Dirac equation in two-component form

In molecular property calculations the same mutual interplay of electron correlation, relativity and perturbation operators (e.g. external fields) occurs. For light until medium atoms relativistic contributions were often accounted for by perturbation theory facilitating quasirela-tivistic approximations to the Dirac-Hamiltonian [114-117]. It is well-known that operators like the Breit-Pauli Hamiltonian are plagued by essential singularities and therefore are not to be used in variational procedures. It can therefore be expected that for heavier elements per-turbational inclusion of relativity will eventually become inadequate and that one has to start from a scheme where relativitiy is included from the beginning. Nevertheless very efficient approximations to the Dirac equation in two-component form exist and will be discussed further below in combination with their relevance for EFG calculations. In order to calculate the different contributions to a first-order property as the EFG, Kello and Sadlej devised a multiple perturbation scheme [118] in which a first-order property is expanded as... 

Quasi-relativistic approximations usually start from re-writing the Dirac equation in two-component form. Starting from the Dirac equation (with rest energy subtracted)... 

The time-independent Dirac equation in two-component form with the rest mass subtracted becomes... 

The historically first reduction of the Dirac equation to two-component form is the Pauli approximation, which can be obtained from Eq. (26) by trancating the series expansion for cu after the first two terms, and eliminating the energy dependence by means of a systematic expansion in c. The result is the familiar Pauli Hamiltonian... 

The reduction of the relativistic many-electron hamiltonian by expansion in powers of the external field is the second-order Douglas-Kroll transformation [29], and has been used with success by Hess and co-workers [30]. The operators which result from this transformation are non-singular, but the integrals over the resulting operators are complicated and have to be approximated, even for finite basis set expansions. The reduction of the Dirac-Coulomb-Breit equation to two-component form using direct perturbation theory has been described by Kutzelnigg and coworkers [26, 27, 31], Rutkowski [32], and van Lenthe et al. [33]. 

However, it is more appropriate to provide theoretical justifications for such use. In this respect, first, we introduce the third category of decoupling of positive and negative states commonly known as the direct perturbation theory . This approach does not suffer from the singularity problems described previously. However, the four-component form of the Dirac equation remains intact. The new Hamiltonian requires identical computational effort as for the Dirac equation itself, hence it is not an attractive alternative to the Dirac equation. However, it is useful to assess the accuracy of approximate two-component forms derived from the Dirac equation such as Pauli Hamiltonian. Consider the transformation... 

At this point, it is appropriate to present a brief discussion on the origin of the FC operator (d function) in the two-component form (Pauli form) of the molecular relativistic Hamiltonian. Many textbooks adopt the point of view that the FC is a relativistic effect, which must be derived from the Dirac equation [50,51]. In other textbooks or review articles it is stressed that the FC is not a relativistic effect and that it can be derived from classical electrodynamics [52,53] disregarding the origin of the gyromagnetic factor g—2. In some textbooks both derivations are presented [54]. The relativistic derivations suffer from the inherent drawbacks in the Pauli expansion, in particular that the Pauli Hamiltonian can only be used in the context of the first-order perturbation theory. Moreover, the origin of the FC term appears to be different depending on whether one uses the ESC method or FW transformation. 

Earlier we mentioned briefly that the electron spin is perfectly consistent with the non-relativistic four-component Levy-Leblond theory [44,45]. The EC type interaction does not manifest in Dirac or Levy-Leblond theory. We shall show that on reducing the four-component Levy-Leblond equation into a two-component form the EC contribution arises naturally. A non-relativistic electron in an electromagnetic radiation field is described by the Levy-Leblond equation given by... 

One-particle wave-functions in a central field are obtained as solutions of the relativistic Dirac equation, which can be written in the two-component form ... 

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. 

Accounting for relativistic effects in computational organotin studies becomes complicated, because Hartree-Fock (HF), density functional theory (DFT), and post-HF methods such as n-th order Mpller-Plesset perturbation (MPn), coupled cluster (CC), and quadratic configuration interaction (QCI) methods are non-relativistic. Relativistic effects can be incorporated in quantum chemical methods with Dirac-Hartree-Fock theory, which is based on the four-component Dirac equation. " Unformnately the four-component Flamiltonian in the all-electron relativistic Dirac-Fock method makes calculations time consuming, with calculations becoming 100 times more expensive. The four-component Dirac equation can be approximated by a two-component form, as seen in the Douglas-Kroll (DK) Hamiltonian or by the zero-order regular approximation To address the electron cor-... 

In this section different methods to decouple the Dirac equation are discussed. A useful prerequisite for this reduction to two-component form is the analysis of the relationship between the large and small components of an exact eigen-solution of the Dirac equation. For every Dirac 4-spinor (f> given by equation... 

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 ... 

We have arrived at the system of three rigorous equations which allow us to solve the Dirac equation in the two-component form to an arbitrary degree of accuracy. Starting from the two-component equation... 

The decoupling of the Dirac equation to the two-component form is a rather complicated process. It manipulates the resolvant operator which is transferred from the denominator to the numerator and then exposed to consecutive commutator relations in order to be shifted to the far right. Finally, a number of one-electron terms of the order 1/c2 is obtained some of them have no classical analogy and cannot be derived from non-relativistic theories. 

This procedure is usually known as the elimination of the small component (ESC), and Eq. (34) is still equivalent to the original Dirac equation. Although the equation has been reduced to a two-component form, nothing is gained since we now have an energy-dependent Hamiltonian, and one must introduce further approximations to transform Eq. (34) into a form useful for actual calculations. The principal difference between the Pauli and the ZORA Hamiltonian is that to obtain the Pauli Hamiltonian, one uses an expansion in c ... 

A key element for the reduction to two-component form is the analysis of the relationship between the large and small components of exact eigenfunctions of the Dirac equation, which we have already encountered in section 5.4.3. This relationship emerges because of the (2 x 2)-superstructure of the Dirac Hamiltonian, see, e.g., Eq. (5.135), which turned out to be conserved upon derivation of the one-electron Fock-type equations as presented in chapter 8. Hence, because of the (2 x 2)-superstructure of Fock-type one-electron operators, we may assume that a general relation. 

Before taking the limit c oo, this equation must be rearranged for two reasons first, because we need to change it to a form where c occurs in some form of denominator—this will provide us with terms that vanish and hopefully other terms that remain finite—and, second, because the nonrelativistic wave function is a scalar function, whereas the Dirac wave function is a four-component vector function. If we use the two-component nonrelativistic Schrodinger equation that we derived in section 4.2, we can write the nonrelativistic wave function in terms of spin-orbitals, which can be transformed to two-component spinors. Then it is only necessary to reduce the Dirac equation from four-component to two-component form. 

Elimination (or isolation) of the small component provides a useful basis for discussion of the properties of the Dirae equation. In particular, in this section we want to develop the relationship between the large- and small-component basis sets. We write the matrix 2-spinor Dirac equation in the form of two coupled matrix equations,... 

We could continue now with the Dirac equation and derive expressions for the molecular properties using standard perturbation theory. However, as stated earlier, the exposition in these notes is restricted basically to a non-relativistic treatment with the exception that we want to include also interactions with the spin of the electrons. The appropriate operator can be found by reduction of the Dirac equation to a non-relativistic two-component form, which can be achieved by several approaches. Here, we want to discuss only the simplest approach, the so-called elimination of the small component. 

An alternative route to the calculation of relativistic effects is the systematic development of a perturbation operator for the relativistic effects. Historically, perturbation approaches were connected with the attempt to reduce the four-component form of the Dirac equation to two pairs of two-component equations. This approach caused problems with singular operators, which will be discussed in the next section. 

A regular alternative to the Foldy-Wouthuysen transformation was given by Douglas and Kroll and later developed for its use in electronic structure calculations by Hess et al. The Douglas-Kroll (DK) transformation defines a transformation of the external-field Dirac Hamiltonian Hq of equation (11) to two-component form which leads, in contrast to the Foldy-Wouthuysen transformation, to operators which are bounded from below and can be used variationally, similarly to those of the regular approximations discussed above. As in the FW transformation, it is not possible in the DK formalism to give the transformation in closed form. Rather, it is... 

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