Symmetrical normal vibration

It is, for example, known that the dehydrohalogenation of vinyl halogenides XXVIII proceeds preferably by the mechanism of trans-elimination. The stereoselectivity can be explained as follows [84]. The structure of bent acetylene emerging upon trans-elimination of the hydrohalogen elements transforms much more readily into an equilibrium linear form than the structure forming in the case of a cis-elimination. The frequency of the symmetrical normal vibration Ttg, which corresponds to the deformations of XXIX, equals 612cm" which is much less than the value of 729 cm for the antisymmetrical vibration 7c of the structure XXX ... 

Each normal mode of vibration can be described by a normal coordinate Qi which is a linear combination of nuclear displacement coordinates of the molecule. For the symmetric stretching vibration vi of C02, the normal coordinate is of the form... 

FIGURE 2 la The three normal vibrational modes of 11,0. Two of these modes are principally stretching motions of the bonds, but mode v2 is primarily bending, (b) The four normal vibrational modes of C02. The first two are symmetrical and antisymmetrical stretching motions, and the last two are perpendicular bending motions. 

According to Eq. (11), the force constant for the normal vibration Q, can be identified with the term in braces and can be negative if the second term, which is positive, exceeds the first term. If the force constant is negative, the energy should be lowered by the nuclear deformation Qi, and the second-order distortion from the symmetrical nuclear arrangement would occur spontaneously. 

It is apparent from Fig. 4 that the normal modes of vibration of the water molecule, as calculated from the eigenvectors, can be described approximately as a symmetrical stretching vibration (Mj) and a symmetrical bending vibration... 

We now allow nuclear motion and seek vibrational wave functions corresponding to states i i and tjfg. We assume throughout that the subunits have the same point group symmetry in both oxidation states (M and N), and then it is only necessary to consider explicitly totally symmetric normal coordinates of the two subunits (4, 5). Let us assume that there are two on each... 

Bemath, P. F. Spectra of Atoms and Molecules. 2nd Ed. Oxford University Press, Oxford (2005). Miyazawa, T. Symmetrization of secular determinant for normal vibration calculation. J. Chem. Phys. 29, 246 (1958). 

There are two totally symmetric ( ,) normal modes and one b2 normal mode. (The convention is to use lowercase letters for the symmetry species of the normal modes.) The symmetry species of the normal modes have been found without solving the vibrational secular equation. Moreover, since there is only one b2 normal mode, the form of this vibration must be determined from symmetry considerations together with the requirement that the vibration have no translational or rotational energy associated with it. Thus (Fig 6.1), any bent XYX molecule has a b2 normal mode with the X atoms vibrating along the X—Y bonds and the Y atom vibrating in the plane of the molecule and perpendicular to the symmetry axis. On the other hand, there are two ax symmetry coordinates and the two ax normal vibrations are linear combinations of the ax symmetry coordinates, where the coefficients are dependent on the nuclear masses and the force constants. Thus the angles between the displacement vectors of the X atoms and the X—Y bonds for the ax modes of a bent XYX molecule vary from molecule to molecule. 

All wave functions for normal vibrations in their ground states, y/, 0), are bases for the totally symmetric representation of the point group of the molecule. 

We encounter here for the first time the occurrence of a normal vibration which is completely inactive as a fundamental. This phenomenon is not commonplace but is encountered occasionally in relatively symmetrical molecules. 

Consider next the water molecule. As we have seen, it has a dipole moment, so we expect at least one IR-active mode. We have also seen that it has CIt, symmetry, and we may use this fact to help sort out the vibrational modes. Each normal mode of iibratbn wiff form a basis for an irreducible representation of the point group of the molecule.13 A vibration will be infrared active if its normal mode belongs to one of the irreducible representation corresponding to the x, y and z vectors. The C2 character table lists four irreducible representations A, Ait Bx, and B2. If we examine the three normal vibrational modes for HzO, we see that both the symmetrical stretch and the bending mode are symmetrical not only with respect to tbe C2 axis, but also with respect to the mirror planes (Fig. 3.21). They therefore have A, symmetry and since z transforms as A, they are fR active. The third mode is not symmetrical with respect to the C2 axis, nor is it symmetrical with respect to the ojxz) plane, so it has B2 symmetry. Because y transforms as Bt, this mode is also (R active. The three vibrations absorb at 3652 cm-1, 1545 cm-1, and 3756 cm-, respectively. 

Both the Raman and the infrared spectrum yield a partial description of the internal vibrational motion of the molecule in terms of the normal vibrations of the constituent atoms. Neither type of spectrum alone gives a complete description of the pattern of molecular vibration, and, by analysis of the difference between the Raman and the infrared spectrum, additional information about the molecular structure can sometimes be inferred. Physical chemists have made extremely effective use of such comparisons in the elucidation of the finer structural details of small symmetrical molecules, such as methane and benzene. But the mathematical techniques of vibrational analysis are. not yet sufficiently developed to permit the extension of these differential studies to the Raman and infrared spectra of the more complex molecules that constitute the main body of both organic and inorganic chemistry. 

The perturbational analysis of the adiabatic surface near the totally symmetric zero point [27] shows that for the realistic values of the electron transfer parameters one of two Alu normal vibrations is much softer than all other vibrations. We can expect that this mode together with two Alg modes play the most important role in the structural distortions and charge redistribution in the cluster. 

---