Off-diagonal tensor components

A significant enhancement of the off-diagonal tensor components, Pzxx rid Pxzx was reported for C2 -symmetric benzene [76] and carbazole derivatives [77]. For the carbazole derivatives, the amino donor is shared by the two acceptor groups, and quantum chemical calculations confirmed the existence of a two-dimensional charge transfer transition. A combination of (depolarized) HRS and EFISHG measurements showed that Pzxx = Pxxz = Pxzx Pzzz = - Pzxx... 

Due to these large off-diagonal components, a poled film of a 3,6-substituted carbazole derivative has J31 = da [78]. A similar enhancement of the off-diagonal tensor components was also verified for C2 -symmetric benzene derivatives with EFISHG and HRS experiments. 

Ruthenium(II) ammines can also give rise to Czv symmetric 2D dipolar chromophores featuring two cis pyridyl-pyridinium ligands (e.g. 30, Figure 1.14). Such V-shaped complexes are interesting and unusual because they have a significant off-diagonal tensor component /S yy in addition to the classical component. They display two MLCT bands in the visible, and TD-DFT calculations have shown that the lowest... 

According to (55) and (58) the two-level and three-level contributions, and to the off-diagonal tensor element p yy can be understood in terms of products of transition dipoles and dipole differences. Remarkably, the off-diagonal element is optimized at = 0.5, where the diagonal element vanishes. This shows that both types of j3 components may not be optimized simultaneously. 

If the 6 a tensor has oidy off-diagonal nonzero components, then the Brugge-man equation leads to the following expression for... 

Fig. 2 Raman spectra of a single-crystal of orthorhombic Sg at three different polarizations in which the off-diagonal elements of the Raman scattering tensor are non-zero big, b2g, b g), after [105]. However, Raman intensities of other polarizations like flg components ( 54 cm ) penetrate in the spectra due to optical anisotropy in the crystal...

We restrict ourselves to the local valence part of the EFG tensor to illustrate the principle. Since the EFG operator is spin-free, there are no off-diagonal elements M M and an inspection of Table 5.6 reveals that there are also no off-diagonal components between different configurations I A J- Hence ... 

One may observe from Eq. (5C) how the resonance condition is a function of the off diagonal components of the g tensor. 

The diagonal components of the Reynolds stress tensor (e.g., u U )) are referred to as the Reynolds normal stresses, while the off-diagonal components are referred to as the... 

It should be noted that the metric factors represent diagonal elements of a transformation matrix. It is therefore prudent to check the off-diagonal components to ensure that the new coordinate system is indeed orthogonal. In general, the elements of the metric tensor are given as [257]... 

Of particular interest is the case of anisotropic tensorial conductivity, which couples together the various field and current components. It is then, of course, impossible to obtain a separate equation for Hz and, in principle, a dynamo is possible with an increment determined by the off-diagonal (in x, y, z coordinates) components of the conductivity tensor. Usually the direction of anisotropy is related to the magnetic field, but then the whole problem becomes nonlinear. 

In low-symmetry molecules, diagonal and off-diagonal matrix elements of the electronic dipolar coupling tensor may contribute to ( h[)0 ) J ssl b ). Therefore, they are specified mostly in terms of their Cartesian components. If symmetry is C2V or higher, the off-diagonal matrix elements of the tensor operator in Eq. [163] vanish (i.e., the principal axes diagonalizing the SCC tensor coincide with the inertial axes). For triplet and higher multiplicity states, one then obtains... 

Inequations (7.155) and (7.158) we have taken the diagonal = 0 component of the second-rank spherical tensors T2(/ . S) and T2(/a, Ia). In general, these interactions and others like them will have off-diagonal terms also, with q = 1 and 2. The q = 2 components are particularly interesting because, for a molecule in a Id electronic state, they connect the A = +1) and A = — 1) components directly. They therefore make additional hyperfine contributions to the /I-doubling of molecules in Id electronic states. As a result, the nuclear hyperfine splitting of one component of a A-doublet is different from that of the other component. The two contributions are ... 

In many cases in this book, particularly when /I-doubling is involved, we also consider the q 0 components of the dipolar interaction, which follow naturally from the irreducible tensor formalism. The matrix elements of such components are off-diagonal in A, and their angular dependence is described by the spherical harmonics Y2) n(0, < />) or Y2, 2(0,0). In such cases, we shall use the constants t and t 2, defined by analogy with to in equation (8.515). To be specific, the parameter d of Frosch and Foley is related to our t2 by the expression... 

One property that makes octopoles interesting is the lack of dipole moment in the ground state. This should increase the probability of non-centrosymmetric crystallization and prevent detrimental dipolar (aggregate) interaction. In addition, the ratio of off-diagonal versus diagonal P tensor components is higher than that for traditional dipolar systems, which opens up the possibility of less stringent polarization schemes for parametric and electro-optic processes [95]. Furthermore, it has been shown that the efficiency-transparency trade-off favors octopolar molecules over traditional dipolar molecules [96]. 

Since e gA p is traceless, there is no corresponding isotropic tensor analogous to a and G. In a principal axis system that diagonalizes aap, the terms in (2.4) involving off-diagonal components of oc vanish. The CIDs (2.3) now become 5,27)... 

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