Tensor irreducible parts

The static hyperpolarizability tensor may be resolved into irreducible parts, 7,(nLJ) [8,9] ... 

Here, y is the unique symmetric and traceless (irreducible) part of Q, whereas D(i) are symmetric and traceless second-order (irreducible) tensors, and v(i) and S are vectors and a scalar. It may be noted that, if we write a general third-order tensor in the form (8 33), there is no loss of generality in assuming that the second-order tensor components are symmetric. The antisymmetric part of any second-order tensor can always be represented by a vector. For example, if D = D,s + D" then the antisymmetric part can always be written as D"= e d where d = — e D" and included in the vector terms of (8-33). 

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. 

To conclude this chapter, let us present the main formulas for sums of unit tensors, necessary for evaluation of matrix elements df the energy operator. They will be necessary in Part 5. The matrix element of any irreducible tensorial operator may be written as follows ... 

The non-relativistic wave function (1.14) or its relativistic analogue (2.15), corresponds to a one-electron system. Having in mind the elements of the angular momentum theory and of irreducible tensors, described in Part 2, we are ready to start constructing the wave functions of many-electron configurations. Let us consider a shell of equivalent electrons. As we shall see later on, the pecularities of the spectra of atoms and ions are conditioned by the structure of their electronic shells, and by the relative role of existing intra-atomic interactions. 

Note that there is more than one way of representing the dipolar interaction in irreducible tensors, as discussed in appendix 8.2. Equation (9.41) can be compared with the coupling choice used in our analysis of the H/ spectrum described in chapter 11. It is partly a matter of choice, but more especially a matter of the basis set used in the analysis. 

Thus far we have only considered one (boson) vector field, namely, the direct product field R Xn of creation and annihilation operators. The coefficients of the creation and annihilation operator pairs in fact also constitute vector fields this can be shown rigorously by construction, but the result can also be inferred. Consider that the Hamiltonian and the cluster operators are index free or scalar operators then the excitation operators, which form part of the said operators, must be contracted, in the sense of tensors, by the coefficients. But then we have the result that the coefficients themselves behave like tensors. This conclusion is not of immediate use, but will be important in the manipulation of the final equations (i.e., after the diagrams have contracted the excitation operators). Also, the sense of the words rank and irreducible rank as they have been used to describe components of the Hamiltonian is now clear they refer to the excitation operator (or, equivalently, the coefficient) part of the operator. 

H should be described in terms of the irreducible tensors A, and then the secular part of H, which is commutable with the Zeeman term, is expressed as [34, 35]... 

In general, symmetry conditions are part of the characterization of a definite type of quantity in a physical space. Tensors and tensor spaces were universal objects for the representation of the linear group transformations that are fundamental for the expansion of the chemical quantum theory of bonding. All the irreducible representations could then be characterized by some symmetry condition inside some tensor power of the state space, symbolized as V. Thus, a broad correspondence between the representations of the symmetric group and the irreducible representations inside the state space (representations of order k ) played an important role for the answer to the first question. 

The real version of the irreducible tensor method, related to the complex representations as mentioned, is highly useful in the ligand-field theory, as will be shown in the second part of this paper. An additional reason for this is the fact that expansion theorems concerning functions and operators achieve apt forms when the tensor method is applied to real spherical harmonics. 

Now Retails is the transition probability from the spin state j3 to the spin state a and Raa/sis = R/3/3aa- The diagonal part of Eq. (5.25) is a second-rank equation of motion for evolution of the density matrix under the effect of a random perturbation. There are two important second-rank relaxation mechanisms the dipole-dipole and the quadrupole interactions. Chapter 2 showed that these interactions and the anisotropic chemical shift can all be written as a scalar product of two irreducible spherical tensors of rank two, that is. 

It is often argued that it is the shape anisotropy which is largely responsible for liquid crystal formation. Two methods have been proposed to introduce this view into the calculation of the interaction tensor for each conformer. In one it is assumed that the tensor is proportional to the moment of inertia tensor which is readily calculated from a knowledge of the molecular geometry [75]. However, it is found that this paramet-rization results in too great a dependence of the N-I transition temperature on the molecular length [76]. This observation was partly responsible for the development of the surface tensor model [77]. In this the interaction tensor is defined in irreducible form as... 

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