# Validity of adiabatic approximation in TDDFT

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?

$$T(t) = \sum_{i} \int |\nabla \phi_{i}(\mathbf{r},t)|^{2} d^{3} \mathbf{r}$$
$$\frac{d T(t)}{d t} \simeq \frac{T(t=0)}{\tau}$$
Where $$\tau$$ is the relaxation time for kinetic energy in the order of period associated with the lowest excitation energy of the system.