Analytical Forces from the Bethe-Salpeter Equation for Large-Scale Excited-State Relaxation

Abstract

We present an efficient plane-wave implementation of analytical nuclear forces for electronic excited states described by the Bethe-Salpeter equation (BSE). The formulation combines density-matrix perturbation theory with a Lagrangian approach, and avoids both explicit empty-state summations and the response calculations for each atomic displacement, required by conventional approaches based on density functional perturbation theory. Together with GPU acceleration, these advances make BSE forces calculations tractable for solid-state systems containing hundreds of atoms. We demonstrate the method on two point defects with distinct dielectric environments: the nitrogen-vacancy center in diamond, where BSE and time-dependent density functional theory (TDDFT) yield consistent excited-state relaxations, and the carbon-dimer defect in two-dimensional hexagonal boron nitride, where the screened electron-hole interaction included in the BSE stabilizes the localized defect excitation and corrects the relaxation pattern predicted by semilocal TDDFT. These results establish a scalable framework for BSE-level studies of excited-state relaxation and vibronic coupling in heterogeneous condensed systems.