Foldy free-particle case

The unitary transformation is known in exact analytic form only in the free particle case, when the operation of charge conjugation gives back the same Dirac operator, and is then known as the free-particle Foldy-Wouthuysen transformation [111]. For a general potential, block diagonalization can only be... 

However, the situation is completely different in the case of a free particle, since the sequence of transformations given by Eq. (48) may be performed in closed form and conveniently summarised in a single operator Uq- The expansion in /m< can thus be avoided, and this so-called free-particle Foldy-Wouthuysen (fpFW) transformation is given by... 

These problems can be solved if one starts from the (untruncated ) Foldy-Wouthuysen transformation for a free particle, the only case for which the transformation is known anal3d ically, and incorporates the effects of the external potential on top. Along these lines, the so-called Douglas-Kroll-HeB (DKH) method [61-64] is constructed which is probably the most successful quasi-relativistic method in wave function based quantum chemistry. No details will be given here since this topic has been extensively discussed in volume 1 [34] of this series. Meanwhile several density functional implementations exist based on the Douglas-Kroll-HeJ3 approach [39-45]. In recent years. 

The historically first attempt to achieve the block-diagonalization of the Dirac Hamiltonian is due to Foldy and Wouthuysen and dates back to 1950 [609]. They derived the very important closed-form expressions for both the unitary transformation and the decoupled Hamiltonian for the case of a free particle without invoking something like the X-operator. Because of the discussion in the previous two sections, we can directly write down the final result since the free-particle X-operator of Eq. (11.10) and hence Uv=o = Uq are known. With the arbitrary phase of Eq. (11.23) being fixed to zero it is given by... 

We have already seen that the free-particle Foldy-Wouthuysen transformation can be expressed in a couple of ways which seem very different at first sight. The only boundary for the explicit choice of a unitary transformation is that the off-diagonal blocks and therefore all odd operators vanish. If many different choices are possible — and we will see in the following that this is actually the case — the question arises how are they related and what this implies for the resulting Hamiltonians. 

Every well-defined two-component method such as the DKH transformation must have a well-defined nonrelativistic limit. In fact, in the DKH case the nonrelativistic limit is solely determined by the even terms of the free-particle Foldy-Wouthuysen transformation given in Eq. (11.37),... 

The innermost or initial transformation of the sequence of Eqs. (11.15) and (12.1), i.e., the free-particle Foldy-Wouthuysen transformation, reads in the N-electron case... 

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). 

The complicated nature of the operator VVi raises the question of whether a simpler alternative can be found. Barysz, Sadlej, and Snijders (1997) devised an approach which starts from the free-particle Foldy-Wouthuysen transformation, just as the Douglas-Kroll transformation does. Whereas the Douglas-Kroll approach seeks to eliminate the lowest-order odd term from the transformed Hamiltonian, their approach seeks to be correct to a particular order in 1 /c, and it provides a ready means for defining a sequence of approximations of increasing order in 1/c. It is important to note that, while the expansion in powers of 1 /c that resulted from the Foldy-Wouthuysen transformation in section 16.2 generates highly singular operators, this is not true per se of expansions in 1/c. What multiplies 1/c is all-important. In the case of the Foldy-Wouthuysen transformation it is and due to the fact that p can become... 

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