Foldy-Wouthuysen transformation, for

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... 

This defines the exact Foldy-Wouthuysen transformation for the free particle. Note that the square root is not expanded here. 

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. 

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. 

An explicit expression for the Breit potential was derived in [2] from the one-photon exchange amplitude with the help of the Foldy-Wouthuysen transformation ... 

We have therefore achieved our objective in that equation (3.83), which is correct to order 1 /c2, contains even operators only. It would, of course, be possible to proceed further with the Foldy Wouthuysen transformation but there is little point in doing so, since the theory is inaccurate in other respects. For example, we have treated the electromagnetic field classically, instead of using quantum field theory. Furthermore, we shall ultimately be interested in many-electron diatomic molecules, for which it will be necessary to make a number of assumptions and approximations. 

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... 

SO coupling is a relativistic effect. The theory of the interaction of the magnetic moments of the electron spin and the orbital motion in one- and two-electron atoms has been formulated independently by Heisenberg and Pauli [12,13], shortly before the advent of the four-component Dirac theory of the electron [14]. Breit later has added the retardation correction [15]. The resulting Breit-Pauli SO operator, which can more elegantly be derived from the Dirac equation via a Foldy-Wouthuysen transformation [16], was thus well known for atoms since the early 1930s [17]. 

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... 

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 ... 

Explicit inclusion of relativistic effects in valence-only calculations has been by far less frequently attempted. Datta, Ewig and van Wazer [135] used a Phillips-Kleinman PP in a study of PbO, whereas Ishikawa and Malli [136] tested PPs of semilocal form in four-component atomic DHF finite difference calculations. This work was extended by Dolg [137] to four-component molecular DHF calculations with a subsequent correlation treatment. In addition a complicated form of Vcv based on the Foldy-Wouthuysen transformation [138] was derived by Pyper [139] and applied in atomic calculations [140]. For all these approaches the computational effort is significantly higher than for the implicit treatment of relativity, and the gain of computational accuracy is not obvious at all. 

Although the Dirac equation in its four-component form can be solved exactly for a few systems (the hydrogen-like atom, electron in a uniform magnetic field), normally a decoupling to the two-component form has to be done [2-6], For this purpose two techniques were developed the Foldy-Wouthuysen transformation and the partitioning method (for a small component). We will follow the second approach, which is based on these steps ... 

Several authors have considered a number of approximate solutions to the Dirac equation. One such method is the use of the Foldy-Wouthuysen transformation (see, for example, Morrison and Moss ). Upon application of a unitary transformation of the form... 

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. 

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 section we consider the first step of the Foldy-Wouthuysen transformation, which eliminates the first odd operator of order c. If we employ the idea presented in Eq. (11.57) for this purpose, we may use the antihermitian operator W to remove the odd term Since this odd term cup does... 

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 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. 

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. 

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