Non-Equilibrium Instantaneous Approximation and Dipole Forbidden d-d Transitions
Time- and angle-resolved photoemission spectroscopy provides direct access to pump-induced changes in the electronic structure of correlated materials, but its theoretical description generally requires computationally demanding two-time non-equilibrium calculations. We introduce an instantaneous approximation for pump-driven correlated systems, based on a separation between the rapid decay of the Green's functions in the relative time variable and their slower evolution in the average time. Combined with a one-shot dynamical mean-field construction at the Hubbard I level, the method incorporates the driven dynamics of the local correlated shell into the lattice Green's function while retaining momentum resolution. We also derive an effective coupling for nominally dipole-forbidden d-d excitations. It arises from a dipole-allowed d-p transition followed by p-d hybridization; beyond the instantaneous limit, the same process produces an energy-dependent vertex correction to the optical response. As a proof of principle, we apply the framework to paramagnetic and antiferromagnetic FePS3, using a density-functional theory derived tight-binding model together with a supercell unfolding procedure. The calculated momentum-resolved spectra reproduce the main qualitative features observed after excitation of the first and second d-d transitions in recent time-resolved photoemission experiments.