  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.
