Time-dependent Kohn-Sham potential for quantum electrodynamical density functional theory: Insights from exact factorization

One of the key challenges of polaritonic chemistry is developing an accurate theoretical description to explain how coupling to the optical cavities modifies the physics of matter. Recent advances in quantum electrodynamical density functional theory (QEDFT), perhaps the most promising method for large and complex systems, have already led to useful ground-state and spectral properties of cavity-embedded molecular systems. Newly developed exchange-correlation functionals for photon-electron Kohn-Sham (KS) systems capture various effects such as the polaritonic Lamb shift due to coupling to vacuum/photonic fluctuations, photon-mediated long-range electron-electron interactions, and cavity-modified many-body dispersion. However, for dynamical properties via time-dependent KS equations, the choice of time-dependent exchange-correlation (TDxc) potential is so far limited to ground-state or mean-field approximations. In this work, we use the exact factorization (XF) framework to explore the features of the TDxc potential for light-matter correlation interactions beyond mean-field. As a starting point, we analyze the properties of the XF-based analytical TDxc potential for a simple single-electron model coupled to a single cavity mode, with the aim of developing some useful photon-electron correlation approximations.