Time-dependent localized Hartree-Fock potential

By minimizing the difference between the left- and the right-hand sides of the many-body time-dependent Schr"{o}dinger equation with the Slater-determinant wave-function, we derive a non-adiabatic and self-interaction free time-dependent single-particle effective potential which is the generalization to the time-dependent case of the localized Hartree-Fock potential. The new potential can be efficiently used within the framework of the time-dependent density-functional theory as we demonstrate by the evaluation of the wave-vector and frequency dependent exchange kernel $f_x(q,\omega )$ and the exchange shear modulus ${\mu }_x$ of the homogeneous electron gas.