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In the time-dependent Kohn-Sham formalism the effective potential on electrons is given by $$ v_s[\rho(\mathbf{r},t)]=v(\mathbf{r},t)+v_H(\mathbf{r},t)+v_{xc}[\rho(\mathbf{r},t)] $$ where, $v(\mathbf{r},t)$ is the external TD potential, $v_H(\mathbf{r},t)$ is the TD Hartree potential and $v_{xc}[\rho(\mathbf{r},t)]$ is the TD exchange-correlation potential. To get accurate results, we need to find a good approximation to $v_{xc}$ and a common starting point is the adiabatic approximation: $$ v_{xc}[\rho(\mathbf{r},t)]= \left.v_{xc}[\rho_0(\mathbf{r})]\right|_{\rho_0(\mathbf{r})=\rho(\mathbf{r},t)} $$ where $v_{xc}[\rho_0(\mathbf{r})]$ is the static XC potential. As said in Ullrich's TDDFT book,

this approximation means that $v_{xc}[\rho(\mathbf{r},t)]$ becomes exact in the limit where the adiabatic theorem of quantum mechanics applies, i.e., a physical system remains in its instantaneous eigenstate if a perturbation that is acting on it is slow enough.

What is the time scale on which the adiabatic approximation works? In other words, when is this approximation valid?

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The time scale is related to time derivative of the kinetic energy of electrons defined as:

$$T(t) = \sum_{i} \int |\nabla \phi_{i}(\mathbf{r},t)|^{2} d^{3} \mathbf{r}$$

You have this for time-derivative of the kinetic energy:

$$\frac{d T(t)}{d t} \simeq \frac{T(t=0)}{\tau}$$

Where $\tau$ is the relaxation time for kinetic energy in the order of period associated with the lowest excitation energy of the system.

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