Symmetry parameters

Electron spin resonance (ESR) studies of the urazole-bridged 1,3 diradicals 64 derived from the azoalkanes 63 confirm a triplet ground state for these species. The nearly zero symmetry parameter, that is, Elhc= 0.0004 0.0001 cm-1, for the triplet diradical 64 of the diphenyl azoalkane 63 establishes a planar conformation <1995JOC308, 1997JA10673>. 

For dissimilar pairs, the parameter ys equals zero and we have Eq. 5.36. Like pairs of zero spin are bosons and all odd-numbered partial waves are ruled out by the requirement of even wavefunctions of the pair this calls for ys = 1. In general, for like pairs, the symmetry parameter ys will be between -1 and 1, depending on the monomer spins (fermions or bosons) and the various total spin functions of the pair. A simple example is considered below (p. 288ff.). If vibrational states are excited, the radial wavefunctions xp must be obtained from the vibrationally averaged potential, Fq(R). The functions gf(R) and gM(R) are similar to the pair distribution function, namely [294]... 

About exchange symmetry. We compute the symmetry parameter ys appearing in Eq. 6.22 for given para-H2 and ortho-H2 densities, np, n0. We assume low temperatures so that all para-H2 is in the j = 0, and all ortho-H2 in the j — 1 state. (Symmetry should not matter at temperatures above roughly 20 K, unless spectral dimer features are considered.)... 

These fractions are related to the symmetry parameter y according to... 

For moment calculations, the accounting for the spectral moments hardly affects the spectral moments at the temperatures considered the extreme variation of the symmetry parameter ys from 0 to 1 modifies the moments M by less than 1% at 40 K, and much less at higher temperatures. It has been previously reported [315] that exchange symmetry matters at temperatures of less than 10 K. At the higher temperatures, one may often neglect symmetry (ys = 0 in Eq. 6.22) unless dimer features are considered the dimer structures of like pairs are in general dependent on symmetry at any temperature. 

Fig. 5 Contour diagram showing the variation of or, the symmetry parameter, with the bond orders of the transition state...

From (28) we can also obtain a value of a the symmetry parameter for the transition state using (32). From this equation we can see that when the... 

A number of simple diarsines have been structurally investigated and, although individual molecules are similar to those of the corresponding distibines, the diarsines do not show the extended chain structures and consequent thermochromism often found for the latter. The geometry at the two arsenic atoms is pyramidal, though the angles at arsenic can be asymmetric. The substituents occupy anti (gauche) conformations and in some cases the molecules have imposed Cj symmetry. Parameters for four of the compounds are summarized in Table 2. 

The directional y and the non-directional G parameters, defined for WHIM descriptors and containing information about the molecular symmetry, are not considered in the frame of the G-WHIM approach as their meaning becomes doubtful, depending heavily upon the point sampling. However, information regarding molecular symmetry can be obtained by directly using the WHIM symmetry parameters. 

G is the geometric mean of the directional symmetries and equals 1 when the molecule shows a central symmetry along each axis, and tends to 0 when there is a loss of symmetry along at least one axis. Different symmetry values are obtained only when unitary, mass and electrotopological weights are used for this reason, only three kinds of symmetry parameters are retained Gu, Gm and Gs. 

In Eq. (42) the parameters Ey occur. These parameters have been proposed as a rational standard for such symmetry parameters [Ref. 6) p. 279] and have been named reduced ligand-field parameters, even though they depend on T. 

WHIM approach as their meaning becomes doubtful, depending heavily upon the point sampling. However, information regarding molecular symmetry can be obtained by directly using the WHIM symmetry parameters. 

The E parameter is usually referred to as the symmetry parameter and its value is zero for molecules with high symmetry (C3 or higher) due to the cancellation of the x2 and y2 terms in Eq. (4) (the z axis is defined along the rotational axis). It is convenient to take the z axis for localized diradicals along the line that connects the two radical centers to obtain the spectroscopic zfs parameters directly from Eq. (4), which obey the relation D 3 E 0. In this case, the E parameter becomes very small due to its inverse dependence on the fifth power of the interspin distance ( oc r-5). The D term, or the distance parameter, depends on the inverse cube of r (D ocr 3 since z r) and in the limit Eq. (5) holds [25]. This equation has been applied extensively for estimating the interspin distances in diradicals. 

The importance of the hyperpolarizability and susceptibility values relates to the fact that, provided these values are sufficiently large, a material exposed to a high-intensity laser beam exhibits nonlinear optical (NLO) properties. Remarkably, the optical properties of the material are altered by the light itself, although neither physical nor chemical alterations remain after the light is switched off. The quahty of nonlinear optical effects is cmciaUy determined by symmetry parameters. With respect to the electric field dependence of the vector P given by Eq. (3-4), second- and third-order NLO processes may be discriminated, depending on whether or determines the process. The discrimination between second- and third-order effects stems from the fact that second-order NLO processes are forbidden in centrosymmetric materials, a restriction that does not hold for third-order NLO processes. In the case of centrosymmetric materials, x is equal to zero, and the nonhnear dependence of the vector P is solely determined by Consequently, third-order NLO processes can occur with all materials, whereas second-order optical nonlinearity requires non-centrosymmetric materials. 

Fig. 3.37 Symbols used in formulae for axial (3.2.1) and rhombic (3.2.2) symmetry. Parameters g, A and D occur along the Z axis, while gj, and Aj, are in the XY plane. Due to the axial symmetry angle 4> does not enter the equations. The g-, A- and D-tensors with rhombic symmetry can be thought of as the lengths and the directions of the x, y, and z axes of an eUip-soid. The principal axes of the different tensors need not coincide with each other or with the X, Y, Z axes...

The greedy (Tables 7.7 and 7.8) and complete combinatorial search algorithm (Tables 7.9 and 7.10) consider acetonitrile a strong outlier and take advantage of the two-valued (f>(P, 1) symmetry parameter that zeroes all indices whose properties have /t = 0, while leaves them unchanged if /t 0, i.e., for /t = 0, 4>-x = 0, while for II 0, (p-x = X- Th optimal full combinatorial descriptor with an improved r but is shown in Tables 7.9 and 7.10, while in Tables 7.11 and 7.12 is shown the corresponding zero-level description, which seems not excessively bad even if the aid of the non-random (f> ad hoc index plays a no minor role. The correlation vector of descriptor in Table 7.9, with whom Fig. 7.11 has been obtained, is,... 

Figure 1. Definition of iigand-field symmetry parameters for a complex of tetragonal symmetry. A represents the cubic field as usual A(e) and A(t2> represent the tetragonal field that requires two indq)endent variables.

S = symmetry parameter from the empirical Van Laar equation B = Zxibi 1... 

Because of the adjustable symmetry parameter, the M-VDW equation proves to be significantly superior to the Mark V. Presumably the Soave equation would give results comparable to the Mark V because they both assume the Redlich-Kwong volume dependence. 

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