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Static and Dynamic Bethe-Salpeter Equations in the $T$-Matrix Approximation

While the well-established $GW$ approximation corresponds to a resummation of the direct ring diagrams and is particularly well suited for weakly-correlated systems, the $T$-matrix approximation does sum ladder diagrams up to infinity and is supposedly more appropriate in the presence of strong correlation. Here, we derive and implement, for the first time, the static and dynamic Bethe-Salpeter equations when one considers $T$-matrix quasiparticle energies as well as a $T$-matrix-based kernel. The performance of the static scheme and its perturbative dynamical correction are assessed by computing the neutral excited states of molecular systems. Comparison with more conventional schemes as well as other wave function methods are also reported. Our results suggest that the $T$-matrix-based formalism performs best in few-electron systems where the electron density remains low.