Time-dependent Density Functional Theory - Linear Response TDDFT

Linear Response TDDFT

Linear-response TDDFT can be used if the external perturbation is small in the sense that it does not completely destroy the ground-state structure of the system. In this case one can analyze the linear response of the system. This is a great advantage as, to first order, the variation of the system will depend only on the ground-state wave-function so that we can simply use all the properties of DFT.

Consider a small time-dependent external perturbation . This gives

and looking at the linear response of the density

\delta \rho(\mathbf{r}t)= \chi(\mathbf{r}t,\mathbf{r'}t') \delta V^{ext}(\mathbf{r'}t')
\delta \rho(\mathbf{r}t)=\chi_{KS}(\mathbf{r}t,\mathbf{r'}t') \delta V^{eff}(\mathbf{r'}t')

where Here and in the following it is assumed that primed variables are integrated.

Within the linear-response domain, the variation of the Hartree (H) and the exchange-correlation (xc) potential to linear order may be expanded with respect to the density variation

\delta V_H(\mathbf{r})=\frac{\delta V_H}{\delta\rho}\delta\rho= \frac{1}{|\mathbf{r}-\mathbf{r'}|}\delta\rho(\mathbf{r'})

and

\delta V_{xc}(\mathbf{r})=\frac{\delta V_{xc}}{\delta\rho}\delta\rho= f_{xc}(\mathbf{r}t,\mathbf{r'}t')\delta\rho(\mathbf{r'})

Finally, inserting this relation in the response equation for the KS system and comparing the resultant equation with the response equation for the physical system yields the Dyson equation of TDDFT:

\chi(\mathbf{r}_1t_1,\mathbf{r}_2t_2)=\chi_{KS}(\mathbf{r_1}t_1,\mathbf{r}_2t_2)+ \chi_{KS}(\mathbf{r_1}t_1,\mathbf{r}_2't_2') \left(\frac{1}{|\mathbf{r}_2'-\mathbf{r}_1'|}+f_{xc}(\mathbf{r}_2't_2',\mathbf{r}_1't_1')\right) \chi(\mathbf{r}_1't_1',\mathbf{r}_2t_2)

From this last equation it is possible to derive the excitation energies of the system, as these are simply the poles of the response function.

Other linear-response approaches include the Casida formalism (an expansion in electron-hole pairs) and the Sternheimer equation (density-functional perturbation theory).

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