Time-dependent Kohn-Sham density-functional theory

Abstract

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 $\ensuremath{-}\frac{1}{2}{\ensuremath{\nabla}}^{2}{\ensuremath{\chi}}_{i}+{v}_{\mathrm{eff}}{\ensuremath{\chi}}_{i}={\ensuremath{\epsilon}}_{i}{\ensuremath{\chi}}_{i}$ and $\frac{\ensuremath{\partial}{\ensuremath{\chi}}_{i}^{2}}{\ensuremath{\partial}t}+\stackrel{\ensuremath{\rightarrow}}{\ensuremath{\nabla}}\ifmmode\cdot\else\textperiodcentered\fi{}({\ensuremath{\chi}}_{i}^{2}\stackrel{\ensuremath{\rightarrow}}{\ensuremath{\nabla}}{S}_{i})=0$ are obtained. The ${\ensuremath{\chi}}_{i}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}},t)$ and ${S}_{i}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}},t)$ are the phase and amplitude, respectively, of the time-dependent Kohn-Sham orbitals, ${v}_{\mathrm{eff}}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}},t)$ is the time-dependent Kohn-Sham effective potential, and ${\ensuremath{\epsilon}}_{i}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}},t)=\ensuremath{-}\frac{\ensuremath{\partial}{S}_{i}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}},t)}{\ensuremath{\partial}t}$. The density $\ensuremath{\rho}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}},t)$ is equal to the sum of the squares of the ${\ensuremath{\chi}}_{i}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}},t)$.