Unitary Transformations of the Dirac Hamiltonian

The unitary transformation of the Dirac Hamiltonian to two-component form is accompanied by a corresponding reduction of the wave function. As discussed in detail in chapters 11 and 12, the four-component Dirac spinor ip will... 

Another major two-component approximation is the Foldy-Wouthuysen transformation (Foldy and Wouthuysen 1950), which makes the large-component and small-component submatrices of the Dirac Hamiltonian matrix, Hd, linear independent by a unitary transformation such as... 

To continue the block-diagonalization of the Dirac Hamiltonian, one needs to find the form of the (7i transformation. The form proposed by Heully et al. [56] was used in the lOTC approach. It is a general technique equally well applied to the initial Dirac Hamiltonian ho or to the free-particle transformed operator (4.32). According to Heully et al., the unitary transformation Ui can be written in the following form ... 

The natural solution of the operator Eq. 4.38 would be in the momentum space due to the presence of momentum p which replaces the standard configuration space formulation by a Eourier transform. The success of the Douglas-Kroll-Hess and related approximations is mostly due to excellent idea of Bernd Hess [53,54] to replace the explicit Eourier transformation by some basis set (discrete momentum representation) where momentum p is diagonal. This is a crucial step since the unitary transformation U of the Dirac Hamiltonian can easily be accomplished within every quantum chemical basis set program, where the matrix representation of the nonrelativistic kinetic energy T = j2m is already available. Consequently, all DKH operator equations could be converted into their matrix formulation and they can be solved by standard algebraic techniques [13]. 

Four-component methods are computationally expensive since one has to deal with small-component integrals. Therefore, various two-component methods in which small-component degrees of freedom are removed have flourished in the literature. We focus the present discussion on the X2c theory at one-electron level (X2c-le). The X2c-le scheme consists of a one-step block diagonalization of the Dirac Hamiltonian in its matrix representation via a Foldy-Wouthuysen-type matrix unitary transformation ... 

The Foldy-Wouthuysen transformation (FWT) (Foldy and Wouthuysen 1950) of the Dirac Hamiltonian uses a sequence of unitary operators f/=exp(i5), with S being hermi-tian, to decouple the upper from the lower components of the wavefunction W = U0 and to remove odd terms in the resulting Hamiltonian H = U HU successively up to a given order of a. (Odd operators like a couple the upper and lower components, whereas even operators like fi do not if the resulting Hamiltonian is even up to a given order of a, its... 

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 previous discussion the semiclassical separation of particles and antiparticles employed projection operators and the associated subspaces of the Hilbert space. By suitable choices of bases such a separation can also be constructed with the help of unitary operators rotating the Hamiltonian into a block-diagonal form. Such a procedure is closely analogous to the Foldy-Wouthuysen transformation that provides a similar separation in a non-relati-vistic limit. A (unitary) semiclassical Foldy-Wouthuysen transformation Usc rotates the Dirac-Hamiltonian Hd into... 

In section 3.2 we pointed out that the Dirac Hamiltonian contains operators which connect states of positive and negative energy. What we now seek is a Hamiltonian which is relativistically correct but which operates on the two-component electron functions of positive energy only. We require that this Hamiltonian contain terms representing electromagnetic fields, and Foldy and Wouthuysen [12] showed, by a series of unitary transformations, that such a Hamiltonian can be derived. The Dirac Hamiltonian... 

It should be mentioned that there are only a few restrictions on the choice of the matrices C/j. Firstly they have to be unitary and analytical (holomorphic) functions on a suitable domain of, and secondly they have to permit a decomposition of Hm in even terms of definite order in the external potential according to Eq. (73). It is thus possible to parametrise them without loss of generality by a power series expansion in an odd and antihermitean operator Wi of ith order in the external potential. In the following, the physical consequences of this freedom in the choice of the unitary transformations will be investigated. Therefore we shall start with a discussion of all possible parametrisations in terms of such power series expansions. Afterwards the most general parametrisation of Ui is applied to the Dirac Hamiltonian in order to derive the fourth-order... 

There are two major approaches to reduce the Dirac Hamiltonian to two-component form. The various regular elimination techniques have been developed to highly sophisticated and very successful methods, which are widely used by the community. The other approach comprises the various unitary transformation methods, which amount either to expansions in 1 /c as for the FW transformation or to expansions in powers of the external potential V as for the DK approximations. In addition, we have presented a very general extension of the traditional DK approximation to arbitrary unitary transformations. 

The two-component methods, though much simpler than the approaches based on the 4-spinor representation, bring about some new problems in calculations of expectation values of other than energy operators. The unitary transformation U on the Dirac Hamiltonian ho (Eq.4.23 is accompanied by a corresponding reduction of the wave function to the two-component form (Eq.4.26). The expectation value of any physical observable 0 in the Dirac theory is defined as ... 

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