Implementation and validation of efficient exact exchange in the real-time TDDFT INQ code

The real-time formulation of TDDFT (RT-TDDFT) has several advantages over linear-response TDDFT, including the natural treatment of coupled electron-ion evolution and the ability to investigate nonlinear material responses to large perturbations. Hybrid exchange-correlation functionals have been shown to improve the accuracy of predicted band gaps and other quantities from ground-state DFT, and are beginning to become commonplace in TDDFT calculations of complex systems. The Hartree-Fock (HF) exchange operator is the critical ingredient in hybrid functionals responsible for this increased accuracy. However, the evaluation of HF exchange requires the computation of 4-point integrals, raising the time complexity of the method compared to local and semi-local exchange approximations. We have implemented in the INQ code the Crank-Nicolson (CN) scheme for time propagation along with the recently-developed adaptive compressive exchange (ACE) algorithm to speed evaluation of HF exchange and validated the implementation against many-body perturbation theory. We also use Ehrenfest electron-ion coupling with HF exchange to approximate the linewidth due to exciton-phonon coupling. To our knowledge, this is the first GPU implementation of ACE for real-time TDDFT in a plane-wave basis with k-points, opening new possibilities for the study of real time dynamics with more sophisticated approximations of electronic interaction in complex systems.