Time-dependent density functional theory (TDDFT)
is a quantum mechanical method used in
physics and chemistry to investigate the proprieties of many-body systems beyond the ground state
structure. It's an extension of density functional theory
(DFT) to the time-dependent domain as
a method to describe such systems when a time dependent perturbation is applied and, as
DFT, it's becoming one of the the most popular and versatile
methods available in condensed matter physics, computational physics, and computational chemistry.
The main ideas of such approach are the same we can find in DFT being the density of the system,
at least in the first formulation of the method, the key quantity. So, respect to the direct
quantum mechanical approach, one has to play with a single variable quantity and not with the
multi-variable wave-function. Still as for the ground state approach one can construct a
Kohn-Sham (KS) time dependent systems of non interacting particles which gives the same density
of the physical interacting system and in which all the effects of the interaction are shifted
in a local effective potential. The main difference here with respect to DFT is that the exact
effective potential in a generic instant will depend on the density of the systems at all the
The main success of TDDFT till now has been its application in the calculation of electron excited
states, mainly for isolated systems, where the method is used in the linear regime domain.
The excited states energies can be computed as the poles of the response function of the system
which can be computed using a Dyson equation. The key ingredients become the KS not interacting
response function and the Hartree plus exchange-correlation kernel which is the functional derivative
of the effective potential with respect to the density.
As for DFT one has to do approximations. The most popular is the adiabatic approximation
which is the respective of the Local Density Approximation (LDA) in the time domain, so that
the effective potential in a generic instant
depends only the density of the systems at that instant; the excitation's energies are usually computed
within Adiabatic + Local Density approximation (ALDA). The results are quite good but still the approach
suffer of some problems, some of which are due to the errors in the DFT/LDA ground state calculation, as the
underestimation ionization energy, some others which are due to the adiabatic approximation, such as the
lack of multi-electron excitations within this approximation.
The equations of TDDFT rely on the Runge-Gross theorem (1984) which is the time-dependent analog
of the Hohenberg-Kohn theorem (1964) for DFT. The complete theorem is valid
only for isolated systems, while for periodic infinite systems one as to use some more general approach as
for example Time Dependet Current Density Functional Theory (TDCDFT) developed by Vignale, in which the
fundamental quantity is the current density.
Formalism of TDDFT
Consider a many body system described by the Hamiltonian:
is the kinetic energy,
is an external potential
is a two body operator which describes the interactions among the particles
of the system.
The basic assumption of DFT uses Kohn-Sham orbitals in the following way. For a fixed nuclear framework, the KS-Hamiltonian contains three terms:
is the kinetic energy
of the electrons,
is the potential due to the nuclei and