# Time-dependent density functional theory

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Time-dependent density functional theory

Time-dependent density functional theory (TDDFT) is a quantum mechanical method used inphysics and chemistry to investigate the proprieties of many-body systems beyond the ground statestructure. It's an extension of density functional theory (DFT) to the time-dependent domain asa method to describe such systems when a time dependent perturbation is applied and, asDFT, it's becoming one of the the most popular and versatilemethods 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 themulti-variable wave-function. Still as for the ground state approach one can construct aKohn-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 shiftedin a local effective potential. The main difference here with respect to DFT is that the exacteffective potential in a generic instant will depend on the density of the systems at all the previous instants.

The main success of TDDFT till now has been its application in the calculation of electron excitedstates, 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 systemwhich 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 derivativeof the effective potential with respect to the density.

As for DFT one has to do approximations. The most popular is the adiabatic approximationwhich is the respective of the Local Density Approximation (LDA) in the time domain, so thatthe effective potential in a generic instantdepends only the density of the systems at that instant; the excitation's energies are usually computedwithin Adiabatic + Local Density approximation (ALDA). The results are quite good but still the approachsuffer of some problems, some of which are due to the errors in the DFT/LDA ground state calculation, as theunderestimation ionization energy, some others which are due to the adiabatic approximation, such as thelack of multi-electron excitations within this approximation.

The equations of TDDFT rely on the Runge-Gross theorem (1984) which is the time-dependent analogof 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 asfor example Time Dependet Current Density Functional Theory (TDCDFT) developed by Vignale, in which thefundamental quantity is the current density.

Formalism of TDDFT

Introduction

Consider a many body system described by the Hamiltonian::$H=T+V\text{'}^\left\{ext\right\}+W$where $T$ is the kinetic energy, $V^\left\{ext\right\}$ is an external potentialand $W$ is a two body operator which describes the interactions among the particlesof 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: $T$ is the kinetic energyof the electrons, $V^\left\{ext\right\}$ is the potential due to the nuclei and$W=sum_\left\{i eq j\right\}frac\left\{e^2\right\}+f_\left\{xc\right\}\left(mathbf\left\{r\right\}_2\text{'}t_2\text{'},mathbf\left\{r\right\}_1\text{'}t_1\text{'}\right) ight\right)chi\left(mathbf\left\{r\right\}_1\text{'}t_1\text{'},mathbf\left\{r\right\}_2t_2\right)$

From this last equation it is possible to derive the excitations energies of the system, as these are simply the poles of the response function.

Key papers

* [http://dx.doi.org/10.1103/PhysRev.136.B864 P. Hohenberg and W. Kohn, Phys. Rev. 136 (1964) B864]
* [http://dx.doi.org/10.1103/PhysRevLett.52.997 E. Runge and E.K.U. Gross, Phys. Rev. Lett. 52 (1984) 997]

Books on TDDFT

* M.A. L.Marques, C.A. Ullrich, F. Nogueira, A. Rubio, K. Burke, and E.K.U. Gross, Time-Dependent Density Functional Theory. (Springer-Verlag, 2006). ISBN 978-3-540-35422-2

* [http://tddft.org tddft.org]
* [http://th.physik.uni-frankfurt.de/~engel/tddft.html Brief introduction of TD-DFT]

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