Second-order many-body perturbation theory with non-canonical and non-orthogonal orbitals

Using non-orthogonal virtual and non-canonical (localized) occupied orbitals, a second-order many-body perturbation theory is formulated. The equations of interest are derived from the first-order Bloch equation by using an approach based on a bi-orthogonal basis and operators expressed in second-quantization. A wide variety of partitions of the Hamiltonian are permitted, since any one-body, non-diagonal, zeroth-order Hamiltonian H0 may be used. A matrix formulation is also presented.