Tensorial property

Note 3 The tensorial properties of a biaxial mesophase have biaxial symmetry unlike the uniaxial symmetries of, for example, the nematic and smectic A mesophases. 

Note 3 In lyotropic systems, biaxial nematic mesophases have been identified from the biaxial symmetry of their tensorial properties. 

Note 1 The long-range biaxial ordering of the mesophase means that the three principal components of a second-rank tensorial property will not normally be the same, hence the two measures of the anisotropy A/ and. ... 

Figure 3. Amount of spectral information for second-rank tensorial properties using different types of sample.

Second-quantization formalism was introduced into the theory of many-electron atoms by Judd [12]. This formalism enables one to give a simple and elegant description of both the rotation symmetry of a system and its permutational symmetry the tensorial properties of wave functions are translated to electron creation and annihilation operators, and the Pauli exclusion principle stems automatically from the anticommutation relations between these operators. 

The methods of theoretical description of many-electron atoms on the basis of tensorial properties of the orbital and spin angular momenta are well established [14, 18] and enable the spectral characteristics of these systems to be effectively found. The relation between the seniority quantum number and quasispin makes it possible to extend the mathematical tools to include the quasispin space and to work out new modifications of the mathematical techniques in the theory of spectra of many-electron atoms that take due account of the tensorial properties of the quasispin operator. 

The mathematical apparatus of the angular momentum theory can be applied to describe the tensorial properties of electron creation and annihilation operators in the space of occupation numbers of a certain definite one-particle state a). It follows from (13.29) and (13.30) that the operators... 

Second-quantization operators as irreducible tensors. Tensorial properties of electron creation and annihilation operators... 

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. 

Utilization of the tensorial properties of the electron creation and annihilation operators allows us to obtain expansions in terms of irreducible tensors of any operators in the second-quantization representation. So, using the Wigner-Eckart theorem (5.15) in (14.11) and (14.12), then coupling ranks of second-quantization operators by (5.12) and utilizing (14.10), we can represent one-shell operators of angular momentum in the irreducible tensor form... 

If we take into account the tensorial properties of creation and annihilation operators, using (13.22) we shall be able to expand in terms of the irreducible tensors... 

This suggests that in the particle-hole representation each occupied one-particle state in the lN configuration can be assigned a value of the z-projection of the quasispin angular momentum 1/4 and each unoccupied (hole) state —1/4. When acting on an AT-electron wave function the operator a s) produces an electron and, simultaneously, annihilates a hole. Therefore, the projection of the quasispin angular momentum of the wave function on the z-axis increases by 1/2 when the number of electrons increases by unity. Likewise, the annihilation operator reduces this projection by 1/2. Accordingly, the electron creation and annihilation operators must possess some tensorial properties in quasispin space. Examination of the commutation relations between quasispin operators, and creation and annihilation operators... 

It has been shown earlier (see Chapters 15 and 16) that the technique relying on the tensorial properties of operators and wave functions in quasispin, orbital and spin spaces is an alternative but more convenient one than the method of higher-rank groups. It is more convenient not only for classification of states, but also for theoretical studies of interactions in equivalent electron configurations. The results of this chapter show that the above is true of more complex configurations as well. 

For the basis (18.27) to be used effectively in practical computations an adequate mathematical tool is required that would permit full account to be taken of the tensorial properties of wave functions and operators in their spaces. In particular, matrix elements can now be defined using the Wigner-Eckart theorem (5.15) in all three spaces, so that the submatrix element will be given by... 

To sum up the potentialities of the isospin method are not exhausted by the results stated above. There is a deep connection between orthogonal transformations of radial orbitals and rotations in isospin space (see (18.40) and (18.41)). This shows that the tensorial properties of wave functions and operators in isospin space must be dominant in the Hartree-Fock method. This issue is in need of further consideration. 

The relationships describing the tensorial properties of wave functions, second-quantization operators and matrix elements in the space of total angular momentum J can readily be obtained by the use of the results of Chapters 14 and 15 with the more or less trivial replacement of the ranks of the tensors l and s by j and the corresponding replacement of various factors and 3nj-coefficients. Therefore, we shall only give a sketch of the uses of the quasispin method for jj coupling, following mainly the works [30, 167, 168]. For a subshell of equivalent electrons, the creation and annihilation operators a and a(jf are the components of the same tensor of rank q = 1/2... 

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... 

As in the case of LS coupling, the tensorial properties of wave functions and second-quantization operators in quasispin space enable us to separate, using the Wigner-Eckart theorem, the dependence of the submatrix elements on the number of electrons in the subshell into the Clebsch-Gordan coefficient. If then we use the relation of the submatrix element of the creation operator to the CFP... 

We shall now turn to the mathematical techniques that rely on the tensorial properties of the quasispin space of two subshells of equivalent electrons. They can be readily generalized to more complex configurations. 

Clearly, for operators corresponding to physical quantities we can derive expansions that enable us to make use of their tensorial properties in quasispin space. Specifically, operator (23.27) will be... 

There is another way of looking at the tensorial properties of operators and wave functions in the quasispin space of the entire two-shell configuration. If now we introduce the basis tensors for two subshells of equivalent electrons... 

Considering the tensorial properties of the electron creation and annihilation operators in quasispin space, we shall introduce the double tensor... 

Operators of electronic transitions, except the third form of the Ek-radiation operator for k > 1, may be represented as the sums of the appropriate one-electron quantities (see (13.20)). Their matrix elements for complex electronic configurations consist of the sums of products of the CFP, 3n./-coefficients and one-electron submatrix elements. The many-electron part of the matrix element depends only on tensorial properties of the transition operator, whereas all pecularities of the particular operator are contained in its one-electron submatrix element. 

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. 

If Ae 0 the reaction/cavity fields, and then the molecular property /, depend on the molecular orientation. Such a dependence affects the physical observables, which are obtained by averaging over the orientational distribution. Considering in general a tensorial property, we can express the average value as ... 

A magnetic field will give rise to tensorial properties for electron-related phenomena... 

There are so many publications in the realm of organic molecules for non-linear optics that this review has focused on second-order polarizabilities. The translation of these properties into bulk structures could only be hinted at and devices could not be mentioned at all. We have tried to develop a formal description of tensorial properties that is consistent with the SI system and would like to suggest to groups working in the area to adopt it in order to... 

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