(i) Write down the interband snd intraband oscillator strengths, in the long wavelength limit, for a system of noninteracting electrons in a solid. Hint: Make use of the exact results of Problem 4.2.
(ii) Obtain an explicit expression for the density-density response funetion in this approximation, $x^{*}(\mathrm{q}, \omega)$.
(iii) The RPA for electrons in a solid is obtsined by setting $\chi_{s c}(q, \omega)=$ $x^{e}(q, \omega)$. Discuss the behavior of
$$
\lim _{q \rightarrow 0} \epsilon_{R P \Lambda}^{f}(q, 0)=1-\frac{4 r e^{2}}{q^{2}} x^{4}(q, 0)
$$
for the following systems:
(a) Insulstor.
(b) Semieonductor, for which conduction electrons form s classical system (the presence of the valence electrons must be taken into account).
(c) Conduction electrons in a metal.
(iv) Give a qualitative discussion of the behavior of $e_{\mathrm{R} P_{A}}(0, \omega)$ for the above cases, ss $\omega$ varies from s frequency less than any interband excitation frequency to one greater than any interband excitation frequency.