Generalized Brillouin theorem and generalized forces in SCF and MC SCF theories and their Hartree-Fock limits

A formula for the change of energy is derived, valid for any infinitesimal change of the hamiltonian and the wave function in CI, SCF and MC SCF theories. A general form of MC Hartree-Fock equations is derived in a somewhat new form. A closed MC theory is defined and a condition of self-consistency is derived for this theory. The role of deorthogonalization of spin orbitals is discussed. The Hellmann-Feynman theorem and the generalized Brillouin theorem are rederived in a unique and simple way. It is shown that in the case of the closed MC SCF theory the Brillouin theorem derived by Levy and Berthier can be strengthened. Formulae for generalized forces are derived, valid for closed-shell, openshell and multiconfigurational SCF theories also far from the Hartree-Fock limit.