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## 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 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 [14,13], (9)

where the mean-field Hamiltonian is defined as the derivative of the total energy with respect to the density matrix, (10)

In the present study we are concerned with the Kohn-Sham approach , whereby the total energy is the sum of the kinetic and potential-energy terms, (11)

where (12)

and is the scalar-isoscalar kinetic density, see, e.g., Ref.  for definitions. The nonlocal density, can be defined in terms of either the fixed-basis orbitals, ,   (13)

or instantaneous Kohn-Sham orbitals, ,   (14)

The mean-field Hamiltonian is the sum of kinetic and potential-energy terms, , where (15)

and (16)

Let us now assume that the potential energy is invariant with respect to a unitary transformation of the density matrix [14,13], , that is, for all parameters we have, (17)

where is the hermitian matrix of a one-body symmetry generator. Then, the first-order expansion in , (18)

gives a condition for the energy to be invariant with respect to this unitary transformation, that is (19)

which allows us to derive the equation of motion for the average value of . Indeed, from the TDDFT equation (9) we then have: (20)

that is, the time evolution of is governed solely by the kinetic term of the mean-field Hamiltonian.

Subsections   Next: Continuity equation for the Up: Continuity equation in the Previous: Time evolution of a
Jacek Dobaczewski 2011-11-11