intermediate type of chemical bonding this method may be either numerically inefficient or fail altogether. Instead, we suggest an exact and numerically efficient method which can be used for orbitals of practically arbitrary localisation. Theory is illustrated by numerical calculations on a model system.