Rigid body rotational oscillations

The next refinement of the model takes into account that the shape of most molecular species differs from being rod-like typical nematogenic molecules are given in Table 4.6-1. The resulting behaviour of such a bi-axial molecule is often associated with hindered rotation, however it can also be understood from a rigid-body model where different moments of inertia lead to oscillations of different angular amplitudes in spite of identical (thermal) excitation and identical repulsive forces (Korte, 1983). This can be summarized by order parameters defined as above but referring to one of the two shorter. 

The first term is evidently the vibrational energy of the molecule, considered as a harmonic oscillator. The second term is the energy of rotation, assuming that the molecule is a rigid body,1 while the third term is the correction which takes account of the stretching of the actual, non-rigid molecule due to the rotation. The terms of higher order are unreliable because of the inaccuracy of the assumed potential function. 

Finally, there is the thermal motion problan in X-ray crystallography." Molecules vibrate internally, and molecules in a aystal lattice also vibrate with what is referred to as the rigid-body motion. That is simply the whole molecule moving back and forth and undergoing rotational types of oscillation in the crystal lattice. (These are motions that correspond to translations and rotations in the isolated molecule that are converted into vibrations by the constraints imposed by the crystal lattice.) The molecule as a whole is quite heavy, so these vibrational frequencies tend to be quite low, and the vibrational amplitudes are large. The most serious part of this problem comes not from the translational motions but rather from the rotational oscillations of the molecule in the crystal lattice. The effect of these is to shorten the apparent bond lengths and compact the apparent size of the molecule. 

The next level of approximation is the so-called RRHO approach. RRHO involves modeling the individual conformations as rigid bodies for rotation, and as harmonic quantum oscillators for vibration. Note that the inclusion of translation is not necessary for CFE differences, because it depends only on the mass of the molecule. An RRHO calculation is identical to that in equations (2) and (3), except that , and 0 also include rotational and vibrational energy as defined by the RRHO harmonic approximation. 

In order to simplify the problem, one identifies from the beginning the well-bound molecules or complex ions in the crystal and treats them as rigid bodies. These rigid bodies will execute not only translational oscillations but also rotational oscillations. Coupling is allowed for between all the various units in the crystal and the resulting vibrations are called external vibrations. The effects of nonrigidity of the molecules or complex ions may be examined separately, thereby treating the effects of the crystal... 

Any body having the possibihty to oscillate freely under a gravitational force around a horizontal axis, not passing through the body s CM, is referred to as a physical pendulum. In this case, all points of a rigid body move along an arc of concentric circles. Consequently, for the description of a physical pendulum s oscillations, the rotational laws of dynamics should be applied. 

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