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Proof of the ionization conjecture in a reduced Hartree-Fock model. (English) Zbl 0732.35066

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 on the maximal possible negative ionization charge and ionization energy. The main tool is a semiclassical renormalization scheme.
Reviewer: J.P.Solovej

MSC:

35P99 Spectral theory and eigenvalue problems for partial differential equations
81Q20 Semiclassical techniques, including WKB and Maslov methods applied to problems in quantum theory
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