Symmetry Coordinates and Normal Modes

The distribution of the iN — 6 vibrational symmetry coordinates of a nonlinear polyatomic molecule among the irreducible representations of its symmetry point group can be determined by standard methods. [7] Ordinarily, not all of the symmetry species will be represented and several of them will include more than one coordinate. If the molecule belongs to a commutative symmetry point group, all of them will be assigned to one-dimensional symmetry species. If its group is non-commutative, and therefore has representations that are two-or three-dimensional, some of its vibrations may be degenerate these are best discussed separately. 

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The whole method of applying group theory to vibrational problems depends upon the fact that the normal coordinates and normal modes of vibration of S3rmme1rical molecules have certain special symmetry... 

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. 

Symmetry coordinates can be generated from the internal coordinates by the use of the projection operator introduced in Chapter 4. Both the symmetry coordinates and the normal modes of vibration belong to an irreducible representation of the point group of the molecule. A symmetry coordinate is always associated with one or another type of internal coordinate—that is pure stretch, pure bend, etc.—whereas a normal mode can be a mixture of different internal coordinate changes of the same symmetry. In some cases, as in H20, the symmetry coordinates are good representations of the normal vibrations. In other cases they are not. An example for the latter is Au2C16 where the pure symmetry coordinate vibrations would be close in energy, so the real normal vibrations are mixtures of the different vibrations of the same symmetry type [7], The relationship between the symmetry coordinates and the normal vibrations can be... 

The calculated harmonic force field given in cartesian coordinates was transformed into a force field defined by symmetry coordinates. A normal coordinate analysis was performed using the geometry and the symmetry force constants from ab initio calculations, resulting in a description of the normal modes by their potential energy distribution (FED). Table 3 presents the potential energy distribution for v SiSi and v, SiCb for both rotamers, and Table 4 summarizes the SiSi, SiCl, and SiC stretching force constants. 

Other qualitative rules for the study of reaction paths have been derived independently. For unimolecular reactions, it has been found that conditions favorable to a given path exist if there is a low-energy excited state of the same symmetry as the normal mode corresponding to the reaction coordinate, the transition density is localized in the region of nuclear motion and the excitation energy decreases along the coordinate 32>. 

Since this transformation to normal coordinates is invertible, one can readily determine the functional dependencies of the terms in Eq. (1) using either the normal or internal coordinates. Interestingly, in our study of vibrational states of the well-known local mode molecule H20 and its deuterated analogs we found only minor differences between the results of CVPT in the internal and normal mode representations (46). The normal mode calculations, however, required significantly less computer time to run, since many terms in the Hamiltonian are constrained to zero by symmetry. For this reason we chose to use the normal mode coordinates for all subsequent studies. 

In this case, analytic formulas can also be obtained for the second derivatives that govern the harmonic vibrations [10]. Table 2 lists the vibrational force constants and normal mode frequencies determined [10, 12] using Wilson s FG matrix treatment, the procedure standard for molecular vibrations [14]. The radial displacements from the minimum are combined in the usual way to form symmetry coordinates,... 

It is accepted that a free CF3SO3 ion in a staggered configuration with a Csv point group symmetry has 18 normal modes with the symmetry representations 5A1+A2+6E. The A1 and E modes are infrared and Raman active. The A2 mode, associated with the internal torsion of the anion, is inactive both in infrared and Raman. Because all these modes are more or less sensitive to coordination effects they can be used to look into the anion local chemical surrounding. ... 

The cyclopentadienyl radical and the cyclopentadienyl cation are two well-known Jahn-Teller problems The traditional Jahn-Teller heatment starts at the D k symmetry, and looks for the normal modes that reduce the symmetry by first-01 second-order vibronic coupling. A Longuet-Higgins treatment will search for anchors that may be used to form the proper loop. The coordinates relevant to this approach are reaction coordinates. 

The integral < vib vib) maY vanish because of symmetry considerations. For example, the C02 normal mode v3 in Fig. 6.2 has eigenvalue — 1 for the inversion operation. Hence (Section 6.4), the v3 factor in the vibrational wave function is an even or odd function of the normal coordinate Q3, depending on whether v3 is even or odd. For a change of 1,3,5,... in the vibrational quantum number v3, the functions p vib and p"ib have different parities and their product is an odd function, so that ( ibl vib) vanishes. Thus we have the selection rule Ac3 = 0,2,4,... for electronic transitions in... 

Qualitative ways of analyzing a problem in molecular vibrations, that is, methods for determining the number of normal modes of each symmetry type which will arise in the molecule as a whole and in each set of equivalent internal coordinates, have been developed. There is also the quantitative problem of how the frequencies of these vibrations, which can be obtained by experiment, are related to the masses of the atoms, the bond angles and bond lengths, and most particularly the force constants of the individual bonds and interbond angles. In this section we shall show how to set up the equations which express these relationships, making maximum use of symmetry to simplify the task at every stage. 

The nature of these six vibrations may be further specified in terms of the contribution made to each of them by the various internal coordinates. We first note that Ag and Bu vibrations must involve only motions within the molecular plane, since the characters of the representations Ag and Bu with respect to ah are positive. The Au vibration will, however, involve out-of-plane deformation, since the character of Au with respect to oh is negative. Thus we may describe the normal mode of Au symmetry as the out-of-plane deformation. In order to treat the remaining five in-plane vibrations we need a set of five internal coordinates so chosen that changes in them may occur entirely within the molecular plane. A suitable set, related to the bonding in... 

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