Relativistic approximations

AuH and Au2 serve as benchmark molecules to test the performance of various relativistic approximations. Figure 4.7 shows predictions for relativistic bond contractions of Au2 from various quantum chemical calculations over more than a decade. In the early years of relativistic quantum chemistry these predictions varied significantly (between 0.2 and 0.3 A), but as the methods and algorithms became more refined, and the computers more powerful, the relativistic bond contraction for Au2 converged and is now at 0.26 A. 

The inclusion of (nonrelativistic) property operators, in combination with relativistic approximation schemes, bears some complications known as the picture-change error (PCE) [67,190,191] as it completely neglects the unitary transformation of that property operator from the original Dirac to the Schrodinger picture. Such PCEs are especially large for properties where the inner (core) part of the valence orbital is probed, for example, nuclear electric field gradients (EEG), which are an important... 

Semiclassical studies of the propagation of coherent states have proven useful in many circumstances see, e.g., (Klauder and Skagerstam, 1982 Perelomov, 1986). Here we consider spin-orbit coupling problems that result from the Dirac equation either in a semiclassical or in a non-relativistic approximation (see, e.g., the Hamiltonians (30) and (31)). The Hamiltonians H that arise in such a context can be viewed as Weyl quantisations of symbols... 

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. 

From here, expanding the operator acting on in terms of powers of one obtains the familiar Pauli approximation as well as numerous two-component quasi-relativistic approximations. 

Non-relativistic (1.14) and relativistic (2.15) wave functions are widely used for theoretical studies of the structure and spectra of many-electron atoms and ions. However, it has turned out that such forms of wave functions in the case of the jj coupling scheme are not optimal. Their utilization, particularly in the relativistic approximation, is rather inconvenient and tedious. 

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]. 

While calculating matrix elements of various items of the energy operator or electron transition quantities in relativistic approximation we shall... 

While calculating the energy spectrum in relativistic approximation, the non-diagonal with respect to the configuration s matrix elements must also be taken into consideration [62]. The necessary expressions for four open subshells, corresponding to two non-relativistic shells of equivalent electrons, are presented in [126]. 

In a non-relativistic approximation the usual fine structure (splitting) of the energy terms is considered as a perturbation whereas the hyperfine splitting - as an even smaller perturbation, and they both are calculated as matrix elements of the corresponding operators with respect to the zero-order wave functions. 

The expressions for Ek-transitions presented here allow one to perform calculations of the respective matrix elements in relativistic approximation for any type of practical configuration and any multipolarity of radiation. 

Let us emphasize that in single-configurational approach the terms of the Hamiltonian describing kinetic and potential energies of the electrons as well as one-electron relativistic corrections, contribute only to average energy and, therefore, are not contained in, which in the non-relativistic approximation consists only of the operators of electrostatic interaction e and the one-electron part of the spin-orbit interaction so, i.e. 

There exist a number of methods to account for correlation [17, 45, 48] and relativistic effects as corrections or in relativistic approximation [18]. There have been numerous attempts to account for leading radiative (quantum-electrodynamical) corrections, as well [49, 50]. However, as a rule, the methods developed are applicable only for light atoms with closed electronic shells plus or minus one electron, therefore, they are not sufficiently general. 

Particular attention is paid to recently worked out methods of theoretical description of highly ionized atoms. They allow us to take into account relativistic effects. Accounting for them both as corrections and in relativistic approximation, as well as the use of various (differing from traditional LS) coupling schemes is considered in detail. 

The main ideas of the book are described in seven Parts divided into 33 Chapters, which are subdivided into Sections. In Part 1 we present the initial formulas to calculate the energy spectrum of a many-electron atom in non-relativistic and relativistic approximations, accounting for the relativistic effects as corrections and use perturbation theory in order to describe the energy spectra of an atom. Radiative and autoionizing... 

It is interesting to note that a similar radiative association process is not possible for the two hydrogen atoms. On the symmetry grounds the dipole moment of the H — H system (which is inversion symmetric) vanishes. In that case the nuclear dipole moment is identically 0 and the electronic dipole moments induced in the two approaching atoms have opposite orientations and cancel each other. For the H — H system (which lacks the inversion symmetry) the dipole moment (in the adiabatic and non-relativistic approximation) is finite. In that case the hadronic moment is e R and the induced leptonic moments of H and H have the same orientations and add together to a non vanishing dipole moment (which tends to 0 in the limit of infinite separation R between the atoms). 

The hyperfine splitting of s-state is determined in the leading non-relativistic approximation by1... 

This vanishes in the leading non-relativistic approximation (1) and so it is sensitive to state-dependent corrections to the hyperfine structure. 

Once again we have introduced the Dirac delta functions in a non-relativistic approximation. The terms in (3.157) represent a correction to the Coulomb potential. 

C. V. Wiillen, Molecular density functional calculations in the regular relativistic approximation Method, application to coinage metal diatomics, hydrides, fluorides and chlorides, and comparison with first-order relativistic calculations, J. Chem. Phys. 109, 392-399 1998. 

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