Elimination of the small component

The central idea of all elimination methods for the small component is to employ relations (13) and (14) and to substitute f in Eq. (5) by an expression for the large component only. This yields a two-component equation for the latter only, which can be written as 

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 

Since a is a scalar operator, even this simple energy-dependent elimination of the small component permits an exact separation of the spin-free and spin-dependent terms of the Dirac Hamiltonian by applying Dirac s relation 

In the mid-eighties another method to eliminate the small component has been developed in order to arrive at regular expansions for the Hamiltonian [13,26]. These regular approximations are based on the general theory of effective Hamiltonians [27,28], where the full problem under consideration is projected onto a smaller, suitably chosen model space with an effective Hamiltonian, which comprises all desired properties of the problem sufficiently well. In the case of the Dirac Hamiltonian the basic idea, being as simple as ingenious, is to rewrite the expression for w in the form 

The latest major achievement in the field of elinnination techniques for the small component is due to Dyall and has been worked out to an efficient computational tool for quantum chemistry within the last few years [36-39]. This method is commonly dubbed normalised elimination of the small component (NESC) and is based on the modified Dirac equation [40,41], where the small component (f of the 4-spinor 4 is replaced by a pseudolarge component defined by the relation 

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. 

We assume that the potentials f) and A r) are time independent and collect the time dependence of the wavefunction in a phase factor 

From Eq. (2.84) we can see that the small component of the wavefunction can be expressed in terms of the large component as 

Inserting this expression in Eq. (2.83) we obtain a single two-component equation for the large component 

This equation together with the expression for the small component in Eq. (2.85) is still the Dirac equation. In order to reduce it to a non-relativistic expression we have to expand ——--------------. If we introduce the non-relativistic energy = E — 

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This non-relativistic equation in terms of four-component spinors has been studied in detail by Levy-Leblond [44,45], who has shown that it results automatically from a study of the irreducible representations of the Gahlei group and that it gives a correct description of spin. It is easy to see that in the absence of an external magnetic field, equation (63) is equivalent to the Schrodinger equation in the sense that after elimination of the small component ... 

Because often only the field-free Pauli Hamiltonian is presented in literature, we shall briefly sketch the derivation of the Hamiltonian hPauh(i) within an external field. For this, we start with the elimination of the small component in the one-electron Dirac equation by substitution of the small component of Eq. (15) to obtain an expression of the large component only... 

An expansion in (E — V)/2me2 is the basis of the method of elimination of the small component, which in its classical version is only of limited use because the expansion... 

The most popular way to derive the non-relativistic limit (nrl) of the Dirac equation is known as the method of the elimination of the small component [38, 39]. One starts by writing the DE as... 

One-component calculations or two-component calculations including also spin-orbit coupling effects provide a firm basis for the calculations of higher-order relativistic corrections by means of perturbation theory. Several quasi-relativistic approximations have been proposed. The most successful approaches are the Douglas-Kroll-Hess method (DKH) [1-7], the relativistic direct perturbation theory (DPT) [8-24], the zeroth-order regular approximation (ZORA) [25-48], and the normalized elimination of small components methods (NESC) [49-53]. Related quasi-relativistic schemes based on the elimination of the small components (RESC) and other similar nonsingular quasi-relativistic Hamiltonians have also been proposed [54-61]. 

Transformed Dirac equations are convenient starting points for the derivation of quasi-relativistic Hamiltonians. The transformed Dirac equations can be obtained by using approximate solutions for the small components as ansatze for the wave function. The ansatz can be deduced from the lower half of the Dirac equation by an approximate elimination of the small component. 

The nonrelativistic Schrodinger equation can be obtained by applying the analogous elimination of the small component transformation on the Levy-Leblond equation [65,66]. The 4-component L6vy-Leblond equation... 

The present idea is to replace the ZORA ansatz, which already is an approximation to the energy-dependent elimination of the small component approach, with another but similar expression that relates the large and the small components. The general ansatz function should have the same shape as the ZORA function close to the nucleus. Its first derivative should also be reminiscent of that of the ZORA function. A general function f[r) that fulfills the desired asymptotic conditions for r —0 and for r -> < can for example consist of one exponential function or of a linear combination of a couple of exponential functions as... 

A straightforward elimination of the small components from the Dirac equation leads to the two-component Wood-Boring (WB) equation [81], which exactly yields the (electronic) eigenvalues of the Dirac Hamiltonian upon iterating the energy-dependent Hamiltonian... 

Before leaving the theoretical formalism section, it is important to note that perturbation theory for relativistic effects can also be done at the fo n-con onent level, i.e. before elimination of the small component by a Foldy-Wouthitysen (FW) or Douglas-Kroll transformation. This is best done with direct perturbation theory (DPT) [71]. DPT involves a change of metric in the Dirac equation and an expansion of this modified Dirac eqtiation in powers of c . The four-component Levy-Leblond equation is the appropriate nonrelativistic limit. Kutzelnigg [72] has recently worked out in detail the simultaneous DPT for relativistic effects and magnetic fields (both external and... 

These problems are avoided if one uses regular Hamiltonians which are bounded from below. Many applications are based on the so-called zero order regular approximation (ZORA), which has been extensively investigated by the Amsterdam group [46-50]. It can be viewed as the first term in a clever expansion of the elimination of the small component, an expansion which already covers, at zeroth order, a substantial part of the relativistic effects. In fact ZORA is a rediscovery of the so-called CPD Hamiltonian (named after the authors. Ref. [60]). 

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 ... 

Some of the other developments in the area have been derived by Dyall et al. [47,48] and improved recently by Filatov et al. [49-51]. The method is known as the normalized elimination of the small component (NESC) method. One has to also mention the recent work of Liu et al. (see the review [52]) and Ilias [15]. 

Elimination of the small component 2a( ) Is ds to a second-order differential equation for the large component Pa( )> the radial Wood-Boring (WB) equation (Wood and Boring... 

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. 

So far no approximation has been introduced at all, and Eq. (13.4) still yields, of course, the exact Dirac eigenvalues and large components. Also the non-relativistic limit (V 2meC, e 0) of Eq. (13.4) is well defined and recovers the Schrodinger equation by setting a = 1. Since a is a scalar operator, even this simple energy-dependent elimination of the small component based on Eqs. (13.4) and (13.5) permits an exact separation of the spin-free and spin-dependent terms of the Dirac Hamiltonian by application of Dirac s relation... 

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