Matthias Huber, Heinz Siedentop
Solutions of the Dirac-Fock Equations and the Energy of the Electron-Positron Field
(302K, pdf)

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 $\gamma$ of the Dirac-Fock equations 
 $[D_{g,\alpha}^{(\gamma)},\gamma]=0$ with the additional property 
 that $\gamma$ is the orthogonal projector onto the first $N$ 
 positive eigenvalues of the Dirac-Fock operator 
 $D_{g,\alpha}^{(\gamma)}$. Moreover, $\gamma$ minimizes the energy 
 of the relativistic electron-positron field in Hartree-Fock 
 approximation, if the splitting of $\gH:=L^2(\rz^3)\otimes \cz^4$ 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 $g:=\alpha Z$ we give 
 quantitative estimates on the maximal value of the fine structure 
 constant $\alpha$ for which the existence can be guaranteed.