Relativistic Hamiltonian for many-electron systems

The perturbation theory of relativistic QED, see for example [47,65], is the source of widely used methods of nonrelativistic many-body perturbation theory (MBPT) [66]. We demonstrate how it can also be used to formulate the theory of relativistic self-consistent fields as the first step in a more elaborate theory of MBPT incorporating radiative corrections. 

The counter term m(x) represents some sort of average effect of the interelec-tronic interaction which has been added to the operator so that 

We shall present a simplified version of the formalism [47, 1 If] which is adequate for our limited purposes. The idea is that for times t —xjl and t +x/2, the interaction is switched off, and the system propagates freely in a state Pq) with Hamiltonian Hq. For —t/2 t t/2, the system evolves according to the equation 

For the moment we shall confine our discussion to the two leading terms. The first order term reduces to the expectation of the counter-term. 

To simplify the discussion, consider a helium atom in a state which, in zero order, is approximated by putting one electron into an eigenstate state a of and the other into a (different) state 

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DHF calculations on molecules using finite basis sets require considerably more computational effort than the corresponding nonrelativistic calculations and cause several problems due to the presence of the Dirac one-particle operator. It is therefore desirable to find (approximate) relativistic Hamiltonians for many-electron systems which are not plagued by unboundedness from below and therefore do not cause problems like the variational collapse at the self-consistent field level or the Brown-Ravenhall disease at the configuration interaction level. It is also desirable to find forms in which the quality of a matrix representation of the kinetic energy is more stable than for the Dirac Hamiltonian, i.e., forms which are not affected by the finite basis set disease . 

Relativistic Hamiltonian for many-electron systems empoying a sum of one-particle Dirac operators and the Coulomb and Breit operators for the electron interaction. 

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