Electron density of periodic systems derived from non-orthogonal localized orbitals

Development of a quantum cluster embedding scheme based on non-orthogonal localized orbitals requires an efficient numerical method for calculating the electron density of the reference periodic crystal. We demonstrate that an existing method due to Loewdin based on the inverse overlap matrix expansion can only be used when the orbitals are highly localized (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 localization. Theory is illustrated by numerical calculations on a model system