Hamiltonian free-particle Foldy-Wouthuysen

The Breit-Pauli Hamiltonian is an approximation up to 1/c2 to the Dirac-Coulomb-Breit Hamiltonian obtained from a free-particle Foldy-Wouthuysen transformation. Because of the convergence issues mentioned in the preceding section, the Breit-Pauli Hamiltonian may only be employed in perturbation theory and not in a variational procedure. The derivation of the Breit-Pauli Hamiltonian is tedious (21). 

This expression is obtained directly from the four-component DKS Hamiltonian, avoiding the free-particle Foldy-Wouthuysen transformation. After application... 

A more explicit formulation of the free-particle Foldy-Wouthuysen transformed Hamiltonian in the presence of scalar potentials, which also highlights the preserved superstructure of this operator, reads... 

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. 

There is, however, one very special parametrization for the transformation Ho that avoids the expansion of the one-electron operator in any way and is therefore free from convergence issues. This particular parametrization yields operators that can be converted into the closed-form free-particle Foldy-Wouthuysen expression defined by Eqs. (11.27) or (11.35), which produces the closed-form Hamiltonian /i derived in section 11.3. Its expansion... 

The different expressions for Uq given by Eqs. (11.27) and (11.35) represent well-defined quantities for all momenta p = p IRq, which are free of any singularities. Accordingly, also the free-particle Foldy-Wouthuysen Hamiltonian /i obtained by this specific transformation,... 

In this numerical approach, the free-particle Foldy-Wouthuysen transformation Uo is chosen as an initial transformation. Then, the sequence of subsequent unitary transformations Ui i > 1) of Eq. (12.1) applied to the free-particle Foldy-Wouthuysen Hamiltonian fi is united to only one transformation step LZi,... 

After insertion of the explicit expressions for the components of the free-particle Foldy-Wouthuysen Hamiltonian fi given by Eqs. (11.36)-(11.39), this equation reads... 

An expansion in terms of V, i.e., the Douglas-Kroll-Hess expansion, is the only valid analytic expansion technique for the Dirac Hamiltonian, where the final block-diagonal Hamiltonian is represented as a series of regular even terms of well-defined order in V, which are all given in closed form. For the derivation, the initial transformation step has necessarily to be chosen as the closed-form, analytical free-particle Foldy-Wouthuysen transformation defined by Eq. (11.35) in order to provide an odd term depending on the external potential that can then be diminished. We now address these issues in the next chapter. 

In order to eliminate the odd operator Oq of the Dirac Hamiltonian as written in Eq. (11.40) order by order in the scalar potential Y, an odd operator that depends on V must be generated. As discussed in section 11.5, only the special closed-form free-particle Foldy-Wouthuysen transformation produces an operator 0 linear in V as indicated by the subscript. This is the mandatory starting point for subsequent transformation steps. 

Because of the structure of the free-particle Foldy-Wouthuysen transformed Hamiltonian... 

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 derivation of the Breit-Pauli Hamiltonian is tedious. It is nowadays customary to follow the Foldy-Wouthuysen approach first given by Chraplyvy [679-681], which has, for instance, been sketched by Harriman [59]. Still, many presentations of this derivation lack significant details. In the spirit of this book, we shall give an explicit derivation which is as detailed and compact as possible. A review of the same expression derived differently was provided by Bethe in 1933 [72]. Compared to the DKH treatment of the two-electron term to lowest order as described in section 12.4.2 we now only consider the lowest-order terms in 1/c. We transform the Breit operator in Eq. (8.19) by the free-particle Foldy-Wouthuysen transformation. 

In the BSS approach, the free-particle Foldy-Wouthuysen transformation in addition to the orthonormal transformation K is applied to obtain the four-component Hamiltonian matrix to be diagonalized. The free-particle Foldy-Wouthuysen transformation Uq is composed of four diagonal block matrices. 

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 initial transformation Uo is the familiar free-particle Foldy-Wouthuysen transformation [609] and therefore independent of the perturbation. Its application to (A) yields the perturbed free-particle Foldy-Wouthuysen Hamiltonian... 

The first step is the application of the free-particle Foldy-Wouthuysen transformation to the Dirac operator. As in the previous section, we use the Dirac Hamiltonian without subtracting the rest mass,... 

The relativistic correction to the potential is no more singular than the potential itself in this limit and therefore will support bound states. In the small momentum limit, when the electron is far from the nucleus, the potential goes as 1 /r and is therefore a short-range potential. It can be seen that the kinematic factors provide a cutoff to the potential that is absent in the Pauli approximation and that permits variational calculations with the free-particle Foldy-Wouthuysen transformed Hamiltonian. 

One issue raised by the use of an expansion in powers of the potential is that of electric gauge invariance. If we add a constant to the potential, we should obtain a constant shift in the energy, if the potential is gauge-invariant. Terms that are of second order in the potential would be expected to give rise to a quadratic term in the added constant. Looking at the free-particle Foldy-Wouthuysen transformed Hamiltonian, (16.42), it is apparent that the added constant only survives in the even operator the odd operator involves a commutator that eliminates the constant. Consequently, the... 

So far we have made no reference to the two-electron terms of the Hamiltonian. Performing the free-particle Foldy-Wouthuysen transformation took the potential from an even operator to a combination of even and odd operators. The same will be true of the two-electron terms, except that a transformation must be performed on both electron coordinates. For the Coulomb interaction the resulting operator is... 

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... 

Hi is defined by the free-particle Foldy-Wouthuysen transformation, (16.42). For the purpose of better analyzing the transformation and the resultant Hamiltonians, we will make some new definitions of various operators, extracting powers of 1/c. First, the free-particle energy will be written... 

Using these definitions, we may write the blocks of the Hamiltonian after the free-particle Foldy-Wouthuysen transformation as... 

This is the equivalent of the second-order Douglas-KroU operator, but it only involves operators that have been defined in the free-particle Foldy-Wouthuysen transformation. As for the Douglas-Kroll transformed Hamiltonian, spin separation may be achieved with the use of the Dirac relation to define a spin-fi ee relativistic Hamiltonian, and an approximation in which the transformation of the two-electron integrals is neglected, as in the Douglas-Kroll-Hess method, may also be defined. Implementation of this approximation can be carried out in the same way as for the Douglas-Kroll approximation both approximations involve the evaluation of kinematic factors, which may be done by matrix methods. 

The complicated nature of the magnetic operators means that the transformed Hamiltonians based on the free-particle Foldy-Wouthuysen transformation are not particularly well suited to calculation of magnetic properties. An alternative is to use the transformed magnetic operator developed in section 13.7, which is also expanded in powers of 1/c, but in terms of even operators, which behave like electric field operators. 

Here we see a modified kinetic energy term that is cut off near the nuclei, a spin-free relativistic correction, and a spin-orbit term, both of which are regularized and behave as 1/r for small r. We may compare this with the regularization of the free-particle Foldy-Wouthuysen or Douglas-Kroll transformed Hamiltonian of section 16.3. The regularization clearly corrects the overestimation of relativistic effects that plagues the Pauli Hamiltonian. There is another consequence of the small r behavior. Since the relativistic correction operator behaves like 1 jr, it ought to be possible to use T zoRA variationally—and in fact we may demonstrate that there is a variational lower bound. 

For the two-electron terms we need to identify the spin-orbit operators in the transformed Hamiltonian. Again, we only consider the free-particle Foldy-Wouthuysen transformation corrections from higher-order transformations are likely to be very small. The transformed Coulomb interaction was written in (16.63) as... 

One can gain some insight into the nature of the Dirac wave equation and the spin angular momentum of the electron by considering the Foldy Wouthuysen transformation for a free particle. In the absence of electric and magnetic interactions, the Dirac Hamiltonian is... 

Its application to the free-particle Dirac Hamiltonian yields the desired block-diagonal form, which has already been comprehensively discussed by Foldy and Wouthuysen [42]. However, the application of the fpFW transformation has proven to be extremely useful also in the presence of an external potential. It can still be performed in closed form and yields the Hamiltonian... 

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 now discuss one of these methods, the Foldy-Wouthuysen (FW) transformation for one particle in an external potential, a situation already far more complicated than the free electron. In this transformation, one tries to find a unitary transformation U = exp[ S] similar to Eq. [76], by which to transform the Dirac Hamiltonian and the 4-component wavefunction. 

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