194 research outputs found
Local defects are always neutral in the Thomas-Fermi-von Weisz\"acker theory of crystals
The aim of this article is to propose a mathematical model describing the
electronic structure of crystals with local defects in the framework of the
Thomas-Fermi-von Weizs\"acker (TFW) theory. The approach follows the same lines
as that used in {\it E. Canc\`es, A. Deleurence and M. Lewin, Commun. Math.
Phys., 281 (2008), pp. 129--177} for the reduced Hartree-Fock model, and is
based on thermodynamic limit arguments. We prove in particular that it is not
possible to model charged defects within the TFW theory of crystals. We finally
derive some additional properties of the TFW ground state electronic density of
a crystal with a local defect, in the special case when the host crystal is
modelled by a homogeneous medium.Comment: 34 page
A mathematical analysis of the GW0 method for computing electronic excited energies of molecules
This paper analyses the GW method for finite electronic systems. In a first
step, we provide a mathematical framework for the usual one-body operators that
appear naturally in many-body perturbation theory. We then discuss the GW
equations which construct an approximation of the one-body Green's function,
and give a rigorous mathematical formulation of these equations. Finally, we
study the well-posedness of the GW0 equations, proving the existence of a
unique solution to these equations in a perturbative regime
Greedy algorithms for high-dimensional eigenvalue problems
In this article, we present two new greedy algorithms for the computation of
the lowest eigenvalue (and an associated eigenvector) of a high-dimensional
eigenvalue problem, and prove some convergence results for these algorithms and
their orthogonalized versions. The performance of our algorithms is illustrated
on numerical test cases (including the computation of the buckling modes of a
microstructured plate), and compared with that of another greedy algorithm for
eigenvalue problems introduced by Ammar and Chinesta.Comment: 33 pages, 5 figure
The Microscopic Origin of the Macroscopic Dielectric Permittivity of Crystals: A Mathematical Viewpoint
The purpose of this paper is to provide a mathematical analysis of the
Adler-Wiser formula relating the macroscopic relative permittivity tensor to
the microscopic structure of the crystal at the atomic level. The technical
level of the presentation is kept at its minimum to emphasize the mathematical
structure of the results. We also briefly review some models describing the
electronic structure of finite systems, focusing on density operator based
formulations, as well as the Hartree model for perfect crystals or crystals
with a defect.Comment: Proceedings of the Workshop "Numerical Analysis of Multiscale
Computations" at Banff International Research Station, December 200
Numerical analysis of the planewave discretization of some orbital-free and Kohn-Sham models
We provide a priori error estimates for the spectral and pseudospectral
Fourier (also called planewave) discretizations of the periodic
Thomas-Fermi-von Weizs\"{a}cker (TFW) model and for the spectral discretization
of the Kohn-Sham model, within the local density approximation (LDA). These
models allow to compute approximations of the ground state energy and density
of molecular systems in the condensed phase. The TFW model is stricly convex
with respect to the electronic density, and allows for a comprehensive
analysis. This is not the case for the Kohn-Sham LDA model, for which the
uniqueness of the ground state electronic density is not guaranteed. Under a
coercivity assumption on the second order optimality condition, we prove that
for large enough energy cut-offs, the discretized Kohn-Sham LDA problem has a
minimizer in the vicinity of any Kohn-Sham ground state, and that this
minimizer is unique up to unitary transform. We then derive optimal a priori
error estimates for the spectral discretization method.Comment: 50 page
Existence of a type of optimal norm-conserving pseudopotentials for Kohn–Sham models
International audienceIn this article, we clarify the mathematical framework underlying the construction of norm-conserving semilocal pseudopotentials for Kohn-Sham models, and prove the existence of optimal pseudopotentials for a family of optimality criteria. Most of our results are proved for the Hartree (also called reduced Hartree-Fock) model, obtained by setting the exchange-correlation energy to zero in the Kohn-Sham energy functional. Extensions to the Kohn-Sham LDA (local density approximation) model are discussed
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