Exact exchange-correlation potential for a time-dependent two-electron system

We obtain a solution of the time-dependent Schrödinger equation for a two-electron system confined to a plane by an isotropic parabolic potential whose curvature is periodically modulated in time. From this solution we compute the “exact” time-dependent exchange correlation potential $v_{\mathrm{xc}},$ which enters the Kohn-Sham equation of time-dependent density functional theory. Our “exact” result provides a benchmark against which various approximate forms for $v_{\mathrm{xc}}$ can be compared. Finally, $v_{\mathrm{xc}}$ is separated in an adiabatic and a pure dynamical part and it is shown that, for the particular system studied, the dynamical part is negligible.