Generalization of Cramer's rule and its application to the projection of Hartree-Fock wave function

Abstract

We generalize the Cramer's rule of linear algebra. We apply it to calculate the spectra of nucleus by applying Hill-Wheeler projection operator to Hartree-Fock wave function, and to derive L\"owdin formula and Thouless theorem. We derive by an elementary method the infinitesimal or L\"owdin projection operators and its integral representation to be useful for the projection of Slater determinant.