where r e (r h ) is for the electron (hole) coordinate, and we drop the label QP for clarity. Here, A vc (B vc ) are the expansion coefficients of the excited state wave function in terms of resonant (anti-resonant) transitions between QP occupied (occ.) states v and unoccupied (unocc.) c, respectively. In this basis, the BSE turns into an effective two-particle Hamiltonian problem of the form   