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
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
and
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:
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).
Read more about this topic: Time-dependent Density Functional Theory
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