Löwdin first presented a formula for the matrix elements of the Hamiltonian between Slater determinants expressed in terms of non-orthogonal spin-orbitals , and after him different methods have been suggested to avoid the need of calculating the inverse, allowing for a rapid evaluation of the cofactors , but in general they do not explicitly consider that the determinants from the set employed in a calculation may differ only in few spin-orbitals from a reference determinant. Hayes and Stone are the first who emphasized the advantages coming from prop-erly taking this peculiarity into account , followed by Figari and Magnasco, who generalized the Hayes' results . Starting from the latter developments, we present here a slightly modified approach, which allows for both computational efficiency and generality.