Kohn-Sham scheme for frequency-dependent linear response

We study the Kohn-Sham scheme for the calculation of the steady-state linear response λ nω (1) (r) cosωt to a harmonic perturbation λ v (1) (r) cosωt that is turned on adiabatically. Although in general the exact exchange-correlation potential vxc (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 λ vs (1) (r) cosωt is given by v xc (1) (r) =δ K xc (2) /δ nω (1) (r). K xc (2) is the exchange-correlation part of the second-order quasienergy Kv (2). The Frenkel variation principle implies a stationary principle for Kv (2) [nω (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).