Oscillatory Behavior of Plasmons

In order to facilitate the understanding of the resonance phenomena given by Eq. (33) it is convenient to analyze the oscillatory character of the plasmon that is obtained if the z component of the full density operator is explicitly considered  

Notice that the operators in Eq. (42) do not commute (cf. Ref. [61]). If the summation over j is carried out the second and third terms on the right-hand side of Eq. (42) compensate  

The RPA has not been applied in the derivation of the general relation (43). Equation (43) shows that the extended Brillouin theorem with the full density operator (cf. Eq. (41)) does not describe the electron-electron interaction at all, since it does not depend on the integrals ( ]) On the contrary, this interaction can be seen only if the double commutator [H, [H, Qf, ( )]] is evaluated with the RPA-like assumption 

In the free-electron model the differences between two excitations A h(j) do not depend 

If we further assume that the term in Eq. (46) that depends on the differences of one-electron integrals is small in comparison with the term containing we have 

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