### 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

A_{xc} 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 V_{1} 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** f_{xc} 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 f

_{xc}
kernel rewriting Eq.(1) as

(with repeated indexes summed). Where

#### TDDFT equation solvers: Bloch representations

When we have a static f

_{xc} (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 f

_{xc} (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 f_{xc}

A few years ago

Reining introduced a simple but
efficient approximation for f

_{xc}
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 et al. Phys. Rev. B
**69**, 155112 (2004) and S. Botti et al. Phys. Rev. B **72**, 125203 (2005).