Totally symmetrical displacement

Moreover, in transforming dj to symmetry displacements the actual values of djiitf) are important only for the totally symmetric displacements since they cancel out for all the others. 

The reaction coordinate along the allowed pathway must therefore include, in addition to totally symmetric displacements, such as decreasing the C1C4 distance and shortening the C2C3 bond, at least one 02 symmetry coordinate Conrotation of the methylene groups evidently fulfills this requirement. 

As in Fig. 5.5, the two totally symmetric CH-bonding orbitals correlate across the diagram, but the other two, both of them bi on one side and 62 on the other, can only be induced to correspond by displacement along an U2 coordinate, viz, a conrotation. The five CC-bonding orbitals correlate across the diagram without requiring any non-totally symmetric displacement at all. Evidently, the reaction has to be characterized formally as allowed in the subgroup C2, the kernel of fl2- This conclusion not only differs from that of the Rules but stands in apparent contradiction to experiment Bicyclobutene is thermodynamically much less stable than butadiene if its conversion to the latter can take place with conservation of orbital symmetry, why is it so difficult ... 

In general, each nomial mode in a molecule has its own frequency, which is detemiined in the nonnal mode analysis [24]- Flowever, this is subject to the constraints imposed by molecular synmietry [18, 25, 26]. For example, in the methane molecule CFI, four of the nonnal modes can essentially be designated as nonnal stretch modes, i.e. consisting primarily of collective motions built from the four C-FI bond displacements. The molecule has tetrahedral synmietry, and this constrains the stretch nonnal mode frequencies. One mode is the totally symmetric stretch, with its own characteristic frequency. The other tliree stretch nonnal modes are all constrained by synmietry to have the same frequency, and are refened to as being triply-degenerate. 

If the initial ground-state wavefunction (/(q is nondegenerate, the first-order term (i. e., the second term) in Eq. (1) is nonzero only for the totally-symmetrical nuclear displacements (note that g, and (dH/dQi) have the same symmetry). Information about the equilibrium nuclear configuration after the symmetrical first-order deformation will be given by equating the first-order term to zero. 

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. 

Consider the symmetry behavior of the normal modes of H20 and COz. For water, the symmetry operations are E, C2 b), ov(ab), and av(bc), where the a and b principal axes lie in the molecular plane the bond angle in water is relatively obtuse, so that the C2 axis is the axis of intermediate moment of inertia. Each of these operations converts the displacement vectors of p, in Fig. 6.1 to an indistinguishable configuration the same is true for v2. Normal modes that are symmetric with respect to all the molecular symmetry operations are called totally symmetric. For H20, the modes and v2 are totally symmetric, but v3 is not. Figure 6.3 shows that v3 is antisymmetric with respect to C2(b). We can tabulate the behavior of the H20 normal modes with respect to the molecular symmetry operations ... 

Fio. 5. Potential curves for a non-linear molecule in a degenerate electronic state. The potential minima correspond to small displacements on a non-totally symmetrical normal coordinate Q, not to the most symmetrical configuration (Q = 0). 

Fig. 2. The bending (Tbc) and stretching (tsc) vibrations of a tetrahedral molecule or complex, involved in the PJT-type Td — C2v vibronic process due to the presence of a lone pair - Tbc and tsc are the displacements according to combinations of the symmetry breaking components of the t2 modes with the angular s(6) component and the totally symmetric ax stretching vibration respectively.

If all the accepting modes (usually the totally symmetric modes) are displaced oscillators [24], then... 

An important special case arises when a valence electron of the central atom is on a degenerate energy level. Breaking the symmetry by some displacement, not of the totally symmetrical type, may lift the degeneracy and lower the total energy. Although such displacement lowers the total en-... 

When the molecule is raised to the electronic excited state, the normal mode Q preserves the same decomposition with respect to the nuclear displacements, except for Jahn-Teller effects, which we exclude in this work. In these conditions, still in the harmonic approximation, two parameters, the equilibrium point and the potential curvature, will change in the excited state (in the particular case of non-totally-symmetric vibrations, the equilibrium point does not change). In the excited state e, the nuclear potential becomes... 

Figure 5. Displacements associated with the Q coordinates which transform as irreducible representations of the group S3 (which is isomorphic with the symmetry point group C3r, itself a subgroup of D J. Q, represents the totally symmetric breathing mode of the equilateral triangle, and Q2 represents the antisymmetric distortion coordinate, which is degenerate with the Q3 bending mode for Dih geometries.

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