Transformations 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). 

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... 

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 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. 

After having carried out an initial free-particle Foldy-Wouthuysen transformation Uo one could try to establish a sequence of further xmitary transformations Ui = ), i = 1,2,3,...) written as... 

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,... 

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. 

For convenience we recall the result of the closed-form free-particle Foldy-Wouthuysen transformation of Eq. (11.35) ... 

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... 

This free-particle Foldy-Wouthuysen-transformed two-electron part has been investigated by various authors [225,623,627,643,653,654]. Still, these are expressions of the four-component framework. Similarly to the procedure for one-electron operators the restriction to the electronic, i.e., upper-left part of... 

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... 

Free-Particle Foldy-Wouthuysen Transformation of Properties... 

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... 

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... 

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