Time-dependent Kohn-Sham density-functional theory

A time-dependent Kohn-Sham theory is presented for obtaining the time-dependent density which has a periodic dependence on time. A set of coupled single-particle equations -1/2∇²χᵢ+veffχᵢ=εᵢχᵢ and ${{∂}{{χ}}ᵢ²}{{∂}t}+{{→}}{{∇}}·{}({{χ}}ᵢ²{{→}}{{∇}}{S}ᵢ)=0$ are obtained. The ${{χ}}ᵢ({{→}}{r},t)$ and ${S}ᵢ({{→}}{r},t)$ are the phase and amplitude, respectively, of the time-dependent Kohn-Sham orbitals, ${v}eff({{→}}{r},t)$ is the time-dependent Kohn-Sham effective potential, and ${{ε}}ᵢ({{→}}{r},t)={-}{{∂}{S}ᵢ({{→}}{r},t)}{{∂}t}$. The density ${ρ}({{→}}{r},t)$ is equal to the sum of the squares of the ${{χ}}ᵢ({{→}}{r},t)$.