Parameterized Attenuated Exchange for Generalized TDHF@$v_W$ Applications

Building upon our previously developed time-dependent Hartree-Fock (TDHF)@$v_W$ method, based on many-body perturbation theory and specifically the Bethe-Salpeter Equation (BSE), we introduce a parameterization scheme for the attenuated exchange kernel, $v_W(|r - r'|)$. In the original method, $v_W$ 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 photochemical 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 $v_W$. Using only 7 parameters, the parameterized $v_W$ is fully grid-independent and broadly applicable within a family of molecules. This enables TDHF@$v_W$ that retains BSE-level accuracy, achieving a mean absolute error of $\sim0.1$ eV compared to experimental optical gaps and representing a five- to ten-fold improvement over conventional TD density functional theory or TDHF while reducing the cost to that of standard TDHF.