Threefold rotational motion

If the barrier energy is lower than this critical value, rotational motion will be fast at 6 K and the STM image will be smeared out and STM simulations will need to include time averaging. If the barrier is larger, then the image may have threefold or sixfold symmetry depending on the energies of the local minima. 

When the quadratic coupling terms in Eq. (3) are included, the rotational symmetry of the potential surfaces in Eq. (4) is lost and replaced by a threefold symmetry inherent to systems with a threefold rotation axis. Consequently, j ceases to be a good quantum number and the spectra of the linear E e JT system for individual j become mixed . This inherently two-dimensional vibronic motion leads to a complicated, erratic line structure (see, for example, Ref. 21) as is typical for other, less symmetric conical intersections discussed throughout this book. The above statements about adiabatic and nonadiabatic behavior for i < 0 and E > 0, and the formation of broad quasi-resonances arising from the upper cone vibrational levels, are not affected by the inclusion of quadratic coupling terms. 

Fig. 5. Types of motion in proteins detected by nmr. Rotation about methyl groups is easily detected from threefold symmetry and is rapid. Rotation or flipping about the C(J—Cy bonds of tyrosine or phenylalanine has been observed readily (see text) because of the twofold symmetry of the aromatic ring. Rotation of more complex side chains is more difficult to define because of the lack of symmetry.

In order to illustrate the power of the Group Theory for Non-Rigid Molecules, let us consider the double internal rotation problem in acetone, solved in [34]. This motion is described by a restricted Hamiltonian operator such as that of pyrocatechin (32) in which only the threefold periodicity of the potential for acetone (47) has been introduced ... 

Levy et al. (38) have performed Si Tj studies on a variety of organosilicon compounds. They observed that in linear polydimethyl-siloxanes motional processes along the chain are quite different. They showed that in MD M systems the relaxation behaviour is not the same for M units and D units. M units are able to spin freely around their threefold axis of symmetry favouring an SR relaxation mechanism, while D units may rotate only through a restricted angle which favours the Si- H DD relaxation mechanism. Table XXXI shows the Ti and NOE data for six linear siloxanes. For the D silicons the spin-rotation... 

The values of AT in the MN crystals are nearly as large as that for benzene (Sect. 5.8.2). This result is also typical the order of magnitude of the van der Waals interactions is similar in all the polyacene crystals. A typical correlation time tc at room temperature can be found e.g. for 1,5 DMN from the values in Table 5.6 and Eq. (5.23) to be (1,5 DMN, 300 K) = 2.4 10 s. Compared with the period of rotation of a free CH3 group around its threefold symmetry axis, similarly to the case of benzene, the corresponding rotational frequency is much lower than that of the free groups or molecules [30]. The reorientation motions of the CH3 groups are thus hindered rotations. 

Pulsed NQR measurements of the spin-lattice relaxation time, T, also give detailed information on the mechanisms and dynamics of molecular motion in solids. For example, quadrupole relaxation times for solid triethylenediamine show a Ti minimum at a temperature close to 260 K. when 7j equals 0.048 On either side of the minimum, 7j depends exponentially on temperature, according to an Arrhenius-type equation with an activation energy of 34 kJ moP . The mechanism of the relaxation is shown to be hindered rotation of the triethylenediamine molecule about its threefold symmetry axis, which modulates the dipolar coupling between N and the adjacent CH2 protons. 

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