 0 (b)  0 -4 -2  0  2  4  0 (a)  0 -4 -2  0  2  4 u$_{11}$ ( x ) u$_{30}$( x ) u$_{20}$ ( x ) u$_{21}$( x ) χ =100 χ = 10 χ = 1 χ = 0 .1 FIG. 4: (Color online) Evolution of the displacement fields of excitations (1 , 1) , (3 , 0) (even) and (2 , 0) , (2 , 1) (odd) as a function of correlation strength χ = ω − 1 / 2 0 , as shown in the top left panel. Notice the variation in the number of nodes as χ increases from the weakly correlated to the strongly corre- lated limit. pronounced for states of small m , and is absent in the n = 0 states. Fig. (4) shows the evolution of the dis- placement field for the even excitations (1 , 1) and (3 , 0) and for the odd excitations (2 , 0) and (2 , 1). We see that in the “non-degenerate” (1 , 1) state, the number of nodes stays constant and equal to 2 as one goes from the weakly correlated to the strongly correlated regime. In the (3 , 0) state the number of nodes grows from 2 to 4 nodes, so that the displacement field of this state becomes pro- portional, in the strong-correlation limit, to that of the much higher in energy (0 , 3) state. The same behavior is observed in state (2 , 0), for which the number of nodes grows from 1 to 3, and in state (2 , 1), for which it grows from 3 to 5. By this mechanism, states of very different energy end up sharing the same displacement field (up to a proportionality constant) in the strong correlation limit. Our discussion has been limited to singlet states (sym- metric wave function in the relative channel). It would be easy to extend the calculation to include triplet states. To this end, we simply replace the density, kinetic energy density, and pair correlation function of the ground-state (a singlet) by the same quantities calculated from the ground state in the triplet ( S = 1) sector of the Hilbert space. The relative wave functions of these states are an- tisymmetric. The correct symmetry of the wave function is automatically taken into account through the ground- state properties, and does not further appear in the elas- tic equation of motion. It remains a challenge to extend the present formal- ism to the nonlinear regime, as well as including external magnetic fields and spin-orbit interactions. The elastic equation of motion derived in this paper relied on the knowledge of the exact density matrices ρ (1) and ρ (2) of the ground-state. In many cases, these ground-state properties can be extracted from Quantum Monte Carlo calculations. When this cannot be done, one can still resort to density functional theory, i.e. apply the QCM formulation directly to the Kohn-Sham system, in which case we do not need the exact ground-state den- sity matrices, but only the ground-state KS orbitals and a reasonable approximation for the exchange-correlation field. While the standard KS method treats the nonin- teracting kinetic stress tensor exactly, our method should be computationally more agile, for large systems, since it does not involve time-dependent orbitals and/or the inversion of large linear response matrices. The elastic approximation is, in a very precise sense, the extension of the well-known collective approximation 39,40 of the homogeneous electron gas to non-homogeneous electronic systems. In the case of two electrons interacting by Coulomb potential in a harmonic trap, we have seen that the elastic approximation re- places groups of excitations characterized by the same displacement field by a single excitation that carries the oscillator strength of the whole group. In more com- plex systems, we do not expect to be able to identify small groups of excitations that share the same displace- ment field. All that can be said is that the displace- ments associated with different excitations will not be linearly independent. Each eigenfunction of the elastic equation of motion will overlap with many different ex- citations. However, the integrated spectral strength of the elastic eigenmodes will still add up to the correct value. For this reason, our approximation should be use- ful in dealing with collective effects which depend on the integrated strength of the excitation spectrum, such as the dipolar fluctuations that are responsible for van-der Waals attraction. 37,38 Other possible applications include possible nonlocal refinements of the plasmon pole ap- proximation in GW theory 41 and studying the dynam- ics of strongly correlated systems, which are dominated by a collective response. As a byproduct we got an ex- plicit analytic representation of the exact xc kernel in the high-frequency (anti-adiabatic) limit. 42 This kernel should help us to study an importance of the space and time nonlocalities in the KS formulation of TD(C)DFT. It would be particularly interesting to try and interpo- late between the adiabatic and anti-adiabatic extremes to construct a reasonable frequency-dependent functional. VI. DISCUSSION AND SUMMARY 17