Excited-State Forces with GW-BSE Through the Hellmann–Feynman Theorem

We introduce a method for calculating the atomic forces of a molecular or extended system in an excited state described through the GW-BSE approach within the Tamm-Dancoff approximation. The derivative of the so-called excitonic Hamiltonian is obtained by finite differences and its application to the excited state is made possible through the use of suitable projectors. The scheme is implemented with the batch representation of the electron-hole amplitudes, allowing for avoiding sums over empty one-particle orbitals. The geometries of small excited molecules, namely, CO and CH₂O, were in excellent agreement with the results from quantum chemistry methods.