Time-dependent Density-functional Response Theory TD-DFRT

Time-dependent Density-functional Response Theory (TD-DFRT) 

Whereas the classic Kohn-Sham (KS) formulation of DFT is restricted to the time-independent case, the formalism of TD-DFT generalizes KS theory to include the case of a time-dependent, local external potential w(t) [27]. 

To evaluate veff (f, t) at a particular time z, the adiabatic approximation is introduced. This approximation is local in time, and thus the Coulomb and exchange-correlation potentials are just those of time-independent DFT, evaluated using the density determined at time z. 

Time-dependent response theory concerns the response of a system initially in a stationary state, generally taken to be the ground state, to a perturbation turned on slowly, beginning some time in the distant past. The assumption that the perturbation is turned on slowly, i.e. the adiabatic approximation, enables us to consider the perturbation to be of first order. In TD-DFT the density response dp, i.e. the density change which results from the perturbation dveff, enables direct determination of the excitation energies as the poles of the response function dP (the linear response of the KS density matrix in the basis of the unperturbed molecular orbitals) without formally having to calculate a(co). 

The quality of the TD-DFT results is determined by the quality of the KS molecular orbitals and the orbital energies for the occupied and virtual states. These in turn depend on the exchange-correlation potential. In particular, excitations to Rydberg and valence states are sensitive to the behavior of the exchange-correlation potential in the asymptotic region. If the exchange-correla- 

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Time-dependent density-functional response theory. An alternative approach to real-time TDDFT as described above is the application of linear-response theory. If the perturbation to the system in its ground-state—in our case, e.g., the exposure to a time-dependent electric field— is only small, the system will response linearly. The formulation of the resulting time-dependent density-functional response theory (TD-DFRT) has been given by Casida. " ... 

The electronic excitation spectra as computed with time-dependent density-functional response theory (TD-DFRT) are shown in Fig. 14 for different ZnSe clusters (both passivated and unpassivated). The interesting feature of the figure is that the lowest excitation energies show a distinct variation with the size of the clusters and also with the nature of surface passivation. For all clusters, lowest excitation energy shows a clear blue shift for passivated clusters (both zinc-blende and wurtzite) as compared to unpassivated clusters. The lowest excitation energy of the passivated clusters of a particular size depends very much on whether the cluster is of zinc-blende or wurtzite type. Therefore, the main crystal structure as well as surface passivation has strong influence on the absorption spectrum of a cluster, particularly on the magnitude of HOMO-LUMO gap. 

The third approach is that used by Salahub and co-workers. They initially used DFT RPA but recendy have reported an implementation of time-dependent density functional response theory (TD DFRT). Their Kohn-Sham linear response function involves a coupling matrix, K, which in the RPA case contains only the response to coulomb terms, but in their present implementation contains exchange and correlation response terms. Their K is time independent as they work within the adiabatic approximation. They calculate the frequency dependent polarizability from a sum-over-states (SOS) formula, and hence have to calculate the excitation spectrum. 

On top of this effective ground-state description a time-dependent extension has been proposed by Niehaus and co-workers, which is usually referred to as a time-dependent density-functional response theory tight-binding (TD-DFRT-TB) scheme. It corresponds to the formulation of Casida s linear-response theory that has been discussed before. The coupling matrix giving the response of the potential with respect to a change in the electron density has to be built as stated in eqn (19), and we use again the adiabatic approximation. 

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