Relativistic approach

Since the inclusion of the higher-order terms caused by electron correlation effects does not change the original selection rules for the non-vanishing transition amplitude (although they modify the values in a tremendous way), it is worthwhile to generalize the description of the/ i— / transitions by intfoducing a relativistic approach. 

In order to derive a relativistic version of the elecfiic-dipole / —) f transitions the concept of Sandras and Beck [59] is applied to include new effects in an effective way. This means that every unit tensor operator analyzed in the non-relativistic approach has to be replaced prior to the partial closure by a double unit tensor operator that acts within the spin (k) - orbital (k) space. The transformation is as follows. 

In this way the relativistic effects are taken into account in an effective way. At the same time the 5 — L coupling, natural for the non-relativistic Schrodinger equation and the standard theory of tri-positive lanthanide ions, is preserved, instead of the j — j basis of a relativistic Dirac equation. As a result of such replacements of all the operators, new angular terms appear and the radial integrals in (10.33) are defined by the small and large 

From a physical point of view this relativistic model is also based on the perturbation approach, and at the second order, similarly as in the case of the standard J-O Theory, the crystal field potential plays the role of a mechanism that forces the electric dipole/ t— f transitions. The only difference is that now the transition amplitude is in effectively relativistic form, as determined by the double tensor operator, but still of one particle nature. Furthermore, the same partitioning of space as in non-relativistic approach is valid here. The same requirements about the parity of the excited configurations are expected to be satisfied. As a final step of derivation of the effective operators, the coupling of double inter-shell tensor operators has to be performed. This procedure is based on the same rules of Racah algebra as presented in the case of the standard J-O theory. However, the coupling of the inter-shell double tensor operators consists of two steps, for spin and orbital parts separately. Thus, the rules presented in equations (10.15) and (10.16) have to be applied twice for orbital and spin momenta couplings, resulting in two 3j— and two 6j— coefficients. 

Taking into account two second-order contributions that differ by the order of the operators [as in (10.11)1 the final expression for the second-order transition amplitude, in fact the relativistic analog of the Judd-Ofelt theory, has the following form  

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It is clear that an ah initio calculation of the ground state of AF Cr, based on actual experimental data on the magnetic structure, would be at the moment absolutely unfeasible. That is why most calculations are performed for a vector Q = 2ir/a (1,0,0). In this case Cr has a CsCl unit cell. The local magnetic moments at different atoms are equal in magnitude but opposite in direction. Such an approach is used, in particular, in papers [2, 3, 4], in which the electronic structure of Cr is calculated within the framework of spin density functional theory. Our paper [6] is devoted to the study of the influence of relativistic effects on the electronic structure of chromium. The results of calculations demonstrate that the relativistic effects completely change the structure of the Or electron spectrum, which leads to its anisotropy for the directions being identical in the non-relativistic approach. 

Ziegler, T, Baerends, E.J., Snijders, J.G., Ravenek, W. and Tschinke, V. (1989) Calculation of bond energies in compounds of heavy elements by quasi-relativistic approach. The Journal of Physical Chemistry, 93, 3050-3056. 

Typical values obtained for po are around 4-6 MeV fm-3, and 0,4 30-36 MeV, i.e., considerably larger than in non-relativistic approaches (a large part of the enhancement can be ascribed to the fact that the kinetic contribution... 

To summarize the present situation in Fig. 4 the resulting density dependence of the SE for the approaches discussed above are compared (excluding the 3NF contribution). One sees that the covariant models predict a much larger increase of the SE with the density than the non-relativistic approaches. The lowest-order BHF method predicts a somewhat higher value for 04 than both the VCS and SCGF methods, which lead to very similar results whether that can be ascribed to a consistent treatment of correlations in these methods, or is fortuitous, is not clear. 

For the non-relativistic case (Schrbdinger equation), T = -V. For relativistic case (Dirac equation), T = c a p + 3mc where m is the rest mass of the electron, c is the velocity of light. We have preferred to write the T operator in a general form, covering both cases, given the importance of the relativistic approach in band calculations for actinide solids - see Chap. F... 

It should be remarked that this is the first practical implementation of the calculation of spin-spin couplings involving heavy atoms applicable to medium sized molecular systems. Formerly, Khandogin and Ziegler101 developed a frozen core scalar relativistic approach in combination with the non-relativistic FC and PSO operators, but the quality of results thus obtained is poorer than those obtained with ZORA. 

The topological approach of Ranada and Trueba, the general relativistic approach of Sachs, and the 0(3) electrodynamics are interlinked and shown to be based on the concept of Faraday s lines of force. 

All the main ideas of Chapter 6 are equally applicable for both non-relativistic and relativistic approaches. 

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. 

In a single-configurational non-relativistic approach, the integrals of electrostatic interactions and the constant of spin-orbit interactions compose the minimal set of semi-empirical parameters. Then for pN and dN shells we have two and three parameters, respectively. However, calculations show that such numbers of parameters are insufficient to achieve high accuracy of the theoretical energy levels. Therefore, we have to look for extra parameters, which would be in charge of the relativistic and correlation effects not yet described. 

For the first configuration in a non-relativistic approach only LS coupling must be used, whereas for the second one all four schemes LS, LK, J K and J] j2 may occur. A submatrix element in the right side of (25.31), presented in form (25.29), becomes equal to... 

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. 

How does one do relativistic calculations Although two formalisms had been in the literature outlining relativistic approaches to shielding calculations, no calculations had been carried out for molecules. 

ZORA Method. Very recently, we began applying a new relativistic approach, the ZORA method of van Lenthe and co-workers (14-16). The ZORA Hamiltonian is given by... 

Abstract An overview is presented of the methodology and computations of nuclear shielding and spin-spin coupling constants of transition-metal complexes. The material presented also includes an outline of relativistic approaches and their applications to heavy transition-metals. 

Different non-relativistic approaches developing these ideas were introduced. The most effective one, used successfully in positronium and muonium calculations was introduced by Caswell and Lepage and is known as NRQED (Non-Relativistic QED) [34]. Some applications to positronium are presented in this edition in Refs. [17,20]. 

The non-relativistic CASPT2 method developed by Anderson et al. [11,12] is one of the most familiar multireference approaches. It is well established and has been applied to a large number of molecular systems with the non- or quasi-relativistic approaches. Because the CASPT2 method treats dynamic correlation effects pertur-batively, it is less expensive than the multireference Cl (MRCI) method. The... 

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