expression for E$_{xc,}$$_{1}$ . The “natural” approximation, in the present context, would be the high-frequency approx- imation, which expresses E$_{xc,}$$_{1}$ as the functional deriva- tive of the exchange-correlation energy functional with respect to u . In practice, since the latter is not known, one has to rely on more or less uncontrolled approxima- tions, such as the high-frequency limit of the local den- sity approximation proposed in Refs. 19,28–30 – see Eq. (117) of Ref. 19 (a general discussion of these approxi- mation can be found in Ref. 31). This approximation is local both in space and time and is obtained by applying an instantaneous geometric deformation to the homoge- neous electron gas. In the second respect it is perfectly consistent with our elastic approximation for the nonin- teracting kinetic force, but we must keep in mind that the latter is fully nonlocal. Unfortunately, the local deformation approximation for the xc potential suffers, like all electron-gas based approximations, from a serious defect: it fails to cancel the unphysical self-interaction that is contained in the Hartree term. This makes it unsuitable for the treat- ment of strongly correlated system, where the spurious self-interaction energy can be very large. More accu- rate approximations 32 do not suffer from this defect, but are more difficult to implement. Furthermore, such ap- proximations would do little (apart from fixing the self- interaction problem) to capture the physics of strongly correlated electrons. Alternatively, one could use func- tionals explicitly designed for electronic systems in the strong coupling limit, such as the ones developed by Seidl, Perdew and Kurth 33 and, more recently by Seidl, Gori-Giorgi, and Savin $^{34}$. An immediate application of our equation of motion is the calculation of excitation energies. 3,4 To this end we turn off the external potential V$_{1}$ and consider the homogeneous equation Fourier-transforming with respect to time and carrying out the indicated expansion of the energy to second order in u we get Although this expression is not the most useful in prac- tice, it does bring forth some important features of the problem. First, because the kernel of the integral equa- tion is a symmetric second functional derivative, we are in where the first term (diamagnetic term) is frequency- independent and where χ$_{µν}$ ( r , r ′ , ω ) is the current-current response func- tion, and notice that, at high frequency, this function has the well-known expansion 35 Additional insight into the significance of the eigen- value problem for the excitation energies is obtained by deriving the equation of motion directly from the lin- ear response of the current density to an external vector potential in the high-frequency regime. To this end we write This means that all the eigenvalues will be real, and eigenfunctions ˜ u corresponding to different eigenvalues are orthogonal with respect to the ordinary scalar prod- uct. It follows that the original eigenfunctions u are or- thogonal with respect to the scalar product defined by Eq. (67). Second, the kernel of the integral equation is positive definite, because E [ u ] has an absolute minimum at u = 0, which corresponds to the ground-state energy. For this reason, it is guaranteed that all the eigenvalues are positive. The square roots of these eigenvalues are the approximate excitation energies of the system, start- ing from the ground-state. The eigenfunctions also have a simple interpretation as approximate matrix elements of the current density operator between the ground-state and the excited state under consideration, divided by the ground-state density. We will show this more clearly in the next section. where f and g are arbitrary functions. This can be seen by rewriting Eq. (66) as an equation for ˜ u ≡ √ n$_{0}$ ( r ) u ( r ) and noting that this equation has the form of a standard eigenvalue problem with a symmetric kernel: the presence of an essentially hermitian eigenvalue prob- lem. More precisely, the problem is hermitian with re- spect to a scalar product with weight n$_{0}$ ( r ), i.e. defined as 9 ( f, g ) ≡ ∫ d r n$_{0}$ ( r ) f ( r ) g ( r ) , (67) ∫ d r ′ 1 √ mn$_{0}$ ( r ) δ $^{2}$E [ u ] δu$_{µ}$ ( r ) δu$_{ν}$ ( r ′ ) ∣ u =0 1 √ mn$_{0}$ ( r ′ ) ˜ u$_{ν}$ ( r ′ , ω ) = ω $^{2}$˜ u$_{µ}$ ( r , ω ) . (68) mn$_{0}$ ( r ) ∂ 2 $_{t}$u ( r , t ) = − δE$_{2}$ [ u ] δ u ( r , t ) . (65) j$_{µ}$ ( r , ω ) = ∫ d r ′ χ$_{µν}$ ( r , r ′ , ω ) A$_{ν,}$$_{1}$ ( r ′ , ω ) , (69) mω $^{2}$n$_{0}$ ( r ) u$_{µ}$ ( r , ω ) = ∫ d r ′ δ $^{2}$E [ u ] δu$_{µ}$ ( r ) δu$_{ν}$ ( r ′ ) ∣ u =0 u$_{ν}$ ( r ′$_{, ω}$$_{)}$$_{.}$ (66) χ$_{µν}$ ( r , r ′ , ω ) = n$_{0}$ ( r ) m δ ( r − r ′ ) δ$_{µν}$ + M$_{µν}$ ( r , r ′ ) m $^{2}$ω2 , (70) M$_{µν}$ ( r , r ′ ) ≡ − m $^{2}$〈 ψ$_{0}$ | [[ ˆ H$_{0}$, ˆ j$_{µ}$ ( r )] , ˆ j$_{ν}$ ( r ′ )] | ψ$_{0}$ 〉 , (71) IV. THE CALCULATION OF EXCITATION ENERGIES A. The eigenvalue problem B. Derivation from linear response theory