The Time Dependent Density Functional Theory
The aim of this section is to give a brief introduction to the TDDFT, introducing its G-space and Bloch representation. Analogously to the standard DFT theory the TDDFT introduces a fictious non-interacting system that moving under the action of an unknown xc-potential
describes the exact time dependent density.
correponding to the Hamiltonian
Axc is the exchange-correlation action.
In the TDDFT the Kohn-Sham equation is replaced with a time-dependent
KS equation
Moreover introducing a time-dependent perturbation V1 as
the response function ( Xd LR ) is readly defined
with
The DFT response function is then introduced, instead, using
and using the chain rule
the key TDDFT equation is obtained, for the exact response function
with the exchange-correlation kernel fxc given by
TDDFT equation solvers: G-space representations
In Xd we have seen the structure of the non interacting response function in the RL space. The same expansion can be applied to the fxc kernel rewriting Eq.(1) as
(with repeated indexes summed). Where
TDDFT equation solvers: Bloch representations
When we have a static fxc (like in the ALDA) the TDDFT equation can be easly solved using the standard Bethe-Salpter techniques described in BSS . Following the notation of BSK Eq.(1) can be expanded in Bloch states leading to a Bethe-Salpter equation where, in addition to the standard exchange part of kernel, we have a TDDFT exchange-correlation part
with
TDDFT equation solvers: Bloch representations
When we have a static fxc (like in the ALDA) the TDDFT equation can be easly solved using the standard Bethe-Salpter techniques described in BSS . Following the notation of BSK Eq.(1) can be expanded in Bloch states leading to a Bethe-Salpter equation where, in addition to the standard exchange part of kernel, we have a TDDFT exchange-correlation part
with
A long range model for fxc
A few years ago Reining introduced a simple but efficient approximation for fxc
the striking agreement between the TDDFT absorption spectrum and the more elaborate BS calculation (see Botti) using static parameters for α have revealed that this simple model captures the key long-range interaction that completely lacks in the local or semilocal approximations, like the ALDA. More recently this approach has been extended using general Many Body arguments, as described in the works of Marini, Adragna, and Sottile.
References
- For a review, see G. Onida, L. Reining and A. Rubio, Rev. Mod. Phys. 74, 601 (2002).
- Runge et al., Phys. Rev. Lett. 52, 997 (1984). Petersilka et al.,Phys. Rev. Lett. 76, 1212 (1996).
- A. Marini, R. Del Sole, and A. Rubio, Phys. Rev. Lett., 91, 256402 (2003),G. Adragna, R. Del Sole, and A. Marini, Phys. Rev. B 68, 165108 (2003), F. Sottile, V. Olevano, and L. Reining, Phys. Rev. Lett. 91, 056402 (2003).
- Reining, V. Olevano, A. Rubio, and G. Onida, Phys. Rev. Lett. 88, 066404 (2002).
- S. Botti, hys. Rev. B 69, 155112 (2004).