Irreducible tensorial sets

One-electron submatrix elements of the spherical functions operator occur in the expressions of any matrix element of a two-electron energy operator and the electron transition operators (except the magnetic dipole radiation), that is why we present in Table 5.1 their numerical values for the most practically needed cases /, / 6. 

This submatrix element has the following fairly simple algebraic expression  

This submatrix element is always positive, and it is non-zero only when l + k + / is even. It is in the following way connected with a special case of the Clebsch-Gordan coefficient  

This is described in more detail in Chapter 6. A relativistic analogue of this quantity will be considered in Chapter 7. 

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. 

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Real irreducible tensorial sets and their application to ligand field theory. S. E. Harnung and C. E. Schaffer, Struct. Bonding (Berlin), 1972,12, 257-295 (26). 

Harnung SE, Schaffer CE (1972) Real Irreducible Tensorial Sets and their Application to the Ligand-Field Theory. 12 257-295... 

Real Irreducible Tensorial Sets and their Application to the Ligand-Field... 

Fano, U., and Racah, G. (1959), Irreducible Tensorial Sets, Academic Press, N.Y. Farrelly, D. (1986), Lie Algebraic Approach to Quantization of Nonseparable Systems with Internal Nonlinear Resonance, J. Chem. Phys. 85, 2119. 

Fano and Racah [5] defined the concept of the irreducible tensorial set as a set of 2k + 1 quantities (k is an integer or half-integer) transforming through each other according to the irreducible tensorial representations of the rotation group... 

Thus, utilizing the concept of irreducible tensorial sets, one is in a position to develop a new method of calculating matrix (submatrix) elements, alternative to the standard way described in many papers [9-11, 14, 18, 21-23]. Indeed, the submatrix element of the irreducible tensorial operator can be expressed in terms of a zero-rank double tensorial (scalar) product of the corresponding operators (for simplicity we omit additional quantum numbers a, a1) ... 

U. Fano and G. Racah. Irreducible Tensorial Sets, Academic Press, New York, 1959. 

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. 

Fano, U. and Racah, G. (1959) Irreducible Tensorial Sets. New York Academic Press. 

FRa59] Fano U and Racah G 1959 Irreducible tensorial sets (Academic Press Inc., New York). [FSt66] Frauenfelder H and Steffen RM 1966 (second printing) Angular correlations, in ... 

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