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Timedependent density functional theory
In the framework of the timedependent
HartreeFock (TDHF) approximation or timedependent
density functional theory (TDDFT), the socalled
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 timedependent
approaches, the starting point is the equation of motion
for the onebody density matrix
[14,13],

(9) 
where the meanfield 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 KohnSham approach [15],
whereby the total energy is the sum of the kinetic and potentialenergy terms,

(11) 
where

(12) 
and
is the scalarisoscalar kinetic density, see, e.g., Ref. [16]
for definitions. The nonlocal density,
can be defined in terms of either the fixedbasis orbitals,
,
or instantaneous KohnSham orbitals,
,
The meanfield Hamiltonian
is the sum of kinetic and potentialenergy 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 onebody symmetry generator.
Then, the firstorder 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 meanfield Hamiltonian.
Subsections
Next: Continuity equation for the
Up: Continuity equation in the
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Jacek Dobaczewski
20111111