Cumulant expansion of the retarded one-electron Green function

The cumulant expansion is a powerful approach for including correlation effects in electronic structure calculations beyond the $GW$ approximation. However, the expansion is not generally valid, as current implementations ignore terms that mix particle and hole states and lead to partial occupation numbers of one-electron states. These limitations are corrected here using a cumulant expansion of the retarded one-electron Green's function that includes both particle and hole contributions. The approach provides a consistent framework that improves on the $GW$ approximation to the spectral function without additional computational effort. The method is illustrated with results for the homogeneous electron gas and comparisons to experiment and other methods.