Alternative time-dependent optimized effective potential

The OEP is known as a single-particle potential minimizing the expectation value of a many-body Hamiltonian on the set of eigen-functions of a single-particle Hamiltonian [1]. The time-dependent (TD) OEP can be constructed with the TD quantum stationary-action principle [2]. Very useful conceptually in DFT and TDDFT, both OEPs are not practicable due to the complexity of their implementations. Here we report a TDOEP by minimizing the difference of LHS and RHS of the TD Schr\"{o}dinger equation [3]. If the orbitals are varied, then the TD Hartree-Fock equations are reproduced. Similarly, we now find the OEP. New OMP does not involve the inversion of the density-response function $\chi_s$, which greatly facilitates implementations. Accordingly, the exchange-correlation kernel $f_{xc}$ involves of $\chi^{-1}_s$ only, not its quadratic counterpart. To show the power of this method, we work out the $f^h_{xc}(q,\omega)$ of the homogeneous electron gas to be used with the nearly-free electrons theory, where $f^h_{xc}$ is the main input [4].\\[4pt] [1]. J. D. Talman et al. Phys. Rev. A 14, 36 (1976).\\[0pt] [2]. C. A. Ullrich et al. Phys. Rev. Lett. 74, 872 (1995).\\[0pt] [3]. V. U. Nazarov, Math. Proc. Cambridge Phil. Soc. 98, 373 (1985).\\[0pt] [4]. V. U. Nazarov et al. Phys. Rev. Lett. 102, 113001 (2009).