Multi-orbital time-dependent spin-density functional theory for strongly correlation systems: Application to Ce and YTiO$_{3}$*

We present a methodology for examining the spectral properties and nonequilibrium response of strongly-correlated electron systems within multi-orbital time-dependent spin-density functional theory. The key element of the theory -- exchange-correlation (XC) kernel - is derived from dynamical mean-field theory (DMFT) expressions for two-particle susceptibilities and the electron self-energy for the effective Hubbard model. We demonstrate that the appropriate description of strongly-correlated materials requires a non-adiabatic (time non-local) XC kernel, though the spatial locality in general is not necessary. We apply the formalism to study the spectral properties of cerium and YTiO$_{3}$, and establish that the method is capable of describing both metallic and insulating systems. In addition, we present results of the nonequilibrium response of YTiO$_{3}$ under an applied short laser pulse. In particular, we analyze the role of inter-orbital interactions in the relaxation dynamics of the system.