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# Time-dependent density functional theory

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 previous instants.

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

### Introduction

Consider a many body system described by the Hamiltonian:

H = T + V'ext + W

where T is the kinetic energy, Vext is an external potential and W is a two body operator which describe the interaction among the particles of your system. We will call this the physical system of our model. To fix the ideas you can imagine that we are describing the electrons of a molecule keeping the ions in a fixed position, so that T is the kinetic energy of the electrons, Vext is the potential due to the ions and $W=\sum_{i\neq j}\frac{e^2}{|\mathbf{r}_i-\mathbf{r_j}|}$ is the coulomb interaction among the electrons. According to DFT there exist a Kohn-Sham (KS) not interacting system described by the hamiltonian

HKSKS] = T + Vext + VHKS] + VxcKS] =

for which the ground-state density of the interacting system is equal to the ground state density of the KS system: $\rho^0_{KS}(\mathbf{r})=\rho^0(\mathbf{r})$. Here the superscript zero stands for ground-state and ρ is the density of the system. (See the article on DFT for more details on this).

Now Runge-Gross theorem extend the validity of this relation even to time dependent domain, so that if we apply a time dependent external perturbation to our system we will have

H'(t) = T + Vext + W + V'ext(t)
H'KSKS](t) = T + Vext + VHKS](t) + VxcKS](t) + V'ext(t)

where now the identity will holds for $\rho_{KS}(\mathbf{r}t)=\rho(\mathbf{r}t)$ assuming that at time t = 0 the two densities are equal and so their first derivative with respect to the time variable. In this way TDDFT can be used to describe the time evolution of our system

### Linear response TDDFT

Linear response TDDFT can be used if we consider the external perturbation to be small in the sense that it does not completely destruct the ground state structure of our system. In this case we can look at the linear response of the system. This is a great advantage as, to first order, the variation of the system will depend only on the ground state wave-function so that we can simply use all the ingredients of DFT.

So consider a small time dependent external perturbation δVext(t). We will have

H'(t) = H + δVext(t)
H'KS[ρ](t) = HKS[ρ] + δVH[ρ](t) + δVxc[ρ](t) + δVext(t)

and looking at the linear response of the density

$\delta \rho(\mathbf{r}t)= \chi(\mathbf{r}t,\mathbf{r'}t') \delta V^{ext}(\mathbf{r'}t')$
$\delta \rho(\mathbf{r}t)=\chi_{KS}(\mathbf{r}t,\mathbf{r'}t') \delta V^{eff}[\rho](\mathbf{r'}t')$

where δVeff[ρ](t) = δVext(t) + δVH[ρ](t) + δVxc[ρ](t) Here and in the following we will intend that primed variables are integrated.

Being in linear response domain we can now expand the variation of the Hartree (H) and the exchange-correlation (xc) potential to linear order respect to the density variation

$\delta V_H[\rho](\mathbf{r})=\frac{\delta V_H[\rho]}{\delta\rho}\delta\rho= \frac{1}{|\mathbf{r}-\mathbf{r'}|}\delta\rho(\mathbf{r'})$

and

$\delta V_{xc}[\rho](\mathbf{r})=\frac{\delta V_{xc}[\rho]}{\delta\rho}\delta\rho= f_{xc}(\mathbf{r}t,\mathbf{r'}t')\delta\rho(\mathbf{r'})$

Finally inserting this relation in the response equation for the KS system and comparing the resultant equation with the response equation for the physical system we get the Dyson equation of TDDFT:

$\chi(\mathbf{r}_1t_1,\mathbf{r}_2t_2)=\chi_{KS}(\mathbf{r_1}t_1,\mathbf{r}_2t_2)+ \chi_{KS}(\mathbf{r_1}t_1,\mathbf{r}_2't_2') \left(\frac{1}{|\mathbf{r}_2'-\mathbf{r}_1'|}+f_{xc}(\mathbf{r}_2't_2',\mathbf{r}_1't_1')\right) \chi(\mathbf{r}_1't_1',\mathbf{r}_2t_2)$

From this last equation it's possible to derive the excitations energies of our system, being them the poles of the response function.

## Key papers

• P. Hohenberg and W. Kohn, Phys. Rev. 136 (1964) B864
• 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