A is a symmetric matrix of dimension N inact ⇥ N inact , which includes the overlap elements between the spin-orbitals that are common to all the product functions, that are the inactive set of spin-orbitals for the metal plus the spin-orbitals included in the y R 0 group functions. The C block collects the overlap terms between the N act orbitals that can vary in the different product functions, that in this paper coincide con with active set of spinorbitals for the metal. Note that N act ⌧ N inact . The U and V block includes the overlap elements between the active and inactive sets of spin-orbitals. While U, V, and C depends on a specific product functions pair, A never changes. Then, S 1 can be written by the use of the Woodbury matrix identity as :