Mean-field models for disordered crystals

Abstract

In this article, we set up a functional setting for mean-field electronic structure models of Hartree-Fock or Kohn-Sham types for disordered crystals. The electrons are quantum particles and the nuclei are classical point-like particles whose positions and charges are random. We prove the existence of a minimizer of the energy per unit volume and the uniqueness of the ground state density of such disordered crystals, for the reduced Hartree-Fock model (rHF). We consider both (short-range) Yukawa and (long-range) Coulomb interactions. In the former case, we prove in addition that the rHF ground state density matrix satisfies a self-consistent equation, and that our model for disordered crystals is the thermodynamic limit of the supercell model.

On stability of ground states for finite crystals in the Schrödinger–Poisson model

“…These results were extended by Catto, Le Bris, Lions and others to Thomas-Fermi and Hartree-Fock models [7,8,9]. Further results in this direction were established by Cancés, Lahbabi, Lewin, Sabin, Stoltz, and others [5,6,22,25,26]. All these results concern either the convergence of the ground state of finite particle systems in the thermodynamic limit or the existence of the ground state for infinite particle systems.…”

Mathematical Modeling of Point Defects in Materials Science

“…We now describe the results of [11,18,19,20,21] on the modeling of local defects in crystals in the framework of the TFW and rHF models. We first consider the case of a single local defect (or of a finite number of such defects) in Section 6.1.…”

“…In the above expressions, Y m (x) = In the second formulation, the usual Laplacian is replaced with the self-adjoint operator ∆ s on L 2 s , called the stationary Laplacian [21], which formally acts as the Laplace operator on the space variable.…”

“…A variational formulation of the rHF model for disordered crystals has been proposed in [21]: Γ sta = γ sta ∈ S sta | 0 ≤ γ sta (ω) ≤ 1 a.s., Tr sta (γ sta ) < ∞, Tr sta (−∆γ sta ) < ∞ . (6.30)…”

“…It is proved in [21] that (6.28) has a ground state. On the other hand, for the reasons mentioned at the end of the previous section, it is not known whether the ground states of (6.28) satisfy the self-consistent equation (6.23).…”