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of which the system is constituted. 10 

That such a description is possible is guaranteed by the very same theorems that lie at the foundation of TDDFT and TDCDFT. 11,12 Indeed, consider a system of particles of mass m described by the time-dependent hamiltonian 

where 

is the sum of kinetic energy ( ˆ T ), interaction potential energy ( ˆ W ), and the energy associated with an external static potential V$_{0}$ ( r ), 

where ˆ n ( r ) is the particle density operator. V$_{1}$ ( r , t ) is an external time-dependent potential. 

The exact Heisenberg equations of motion for the den- sity and the current density operators, averaged over the quantum state, lead to equations of motion for the aver- age particle density n ( r , t ) and the average particle cur- rent density j ( r , t ): 

and 

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where ∂$_{t}$ denotes the partial derivative with respect to time and ∂$_{ν}$ is a short-hand for the derivative with re- spect to the cartesian component ν of the position vector r . Here and in the following we adopt the convention that repeated indices are summed over. These equations simply express the local conservation of particle number (Eq. (4)) and momentum (Eq. (5)). The key quantity on the right hand side of Eq. (5) is the stress tensor P$_{µν}$ ( r , t ) – a symmetric tensor which will be defined in the next section as the expectation value of a hermitian operator, and whose divergence with respect to one of the indices yields the force density arising from internal quantum- kinetic and interaction effects. 

Now the Runge-Gross theorem of TDDFT guarantees that the stress tensor, like every other observable of the system, is a functional of the current density and of the initial quantum state. Thus, Eq. (5) is in principle a closed equation of motion for j – the only missing piece of information being the explicit expression for P$_{µν}$ in terms of the current density. 

In recent years much effort has been devoted to con- structing an approximate QCM 13–21 and several applica- tions have appeared in the literature (see Ref. 22 for some representative examples). All approximation schemes so far have been based on the local density approximation 

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and generalizations thereof. The objective of this pa- per is to derive and discuss an approximate expression for P$_{µν}$ ( r , t ), and, more importantly, for the associated force density − ∂$_{ν}$P$_{µν}$ ( r , t ), as functionals of the current density and the initial state. We will do this in the lin- ear response regime , i.e. for systems that start from the ground-state of the static hamiltonian ˆ H$_{0}$ and perform small-amplitude oscillations about it. The external po- tential V$_{1}$ ( r , t ) will be treated as a small perturbation. In this regime the equations of motion (4) and (5) are conveniently expressed in terms of the displacement field u ( r , t ), defined by the relation 

where n$_{0}$ ( r ) is the ground-state density. It is also conve- nient to write the density and the stress tensor as sums of a large ground-state component and a small time- dependent part, in the following manner 

where the equilibrium components, marked by the sub- script 0, satisfy the equilibrium condition 

Then the two equations (4) and (5) take the form 

and 

Our task is to find an expression for the force density ∂$_{ν}$P$_{µν,}$$_{1}$ ( r , t ) as a linear functional of u ( r , t ). If this can be achieved, then the excitation energies of the system will be obtained from the frequencies of the time-periodic solutions of Eq. (10) in the absence of external field (i.e., with V$_{1}$ = 0). 

It is easy to see that the spatial dependence of these so- lutions will be proportional to the matrix element of the current density operator between the ground-state and the excited state in question. This is because, in a many- body system with stationary states | ψ$_{0}$ 〉 , | ψ$_{1}$ 〉 , ..., | ψ$_{n}$ 〉 , ... ( | ψ$_{0}$ 〉 is the ground-state), and corresponding energies E$_{0}$, E$_{1}$, ..., E$_{n}$... , the n-th linear excitation is described by the time-dependent state 

where ε is an arbitrarily small “mixing parameter”. The expectation value of the current density operator in this state is 

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