Relativistic many-electron theory treatments

This potential is referred to in electromagnetism texts as the retarded potential. It gives a clue as to why a complete relativistic treatment of the many-body problem has never been given. A theory due to Darwin and Breit suggests that the Hamiltonian can indeed be written as a sum of nuclear-nuclear repulsions, electron-nuclear attractions and electron-electron repulsions. But these terms are only the leading terms in an infinite expansion. 

A fully relativistic treatment of more than one particle has not yet been developed. For many particle systems it is assumed that each electron can be described by a Dirac operator (ca ir + p mc ) and the many-electron operator is a sum of such terms, in analogy with the kinetic energy in non-relativistic theory. Furthermore, potential energy operators are added to form a total operator equivalent to the Hamilton operator in non-relativistic theory. Since this approach gives results which agree with experiments, the assumptions appear justified. 

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. 

For the lanthanides and actinides the adequate treatment of relativistic corrections is absolutely essential. Indeed these atoms and their compounds may provide the ultimate testing ground for new theories of "many-electron relativistic quantum... 

While a great deal of progress has proved possible for the case of the hydrogen atom by direct solution of the Schrodinger wave equation, some of which will be summarized below, at the time of writing the treatment of many-electron atoms necessitates a simpler approach. This is afforded by the semi-classical Thomas-Fermi theory [4-6], the first explicit form of what today is termed density functional theory [7,8]. We shall summarize below the work of Hill et al. [9], who solved the Thomas-Fermi (TF) equation for heavy positive ions in the limit of extremely strong magnetic fields. This will lead naturally into the formulation of relativistic Thomas-Fermi (TF) theory [10] and to a discussion of the role of the virial in this approximate theory [11]. 

Needless to say, many-electron atoms and molecules are much more complicated than one-electron atoms, and the realization of the nonrelativistic limit is not easily accomplished in these cases because of the approximations needed for the description of a complicated many-particle system. However, the signature of relativistic effects (see, for example, Chapter 3 in this book) enables us to identify these effects even without calculation from experimental observation. Two mainly experimentally oriented chapters will report astounding examples of relativistic phenomenology, interpreted by means of the methods of relativistic electronic structure theory. These methods for the theoretical treatment of relativistic effects in many-electron atoms and molecules are the subject of most of the chapters in the present volume, and with the help of this theory relativistic effects can be characterized with high precision. 

In the preceding sections we delineated how the many-electron problem was made tractable by use of an effective one-electron operator in the Hartree-Fock approximation. A similar treatment is possible in relativistic theory, in... 

The technique for dealing with this problem is well known from nonrelativistic calculations on many-electron systems. One-particle basis sets are developed by considering the behavior of the single electron in the mean field of all the other electrons, and while this neglects a smaller part of the interaction energy, the electron correlation, it provides a suitable starting point for further variational or perturbational treatments to recover more of the electron-electron interaction. It is only natural to pursue the same approach for the relativistic case. Thus one may proceed to construct a mean-field method that can be used as a basis for the perturbation theory of QED. In particular, the inclusion of the Breit interaction in the mean-field calculations ensures that the terms of O(a ) are included to infinite order in QED. 

Other electron nuclear interaction terms involving 7ra rather than Ia arise from this treatment. However, these terms have all been dealt with in the previous chapter and we do not repeat them here.) The terms in (4.23) are the same as those obtained previously starting from the Dirac equation. Equation (3.244) will yield both the electron and nuclear Zeeman terms and a Breit equation for two nuclei, reduced to non-relativistic form, would yield the nuclear-nuclear interaction terms. Although many nuclei have spins other than 1/2, and even the proton with spin 1 /2 has an anomalous magnetic moment which does not fit the simple Dirac theory, the approach outlined here is fully endorsed by quantum electrodynamics provided that only terms involving M l are retained (see equation (4.23)). The interested reader is referred to Bethe and Salpeter [11] for further details. In our present application we see that the expressions for both... 

A straightforward relativistic ab initio AE calculation might appear to be the most rigorous approach to a problem in electronic stmcture theory, however, one has to keep in mind that the methods to solve the Schrodinger equation used in connection with both the AE Hamiltonian and the ECP VO model Hamiltonian usually also rely on approximations, e.g., the choice of the one- and many-particle basis sets, and therefore lead to more or less significant errors in the results. In some cases the introduction of ECPs even helps to avoid or reduce errors, e.g., the basis set superposition error (BSSE), or allows a higher quality treatment of the chemically relevant valence electron subsystem compared to the AE case. 

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