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.