In order to properly understand the utility of many-body perturbation theory as applied to finite systems, we use the configuration interaction approach to compute the electronic self-energy. The validity of the commonly used $GW$ approximation from many-body perturbation theory is tested by comparing the self-energy explicitly computed according to the diagrammatic expansions to that from the inversion of the Dyson equation. It is constructed as a functional of the interacting Green function $G$ and screened Coulomb interaction $W$. $W$ is explicitly computed through the diagonalization of the many-body Hamiltonian and, thus, takes polarization effects into account beyond the random phase approximation. The $\mathrm{Na}_9{}^{+}$ cluster and the dissociation of the ${\mathrm{C}}_2$ molecule are discussed as examples. We find that the $GW$ approximation yields accurate results for weakly correlated systems with a predominantly single-determinant ground state such as $\mathrm{Na}_9{}^{+}$ clusters. The ${\mathrm{C}}_2$ molecule is a pathological system with multideterminantal ground state where the $GW$ approach ceases to be valid.

• Received 19 December 2006

DOI:https://doi.org/10.1103/PhysRevB.75.205129