Prolate molecules

Two different classes of distortion may be recognized, namely oblate and prolate. A prolate molecule, like CH3F, has two equal principal moments of inertia, which are greater than the third, while an oblate molecule, like NH3, has two equal moments of inertia, which are less than the third. In each case, the unique axis corresponds to a rotational axis of order 3 or more. The classification may be loosely extended to less symmetrical species, where two of the principal moments of inertia are similar and different from the third. 

Very few data exist for the viscosities or Frank constants of discotic nematics—that is, nematics composed of disc-Uke particles or molecules (Chandrasekhar 1992). One can estimate values of the Leslie viscosities from the Kuzuu-Doi equations (10-20) by setting the aspect ratio p equal to the ratio of the thickness to the diameter of the particles thus /j — 0 for highly anisotropic disks. This implies that R(p) —1, and Eq. (10-20b) implies that the viscosity o 2 is large and positive for discoidal nematics, while it is negative for ordinary nematics composed of prolate molecules or particles. If, as expected, is much smaller in magnitude than 0 2. the director (which is orthogonal to the disks) will tend... 

The entropy production is consequently dependent on the orientation of the director. In a nematic liquid crystal consisting of prolate molecules A II >Aj l. The entropy production is consequently minimal in the perpendicular orientation. In a system consisting of oblate molecules the reverse is true, Ajj > A. Thus the entropy production is minimal in the parallel orientation. 

Kuzuu and Doi (1983, 1984) derived the six coefficients (in the unit of fj) in the limit of low flow rates for prolate molecules... 

The relations between the different sets of parameters are given in [77Wat, 84Gor]. The notation of the centrifugal distortion constants permits to know which reduction is used, and therefore the rotational constants are simply called A, B, C (without the superscript A or S). There are six different ways (representations) to identify the (x, y, z) reference system with the (a, b, c) principal axis system. In practice two different representations are used F where x=b, y=c, z=a and which is best for prolate molecules (Ray s asymmetiy parameter k = 1B-A-C)I A-C) <0), and IIF where x=a, y=b, z=c which is thought to be better for oblate molecules (k > 0). Representation IlF is also used where x = a, y = c, z = b ) xXiX is equivalent to representation mf Maity authors use codes written in F representation for oblate molecules, in particular for the analysis of inCnared spectra. 

This potential has been studied in connection with two prolate molecules, CO2 and N2, and one oblate molecule, benzene. The potential parameters were chosen to optimize agreement with the temperature dependence of measured second virial coefficients. These potentials were then used to compute the cohesive energy of the respective solids with some success. " ... 

The elastic constants kn, 22, and 33 pertain to the three basic deformations splay, twist, and bend, respectively. For typical nematics with prolate molecules one has hi > hi > k22 and hi 10 " N. 

Table 7-4. The rotational constants in MHz along with the Ray s asymmetry parameter, ic, for some near prolate molecules (A > B = C) are shown. Data obtained from W.H. Flygare, Molecidar Structure and Dynamics, Prentice-Hall, New Jersey, 1978.

For prolate (7c = 7b > la) and oblate (7a = 7 >7c) molecules, two of the moments of inertia are the same canceling one term from the Hamiltonian in Equation 7-17. This results in two different Hamiltonians, one for prolate molecules and the other for oblate molecules. 

The eigenvalues needed to completely specify the rotational state of an oblate or prolate molecule are as follows (Afy is degenerate). 

Molecules for which two of the three principal moments of inertia are equal are called symmetric tops. Those for which the unique moment of inertia is smaller than the other two are termed prolate symmetric tops if the unique moment of inertia is larger than the others, the molecule is an oblate symmetric top. 

Figure 5.5 The rotational angular momentum vector P for (a) a linear molecule and (b) the prolate symmetric rotor CH3I where is the component along the a axis...

The rotational energy levels for a prolate and an oblate symmetric rotor are shown schematically in Figure 5.6. Although these present a much more complex picture than those for a linear molecule the fact that the selection mles... 

For a symmetric rotor molecule such as methyl fluoride, a prolate symmetric rotor belonging to the C3 point group, in the zero-point level the vibrational selection mle in Equation (6.56) and the character table (Table A. 12 in Appendix A) show that only... 

Whether the molecule is a prolate or an oblate asymmetric rotor, type A, B or C selection mles result in characteristic band shapes. These shapes, or contours, are particularly important in gas-phase infrared spectra of large asymmetric rotors, whose rotational lines are not resolved, for assigning symmetry species to observed fundamentals. 

This general behaviour is characteristic of type A, B and C bands and is further illustrated in Figure 6.34. This shows part of the infrared spectrum of fluorobenzene, a prolate asymmetric rotor. The bands at about 1156 cm, 1067 cm and 893 cm are type A, B and C bands, respectively. They show less resolved rotational stmcture than those of ethylene. The reason for this is that the molecule is much larger, resulting in far greater congestion of rotational transitions. Nevertheless, it is clear that observation of such rotational contours, and the consequent identification of the direction of the vibrational transition moment, is very useful in fhe assignmenf of vibrational modes. 

The way forward was proposed by Berne and Pechukas [11] many years later. Their important idea was to consider the overlap between two prolate ellipsoidal gaussian distributions. From the expression for this overlap they evaluated a range parameter which was taken to be the contact distance g and a strength parameter which was set equal to the well depth, e. If the orientations of the two rod-like molecules in the laboratory frame are represented by the unit vectors Ui and Uj and the orientation of the intermolecular vector by the unit vector f then the expression for the angular dependence of the contact distance is... 

The quantity riV/RT is equal to six times the rotational period. The rotational relaxation time, p, should he shorter than the fluorescence lifetime, t, for these equations to apply. It is possible to perform calculations for nonspherical molecules such as prolate and oblate ellipsoids of revolution, but in such cases, there are different rotational rates about the different principal axes. 

Another curvilinear coordinate system of importance in two-centre problems, such as the diatomic molecule, derives from the more general system of confo-cal elliptical coordinates. The general discussion as represented, for instance by Margenau and Murphy [5], will not be repeated here. Of special interest is the case of prolate spheroidal coordinates. In this system each point lies at the intersection of an ellipsoid, a hyperboloid and and a cylinder, such that... 

Our strategy is to incorporate a prolate redox molecule into the hydrocarbon phase of the organized monolayer. Steric restraints imposed by a dose-packed monolayer would presumably force the redox molecule to adopt an orientation parallel to the hydrocarbon tails. Spacing can then be controlled by a short hydrocarbon chain between the redox center and the metal. A class of molecules fitting these requirements are the assymetric 4,4 -bipyridiniums ... 

Fig. 15a,b. a Oblate configuration of adsorbed dendrimers in a dense monolayer, b The prolate configuration of dendrimers was obtained by adsorption from solution containing alkanetiols molecules. Redrawn after [98]... 

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