Lagrangian Z-vector approach to Bethe-Salpeter analytic gradients: Assessing approximations

We present an implementation of excited-state analytic gradients within the Bethe-Salpeter equation formalism using an adapted Lagrangian Z-vector approach with a cost independent of the number of perturbations. We focus on excited-state electronic dipole moments associated with the derivatives of the excited-state energy with respect to an electric field. In this framework, we assess the accuracy of neglecting the screened Coulomb potential derivatives, a common approximation in the Bethe-Salpeter community, as well as the impact of replacing the GW quasiparticle energy gradients by their Kohn-Sham analogs. The pros and cons of these approaches are benchmarked using both a set of small molecules for which very accurate reference data are available and the challenging case of increasingly extended push-pull oligomer chains. The resulting approximate Bethe-Salpeter analytic gradients are shown to compare well with the most accurate time-dependent density-functional theory (TD-DFT) data, curing in particular most of the pathological cases encountered with TD-DFT when a nonoptimal exchange-correlation functional is used.