Time-dependent generalized Kohn-Sham theory

Generalized Kohn-Sham (GKS) theory extends the realm of density functional theory (DFT) by providing a rigorous basis for non-multiplicative potentials, the use of which is outside original Kohn{Sham theory. GKS theory is of increasing importance as it underlies commonly used approximations, notably (conventional or range-separated) hybrid functionals and meta-generalized-gradient-approximation (meta-GGA) functionals. While this approach is often extended in practice to time-dependent DFT (TDDFT), the theoretical foundation for this extension has been lacking, because the Runge-Gross theorem and the van Leeuwen theorem that serve as the basis of TDDFT have not been generalized to non-multiplicative potentials. Here, we provide the necessary generalization. Specifically, we show that with one simple but non-trivial additional caveat - upholding the continuity equation in the GKS electron gas - the Runge-Gross and van Leeuwen theorems apply to time-dependent GKS theory. We also discuss how this is manifested in common GKS-based approximations.