B Symmetry species of vibrations

APPENDIX B SYMMETRY SPECIES OF VIBRATIONS Table B.2 continued)... 

Asymmetric tops have three different moments of inertia, and up to three different symmetry species of vibration could give rise to IR bands. These correspond to dipole changes along the x, y and z Cartesian axes. If these axes coincide with the three principal inertial axes A, B and C (Section 7.2.1), either because of... 

Under certain conditions, these two expressions will give rise to simple relations between the intensities of isotopic molecules, the latter analogous to the product rule, the former to the sum rule for the frequencies. The conditions are (a) that the molecule should have no dipole moment and/or (b) that the symmetry species of the vibrations over which the summation of intensities is carried out should not be the same as that of any rotation of the molecule which moves the permanent dipole moment. If one or both of these conditions are satisfied, dyu/dS. is the same for all isotopic molecules. It then immediately follows that the intensity sum... 

In Table B. 1 in Appendix B are given formulae, analogous to those derived for the C2 point group, for determining the number of normal vibrations belonging to the various symmetry species in all non-degenerate point groups. 

Having assigned symmetry species to each of the six vibrations of formaldehyde shown in Worked example 4.1 in Chapter 4 (pages 90-91) use the appropriate character table to show which are allowed in (a) the infrared specttum and (b) the Raman specttum. In each case state the direction of the transition moment for the infrared-active vibrations and which component of the polarizability is involved for the Raman-active vibrations. 

Vibrations of the symmetry class Ai are totally symmetrical, that means all symmetry elements are conserved during the vibrational motion of the atoms. Vibrations of type B are anti-symmetrical with respect to the principal axis. The species of symmetry E are symmetrical with respect to the two in-plane molecular C2 axes and, therefore, two-fold degenerate. In consequence, the free molecule should have 11 observable vibrations. From the character table of the point group 04a the activity of the vibrations is as follows modes of Ai, E2, and 3 symmetry are Raman active, modes of B2 and El are infrared active, and Bi modes are inactive in the free molecule therefore, the number of observable vibrations is reduced to 10. 

Before concluding the discussion on the notation of the irreducible representations, we use C2v point group as an example to repeat what we mentioned previously since this point group has only four symmetry species, A, A2, B, and B2, the electronic or vibrational wavefunctions of all C2V molecules (such as H2O, H2S) must have the symmetry of one of these four representations. In addition, since this group has only one-dimensional representations, we will discuss degenerate representations such as E and T in subsequent examples. 

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,. ...

The spectra of the anti conformer will follow the rule of mutual exclusion, as the molecules possess an inversion centre. The selection rules predict five Raman active skeleton deformations for the anti conformer. For the crystalline state (Fig. 1), five deformations are observed, leaving no doubt that the anti conformer is energetically favored. Table 2 summarizes the calculated and observed skeleton stretching vibrations of both isotopomers. According to the correlation tables, Ag and Au vibrations of the anti conformer (point group 21) combine into A vibrations of point group C2, Bg, and B vibrations into vibrations belonging to symmetry species B. 

Two structures have been proposed for (Gly) I an antiparallel-chain pleated sheet (APPS) and a similar rippled sheet (APRS) (see Section III,B,1). These structures have different symmetries the APPS, with D2 symmetry, has twofold screw axes parallel to the a axis [C (a)] and the b axis [C (b)], and a twofold rotation axis parallel to the c axis [62(0)] the APRS, with C2h symmetry, has a twofold screw axis parallel to the b axis ( 2(6)], an inversion center, i, and a glide plane parallel to the ac plane, o-Sj. Once these symmetry elements are known, together with the number of atoms in the repeat, it is possible to determine a number of characteristics of the normal modes the symmetry classes, or species, to which they belong, depending on their behavior (character) with respect to the symmetry operations the numbers of normal modes in each symmetry species, both internal and lattice vibrations their IR and Raman activity and their dichroism in the IR. These are given in Table VII for both structures. 

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