Relativistic many-electron theory

Highly-ionized atoms DHF calculations on isoelectronic sequences of few-electron ions serve as the starting point of fundamental studies of physical phenomena, though many-body corrections are now applied routinely using relativistic many-body theory. Relativistic self-consistent field studies are used as the basis of investigations of systematic trends in ionization energies [137-144], radiative transition probabilities [145-148], and quantum electrodynamic corrections [149-151] in few-electron systems. Increased experimental precision in these areas has driven the development of many-body methods to model the electron correlation effects, and the inclusion of Breit interaction in the evaluation of both one-body and many-body corrections. 

For heavy atoms and molecules, many-electron theory can be made to start with relativistic equations. Though the exact relativistic Hamiltonian is not known it seems a good approximation to base the theory on the relativistic Hartree-Fock Hamiltonian corrected by the non-relativistic 1/r,-, terms. 

The relativistic many-electron theory can then be formulated in just the same way as in the non-relativistic case above the relativistic x can be obtained and various shells and electron groups separated in it. Because of their strong Z (effective nuclear charge) dependence, relativistic effects will then be confined mainly to the inner shells and will cancel out in the calculations of molecular binding energies and other vedence electron properties. Further approximations may then be made in the formal relativistic theory for the outer shell parts of Xrei and rei to get the non-relativistic equations of this article. 

The spectre of the Bethe-Salpeter equation was raised in the discussion at this meeting, in the context of what a covariant relativistic many-electron theory should look like. In 427 [20], however, Bethe and Salpeter note that... 

Relativity adds a new dimension to quantum chemistry, which is the choice of the Hamiltonian operator. While the Hamiltonian of a molecule is exactly known in nonrelativistic quantum mechanics (if one focuses on the dominating electrostatic monopole interactions to be considered as being transmitted instantaneously), this is no longer the case for the relativistic formulation. Numerical results obtained by many researchers over the past decades have shown how Hamiltonians which capture most of the (numerical) effect of relativity on physical observables can be derived. Relativistic quantum chemistry therefore comes in various flavors, which are more or less well rooted in fundamental physical theory and whose relation to one another will be described in detail in this book. The new dimension of relativistic Hamiltonians makes the presentation of the relativistic many-electron theory very complicated, and the degree of complexity is far greater than for nonrelativistic quantum chemistry. However, the relativistic theory provides the consistent approach toward the description of nature molecular structures containing heavy atoms can only be treated correctly within a relativistic framework. Prominent examples known to everyone are the color of gold and the liquid state of mercury at room temperature. Moreover, it must be understood that relativistic quantum chemistry provides universal theoretical means that are applicable to any element from the periodic table or to any molecule — not only to heavy-element compounds. 

The theory of molecular science has to be a custom-made theory of matter on a molecular scale and at energies accessible by thermal motion or photoexcitation. Specifically, we may choose as elementary particles, which compose molecular aggregates, electrons and atomic nuclei to be treated by this theory. It will thus not be necessary to explicitly consider protons or neutrons or even more fundamental particles such as quarks. By choosing electrons and atomic nuclei as the elementary particles of relativistic quantum chemistry, in this book we will elevate the rather general and fundamental quantum mechanical many-electron theory of quantum electrodynamics to a level that allows us to perform actual calculations and simple qualitative considerations of specific systems. Despite these restrictions, a theory for processes at a molecular and sub-molecular scale still has to be quantum mechanical in nature. 

Several possibilities of formulating these postulates exist. The collection and presentation of postulates which we consider useful for the development of a relativistic many-electron theory are given in the following sections. Also, we avoid any discussion of the wave-particle dualism of matter since we are interested in the presentation of a formalism that eventually allows us to describe and predict matter on a molecular scale. It is thus most convenient to think of electrons and atomic nuclei simply as particles rather than as waves. The following sections introduce five basic axioms of quantum theory. 

This relation between the components of the spinor ensures that states below -2meC are omitted (otherwise ihd/dt E would not be small compared to the rest energy). This approximation will turn out to be very important in the relativistic many-electron theory so that a few side remarks might be useful already at this early stage. Eq. (5.137) will become important in chapter 10 as the so-called kinetic-balance condition (in the explicit presence of external vector potentials also called magnetic balance). It shows that the lower component of the spinor Y is by a factor of 1/c smaller than Y (for small linear momenta), which is the reason why Y is also called the large component and Y the small component. In the limit c oo, the small component vanishes. 

In the molecular sciences it is most appropriate to adopt a pragmatic attitude toward the Dirac equation in order to set up a theory which closely resembles nonrelativistic many-electron theory. We will see that we can afford a number of approximations designed such that the numerical effect on physical observables still resembles that of a truly relativistic many-electron theory. Hence, we proceed from the fundamental physical principles of Einstein s special theory of relativity to approximations of different degree. As a matter of fact this is exactly the program of relativistic quantum chemistry that we shall start to develop in this chapter. 

The first step toward a practical relativistic many-electron theory in the molecular sciences is the investigation of the two-electron problem in an external field which we meet, for instance, in the helium atom. Salpeter and Bethe derived a relativistic equation for the two-electron bound-state problem [135,170-173] rooted in quantum electrod)mamics, which features two separate times for the two particles. If we assume, however, that an absolute time is a good approximation, we arrive at an equation first considered by Breit [101,174,175]. The Bethe-Salpeter equation as well as the Breit equation hold for a 16-component wave function. From a formal point of view, these 16 components arise when the two four-dimensional one-electron Hilbert spaces are joined by direct multiplication to yield the two-electron Hilbert space. 

Historically, the nonrelativistic many-electron theory was developed and computationally tested first. For molecules containing only light atoms, this approach turned out to be remarkably successful in chemistry. Consequently, we are well advised to be inspired by this success if we look for a relativistic theory that is to be computationally as feasible as its nonrelativistic relative. 

This book represents an excellent account of the basics of nonrelativistic quantum chemistry. All essential concepts of many-electron theory are introduced. It is extremely useful, as relativistic, first-quantized quantum chemistry heavily exploits the historically older nonrelativistic quantum chemistry. 

Kato derived analytic properties for the exact many-electron wave function with a focus on their behavior at an atomic nucleus with Coulomb singularity [482]. Some of these results were later considered from the point of view of a relativistic first-quantized many-electron theory [483]. 

Many attempts have been undertaken to rewrite the one-electron Dirac equation — of hydrogen-like atoms and also the mean-field SCF type derived in chapter 8 and in matrix form in chapter 10 — to obtain a form that is most suitable for numerical computations. Historically, the transformation and elimination techniques first emerged from such endeavors and were only later studied from a formal point of view as an essential part of the complete picture of relativistic many-electron theory. For instance, the DKH theory was first developed as an efficient low-order approximation to the Dirac equa-... 

Finally, it is a pleasure to mention that our view of relativistic many-electron theory has been shaped over a period of more than a decade in which we had the opportunity to sharpen our understanding by comparison with the views of colleagues who shared their knowledge to various extents with us these are (in alphabetical order) PD Dr. D. Andrae, D. Dath, Prof. E. Eliav, PD Dr. T. Fleig, Prof. L. Gagliardi, Prof. B. A. Hess, Prof. H.-J. Himmel, Prof. J. Hinze, Prof. J. Hutter, Prof. H. J. A. Jensen, Prof. G. Jeschke, Prof. U. Kaldor, Dr. D. Kfdziera, Prof. B. Kirchner, Dr. T. Koch (it took only 13 years), Dr. A. Landau, Prof. R. Lindh, Prof. P.-A. Malmqvist, Prof. B. Meier, Prof. F. Merkt, Prof. U. 

So far, our discussion of relativistic effects has been restricted to systems with a single electron. Electrons in a many-electron system not only will experience the external nuclear field but will also interact with each other. There is no straightforward way by which the Dirac equation (Eq. [68]) can be extended to a many-electron theory. By analogy with the nonrelativistic Schrodinger equation (Eq. [8]), we may formulate an equation of the form... 

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. 

A systematic development of relativistic molecular Hamiltonians and various non-relativistic approximations are presented. Our starting point is the Dirac one-fermion Hamiltonian in the presence of an external electromagnetic field. The problems associated with generalizing Dirac s one-fermion theory smoothly to more than one fermion are discussed. The description of many-fermion systems within the framework of quantum electrodynamics (QED) will lead to Hamiltonians which do not suffer from the problems associated with the direct extension of Dirac s one-fermion theory to many-fermion system. An exhaustive discussion of the recent QED developments in the relevant area is not presented, except for cursory remarks for completeness. The non-relativistic form (NRF) of the many-electron relativistic Hamiltonian is developed as the working Hamiltonian. It is used to extract operators for the observables, which represent the response of a molecule to an external electromagnetic radiation field. In this study, our focus is mainly on the operators which eventually were used to calculate the nuclear magnetic resonance (NMR) chemical shifts and indirect nuclear spin-spin coupling constants. 

The numerical determination of E grr by the use of many-body theory is a formidable task, and estimates of it based on E j and E p serve as important benchmarks for the development of methods for calculating electron correlation effects. The purpose of this work is to obtain improved estimates of Epp by combining the leading-order relativistic and many-body effects which have been omitted in Eq. (1) with experimentally determined values of the total electronic energy, and precise values of Epjp. We then obtain empirical estimates of E grr for the diatomic species N2, CO, BF, and NO using Epip and E p and the definition of E g in Eq. (1). 

As we said in the introduction, the only consistent framework for a relativistic many-electron system is QED. By means of the Hartree-Fock limit of this theory, after renormalization, and using gradient techniques, Engel and Dreizler [22] found a complete energy functional where both terms of the two previous sections appear naturally. 

We demonstrated that by the selection of a representation of the Dirac Hamiltonian in the spinor space one may strongly influence the performance of the variational principle. In a vast majority of implementations the standard Pauli representation has been used. Consequently, computational algorithms developed in relativistic theory of many-electron systems have been constructed so that they are applicable in this representation only. The conditions, under which the results of these implementations are reliable, are very well understood and efficient numerical codes are available for both atomic and molecular calculations (see e.g. [16]). However, the representation of Weyl, if the external potential is non-spherical, or the representation of Biedenharn, in spherically-symmetric cases, seem to be attractive and, so far, hardly explored options. 

The elements of the theory of angular momentum and irreducible tensors presented in this chapter make a minimal set of formulas necessary when calculating the matrix elements of the operators of physical quantities for many-electron atoms and ions. They are equally suitable for both non-relativistic and relativistic approximations. More details on this issue may be found in the monographs [3, 4, 9, 11, 12, 14, 17]. 

The non-relativistic wave function (1.14) or its relativistic analogue (2.15), corresponds to a one-electron system. Having in mind the elements of the angular momentum theory and of irreducible tensors, described in Part 2, we are ready to start constructing the wave functions of many-electron configurations. Let us consider a shell of equivalent electrons. As we shall see later on, the pecularities of the spectra of atoms and ions are conditioned by the structure of their electronic shells, and by the relative role of existing intra-atomic interactions. 

As was mentioned in Chapter 2, there exists another method of constructing the theory of many-electron systems in jj coupling, alternative to the one discussed above. It is based on the exploitation of non-relativistic or relativistic wave functions, expressed in terms of generalized spherical functions [28] (see Eqs. (2.15) and (2.18)). Spin-angular parts of all operators may also be expressed in terms of these functions (2.19). The dependence of the spin-angular part of the wave function (2.18) on orbital quantum number is contained only in the form of a phase multiplier, therefore this method allows us to obtain directly optimal expressions for the matrix elements of any operator. The coefficients of their radial integrals will not depend, except phase multipliers, on these quantum numbers. This is the case for both relativistic and non-relativistic approaches in jj coupling. 

A comprehensive review of non-relativistic and relativistic versions of the random phase approximation may be found in [217]. Some applications are described in [218]. A survey of various aspects of the modern theory of many-electron atoms is presented in [219]. 

The use of the tensorial properties of both the operators and wave functions in the three (orbital, spin and quasispin) spaces leads to a new very efficient version of the theory of the spectra of many-electron atoms and ions. It is also developed for the relativistic approach. 

In fact, it is the only book in which you can find successive general non-relativistic and relativistic descriptions of the theory of energy spectra and transition probabilities in complex many-electron atoms and ions. The formulas and tables presented give the possibility, at least in principle, of calculating the energy spectra and electronic transitions of any multipolarity for any atom or ion of the Periodical Table. This book contains the bulk of new achievements in the non-relativistic and relativistic theory of an atom, especially as concerns the many-particle aspects of the non-relativistic and relativistic problem. It therefore complements books already available. 

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