Proof of the ionization conjecture in a reduced Hartree-Fock model
type: Item
instance of: scholarly article
title: Proof of the ionization conjecture in a reduced Hartree-Fock model
zbMATH Open document ID: 0732.35066
DOI: 10.1007/BF01245077
author: J. P. Solovej
published in: Inventiones Mathematicae
publication date: 1991-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/143884
review text: We study as a model of an atom a functional \({\mathcal E}\) defined on a class of bounded selfadjoint trace class operators \(\gamma\) on the Hilbert space \({\mathcal H}=L^ 2({\mathbb{R}}^ 3;{\mathbb{C}}^ 2)\) satisfying the operator inequality \(0\leq \gamma \leq 1\). Such operators are called admissible density matrices. The functional is defined by \[ {\mathcal E}(\gamma)=Tr_{{\mathcal H}}[(-\Delta -Z| x|^{- 1})\gamma]+1/2\iint \rho_{\gamma}(x)\rho_{\gamma}(y)| x- y|^{-1} dx dy, \] where the parameter \(Z>0\) is the nuclear charge and the density \(\rho_{\gamma}(x)=\gamma (x,x)\). (\(\gamma\) is here identified with its integral kernel. The diagonal values of the kernel are well-defined since \(\gamma\) is trace class.) The functional is of course restricted to those admissible density matrices for which each term above is bounded. In quantum mechanics one can associate to the ground state of an atom with N electrons an admissible density matrix \(\gamma\) with \(Tr_{{\mathcal H}}[\gamma]=N\) (it is called the one particle density matrix and is obtained by taking the partial trace (properly normalized), with respect to the space corresponding to N-1 of the electrons, of the many body state). The true quantum energy, however, does not depend only on \(\gamma\). It can be shown that for large N and Z the true energy can be approximated by \({\mathcal E}(\gamma)\). This basically amounts to neglecting correlations. The functional \({\mathcal E}\) is similar to the famous Hartree-Fock functional except that the exchange term (which is of smaller order) has been neglected. We show in the paper that if \(\gamma\) is a global minimizer for \({\mathcal E}\) (existence of such a minimizer is also established) then for the screened nuclear charge \(\nu_ Z(R)=Z-\int_{| x| <R}\rho_{\gamma}(x) dx\) we have \[ \lim_{R\to 0}\liminf_{Z\to \infty}(\nu_ Z(R)R^ 3)=\lim_{R\to 0}\limsup_{Z\to \infty}(\nu_ Z(R)R^ 3)=324\pi^ 2. \] This result is then used to give bounds uniform in Z …
Mathematics Subject Classification ID: 35P99
Mathematics Subject Classification ID: 81Q20
zbMATH DE Number: 4210677
zbMATH Keywords: selfadjoint trace class operators
zbMATH Keywords: density matrices
zbMATH Keywords: Hartree-Fock functional
zbMATH Keywords: semiclassical renormalization scheme
MaRDI profile type: Publication
Wikidata QID: Q59255846
cites work: Asymptotic neutrality of large ions
cites work: On the energy of a large atom
cites work: Thomas-fermi and related theories of atoms and molecules
cites work: The Thomas-Fermi theory of atoms, molecules and solids
cites work: Bound on the ionization energy of large atoms
cites work: On some basic properties of density functionals for angular momentum channels
cites work: Geometric methods in the quantum many-body problem. Nonexistence of very negative ions
cites work: Universality in the Thomas-Fermi-von Weizsäcker model of atoms and molecules
OpenAlex ID: W2051979557
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description: scientific article; zbMATH DE number 4210677