Green's function approach to the direct perturbation calculation of the excitation energies of closed shell fermion systems

General expressions to the third order of perturbation theory for a direct calculation of the excitation energies of closed shell atomic and molecular systems are presented. The Hartree-Fock ground state orbitals are used as single particle states, but the method may be easily generalized for an arbitrary one-electron basis. The derivation is based on the Green function and diagrammatic approaches. A ``double projection'' scheme is developed which enables not only a very effective reduction of the number of Feynman diagrams which must be explicitly considered but also yields very compact formulas for the excitation energies, which may be easily implemented on a computer. The method is illustrated and checked on a simple six-electronic model system for which the exact solutions are available.