Racah algebra

In a similar way one can compute matrix elements of any interbond interaction. The use of recoupling techniques (Racah algebra) allows one to reduce calculations of properties of molecules with n bonds to those of molecules with 2 bonds. 

While finding the numerical values of any physical quantity one has to express the operator under consideration in terms of irreducible tensors. In the case of Racah algebra this means that we have to express any physical operator in terms of tensors which transform themselves like spherical functions Y. On the other hand, the wave functions (to be more exact, their spin-angular parts) may be considered as irreducible tensorial operators, as well. Having this in mind, we can apply to them all operations we carry out with tensors. As was already mentioned in the Introduction (formula (4)), spherical functions (harmonics) are defined in the standard phase system. 

During the last two decades a number of new versions of the Racah algebra or its improvements have been suggested [27]. So, the exploitation of the community of transformation properties of irreducible tensors and wave functions allows one to adopt the notion of irreducible tensorial sets, to deduce new relationships between the quantities considered, to simplify further on the operators already expressed in terms of irreducible tensors, or, in general, to offer a new method of calculating the matrix elements, as an alternative to the standard Racah way. It is based on the utilization of tensorial products of the irreducible operators and wave functions, also considered as irreducible tensors. 

Part 2 is devoted to the foundations of the mathematical apparatus of the angular momentum and graphical methods, which, as it has turned out, are very efficient in the theory of complex atoms. Part 3 considers the non-relativistic and relativistic cases of complex electronic configurations (one and several open shells of equivalent electrons, coefficients of fractional parentage and optimization of coupling schemes). Part 4 deals with the second-quantization in a coupled tensorial form, quasispin and isospin techniques in atomic spectroscopy, leading to new very efficient versions of the Racah algebra. 

Coupling of three or more angular momenta Racah algebra, Wigner 6-j and 9-j symbols... 

T. Bancewiez. Molecular-statistical theory of the influence of molecular fields in liquids on the spectral distribution and intensity of Rayleigh scattered light in the approach of Racah algebra. Acta Phys. Polonica, A, 56 431-438 (1979). 

From the standpoint of Atomic Physics, the study of the Q-elements has historically been concentrated on the complexity of their spectral structure, arising from the presence of incomplete d and f shells. This aspect was extremely important for the development of group-theoretical techniques, perfected by Racah (1942, 1943) and others, and usually referred to as Racah algebra . Since these mathematical techniques find wide application in science, this aspect of the physics of these elements is well documented. Readers interested in this subject can consult the excellent text by Wyboume (1965) entitled Spectroscopic Properties of Rare Earths. 

In a recent voluminous monograph. Grant presented relativistic atomic structure theory from the perspective of aU developments by himself and his collaboratos in the second half of the 20th century. The total electronic wave function is elegantly constructed in terms of Racah algebra instead of the pedestrian way chosen for the sake of simplicity here. Also, the numerical solution methods for the SCF equations are different from the matrix approach presented here. 

I now give, for completeness, a brief account of the derivation of the spherical tensor formulation of the multipole expansion. This does require an understanding of spherical tensor methods and of Racah algebra, but it may be omitted by readers who are unfamiliar with these techniques, who should skip to the beginning of the next section. 

These are the numbers that evidently show the role of the J-O Theory in the field with undefined limits due to the broad applications of its achievements. In order to understand why this theory is so important, its physical background must be presented. It is possible to conclude briefly that the J-O Theory is a simple application of the outstanding beauty of tri-positive lanthanide ions, and in particular their unusual electronic structure. Its features are defined in the language of Racah algebra applied for the concept of effective tensor operators. The simplicity and clarity of this approach, including the well-known Judd-Ofelt parametrization scheme of the/-spectra based on (10.1), when successfully applied to very complex systems makes one wonder how is it possible that this tool works so well in fact this query is its power. 

Using the tools of Racah algebra, it is rather straightforward to derive the final expression for the amplitude of one photon electric dipole / / transitions. This is the original... 

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. 

---