Molecular axes

For the interaction between a nonlinear molecule and an atom, one can place the coordinate system at the centre of mass of the molecule so that the PES is a fiinction of tlie three spherical polar coordinates needed to specify the location of the atom. If the molecule is linear, V does not depend on <() and the PES is a fiinction of only two variables. In the general case of two nonlinear molecules, the interaction energy depends on the distance between the centres of mass, and five of the six Euler angles needed to specify the relative orientation of the molecular axes with respect to the global or space-fixed coordinate axes. 

I done as follows [Essex etal. 1994]. Molecular axes are defined for each CH2 unit in the as shown in Figure 7.19. These molecular axes are defined for the nth CH2 unit as s ... 

Figure 4-11. INDQ/SCI-caleulalcd evolution of the transition energies (upper pan) and related intensities (bottom pan) of the lowest two optical transitions of a cofacial dimer formed by two stilbenc molecules separated by 4 A as a function of the dihedral angle between the long molecular axes, when rotating one molecule around the stacking axis and keeping the molecular planes parallel (case IV of Figure 4-10). Open squares (dosed circles) correspond to the S(J - S2 (S0 — S, > transition.

Figure 6-3. Top Structure of the T6 single crystal unit cell. The a, b, and c crystallographic axes are indicated. Molecule 1 is arbitrarily chosen, whilst the numbering of the other molecules follows the application of the factor group symmetry operations as discussed in the text. Bottom direction cosines between the molecular axes L, M, N and the orthogonal crystal coordinate system a, b, c. The a axis is orthogonal to the b monoclinic axis.

Figure 16-21. Molecular packing of Omc-OPV3. Tire four molecules lhal make up one unit cell arc shown, viewed al a slight angle with respect lo the long molecular axes (let ) and perpendicular to the plane of the central ring (right).

Table 6-1. C2(l molecular poinl group. The electronic stales of the flat T6 molecule are classified according lo the lwo-1 old screw axis (C2). inversion (/). and glide plane reflection (o ) symmetry operations. The A and lt excited slates transform like translations Oi along the molecular axes and are optically allowed. The Ag and Bg stales arc isoniorphous with the polarizability tensor components (u), being therefore one-photon forbidden and Iwo-pholon allowed.

Stacked together along the a-axis. The long molecular axes are tilted by 13° within the layers. By Raman spectral measurements Tashiro et al. found that the biphenyl group of the compound has a twisted structure in the liquid crystalline state as well as in the a-form. 

The compounds crystallise in noncentrosymmetric space groups namely PI, P2i, C2, and P2i2i2i (but with priority of P2i) due to the chirality of the molecules. Most of the compounds have a tilted layer structure in the crystalline state. The tilt angle of the long molecular axes with respect to the layer normal in the crystal phase of the compounds is also presented in Table 18. Some compounds show larger tilt angles in the crystalline state than in the smectic phase. In the following only the crystal structures of some selected chiral liquid crystals will be discussed. 

Tilt angle of the long molecular axes with respect to the layer normal in the crystal structures. These compounds show a bent structure. 

Ito et al. [152] described the crystal structure of 4-[(S)-2-methylbutyl]phe-nyl 4 -hexylbiphenyl-4-carboxylate which shows a smectic A phase and a cholesteric phase. The molecules are arranged in a tilted smectic-like layer structure. Within the layers, the long molecular axes are tilted (30°). However, the compound exhibits no smectic C phase. 

A nematic phase of discotic molecules exists where the short molecular axes are correlated directionally but this phase is still rather rare. By far and away the most common behaviour is for the molecules to stack in columns, which are then arranged in a particular way with respect to one another [7]. Examples are given in Fig. 4. 

Figure 4 ESR of nitroxide radicals (a) Molecular axes of the nitroxide (b) ESR spectra of oriented nitroxide radicals, with the molecular axes x, y, and z along the external magnetic field, respectively and (c) ESR spectra of randomly oriented nitroxide radicals motionally averaged (upper) and rigid limit (lower) regimes.

At first glance, it would appear that all orientation dependence should be lost in the spectrum of a randomly oriented sample and that location of the g- and hyperfine-matrix principal axes would be impossible. While it is true that there is no way of obtaining matrix axes relative to molecular axes from a powder pattern, it is frequently possible to find the orientation of a set of matrix axes relative to those of another matrix. 

If the radical is square pyramidal (C4 ) Fe(CO)5+ (1), the principal axes of the g-matrix must be the molecular axes (the C4 axis and normals to the reflection planes). The iron atom and the carbon of the axial CO group would have the full symmetry of the group and so these hyperfine matrices would share principal axes with the g-matrix. The four equatorial carbonyl carbons, on the other hand, lie in reflection planes, but not on the C4-axis and so are symmetry-required to share only one principal axis with the g-matrix. In fact, the major matrix axes for the equatorial carbons are tilted slightly in the -z direction from the ideal locations along the x and y axes. The g-matrix suggests that the metal contribution is dz2 and the iron hyperfine matrix then can be used to estimate about 55% iron 3d and 34% axial carbon 2pz spin density. The spin density on the equatorial carbons then is mostly negative and due to spin polarization. 

In any metalloprotein, be it tumbling in water or fixed in a frozen solution, not only the Zeeman interaction but also the hyperfine interaction will be anisotropic, so the resonance held in Equation 5.10 becomes a function of molecular orientation in the external held (or alternatively of the orientation of B in the molecular axes system) ... 

FIGURE 6.4 Vectors to describe a walk on the unit sphere. The orientation of a vector b of unit length along the dipolar magnetic field vector B in a Cartesian molecular axes system xyz is defined by the two polar angles 9 between b and the z-axis, and cp between the projection of b on the x-y plane and the x-axis. 

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