Modified potentials in many-body perturbation theory: Three-body and four-body contributions

Three-body and four-body contributions to the third-order diagrammatic perturbative energy expansion are evaluated for the electronic energy, using both the standard self-consistent-field potential $V^N$ and a modified potential $V^{N-1\mathrm{}}$ to determine the unoccupied states. The [$\frac{2}{1}$] Padé approximants to the energy are constructed and first-order perturbative wave functions are inserted into the Rayleigh quotient to yield rigorous upper bounds for the energy. The results indicate that the standard $V^N$ potential generates a more suitable virtual orbital basis for the calculations.