The Free-Particle Foldy-Wouthuysen Transformation

Unfortunately, this X-operator and hence the unitary transformation U are not available in closed form, except for a few very special cases, e.g., the free particle or an electron being exposed to a homogeneous magnetic field [605]. It is, however, possible to solve equations of the type shown in Eq. (11.6) purely numerically in a given one-particle basis (see section 11.6 below). 

In addition, approximate decoupling schemes can be envisaged in order to arrive at the block-diagonal Hamiltonian of Eq. (11.15), which will be particularly valuable in cases with complicated expressions for the potential V. 

This can either be achieved by a systematic analytic decomposition of the transformation U into a sequence of unitary transformations, each of which is expanded in an fl priori carefully chosen parameter. These issues will be addressed in detail in chapter 12, and we shall now stick to the long-known unitary transformation scheme for free particles, namely the free-particle Foldy-Wouthuysen transformation. 

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 

In the literature, many forms of the unitary matrix in Eq. (11.27) can be found and are not easy to be recognized as the same — often also because the notation is varied and authors chose not to stick to historically older conventions. To demonstrate how these different forms can be interconverted we may consider some of them explicitly in the following. An often employed version of Uq, identical to the one above, is the exponential form. 

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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... 

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

The free-particle Foldy-Wouthuysen transformation can still be performed in closed form even in the presence of a scalar potential V of any form. 

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... 

It must be emphasized that these expressions are still exact when compared to the original untransformed operator. The transformed operator /i would be completely decoupled if V = 0, i.e., if the particle were moving freely, and hence only the kinetic energy operator (apart from the rest energy term) remains. Thus, the even terms of the free-particle Foldy-Wouthuysen transformation already account for all so-called kinematic relativistic effects. 

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. 

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,... 

This section demonstrates how the first three unitary matrices are explicitly constructed and applied to the one-electron operator / (or to some of its parts such as + V uc)- The first transformation has necessarily to be the free-particle Foldy-Wouthuysen transformation Uo, which is followed by the transformation Ui. The third transformation U2 turns out to produce even operators that depend on the parametrization chosen for Uz- Afterwards the infinite-order, coefficient-dependence-free scheme is discussed. 

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

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... 

The same discussion applies equally well for the Breit interaction. Only the four terms of the second line of Eq. (12.70) yield even operators for g i,j) = The corresponding two-component form of the free-particle Foldy-Wouthuysen-transformed frequency-independent Breit interaction Bq reads... 

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. 

KB-transformed operators are crucial for the implementation of exact-decoup-ling methods. The free-particle Foldy-Wouthuysen transformation (cf. chapter 11)... 

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. 

Since Uq is a unitary matrix, it preserves the orthonormality condition. Ho is then diagonalized by a standard hermitean eigenvalue solver. The eigenvalue equation has the same structure as Eq. (14.35) with the primed labels (L) and (S) replaced by doubly primed ones, ( )" and (S)", to indicate the change of basis by the free-particle Foldy-Wouthuysen transformation. The X matrix in this basis representation is obtained by Eq. (14.36) with the same label replacement. Analogously, the renormalization matrix reads R" = I + The final decoupling transformation... 

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... 

Since all unitary decoupling transformations have necessarily to start with the free-particle Foldy-Wouthuysen transformation Uo, it is convenient to introduce the free-particle Foldy-Wouthuysen-transformed operator... 

We recall that it was the desire to find an expansion of this square root operator that led to the development of the Dirac equation (see chapter 4). We see also that the assumption that X commutes with (a p) was justified. The free-particle Foldy-Wouthuysen transformation can now be written... 

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. 

Notice that the kinetic relativistic correction applies to both the spin and orbital Zeeman terms, but the relativistic correction from the potential only affects the spin Zeeman term. These corrections are only the first in a truncated expansion, and for a better description of relativistic corrections we turn to the free-particle Foldy-Wouthuysen transformation and the higher transformations that are based on it. 

The magnetic terms for the free-particle Foldy-Wouthuysen transformation are more complicated because the operator is not linear in the momentum. It is possible nevertheless to define the field-dependent transformation by the same procedure used in section 16.1. We arrive at a two-component field-dependent operator corresponding to Cp, which we will call p,... 

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. 

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