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 (γ) g,α, γ] = 0 with the additional property that γ is the orthogonal projector onto the first N positive eigenvalues of the Dirac-Fock operator D (γ) g,α. Moreover, γ minimizes the energy of the relativistic electron-positron field in Hartree-Fock approximation, if the splitting of H := L2(R3) ⊗ C4 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 (γ) g,α. For fixed electron-nucleus coupling constant g := αZ we give quantitative estimates on the maximal value of the fine structure constant α for which the existence can be guaranteed.