A New Approach to the Modeling of Local Defects in Crystals: The Reduced Hartree-Fock Case

The crystallization conjecture: a review

“…It is also possible to consider unitary or orthogonal matrices (CUE for Circular Unitary Ensemble, and COE for Circular Orthogonal Ensemble, respectively), using the uniform law on this compact subset of matrices. Then, the eigenvalues are distributed according to the Gibbs measure (47) with…”

“…The term N j=1 |p j | 2 /(2m) in (1) disappears in the minimization problem, and factors out and gives a Gaussian at positive temperature in (47). The situation is different in quantum mechanics, in which there is a link between velocity and position, in order to respect Heisenberg's uncertainty principle.…”

“…In the Coulomb case, assuming that the nuclei are classical particles with positive charge and are distributed on a lattice, it has been proved for convex models that the electrons are periodically arranged [162,49,50,51,47]. If in addition one optimizes over the positions of the nuclei, then crystallization is only known in 1D for Thomas-Fermi type models [26].…”

Mathematical Modeling of Point Defects in Materials Science

“…It is possible to provide more explicit expressions for Tr per (γ per ) and Tr per (−∆γ per ) by means of Bloch theory, but we will not elaborate further on this point here and refer the reader to [26,18] for more details. We however mention that numerical methods to solve (5.36) heavily rely on the Bloch decomposition of the trial density matrices γ per .…”

“…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.…”