A generalization of the reduced density matrix theory when the number of particles is not fixed is given here. The first and second order reduced density matrices are defined with respect to a Bardeen-Cooper-Schrieffer ground state for molecules with an even number of electrons. This formalism is extended to molecules with an odd number of electrons. Valatin’s generalized Hartree-Fock method is applied to the functional of energy. A new one-particle operator, (V) over cap(q), is thus obtained. This operator has two new terms. It takes into account all the correlation effects mainly through a new non-local potential (L) under bar(q) -- the Exchange and Time Inversion Potential. The operator (V) over cap(q) thus correlates the motion of two particles with anti-parallel spins and determines a new set of spin-orbitals. The new set of equations is established and solved by using the MO-LCAO approximation. The method was tested for several molecules with good results. The calculated vertical ionization potentials compare satisfactorily with the experimental data.