Polar normal coordinates

To see the significance of these functions, we define the plane polar normal coordinates p and

[Pg.141]

Fig. 6.9 Plane polar normal coordinates. The QxQy pl c is not a plane in physical space, but rather a plane in the abstract normal-coordinate space. ...

Pi orbitals, 68, 310, 317 Pi-star orbital, 309, 310 Planarity, 224-225 Planck, M., 122 Plane of symmetry, 53 Plane-polarized wave, 114 Plane polar normal coordinates, 272 Point groups, 53-56,388-389 character tables of, 458-462 see also Group theory... 

Thus the polar normal coordinate

linear molecule. Substitution of (6.89) into (6.88) and use of trigonometric identities gives... 

While s-polarized radiation approaches a phase change near 180° on reflection, the change in phase of the p-polarized light depends strongly on the angle of incidence [20]. Therefore, near the metal surface (in the order of the wavelength of IR) the s-polarized radiation is greatly diminished in intensity and the p-polarized is not [9]. This surface selection rule of metal surfaces results in an IR activity of adsorbed species only if Sfi/Sq 0 (/i = dipole moment, q = normal coordinate) for the vibrational mode perpendicular to the surface. 

The normal vibrations q and q are related to the shifts of the ions Y and X . The low-frequency part of the inertial polarization of the medium, k(cok co 9 co ), cannot follow these shifts. The high-frequency part of the inertial polarization, /(a>/ co 1, co )9 adiabatically follows the shifts of the ions Y" and X-, and the equilibrium coordinates of the effective oscillators describing this part of the polarization depend on the normal coordinates of the corresponding normal vibrations, viz. /0i(gl), (iof(q )-... 

The obtained results are in agreement with our previous hypothesis [2-4] that absolute intensities and the distribution of relative intensities of IR bands in the spectra of adsorbed species are sensitive to the chemical activation of the corresponding bonds arising from polarization by adsorption sites. Hence, in addition to the low frequency shifts, intensites can be used as a criterion for chemical activation. Indeed, according to the fundamentals of IR spectroscopy, the intensities of IR stretching bands are proportional to the square of the dipole moment changes (dp) created by the stretching vibrations over the normal coordinates q of these vibrations [6] I °c [dp/d q]2. 

Here, we discuss the motion of a system of three identical nuclei in the vicinity of the Du, configuration. The conventional coordinates for the in-plane motion are employed, as shown in Figure 5. The normal coordinates (Qx, Qy, Q-), the plane polar coordinates (p,(p,z), and the Cartesian displacement coordinates (Xi,yi,Zi) of the three nuclei (i = 1,2,3) are related by [20,94]... 

Fig. 4. Schematic diagram of the layered model for a pore (47). The two nuclear spins diffuse in an infinite layer of finite thickness d between two flat surfaces. The M axes are fixed in the layer system. The L axes are fixed in the laboratory frame, with Bq oriented at the angle P from the normal axis n. The cylindrical polar relative coordinates p, (p, and z are based on the M axis. The smallest value of p corresponding to the distance of minimal approach between the two spin bearing molecules is 5.

This model has the advantage that the atomic polar tensor elements can be determined at the equilibrium geometry from a single molecular orbital calculation. Coupled with a set of trajectories (3R /3G)o obtained from a normal coordinate analysis, the IR and VCD intensities of all the normal modes of a molecule can be obtained in one calculation. In contrast, the other MO models require a separate MO calculation for each normal mode, since the (3p,/3G)o contributions for each unit are determined by finite displacement of the molecule along each normal coordinate. Both the APT and FPC models are useful in readily assessing how changes in geometry or refinements in the vibrational force field affect the frequencies and intensities of all the vibrational modes of a molecule. 

Here, and in eqn. (50), summing runs with the fixed k value by the frequencies from various spectrum bands of the normal coordinates. The full polarization can also be expressed through the dielectric permeability s(k,ao)... 

Formic acid H.COOH and acetic acid CH3.COOH were studied in the crystalline state by Carlson, Witkowski and Fateley, 66> in the far infra-red region, while Fukushima and Zwolinski, 67> gave a normal coordinate calculation of acetic acid dimers (CH3.COOH)2 the acetate ion in Li(CH3.C00).2H20 was studied by Cadene 168>, while DiLauro, Califano and Adembri 169> studied the crystal spectra and normal modes of the anhydrides of maleic and succinic acids in polarized light. 

Here p and dp refer to polarized and depolarized, respectively. Data taken from (4). Assignments are not based on normal coordinate analysis and are made on the basis of polarizations and by analogy with the IR data. 

Bradbury and Elliot have reported64 the infrared spectrum of crystalline NMA. Polarized spectra were run with the E vector parallel to each of the crystal axes in turn. The temperature dependence of the spectra was quite marked (see also Ref.9 ) and this was attributed, at least in part, to the previously mentioned solid phase transition. Apparently the atomic motions in NMA are of considerable complexity. The far-infrared spectrum has been reported by Itoh and Shimanouchi65. Schneider and co-workers66-68 have done an extensive study of the vibrational spectra of solid NMA and its deuterated analogues and have done complete normal coordinate analyses of these compounds69. ... 

Normal-coordinate analyses were carried out for SeFsCl. There is a decrease of the stretching force constant values from SeFe to SeFsCl, demonstrating that the substitution of one fluorine atom in SeFe by the less electronegative chlorine atom causes an increased polarity of the remaining Se F bonds. 

Equation 3.6 indicates that only vibrational modes that are associated with a change in the dipole moment p of the molecule between the extremes of the atomic displacements [Q is the normal coordinate) different from zero (i.e. the polar modes) can be directly excited. 

As written, the CIDs (2.3) and (2.5) apply to Rayleigh scattering. The same expression can be used for Raman optical activity if the property tensors are replaced by corresponding vibrational Raman transition tensors. This enables us to deduce the basic symmetry requirements for natural vibrational ROA 15,5) the same components of aap and G p must span the irreducible representation of the particular normal coordinate of vibration. This can only happen in the chiral point groups C , Dn, O, T, I (which lack improper rotation elements) in which polar and axial tensors of the same rank, such as aaP and G (or e, /SAv6, ) have identical transformation properties. Thus, all the Raman-active vibrations in a chiral molecule should show Raman optical activity. 

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