Parameterized attenuated exchange for generalized TDHF@vW applications

Building upon our previously developed time-dependent Hartree-Fock (TDHF)@vW method, based on many-body perturbation theory and, specifically, the Bethe-Salpeter Equation (BSE), we introduce a parameterization scheme for the attenuated exchange kernel, vW(|r - r'|). In the original method, vW was determined individually for each system via an efficient stochastic short-time TD Hartree propagation for the screened Coulomb interaction, W(r, r'). The new parameterization leverages photophysical similarities in exciton binding energies (or exchange interaction attenuation) among molecules with comparable static dielectric responses. We parameterize the inverse dielectric function using a low-order polynomial with error function apodization, calibrated on a few representative molecules, each with its own vW. Using only seven parameters, the parameterized vW is fully grid-independent and broadly applicable within a family of molecules. This enables TDHF@vW that retains BSE-level accuracy, achieving a mean absolute error of ∼0.1 eV compared to experimental optical gaps and representing a five- to tenfold improvement over conventional TD density functional theory or TDHF while reducing the cost to that of standard TDHF.