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energies ω$_{n}$$_{0}$ . The sum rule reads: 

where the “oscillator strengths” 

and where u$_{λ}$ ( r ) is the solution of the elastic eigen- value problem normalized with respect to the scalar product (67), ω$_{λ}$ is its eigenvalue, and the sum runs over all the exact eigenfunctions. Further, the oscillator strengths satisfy the sum rule 

for all λ . The proof of these results is presented in Ap- pendix E. 

From this vantage point we see that the elastic ap- proximation is the extension of the well-known collective approximation 40 of the homogeneous electron gas to in- homogeneous systems. Each eigenvalue ω$_{λ}$ of the elastic equation of motion is a weighted average of exact exci- tation energies, with a weight controlled by the overlap of the exact current matrix element with the eigenmode u$_{λ}$ ( r ). Since u$_{λ}$ ( r ) form an orthogonal basis in the space of displacements, we can say that ω$_{λ}$ represents the av- erage energy of exact excitations in the “direction” λ . Thus, the full excitation spectrum is replaced by a set of spectral lines, one for each orthogonal direction in dis- placement space, and each one carrying the entire and exact spectral weight for that particular direction. 

For orientation we now examine the application of our theory to a few simple models. 

In a homogeneous electron gas the ground-state den- sity n$_{0}$ = n is independent of position. The equilibrium kinetic stress tensor has a constant value 

where t ( n ) is the kinetic energy per particle. The two particle density matrix ρ$_{2}$ ( r , r ′ ) is a function of | r − r $^{′}$| . In such a homogeneous system the displacement eigen- functions are simply plane waves 

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## V. MODEL APPLICATIONS 

## A. Homogeneous electron gas 

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characterized by a wave vector q . Of these there are two kinds: longitudinal, in which ˜ u is parallel to q , and transverse, in which ˜ u is perpendicular to q . The expres- sion (53) for the kinetic force density reduces to 

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The force density from potential energy, Eq. (58), is given by 

where 

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where v ( q ) = 4 πe $^{2}$/q2 is the Fourier transform of the Coulomb potential in three dimensions, and ρ$_{2}$ ( q ) – the Fourier transform of ρ$_{2}$ ( r − r ′ ) – is related to the static structure factor S ( q ) in the following manner: 35 

Thus, we get 

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Finally, in order to take into account the neutralizing background of positive charge (required for the stability of the electron gas), we add the external potential 

where ω 2 p = 4 πne $^{2}$/m is the square of the plasmon fre- quency and ˆ q is the unit vector in the direction of q . Notice that this potential is assumed to vary only in the direction of ˆ q , because it is only in this direction that the displacement generates boundary charges: the system re- mains perfectly homogeneous in the direction perpendic- ular to q . 45 The corresponding force in the equation of motion (15) is 

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