Direct-perturbation theory

We may consider making the perturbation partitioning in the Dirac equation itself, without eliminating the small component, but first we have to write the Dirac equation in a form involving 1/c rather than c. This is simply achieved by redefining the small component, much as we did in the modified Dirac equation. Here we make the replacement 

Applying the transformation to the Dirac Hamiltonian, we may now partition it into a zeroth-order term and a perturbation of 

Here we are explicitly taking into account the fact that the metric has a relativistic correction, a feature that in the Pauli approximation was avoided with the consequence of the appearance of pathological operators. The metric in fact is a combination of projectors onto the large and small components. Using the definitions from (15.40), 

Using these projection operators, the perturbation H2 may be written in the form 

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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... 

The example of neon, where relativistic contributions account for as much as a0.5% of 711, shows that relativistic effects can turn out to be larger for high-order NLO properties and need to be included if aiming at high accuracy. Some efforts to implement linear and nonlinear response functions for two- and four-component methods and to account for relativity in response calculations using relativistic direct perturbation theory or the Douglas-Kroll-Hess Hamiltonian have started recently [131, 205, 206]. But presently, only few numerical investigations are available and it is unclear when it will become important to include relativistic effects for the frequency dispersion. 

Hamiltonian (PH), Douglas-KroU-Hess (DKH), and direct perturbation theory (DPT)—give similar results as the much cheaper, quasi-relativistic ECP calculations. There is no reason for not using ECPs for the calculation of geometries, energies, and vibrational spectra of TM compounds. We want to point out that relativistic aU-electron methods become important for the calculation of NMR chemical shifts and coupling constants, because the effect of the core electrons is not negligible anymore. This is discussed in more detail later. 

Table 8 Maximum box radii Rpj and Rpa( e for, respectively, direct perturbation theory expansion and Pade approximant, which guarantee a relative accuracy of at least 10 6 in the energy...

The starting point for direct perturbation theory, to be discussed in detail in chapter 12, is equation (123) with the L6vy-Leblond equation as the zeroth-order equation. Solving for the small components we obtain the following equation... 

But we should not proceed without mentioning two other methods in order to arrive at workable and regular formulations. The first method is nowadays dubbed Direct Perturbation Theory (DPT) and guarantees a valid limiting procedure for 1/c —0 [43,44,47-51]. Here, it is very important to consider the non-relativistic limit of the metric (essentially the normalisation requirement) and of the operator itself separately. This is most conveniently achieved by formulating the Dirac equation in terms of a scaled small component c(f>. This approach will comprehensively be discussed in the article of Kutzelnigg in the next chapter and thus not further be described here. 

No such singularities arise if one uses the direct perturbation theory (DPT) [12, 13, 15], which starts directly from the Dirac equation, with the Levy-Leblond equation [16] as zeroth-order (non-relativistic) approximation. Unfortunately, most practical applications of DPT were so far limited to the leading order, which is particularly easily implemented. This has sometimes led to the unjustified identification of DPT with its lowest order. Higher orders of DPT are straightforward, but have only occasionally been evaluated [17, 18, 19]. Even an infinite-order treatment of DPT is possible [12, 20], where one starts with a non-relativistic calculation and improves it iteratively towards the relativistic result. 

We start this chapter with a discussion of the non-relativistic limit (nrl) of relativistic quantum theory (section 2). The Levy-Leblond equation will play a central role. We also discuss the nrl of electrodynamics and study how properties differ at their nrl from the respective results of standard non-relativistic quantum theory. We then present (section 3) the Foldy-Wouthuysen (FW) transformation, which still deserves some interest, although it is obsolete as a starting point for a perturbation theory of relativistic corrections. In this context we discuss the operator X, which relates the lower to the upper component of a Dirac bispinor, and give its perturbation expansion. The presentation of direct perturbation theory (DPT) is the central part of this chapter (section 4). We discuss the... 

We can now apply PT as in subsection 3.3 and get results without divergences. We are not going to pursue this possibility, since the direct perturbation theory to be outlined later (section 4), is much easier, and since nonhermitean effective operators are not very convenient. 

We make a change of the metric in the Dirac equation in the sense of direct perturbation theory i.e. we take again Eq. (17) as our starting point. As shown in subsection (4.1) expansion of (17) in powers of c leads to direct perturbation theory (DPT). 

The functionals Fik i>2k) play a central role in stationary direct perturbation theory. Fq iPq) has been called the Levy-Leblond functional [23], since its stationarity condition is the LLE. For 4( 2) th name Hylleraas-Rutkowski functional has been suggested [23], since this belongs to the class of Hylleraas functionals of second-order perturbation theory, and since it has first been proposed by Rutkowski [73, 74] in a slightly different form. 

DIRECT PERTURBATION THEORY USING ENERGY GRADIENTS OR FINITE PERTURBATIONS... 

CONCLUSIONS. MERITS AND DRAWBACKS OF DIRECT PERTURBATION THEORY... 

The direct perturbation theory (DPT) of relativistic effects has a few nice features. 

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]. 

In this Chapter, we will show how a whole family of one- and two-component quasi-relativistic Hamiltonians can conveniently be derived. The operator difference between the quasi-relativistic Hamiltonians and the Dirac equation can be explicitly identified and used in perturbation expansions. Expressions are derived for a direct perturbation theory scheme based on quasi-relativistic two-component Hamiltonians. The remaining difference between the variational energy obtained using quasi-relativistic Hamiltonians and the energy of the Dirac equation is estimated numerically by applying the direct perturbation theory ap-... 

The transformation of the Dirac equation into two-component equations are also discussed in Chapter 11, whereas Chapter 12 deals with the relativistic direct perturbation theory. 

As seen in equation (26), the quasi-relativistic Hamiltonian and the operators describing the difference between the exact Dirac Hamiltonian and the quasi-relativistic one are now explicitly separated and the direct perturbation theory method can be applied. In the direct perturbation theory approach, the metric is also affected by the perturbation [12]. Note that the interaction matrix is block diagonal at the lORA level of theory, whereas the coupling between the upper and the lower components still appears in the metric. 

In the spirit of the direct perturbation theory approach [8-23,38], equation (26) can be expanded in an infinite perturbation series expansion as... 

As shown in Section 4, quasi-relativistic Hamiltonians such as the lORA, ERA, MIORA and MERA ones can be used as a zeroth-order approximation to the Dirac Hamiltonian. The operator difference between the quasi-relativistic and fully relativistic equations can be used as a perturbation operator and the corresponding energy difference can be considered by using a direct perturbation theory approach. 

The first-order perturbation theory corrections to the quasi-relativistic energies obtained with the lORA and the ERA Hamiltonian as the zeroth-order approximation to the Dirac equation are given in Table 6 and Table 7, respectively. Direct perturbation theory calculations on top of the MIORA and MERA Hamiltonians has not been studied computationally. 

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