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 $-\frac{1}{2}{\nabla }^2{\chi }_i+v_{\mathrm{eff}}{\chi }_i={\epsilon }_i{\chi }_i$ and $\frac{\partial {\chi }_i^2}{\partial t}+\overrightarrow{\nabla }·\left({\chi }_i^2\overrightarrow{\nabla }{S}_i\right)=0\mathrm{}$ are obtained. The ${\chi }_i(\overrightarrow{\mathrm{r}},t)$ and $S_i(\overrightarrow{\mathrm{r}},t)$ are the phase and amplitude, respectively, of the time-dependent Kohn-Sham orbitals, $v_{\mathrm{eff}}(\overrightarrow{\mathrm{r}},t)$ is the time-dependent Kohn-Sham effective potential, and ${\epsilon }_i(\overrightarrow{\mathrm{r}},t)=-\frac{\partial S_i(\overrightarrow{\mathrm{r}},t)}{\partial t}$. The density $\rho (\overrightarrow{\mathrm{r}},t)$ is equal to the sum of the squares of the ${\chi }_i(\overrightarrow{\mathrm{r}},t)$.

• Received 1 April 1982

DOI:https://doi.org/10.1103/PhysRevA.26.2243