The Dielectric Permittivity of Crystals in the Reduced Hartree–Fock Approximation

In a recent article (Cancès et al. in Commun Math Phys 281:129–177, 2008), we have rigorously derived, by means of bulk limit arguments, a new variational model to describe the electronic ground state of insulating or semiconducting crystals in the presence of local defects. In this so-called reduced Hartree–Fock model, the ground state electronic density matrix is decomposed as $${\gamma = \gamma^0_{\rm per} + Q_{\nu,\varepsilon_{\rm F}}}$$, where $${\gamma^0_{\rm per}}$$ is the ground state density matrix of the host crystal and $${Q_{\nu,\varepsilon_{\rm F}}}$$ the modification of the electronic density matrix generated by a modification ν of the nuclear charge of the host crystal, the Fermi level εF being kept fixed. The purpose of the present article is twofold. First, we study in more detail the mathematical properties of the density matrix $${Q_{\nu,\varepsilon_{\rm F}}}$$ (which is known to be a self-adjoint Hilbert–Schmidt operator on $${L^2(\mathbb{R}^3)}$$). We show in particular that if $${\int_{\mathbb{R}^3}\,\nu \neq 0, Q_{\nu,\varepsilon_{\rm F}}}$$ is not trace-class. Moreover, the associated density of charge is not in $${L^1(\mathbb{R}^3)}$$ if the crystal exhibits anisotropic dielectric properties. These results are obtained by analyzing, for a small defect ν, the linear and nonlinear terms of the resolvent expansion of $${Q_{\nu,\varepsilon_{\rm F}}}$$. Second, we show that, after an appropriate rescaling, the potential generated by the microscopic total charge (nuclear plus electronic contributions) of the crystal in the presence of the defect converges to a homogenized electrostatic potential solution to a Poisson equation involving the macroscopic dielectric permittivity of the crystal. This provides an alternative (and rigorous) derivation of the Adler–Wiser formula.