Solutions of the dirac-fock equations and the energy of the electron-positron field

摘要

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,. minimizes the energy of the relativistic electron-positron field in Hartree-Fock approximation, if the splitting of H := L-2(R-3) circle times 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 := alpha Z we give quantitative estimates on the maximal value of the fine structure constant alpha for which the existence can be guaranteed.