  We study the Kohn-Sham scheme for the calculation of the steady state linear
response to a harmonic perturbation that is turned on adiabatically. Although
in general the exact time dependent exchange-correlation potential cannot be
expressed as the functional derivative of a universal functional due to the
so-called causality paradox, we show that for a harmonic perturbation the
exchange-correlation part of the first-order Kohn-Sham potential $v_s^{(1)}(r)
\cos(\omega t)$ is given by $v_{xc}^{(1)}(r) = \delta K_{xc}^{(2)}/\delta
n^{(1)}(r)$. $K_{xc}^{(2)}$ is the exchange-correlation part of the
second-order quasienergy $K_v^{(2)}$. The Frenkel variation principle implies a
stationary principle for the second-order quasienergy. We also find an
analogous stationary principle and KS scheme in the time dependent extension of
one-matrix functional theory, in which the basic variable is the one-matrix
(one-body reduced density matrix).
