Vector additions, vibration-rotation

In the ground vil tional state Mg =gg =0 while all the oth quantities do not change significantly. In symmetric-top molecules, the vector addition of the vibrational and the skeleton-rotational induced magnetic moments along the figure axis must be considered. There are four possible relative orientations of the total and vibrationally induced angular momenta, J, and p , which obey the relations... 

If the basis captures molecular movement or rotation, then remove the corresponding representations. For example, if three vectors per atom are used for a full vibrational analysis of a nonlinear molecule, then we will find six irreducible representations that do not describe vibration, one each from the group s standard representations for x, y, z and Rx, Ry, Rz- For a linear molecule, rotation around the molecular axis Rz is not a degree of freedom and an additional vibrational mode will be found. 

Consider the waves scattered by isotropic dipole oscillators in the thin slab of matter shown in Fig. 9.3 only part a is of concern at the moment. These waves add vectorially at point P to produce the resultant forward-scattered wave s the important point, which is by no means obvious yet, is that this resultant scattered wave is phase shifted 90° relative to the incident wave (in addition to the phase shift between the oscillators and the incident wave). The background necessary to show this has been presented in Chapter 3 Fig. 3.8 is similar to Fig. 9.3 except that in the former the scatterers were arbitrary particles. The transmitted field t at P is the vector sum of the incident field and the fields scattered by all the oscillators. If we assume that the direction of vibration of the incident wave is not rotated as it propagates through the slab, the transmitted field is given by (3.39) ... 

Rotational Motions. In addition to molecular vibrational motions, one can use neutrons to characterize the rotation of a molecule about its center of mass. If the position of the hydrogen atoms in a molecule rotates as a function of time in such a way that the relative positions of the atoms move with a constant radius d, but the motion is such that the terminus of the vector d moves randomly or ergotically on the surface of the sphere this motion is referred to as rotational diffusion. This motion can also provide insight into the nuclear density. An example of this is shown in Figure 10. 

The fundamental requirement for the existence of molecular dissymmetry is that the molecule cannot possess any improper axes of rofation, the minimal interpretation of which implies additional interaction with light whose electric vectors are circularly polarized. This property manifests itself in an apparent rotation of the plane of linearly polarized light (polarimetry and optical rotatory dispersion) [1-5], or in a preferential absorption of either left- or right-circularly polarized light (circular dichroism) that can be observed in spectroscopy associated with either transitions among electronic [3-7] or vibrational states [6-8]. Optical activity has also been studied in the excited state of chiral compounds [9,10]. An overview of the instrumentation associated with these various chiroptical techniques is available [11]. 

The radiation produced by the coil of an R - oscillator, which serves as tlie source in NMR instruments, is plane polarized. Plane-polarized radi.llion, however, consists of d and I circularly polarized radiation. As sliown in Figure l9-4b, the vector of the d component rotates cliKrkwise as the radiation approaches the observer the vector of the / component rotates in the opposite sense. Addition of the two vectors leads lo a vector sum that vibrates in a single plane (Figure l9-4a). 

Fig. 6.5. Space- and Body-Fixed Cooidinate Systems (SFCS and BFCS). (a) SFCS is a Cartesian coordinate system arbitrarily chosen in space (left). The origin of the BFCS is located in the centre of mass of the molecule (right). The centre of mass is shown by the vector Rcm front the SFCS. The nuclei of the atoms are indicated by vectors ri,r2,rj... from the BFCS. Fig. (b) shows what happens to the velocity of atom a, when the system is rotating with the angular velocity given as vector ta. In such a case the atom acquires additional velocity ta x r . Fig. (c) shows that if the molecule vibrates, then atomic positions ta differ from the equilibrium positions a by the displacements...

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