Kohn-Sham scheme for frequency-dependent linear response

We study the Kohn-Sham scheme for the calculation of the steady-state linear response lambda n(omega)((1))(r)cos omega t to a harmonic perturbation lambda v((1))(r)cos omega t that is turned on adiabatically. Although in general the exact exchange-correlation potential v(xc)(r,t) 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 lambda v(s)((1))(r)cos omega t is given by v(xc)((1))(r)=delta K-xc((2))/delta n(omega)((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 K-v((2))[n(omega)((1))]. We also find an analogous stationary principle and Kohn-Sham 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).