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

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. [ABSTRACT FROM AUTHOR]