Perturbation Theory of Relativistic Effects

Perturbation Theory of Relativistic Effects Werner Kutzelnigg  

In the introduction of his probably most-quoted paper Dirac [1] claims that relativistic effects are unimportant for chemistry and most of physics. It is hard to understand [2] that the father of relativistic quantum theory underestimated so much the importance of his own fundamental work [3]. Nevertheless Dirac s claim keeps some meaning if we attenuate it to  

For most molecules, built up from light atoms, relativistic effects are so small, that they can be treated safely and in a cheap way by means of perturbation theory. 

In the community of Relativistic Quantum Chemistry one often hears the opposite statement, that there is no need for a perturbation theory of relativistic effects, since it is straightforward (though admittedly not cheap ) to perform fully relativistic calculations [4]. 

The following arguments can be put forward in favor of perturbation theory  

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General two-component methods have been discussed in various chapters of the first part of this book, for instance in chapter 11 on Two-Component Methods and the Generalised Douglas-Kroll Transformation by Wolf, Reiher and Hess [165], in chapter 12 by Kutzelnigg on Perturbation Theory of Relativistic Effects [166] and in chapter 13 by Sundholm on Perturbation Theory Based on Quasi-Relativistic Hamiltonians [167]. 

W. Kutzelnigg, Perturbation theory of relativistic effects, in P. Schw-erdtfeger (Ed.), Relativistic Electronic Structure Theory, Part 1, Fundamentals, Elsevier, Netherlands, 2002, pp. 669-762. 

V. Kutzelrdgg. Perturbation Theory of Relativistic Effects. In P. Schwerdtfeger, Ed., Relativistic Electronic Structure Theory — Part I. Fundamentals, p. 664-757, Amsterdam, 2002. Elsevier. 

W. Kutzelnigg. Effective Hamiltonians for degenerate and quasidegenerate direct perturbation theory of relativistic effects. 

Perturbation theory, including relativistic effects without the contribution of spin-orbit coupling from Reference 32. 

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... 

A review of the problems associated with the Dirac Hamiltonian as well as a classification of attempts to avoid them has been given by Kutzelnigg (1984). Recently, the same author developed a direct perturbation theory for relativistic effects (Kutzelnigg 1990, Kutzelnigg et al. 1995, Ottschofski and Kutzelnigg 1995). Earlier work in this direction was published by Rutkowski (1986a-c) and Jankowski and Rutkowski (1987). A modification of the Dirac-Coulomb-Breit Hamiltonian which allows the exact separation of spin-free and spin-dependent terms has been proposed by Dyall (1994b). 

As the fully relativistic (four-component) calculations demand severe computational efforts, several quasirelativistic (two-component) approximations have been proposed in which only large components are treated explicitly. The approaches with perturbative treatment of relativistic effects [507] have also been developed in which a nonrelativistic wavefunction is used as reference. The Breit-Pauli (BP) approximation uses the perturbation theory up to the (p/mc) term and gives reasonable results in the first-order perturbation calculation. Unfortunately, this method cannot be used in... 

Ab initio calculations with the help of finite perturbation theory on relativistic effects have recently been done by Nakatsuji et al. Relativistic DFT calculations on chemical shifts were performed by Malkin et al. ... 

Perhaps the simplest and most cost-effective way of treating relativistic contributions in an all-electron framework is the first-order perturbation theory of the one-electron Darwin and mass-velocity operators [46, 47]. For variational wavefunctions, these contributions can be evaluated very efficiently as expectation values of one-electron operators. 

The first-order perturbation theory estimate of relativistic effects (inclusion of the mass-velocity and one-electron Darwin terms as suggested by Cowan and Griffin) is cheap and easy to compute as a property value at the end of a calculation. It is therefore very valuable as a check on the importance of relativistic effects, and should certainly be included in accurate calculations on, for example, transition-metal compounds. For even heavier elements relativistic effective core potentials should be used. 

Since the Dirac equation is written for one electron, the real problem of ah initio methods for a many-electron system is an accurate treatment of the instantaneous electron-electron interaction, called electron correlation. The latter is of the order of magnitude of relativistic effects and may contribute to a very large extent to the binding energy and other properties. The DCB Hamiltonian (Equation 3) accounts for the correlation effects in the first order via the Vy term. Some higher order of magnitude correlation effects are taken into account by the configuration interaction (Cl), the many-body perturbation theory (MBPT) and by the presently most accurate coupled cluster (CC) technique. 

We refer the interested reader to our previous report1 for a review of the literature on many-body perturbation theory studies of relativistic effects molecules upto 1999. Here the background to the relativistic many-body problem in molecules was given in Section 2.1 and a review of the relativistic many-body perturbation theory was given in Section 2.3. 

That the perturbation theory (PT) of relativistic effects has not yet gained the popularity that it deserves, is mainly due to the fact that early formulations of the perturbation expansion in powers of were based on the Foldy-Wouthuysen transformation [11]. In this framework PT is not only formally rather tedious, it also suffers from severe singularities [12, 13], the controlled cancellation of which is only possible at low orders... 

Although a perturbation theory based on the FW transformation is obsolete - it is not only very laborious, but is also suffers from serious unphysical singularities - one cannot present the PT of relativistic effects, without demonstrating why the approach based on the FW transformation is bound to fail. 

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

Recent advances in electronic structure theory achieved in our group have been reviewed. Emphasis is put on development of ab initio multireference-based perturbation theory, exchange and correlation functionals in density functional theory, and molecular theory including relativistic effects. 

Bogumil Jeziorski received his M.S. degree in chemistry from the University of Warsaw in 1969. He conducted his graduate work also in Warsaw under the supervision of W. Kolos. After a postdoctoral position at the University of Utah, he was a research associate at the University of Florida and a Visiting Professor at the University of Waterloo, University of Delaware and University of Nijmegen. Since 1990 he has been a Professor of Chemistry at the University of Warsaw. His research has been mainly on the coupled-cluster theory of electronic correlation and on the perturbation theory of intermolecular forces. His other research interests include chemical effects in nuclear beta decay, theory of muonic molecules and relativistic and radiative effects in molecules. 

Relativistic Many-body Perturbation Theory. - Since the early 1980s, we have witnessed a growing interest in the effects of relativity on the electronic structure of atoms and molecules. Over the past decade the theoretical and computational machinery has been put in place for a relativistic many-body perturbation theory of atomic and molecular electronic structure.29,32-36,38... 

Also in the molecular case perturbation theory helps to clarify individual contributions to the property ax cording to their order [120] but more crucial is the fact that perturbation theory becomes an inadequate means for the proper description of relativistic effects in heavy atoms. Furthermore spin-orbit effects can substantially influence the result and methods using multi-component wave functions incorporating spin-orbit coupling from the beginning are favorable. 

Vaara, J. Manninen, P. Lantto, P. Perturbational and ECP Calculation of Relativistic Effects in NMR Shielding and Spin-Spin Coupling. In Calculation of NMR and EPR Parameters Theory and Applications-, Kaupp, M., Buhl, M., Malkin, V. G., Eds. Wiley-VCH Verlag GmbH Weinheim, 2004 Chapter 13, pp 209-226. 

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