Time-dependent density functional theory

In the framework of the time-dependent Hartree-Fock (TDHF) approximation or time-dependent density functional theory (TDDFT), the so-called memory effects are often neglected and it is assumed that the potential at time $t$ is just the static potential evaluated at the instantaneous density [G. F. Giuliani and G. Vignale,The Quantum Theory Of The Electron Liquid, (Cambridge University Press, Cambridge, 2005)()]. For these two time-dependent approaches, the starting point is the equation of motion for the one-body density matrix $\rho_{\alpha\beta}$ [14,13],

$$i\hbar \frac{{\rm d}}{{\rm d} t}\rho = [h,\rho] ,$$ (9)

where the mean-field Hamiltonian $h_{\alpha\beta}$ is defined as the derivative of the total energy $E \{\rho\}$ with respect to the density matrix,

$$h_{\alpha\beta} = \frac{\partial E \{\rho\}}{\partial \rho_{\beta\alpha}} .$$ (10)

In the present study we are concerned with the Kohn-Sham approach [15], whereby the total energy is the sum of the kinetic and potential-energy terms,

$$E \{\rho\} = E_k \{\rho\} + E_p \{\rho\} ,$$ (11)

where

$$E_k \{\rho\} = \frac{\hbar^2}{2m}\int {\rm d}^3\bm{r} \tau_0^0({\bm r},t)$$ (12)

and $\tau_0^0({\bm r},t)=\Big(\sum_{\sigma\tau}{\bm\nabla}\cdot{\bm\nabla'} \rho({\bm r}\sigma\tau,{\bm r}'\sigma\tau,t)\Big)\vert _{{\bm r}={\bm r}'}$ is the scalar-isoscalar kinetic density, see, e.g., Ref. [16] for definitions. The nonlocal density, can be defined in terms of either the fixed-basis orbitals, $\psi_\alpha({\bm r}\sigma\tau)$,

$$\rho({\bm r}\sigma\tau,{\bm r}'\sigma'\tau',t) =\!\! \sum_{\beta\alpha}\psi_\beta({\bm r}\sigma\tau)\rho_{\beta\alpha}(t) \psi^*_\alpha({\bm r}'\sigma'\tau') ,$$ (13)

or instantaneous Kohn-Sham orbitals, $\phi_i({\bm r}\sigma\tau,t)$,

$$\rho({\bm r}\sigma\tau,{\bm r}'\sigma'\tau',t) =\!\! \sum_{i=1}^A\phi_i({\bm r}\sigma\tau,t) \phi^*_i({\bm r}'\sigma'\tau',t) .$$ (14)

The mean-field Hamiltonian is the sum of kinetic and potential-energy terms, $h_{\alpha\beta}=T_{\alpha\beta}+\Gamma_{\alpha\beta}$, where

$$T_{\alpha\beta} = \int {\rm d}^3\bm{r}\sum_{\sigma\tau}\psi^... ...a\tau) \frac{- \hbar^2}{2m}\Delta\psi_\beta({\bm r}\sigma\tau)$$ (15)

and

$$\Gamma_{\alpha\beta} = \frac{\partial E_p \{\rho\}}{\partial \rho_{\beta\alpha}} .$$ (16)

Let us now assume that the potential energy is invariant with respect to a unitary transformation of the density matrix [14,13], $U=\exp(i\eta G)$, that is, for all parameters $\eta$ we have,

$$E_p \{\rho\} = E_p \{U\rho U^+\} ,$$ (17)

where $G_{\alpha\beta}$ is the hermitian matrix of a one-body symmetry generator. Then, the first-order expansion in $\eta$,

$$E_p \{U\rho U^+\} \simeq E_p \{\rho\} +\eta \sum_{\beta\alp... ...al (U\rho U^+)_{\beta\alpha}}{\partial \eta}\right]_{\eta=0} ,$$ (18)

gives a condition for the energy to be invariant with respect to this unitary transformation, that is

$$\mbox{Tr}\Gamma[G,\rho] \equiv \mbox{Tr}G[\Gamma,\rho] = 0 ,$$ (19)

which allows us to derive the equation of motion for the average value of $\langle G\rangle=\mbox{Tr}G\rho$. Indeed, from the TDDFT equation (9) we then have:

$$i\hbar \frac{{\rm d}}{{\rm d} t}\langle G\rangle = i\hbar \m... ...d}}{{\rm d} t}\rho = \mbox{Tr}G[h,\rho] = \mbox{Tr}G[T,\rho] ,$$ (20)

that is, the time evolution of $\langle G\rangle$ is governed solely by the kinetic term of the mean-field Hamiltonian.