The Bethe-Salpeter equation in chemistry: relations with TD-DFT, applications and challenges

We review the many-body Green's function Bethe-Salpeter equation (BSE) formalism that is rapidly gaining importance for the study of the optical properties of molecular organic systems. We emphasize in particular its similarities and differences with time-dependent density functional theory (TD-DFT), both methods sharing the same formal O(N-4) computing time scaling with system size. By comparison with higher level wavefunction based methods and experimental results, the advantages of BSE over TD-DFT are presented, including an accurate description of charge-transfer states and an improved accuracy for the challenging cyanine dyes. We further discuss the models that have been developed for including environmental effects. Finally, we summarize the challenges to be faced so that BSE reaches the same popularity as TD-DFT.