  Density matrix perturbation theory [Phys. Rev. Lett. Vol. 92, 193001 (2004)]
provides an efficient framework for the linear scaling computation of response
properties [Phys. Rev. Lett. Vol. 92, 193002 (2004)]. In this article, we
generalize density matrix perturbation theory to include properties computed
with a perturbation dependent non-orthogonal basis. Such properties include
analytic derivatives of the energy with respect to nuclear displacement, as
well as magnetic response computed with a field dependent basis. The
non-orthogonal density matrix perturbation theory is developed in the context
of recursive purification methods, which are briefly reviewed.
