Time-Dependent Optimized Effective Potential

Given an expression for the quantum mechanical action $A[{\ensuremath{\varphi}}_{j\ensuremath{\sigma}}]$ of an $N$-electron system as a functional of $N$ time-dependent spin orbitals, we present a method of constructing the variationally best local time-dependent single-particle potentials ${v}_{\ensuremath{\sigma}}(\mathrm{r}t)$ which, when inserted in time-dependent single-particle Schr\"odinger equations for the spin-up and spin-down electrons yield orbitals ${{\ensuremath{\varphi}}_{j\ensuremath{\sigma}}(\mathrm{r}t)}$ that make $A[{\ensuremath{\varphi}}_{j\ensuremath{\sigma}}]$ stationary. We also propose a simplification of this scheme leading to a time-dependent generalization of the static optimized effective potentials recently introduced by Krieger, Li, and Iafrate [Phys. Lett. A 146, 256 (1990)]