Vertex corrections and time-dependent GW approximation

In a many-body perturbation treatment of electronic excitations, one has to solve the so-called Bethe-Salpeter equation (BSE) for the kernel (vertex) describing the interaction between electrons. In general, the BSE exhibits a non-local and frequency-dependent, screened interaction and consequently it is extremely difficult to solve. We have developed a scheme that solves the BSE equation iteratively, including dynamically screening, by considering the change in the electron self-energy upon applying a time-dependent field. The BSE is obtained from the self-energy in the GW approximation (GWA) using the Schwinger functional technique. We apply our formalism to a simple model system and discuss briefly changes in the charge response, when dynamical screening is taken into account.