Article

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

Archive for Rational Mechanics and Analysis (Impact Factor: 2.29). 03/2007; 184(1):1-22. DOI: 10.1007/s00205-006-0016-6

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
[Dg,a(g),g]=0[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 Dg,a(g)D_{g,\alpha}^{(\gamma)}. Moreover, γ minimizes the energy of the relativistic electron-positron field in Hartree–Fock approximation, if the splitting
of

\mathfrakH:=L2(\mathbbR3)Ä\mathbbC4\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 ofDg,a(g)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.

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