Irreducible tensor product

This set of operators is in general reducible but a proper linear combination of them can be taken irreducible. It can be used in constructing an irreducible tensor product of order K with (2K + 1) elements indexed as -K < M < K by using Clebsch-Gordan coefficients... 

The irreducible tensor product between the brackets represents the usual angular momentum coupling... 

The irreducible tensor product between two (spherical) vectors is defined in Eq. (37). An important feature of this Hamiltonian is that it explicitly describes the dependence of the coupling constants J, Am, and T, on the distance vectors rPP between the molecules and on the orientations phenomenological Hamiltonian (139). Another important difference with the latter is that the ad hoc single-particle spin anisotropy term BS2y, which probably stands implicitly for the magnetic dipole-dipole interactions, has been replaced by a two-body operator that correctly represents these interactions. The distance and orientational dependence of the coupling parameters J, A, , and Tm has been obtained as follows. 

In this picture, the correspondence between irreducible representations of F (except the trivial representation) and irreducible components of the exceptional set becomes concrete. It is realized by the tautological bundles V s. In [66, 5.8], we have shown the correspondence respects the multiplicative structures, one given by the tensor product and one given by the cup product. In fact, using (4.11), we can show that two matrices... 

The results of this section, even with their limitations, are the punch line of our story, the particularly beautiful goal promised in the preface. Now is a perfect time for the reader to take a few moments to reflect on the journey. We have studied a significant amount of mathematics, including approximations in vector spaces of functions, representations, invariance, isomorphism, irreducibility and tensor products. We have used some big theorems, such as the Stone-Weierstrass Theorem, Fubini s Theorem and the Spectral Theorem. Was it worth it And, putting aside any aesthetic pleasure the reader may have experienced, was it worth it from the experimental point of view In other words, are the predictions of this section worth the effort of building the mathematical machinery ... 

Let us present the main definitions of tensorial products and their matrix or reduced matrix (submatrix) elements, necessary to find the expressions for matrix elements of the operators, corresponding to physical quantities. The tensorial product of two irreducible tensors and is defined as follows ... 

In the second-quantization representation the atomic interaction operators are given by relations (13.22) and (13.23), which do not include the operators themselves in coordinate representations, but rather their one-electron and two-electron matrix elements. Therefore, in terms of irreducible tensors in orbital and spin spaces, we must expand the products of creation and annihilation operators that enter (13.22) and (13.23). In this approach, the tensorial properties of one-electron wave functions are translated to second-quantization operators. 

This differs only by sign from irreducible tensorial product (14.22) which makes especially convenient the use of tensor in the second-... 

Apart from irreducible tensors (14.30) we can also introduce other operators that are expressed in terms of irreducible tensorial products of second-quantization operators, and establish commutation relations for them. As was shown in [12, 102, 103], using relations of this kind, we can relate standard quantities of the theory, which at first sight seem totally different. Consider the operator... 

Since affij are irreducible tensors, we can define their irreducible tenso-rial product, i.e. the operator... 

Tensors (15.52) in the quasispin method are the main operators. Their irreducible tensorial products are used to expand operators corresponding to physical quantities. That is why we shall take a closer look at their properties. Examining the internal structure of tensor Wyields... 

Let us now return to the Casimir operators for groups Spy+2, SU21+1, R21+1, which can also be expressed in terms of linear combinations of irreducible tensorial products of triple tensors WiKkK To this end, we insert into the scalar products of operators Uk (or Vkl), their expressions in terms of triple tensors (15.60) and then expand the direct product in terms of irreducible components in quasispin space. As a result, we arrive at... 

When dealing with complex configurations, it is necessary, apart from one-shell tensors, to consider the irreducible tensorial products of creation... 

Operators corresponding to physical quantities can also be expanded in terms of irreducible tensors in the quasispin space of each individual shell. To this end, it is sufficient to go over to tensors (17.43) and next to provide their direct product in the quasispin space of individual shells. This procedure can conveniently be carried out for a representation of operators such that the orbital and spin ranks of all the one-shell tensors are coupled directly. Here we shall provide the final result for the two-particle operator of general form (14.57)... 

Any products of creation and annihilation operators for electrons in a pairing state can be expanded in terms of irreducible tensors in the space of quasispin and isospin. So, for the operators (18.10) and (15.35) we have, respectively,... 

In this method tensors of such a type are the basic operators, and the operators for the pertinent physical quantities will be given in terms of irreducible tensorial products of these tensors. Specifically, unit tensor Tp (7.4) will be... 

Utilizing the tensorial properties of operators in quasispin space, we can, in particular, find an expansion of the scalar products of the operators Tk in terms of irreducible tensors in quasispin space... 

The irreducible tensorial products of the double tensors can be found... 

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. 

Of frequent interest is the need of evaluation of matrix elements for a tensor product of two irreducible tensor operators... 

Up to this point, our main concern was to reformulate the results of the LD ligand influence theory in the DMM form. Its main content was the symmetry-based analysis of the possible interplay between two types of perturbation substitution and deformation, controlled by the selection rules incorporated in the polarization propagator of the CLS. The mechanism of this interplay can be simply formulated as follows substitution produces perturbations of different symmetries which are supposed to induce transition densities of the same symmetries. In the frontier orbital approximation, only those densities among all possible ones can actually appear, which have the symmetry which enters into decomposition of the tensor product TH TL to the irreducible representations. These survived transition densities then induce the geometry deformations of the same symmetry. 

Thus it may produce the transitional densities of the alg, egc, and tluz symmetries. At this point selection rules pertinent to the frontier orbitals approximation enter for the 12-electron complexes the symmetries of the frontier orbitals are Th = eg and Tl = ai3, the tensor product Th <8> TL = eg aig = eg contains only the irreducible representation eg so that the selection rules allow only the density component of the egc symmetry to appear. In its turn this density induces additional deformation of the same symmetry. That means that in the frontier orbitals approximation, only the elastic constant for the vibration modes of the symmetry eg is renormalized. This result is to be understood in terms of individual nuclear shifts of the ligands in the trans- and cis-positions relative to the apical one. They, respectively, are ... 

In a tensor product 0(k) each component of is to be multiplied with each component of The resulting 2(k + f) -2( + -dimensional tensor is in general not irreducible. Reduction yields irreducible tensor operators with ranks ranging from k + j, k + j — 1, , k—j. ... 

As an example, consider the product of two arbitrary first-rank tensor operators 0 and It is nine-dimensional and can be reduced to a sum of compound irreducible tensor operators of ranks 2, 1, and 0, respectively. Operators of this type play a role in spin-spin coupling Hamiltonians. In terms of spherical and Cartesian components of 0 and J2, the resulting irreducible tensors are given in Tables 8 and 9, respectively.70... 

In a rotating molecule containing one quadrupolar nucleus there is an interaction between the angular momentum J of the molecule and the nuclear spin momentum I. The operator of this interaction can be written as a scalar product of two irreducible tensor operators of second rank. The first tensor operator describes the nuclear quadrupole moment and the second describes the electrical field gradient at the position of the nucleus under investigation. 

This may be written as the scalar product of two second-rank irreducible tensors,... 

We work in the basis set 177,./, Q, I, F, Mp) where F= J+1 and i2 is the component of electronic angular momentum along the intemuclear axis. We shall ignore any possibility of i -degeneracy r refers to any other unspecified quantum numbers. Using the standard results for the matrix elements of the scalar product of two irreducible tensor operators, we obtain... 

The Wigner 3j-symbol is often defmed as the coefficient coupling a product of three irreducible tensors (of the same variance) to an invariant Invoking this definition, it immediately follows that the function c, Q) is an invariant. 

In degenerate states the situation is essentially changed. In these cases in the decomposition of the product [P] into irreducible representations there are other representations in addition to the totally symmetric representation. As a result, there may be nonzero matrix elements for nonto-tally symmetric components of irreducible tensor operators of the polarizability and multipole moments. In particular, in the decomposition of [P] there may be representations contained in D with an / value less than lA, for which D a contains the totally symmetric representation. 

The restrictions listed until now are of the type of necessary conditions. In order to obtain sufficient conditions for the observation of the effect of magnetic optical activity of molecules in T states, it is necessary to pass under the sign of the spur in the expression Sp pLK-K Cl J]Clf. J for allowed values of the pairs (JJ ) to appropriate products of irreducible tensor operators for the point group of the molecule. Using the decomposition of the representation DJ of the full spherical group into irreducible representations of the molecular point group, one can present the result in the form of a sum of following terms ... 

Having established that creation and annihilation operators are rank 1 covariant and contravariant tensors, respectively, with respect to the operator ( )L,S, we can define an rath-rank boson operator as consisting of a like number of fermion creation and annihilation operators. Then the normal product of an rath-rank boson operator is a natural definition for the irreducible tensor. 

The collisional operator is obtained by taking the product with the zero and second rank irreducible tensors built from U, that is,... 

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