Figure axis

In linear molecules only the component of orbital momentum normal to the figure axis is destroyed, that along the figure axis being retained. In non-linear molecules with strong interatomic interactions the concept of orbital angular momentum loses its significance. 

It is convenient to regard R 1 = (— 3, — 2, — i) as coordinates which specify the detector position in the laboratory frame, and then to replace these Euler angles by the polar and azimuthal angles and which describe the detector axis in the laboratory frame and an angle x which gives a reorientation of the detector about its figure axis ... 

Fig. 6.5. (a) Angular momentum orientation via beam deflection in a magnetic field. (6) Figure axis orientation for symmetric top molecules via beam deflection in an electric field, (c) Figure axis orientation in hexapole focussing field. 

Let us consider this case in some detail. If collisions are eliminated in a molecular beam, it is possible to orient molecules (their figure axis) by removing the particles possessing unwanted orientation (analogous to the Stern-Gerlach experiment with a magnetic field). Then, classically, the interaction energy with external electric field is simply... 

In addition to the figure axis internal rotation, each methyl group is allowed free tumbling about its own axis, which also has a moment of inertia of 2.92 X 10-40 g. cm.2 although the moments of inertia are equal, the energy level spacings for the two kinds of rotations are not quite the same. 

Vibrational frequencies of the alkyl radicals have been assigned by comparison to the hydrocarbon molecules. All vibrational and internal rotational degrees of freedom are taken as active and, for ethyl radical, the figure axis rotation is also taken as active for reasons given in Sec. III-C,2. As for the alkanes, internal rotational motions of the radicals (one for ethyl, two for propyl, and two for butyl) were considered to be unhindered. 

Figure 6.9 The H-bond configuration in the ferroelectric phase of KDP (left). New configuration after transfers of protons (right). Cations are not represented. O-atoms are neither represented, but they are positioned at all summits of tetrahedra. Axis c is the vertical axis in the plane of the figure. Axis a is horizontal and also in the plane of the figure, while axis b, perpendicular to the plane of the figure, is drawn in perspective.

As a first example the states of any pure rotor series should all have maximum probability for r, = r2 and 0l2 = n, and the probability densities for members of a given rotor series should have very similar spatial distributions in their internal coordinate system (called the intrinsic coordinate system in the context of nuclear physics). The total wavefunctions of different rotor states in any series should differ primarily only in the parts that describe the rotation of the figure axis in space these parts do not affect the distributions in their internal coordinate systems. At a higher level of approximation, the distributions for the states of a given series may be expected to differ a little because of centrifugal distortion such differences, of course, are apparent in the internal coordinate system. 

Figure 10. Initial anthracene crystal configuration for shock impact along a axis. Crystal segments are projected toward each other along the horizontal figure axis, closing the gap and initiating shock compression.

The rigid rotator in space can be described by polar coordinates of the figure axis,

quantum rules it is found that the total angular momentum is given by Equation 6-8, and the component of angular momentum along the z axis by... 

Compare Appendix II and note that fi i is aligned along the direction of the figure axis). The matrix elements which are off-diagonal in J and K may be neglected from order of magnitude considerations as discussed earlier in the case of the asymmetric top molecules. 

FIGURE 22.4. The tangential orbitals of the square and triangle. These break down into two types. Those parallel (li) and those perpcnclieular (i) to the figure axis of the molecule. 

As in our discussion of (2.31), it is usual to expand the nonconstant coefficient cos a(/fc) about the equilibrium value of a which is a = 0° or cos a = 1 for the case where the molecular figure axis at equilibrium is taken as the 2-axis and the angles a and are measured from this equilibrium axis. Equation (2.65) then takes the form to be expected from (2.62). It is, of course, clear that the displacements m, Up are defined in terms of the local axes appropriate for each molecule (ft). 

Here again the molecular figure axis is taken as parallel to the z-axis and the sign convention of rotational displacements is such that it is positive for a right-handed screw advancing in the direction of the positive coordinate axis (a right-handed coordinate frame is assumed). 

The tunneling interconversion of HF...HF to FH...FH was first observed by Dyke et a/.iii in molecular beam electric resonance studies of the radiofrequency (AMj = 1) and microwave (A J = + 1, AMj = 0) spectra. They also observed the splitting for the perdeutero dimer. (The mixed dimer, HF...DF, has two different isotopomers 14,115 rather than a splitting.) The splittings which have been observed so far for HF...HF are summarized in Table IV.93 1 "108 111 H3.115 (Note that HF dimer is a nearly symmetric top so the figure-axis angular momentum quantum number K is a useful quantum number.)... 

Table IV. Tunneling splittings (cm 1) as a function of vibrational quantum numbers (row headings) and the figure-axis angular momentum quantum number K... 

Figure 5.3 Examples of a prolate symmetric top, 2-butyne (left) and an oblate symmetric top, benzene (right). By convention, the figure axis is labeled the a axis and the c axis in prolate and oblate tops, respectively.

In the oblate symmetric top, the rotational Hamiltonian is given by Eq. 5.10. The c axis is denoted the figure axis. According to the commutation rules obtained in Section 5.3, one possible commuting set of observables is J, J, and J. It is then possible to formulate rotational states JKM which simultaneously obey the eigenvalue equations... 

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