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 γ = γ_per⁰ + Q_v,εF where γ_per⁰ is the ground state density matrix of the host crystal and Q_ν,εF the modification of the electronic density matrix generated by a modification ν of the nuclear charge of the host crystal, the Fermi level ε_F 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_ν,εF (which is known to be a self-adjoint Hilbert-Schmidt operator on L²(ℝ³)). We show in particular that if is not trace-class. Moreover, the associated density of charge is not in L¹(ℝ³ 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_ν,εF. 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.