Second- and third-order MBPT for closed-shell atoms

Second- and Third-Order MBPT for Closed-Shell Atoms 

The rules of perturbation theory associated with the relativistic no-pair Hamiltonian are identical to the well-known rules of nonrelativistic many-body perturbation theory, except for the restriction to positive-energy states. The nonrelativistic rules are explained in great detail, for example, in Lindgren and Morrison [30]. Let us start with a closed-shell system such as helium or beryllium in its ground state, and choose the background potential to be the Hartree-Fock potential. Expanding the energy in powers of V) as 

In the above equations, we denote sums over occupied levels by letters (o, 6, ) at the beginning of the alphabet, virtual states by letters (m, n, ) in the middle of the alphabet. Later, we use indices i or j to designate sums over both occupied and virtual states. The restriction to positive-energy states in the no-pair Hamiltonian leads to the restriction that virtual states be bound states and positive-energy continuum states in the expressions for the second- and third-order energy. Owing to the relatively small size of the Breit interaction, only terms linear in 6y/t( are important for most applications. The second-order correction from one Breit and one Coulomb interaction is easily found to be 

To evaluate the multiple sums in the expressions above, we first do a reduction to sums of radial integrals and then use make use of finite basis sets to put the resulting expressions into a form suitable for numerical evaluation. 

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