Linear and Nonlinear Optical Properties of Molecules from Real-Time Propagation Based on the Bethe-Salpeter Equation

We present a real-time propagation method for computing linear and nonlinear optical properties of molecules based on the Bethe-Salpeter equation. The method follows the time evolution of the one-particle density matrix under an external electric field. We include electron-electron interaction effects through a self-energy based on the screened exchange approximation. Quasiparticle energies are taken from a prior $GW$ calculation to construct the effective single-particle Hamiltonian and we represent all operators and wavefunctions in an atom-centered Gaussian basis. We benchmark the accuracy of the real-time propagation against the standard linear-response Bethe-Salpeter equation using a set of organic molecules. We find very good agreement when computing linear-response isotropic polarizability spectra from both approaches, with a mean absolute deviation of 30~meV in peak positions. Beyond linear response, we simulate second harmonic generation and optical rectification in a non-centrosymmetric molecule. These phenomena are not captured by the commonly used linear-response Bethe-Salpeter equation. We foresee broad applicability of real-time propagation based on the Bethe-Salpeter equation for the study of linear and nonlinear optical properties of molecules as the method has a similar computational cost as time-dependent density functional theory with hybrid functionals.