Time-Dependent Optimized Effective Potential

Given an expression for the quantum mechanical action $A\left[{\phi }_{j\sigma }\right]$ 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 $v_{\sigma }\left(\mathrm{r}t\right)$ which, when inserted in time-dependent single-particle Schrödinger equations for the spin-up and spin-down electrons yield orbitals $\left\{{\phi }_{j\sigma }\right(\mathrm{r}t\left)\right\}$ that make $A\left[{\phi }_{j\sigma }\right]$ 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)]