Foldy-Wouthuysen transformation expansion

Our approach is based on a systematic semiclassical study of the Dirac equation. After separating particles and anti-particles to arbitrary powers in h, a semiclassical expansion of the quantum dynamics in the Heisenberg picture is developed. To leading order this method produces classical spin-orbit dynamics for particles and anti-particles, respectively, that coincide with the findings of Rubinow and Keller Hamiltonian relativistic (anti-) particles drive a spin precession along their trajectories. A modification of that method leads to a semiclassical equivalent of the Foldy-Wouthuysen transformation resulting in relativistic quantum Hamiltonians with spin-orbit coupling. 

The weakly relativistic limit of the Hamiltonian (2.20) for fermions in external electric and magnetic fields can be derived with standard techniques, either by direct expansion or by a low order Foldy-Wouthuysen transformation. One obtains... 

In the nonrelativistic context current-density functional theory is based on the nonrelativistic limits of the paramagnetic current (87) and/or the magnetization density (89) [128,129]. In the relativistic situation, however, a density functional approach relying on jp or m can only be considered an approximation, as long as the external magnetic field does not vanish. In order to clarify the relation between these two points of view the weakly relativistic limit of RDFT has to be analyzed. The weakly relativistic limit of the Hamiltonian (23) can be derived either by a direct expansion in 1/c or by a low order Foldy-Wouthuysen transformation,... 

That the perturbation theory (PT) of relativistic effects has not yet gained the popularity that it deserves, is mainly due to the fact that early formulations of the perturbation expansion in powers of were based on the Foldy-Wouthuysen transformation [11]. In this framework PT is not only formally rather tedious, it also suffers from severe singularities [12, 13], the controlled cancellation of which is only possible at low orders... 

Of course, the Dirac operator for H-like ions has continuum states as well, including ultrarelativistic ones. One can therefore neither expect that all eigenstates are analytic in c , nor that the entire Dirac operator allows an expansion in powers of c. This can at best be the case for the projection of D to positive-energy non-ultrarelativistic states. The paradigm, on which the Foldy-Wouthuysen transformation is based, to construct a Hamiltonian, related to the Dirac operator by a unitary transformation, in an expansion in c ... 

One seemingly sensible approach to the relativistic electronic structure theory is to employ perturbation theory. This has the apparent advantage of representing supposedly small relativistic effects as corrections to a familiar non-relativistic problem. In Appendix 4 of Methods of molecular quantum mechanics, we find the terms which arise in the reduction of the Dirac-Coulomb-Breit operator to Breit-Pauli form by use of the Foldy-Wouthuysen transformation, broken into electronic, nuclear, and electron-nuclear effects. FVom a purely aesthetic point of view, this approach immediately looks rather unattractive because of the proliferation of terms at the first order of perturbation theory. To make matters worse, many of the terms listed are singular, and it is presumably the variational divergences introduced by these operators which are referred to in [2]. Worse still, higher-order terms in the Foldy-Wouthuysen transformation used in this way yield a mathematically invalid expansion. 

It must be stressed again — following the original paper by Foldy and Wouthuysen — that the phrase Foldy-Wouthuysen transformation is strictly reserved for denoting a 1/c expansion of fu rather than any arbitrary decoupling transformation of the Dirac Hamiltonian as it is sometimes applied in the literature. 

According to Eq. (11.93), the decoupled Hamiltonian within the Foldy-Wouthuysen framework is formally given as a series of even terms of well-defined order in 1/c. In most presentations of the Foldy-Wouthuysen transformation the exponential function parametrization Hjj] = exp(W[j]) is applied for each transformation step. However, in the light of the discussion in section 11.4 the specific choice of this parametrization does not matter at all, since one necessarily has to expand Ui into a power series in order to evaluate the Hamiltonian. Consequently, in order to guarantee a most general analysis, the most general parametrization for the Foldy-Wouthuysen transformation should be employed [610]. Thus, li is parametrized as a power series expansion in an odd and antihermitean operator W, , which is of (2/+l)-th order in 1/c, (cf. section 11.4). After n transformation steps, the intermediate, partially transformed Hamiltonian f has the following structure. 

The intrinsic failure of the Foldy-Wouthuysen protocol is therefore without doubt related to the ill-defined 1/c expansion of the kinetic term Ep, which does not bear any reference to the external potential V. However, in the literature the ill-defined behavior of the Foldy-Wouthuysen transformation has sometimes erroneously been attributed to the singular behavior of the Coulomb potential near the nucleus, and even the existence of the correct nonrelativistic limit of the Foldy-Wouthuysen Hamiltonian is sometimes the subject of dispute. Because of Eqs. (11.82) and (11.83) and the analysis given above, the nonrelativistic limit c —> oo, i.e., X —> 0 is obviously well defined, and for positive-energy solutions given by the Schrodinger Hamiltonian /nr = / 2me + V. 

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. 

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). 

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... 

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... 

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... 

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. 

This chapter is devoted to the development of perturbation expansions in powers of 1 /c from the Dirac equation. In the previous chapter, the Pauli Hamiltonian was developed using the Foldy-Wouthuysen transformation. While this is an elegant method, it is probably simpler to make the derivation from the elimination of the small component with expansion of the denominator, and it is this approach that we use here. Another convenient approach is to make use of the modified Dirac equation in the limit of equality of the large and pseudo-large components. This approach enables us to draw on results from the modified Dirac approach in developing the two-electron terms of the Breit-Pauli Hamiltonian. We then demonstrate how the use of perturbation theory for relativistic corrections requires that multiple perturbation theory be employed for correlation effects and for properties. The last sections of this chapter are... 

Relativity becomes important for elements heavier than the first row transition elements. Most methods applicable on molecules are derived from the Dirac equation. The Dirac equation itself is difficult to use, since it involves a description of the wave function as a four component spinor. The Dirac equation can be approximately brought to a two-component form using e.g. the Foldy-Wouthuysen (FW) transformational,12]. Unfortunately the FW transformation, as originally proposed, is both quite complicated and also divergent in the expansion in the momentum (for large momenta), and it can thus only be carried out approximately (usually to low orders). The resulting equations are not variationally stable, and they are used only in first order perturbation theory. 

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... 

We start this chapter with a discussion of the non-relativistic limit (nrl) of relativistic quantum theory (section 2). The Levy-Leblond equation will play a central role. We also discuss the nrl of electrodynamics and study how properties differ at their nrl from the respective results of standard non-relativistic quantum theory. We then present (section 3) the Foldy-Wouthuysen (FW) transformation, which still deserves some interest, although it is obsolete as a starting point for a perturbation theory of relativistic corrections. In this context we discuss the operator X, which relates the lower to the upper component of a Dirac bispinor, and give its perturbation expansion. The presentation of direct perturbation theory (DPT) is the central part of this chapter (section 4). We discuss the... 

The Breit-Pauli (BP) approximation [140] is obtained truncating the Taylor expansion of the Foldy-Wouthuysen (FW) transformed Dirac Hamiltonian [141] up to the (p/mc) term. The BP equation has the well-known mass-velocity, Darwin, and spin-orbit operators. Although the BP equation gives reasonable results in the first-order perturbation calculation, it cannot be used in the variational treatment. 

The Foldy-Wouthuysen (FW) transformation [67] offers a decoupling, which in principle is exact, but it is impractical and leads to a singular expansion in 1/c in the important case of a Coulomb potential [68]. Douglas and Kroll (DK) suggested an alternative decoupling procedure based on a series of appro-... 

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... 

Chapter 11 introduced the basic principles for elimination-of-the-small-component protocols and noted that the Foldy Wouthuysen scheme applied to one-electron operators including scalar potentials yield ill-defined 1 /c-expansions of the desired block-diagonal Hamiltonian. In contrast, the Douglas Kroll-Hess transformation represents a unique and valid decoupling protocol for such Hamiltonians and is therefore investigated in detail in this chapter. 

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