Relativistic description of few-electron systems

The nuclear current consists of a classical, external part (jmc) == jcxt describing the nucleus in its ground state and a second quantized part j uc describing internal nuclear degrees of freedom. Specification of this fluctuating current employs nuclear models. The action principle yields equations of motion for the coupled Dirac-Maxwell helds and A M  

Maxwell s equations (1.9) can be formally integrated with the aid of the free-photon propagator According to the total current introduced above, the resulting total radiation held may be decomposed as 

The total Hamiltonian describing the total interacting many-body problem—Dirac particles + radiation field + nucleus—may be obtained from the T 00-component of the energy-momentum tensor. The part of the Hamiltonian relevant for the relativistic description of the atomic many-body problem in the presence of the external electromagnetic held of the nucleus including radiative corrections and possible interactions 

A proper definition of (quasi-)particle-creation and (quasi-)particle-annihilation operators an and a is provided by diagonalization of the (time-independent) unperturbed part Ho = //ext+Z/e-e of the total Hamiltonian. After the iteration is performed (e.g. on the Dirac-Fock level) the latter may be cast into the form 

The creation/annihilation operators aj /a, denote the one-particle operators which diagonalize the Hamiltonian Hen. The summation indices i, j, k, l denote the usual set of one-electron quantum numbers and run over positive-energy states only. The quantities Vjju are two-electron Coulomb matrix elements and the quantities biju denote two-electron Breit matrix elements, respectively. We specify their static limit (neglecting any frequency dependence)  

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