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?


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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