Hamiltonian exact Foldy-Wouthuysen-transformed

This operator is the two-component analog of the Breit operator derived in section 8.1. The reduction has already been considered by Breit [101] and discussed subsequently by various authors (in this context see Refs. [225,626]). We come back to this discussion when we derive the Breit-Pauli Hamiltonian in section 13.2 in exactly the same way from the free-particle Foldy-Wouthuysen transformation. However, there are two decisive differences from the DKH terms (i) in the Breit-Pauli case the momentum operators in Eq. (12.74) are explicitly resolved by their action on all right-hand side terms, whereas in the DKH case they are taken to operate on basis functions in the bra and ket of matrix elements instead, and (ii) the Breit-Pauli Hamiltonian then results after (ill-defined) expansion in terms of 1 /c (remember the last chapter for a discussion of this issue). 

If n is large (strictly, if it approaches infinity), exact decoupling will be achieved. Usually, a very low value of n is sufficient for calculations of relative energies and valence-shell properties. If the total DKH decoupling transformation, i.e., the product of a sequence of transformations required for the nth order DKH Hamiltonian (without considering the free-particle Foldy-Wouthuysen transformation), is written as... 

The process of making approximations starts with either a partitioning of the Hamiltonian and the metric, as in direct perturbation theory, or, for variational approximations, the elimination of the small component. This can be done either directly or via a Foldy-Wouthuysen transformation. Here, we are interested first and foremost in variational approximations, so we will focus on the elimination of the small component and the Foldy-Wouthuysen transformation. Before considering the approximations, we first outline some theory for the exact solutions. 

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