Point groups Polarizability

The vibrations of acetylene provide an example of the so-called mutual exclusion mle. The mle states that, for a molecule with a centre of inversion, the fundamentals which are active in the Raman spectmm (g vibrations) are inactive in the infrared spectmm whereas those active in the infrared spectmm u vibrations) are inactive in the Raman spectmm that is, the two spectra are mutually exclusive. Flowever, there are some vibrations which are forbidden in both spectra, such as the torsional vibration of ethylene shown in Figure 6.23 in the >2 point group (Table A.32 in Appendix A) is the species of neither a translation nor a component of the polarizability. 

For a molecule belonging to the D2h point group deduce whether the following vibrational transitions, all from the zero-point level, are allowed in the infrared spectmm and/or Raman spectmm, stating the direction of the transition moment and/or the component of the polarizability involved ... 

Symmetry selection rules for Raman spectrum can be derived by using a procedure similar to that for the IR spectrum. One should note, however, that the symmetry property of symmetry species of six components of polarizability are readily found in character tables. In point group C3V, for example, normal vibrations of the NH3 molecule (2A1 and 2E) are Raman-active. More generally, the vibration is Raman-active if the component(s) of the polarizability belong(s) to the same symmetry species as that of the vibration. 

Although the spherical form of the multipole expansion is definitely superior if the orientational dependence of the electrostatic, induction, or dispersion energies is of interest, the Cartesian form171-174 may be useful. Mutual transformations between the spherical and Cartesian forms of the multipole moment and (hyper)polarizability tensors have been derived by Gray and Lo175. The symmetry-adaptation of the Cartesian tensors of quadrupole, octupole, and hexadecapole moments to all 51 point groups can be found in Ref. (176) while the symmetry-adaptation of the Cartesian tensors of multipole (hyper)polarizabilities to simple point groups has been considered in Refs. (172-175). 

The different symmetry properties considered above (p. 131) for macroscopic susceptibilities apply equally for molecular polarizabilities. The linear polarizability a - w w) is a symmetric second-rank tensor like Therefore, only six of its nine components are independent. It can always be transformed to a main axes system where it has only three independent components, and If the molecule possesses one or more symmetry axes, these coincide with the main axes of the polarizability ellipsoid. Like /J is a third-rank tensor with 27 components. All coefficients of third-rank tensors vanish in centrosymmetric media effects of the molecular polarizability of second order may therefore not be observed in them. Solutions and gases are statistically isotropic and therefore not useful technically. However, local fluctuations in solutions may be used analytically to probe elements of /3 (see p. 163 for hyper-Rayleigh scattering). The number of independent and significant components of /3 is considerably reduced by spatial symmetry. The non-zero components for a few important point groups are shown in (42)-(44). 

The simplest system exhibiting two-dimensional polarizabilities belongs to the point group C2 and consists of three centres. These may include either one donor (D) and two equivalent acceptors (A), DA2, or two equivalent donors (D) and one acceptor (A), D2A. For simplicity, it is assumed that the three centres occupy the corners of an equilateral triangle of side length / (cf. Scheme 2). TTien, if the three centres are equivalent, the system assumes point group symmetry There are three MO, ij/i, if/2 and that can be constructed by linear combination of the three AO < i, and As in the one-dimensional case above, the normalized MO can be represented as functions of a single parameter c [0,1] by (75)-(77). 

Figure 2.7-6 A Assignment of the Cartesian coordinate axes and the symmetry operations of a planar molecule of point group C2,.. B Character table, 1 symbol of the point group after Schoen-flies 2 international notation of the point group 3 symmetry species (irreducible representations) 4 symmetry operations 5 characters of the symmetry operations in the symmetry species +1 means symmetric, -1 antisymmetric 6 x, y, z assignment of the normal coordinates of the translations, direction of the change of the dipole moment by the infrared active vibrations, R, Ry, and R stand for rotations about the axes specified in the subscript 7 x, xy,. .. assign the Raman active species by the change of the components of the tensor of polarizability, aw, (Xxy,. ...

Four thiourea molecules at sites are the building blocks of the unit cell of the crystal of the space group >2. The point group which is relevant for the selection rules is found by deleting the superscript, which yields >2a- Table 2.7-4 lists the details and results of the application of Eqs. I and II from Table 2.7-1. As shown in Fig. 2.7-8, the results are assigned to the components of the polarizability tensor and the dipole moment vector of the crystal, with x and y = a, b, c, which explains the Raman and infrared activity. We see that in addition to the translations of the whole crystal most... 

Electric and magnetic properties of microsystems. Definition of multipoles electrostatics of permanent multipoles interaction energies for two multipoles induced molecular multipoles interaction energies of induced multipoles. Tables of point groups tensor elements of multipoles vector elements of multipoles tensor elements of polarizabilities. 

The numbers of non-zero and mutually independent demits of the polarizability tensors a, p, and y for all point groups are giv in Tables 10—12. 

The key point then was to assess how well these mean V/7S ave represent the relative polarizabilities of the atoms and groups. For this purpose, we took Miller s atomic (valence state) and group polarizabilities as the standards [175], Excluding N02, which is not included in his tabulation, R2 for the V/7Save ra. aMiller correlation is 0.963. This allows us to predict aMiller for the N02 group 2.7A3. 

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