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In a recent article (Canc\`es, Deleurence and Lewin, Commun. Math. Phys., 281(2008), pp. 129-177), we have rigorously derived, by means of bulk limitarguments, a new variational model to describe the electronic ground state ofinsulating or semiconducting crystals in the presence of local defects. In thisso-called reduced Hartree-Fock model, the ground state electronic densitymatrix is decomposed as $\gamma = \gamma^0_{\rm per} + Q_{\nu,\epsilon_{\rmF}}$, where $\gamma^0_{\rm per}$ is the ground state density matrix of the hostcrystal and $Q_{\nu,\epsilon_{\rm F}}$ the modification of the electronicdensity matrix generated by a modification $\nu$ of the nuclear charge of thehost crystal, the Fermi level $\epsilon_{\rm F}$ being kept fixed. The purposeof the present article is twofold. First, we study more in details themathematical properties of the density matrix $Q_{\nu,\epsilon_{\rm F}}$ (whichis known to be a self-adjoint Hilbert-Schmidt operator on $L^2(\R^3)$). We showin particular that if $\int_{\RR^3} \nu \neq 0$, $Q_{\nu,\epsilon_{\rm F}}$ isnot trace-class. Moreover, the associated density of charge is not in$L^1(\R^3)$ if the crystal exhibits anisotropic dielectric properties. Theseresults are obtained by analyzing, for a small defect $\nu$, the linear andnonlinear terms of the resolvent expansion of $Q_{\nu,\epsilon_{\rm F}}$.Second, we show that, after an appropriate rescaling, the potential generatedby the microscopic total charge (nuclear plus electronic contributions) of thecrystal in the presence of the defect, converges to a homogenized electrostaticpotential solution to a Poisson equation involving the macroscopic dielectricpermittivity of the crystal. This provides an alternative (and rigorous)derivation of the Adler-Wiser formula.

The Dielectric Permittivity of Crystals in the reduced Hartree-Fock approximation