  We present a generalization of the Li, Nunes and Vanderbilt density-matrix
method to the case of a non-orthogonal set of basis functions. A representation
of the real-space density matrix is chosen in such a way that only the overlap
matrix, and not its inverse, appears in the energy functional. The generalized
energy functional is shown to be variational with respect to the elements of
the density matrix, which typically remains well localized.
