, Volume 184, Issue 1, pp 1-22
Date: 07 Jun 2006

Solutions of the Dirac–Fock Equations and the Energy of the Electron-Positron Field

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Abstract

We consider atoms with closed shells, i.e. the electron number N is 2, 8, 10,..., and weak electron-electron interaction. Then there exists a unique solution γ of the Dirac–Fock equations \([D_{g,\alpha}^{(\gamma)},\gamma]=0\) with the additional property that γ is the orthogonal projector onto the first N positive eigenvalues of the Dirac–Fock operator \(D_{g,\alpha}^{(\gamma)}\) . Moreover, γ minimizes the energy of the relativistic electron-positron field in Hartree–Fock approximation, if the splitting of \(\mathfrak{H}:=L^2(\mathbb{R}^3)\otimes \mathbb{C}^4\) into electron and positron subspace is chosen self-consistently, i.e. the projection onto the electron-subspace is given by the positive spectral projection of \(D_{g,\alpha}^{(\gamma)}\) . For fixed electron-nucleus coupling constant g:=α Z we give quantitative estimates on the maximal value of the fine structure constant α for which the existence can be guaranteed.

Communicated by G. Friesecke