Totally symmetric species

It was explained in Section 4.3.2 that the direct product of two identical degenerate symmetry species contains a symmetric part and an antisymmetric part. The antisymmetric part is an A (or 2") species and, where possible, not the totally symmetric species. Therefore, in the product in Equation (7.79), Ig is the antisymmetric and Ig + Ag the symmetric part. 

T( p") in Equation (7.126) and the product x T( p") is totally symmetric (or contains the totally symmetric species). There may be several Au, = 0 sequences in various vibrations i but only those vibrations with levels of a sufficiently low wavenumber to be appreciably populated (see Equation 7.83) will form sequences. In a planar molecule the lowest wavenumber vibrations are usually out-of-plane vibrations. 

Once the state symmetries have been established it only remains to be shown that the direct product of the species of the ground state symmetry, the coordinate translational symmetries and the excited state symmetry belong to the totally symmetric species A. Let us take the n-wr transition in formaldehyde. The ground state total wave function has the symmetry Ax. The coordinate vectors x, y and z transform as By, Bf and Ax respectively (refer Character Table for C2k Section 2.9, Table 2.2), The excited state transforms as symmetry species A2. The direct products are ... 

As an example, we take H20. In Section 6.3 we tabulated the symmetry behavior of the normal modes. The totally symmetric species (to which vx and v2 belong) is Ax and the symmetry species to which j>3 belongs is designated B2 see (1.283). From the preceding discussion, we tabulate the symmetry behavior of a few vibrational states ... 

Examining the six quadratic functions displayed in Fig. 6.3.3, it is clear that x2 + y2, x2 - y2, and z2 are symmetric with respect to all four operations of the C2v point group. Hence they belong to the totally symmetric species Ai, as do x2 and y2 since they are merely linear combinations of x2 + y2 and x2 -y2. Similarly, the xy function is symmetric with respect to E and C2 and antisymmetric with respect to the reflection operations, and hence it has A2 symmetry. The symmetries of the xz and yz functions can be determined easily in an analogous manner. 

Vibrations of totally symmetric species (defined by the first row of the character tables) emit Raman lines which are polarized, the depolarization ratio pk can assume, according to Eqs. 2.4-11. .. 13 values of 0 < p < 6/7. All other Raman-active vibrations are emitting lines which are depolarized, they have a depolarization ratio of 6/7. The value 6/7 is appropriate for an arrangement, where the Raman radiation s investigated without an analyzer. If an analyzer is used 3/4 has to be taken instead. Cubic and icosahedral point groups are a special case the depolarization ratio for totally symmetric vibrations is 0. 

In order to recover the correct symmetry in the integrand for the degenerate case, we apply the projection operator of the totally symmetric species for the integrand. The projection operator is defined by... 

Overtone and combination bands belong to symmetry species determined by the species of their fundamentals. We can determine this symmetry by multiplying the characters for the fundamentals. Thus, overtones of all species have the character of the totally symmetric species, A for D2 symmetry and Ag for 21, symmetry (see Table VII). For combinations, this rule implies that, for >2 symmetry, a Bi mode combining with a Bs mode produces a combination of B2 symmetry, etc., while for C2h, Bu combining with Bg results in a band of Au symmetry, etc. (see Table VII). 

To see whether the totally symmetric species A g is present, we form the sum over classes of the number of operations times the character of the integrand... 

P12.22 Can the Eu excited state be reached by a dipole transition from the A g ground state Only if the representation of the product includes the totally symmetric species A[g. The z component of... 

The symbol <... > represents integration over the electron coordinates, covering all space. We can now use a group theory rule to decide whether the integrals in Eq. (3) are exactly zero or not. The rule is that the direct product of three functions must contain the totally symmetric species, or the integral over all space is zero. 

It is an empirical fact that for most molecules in their electronic ground states the electronic wave function belongs to the (qondegenerate) totally symmetric species. Also the electronic spins are usually all paired in the ground state, and the ground state is a singlet. For water the ground electronic state is Au for benzene it is... 

The carbon atom is at the center of the molecule, and the carbon Ij and Is AOs are each sent into themselves by every symmetry operation. These AOs transform according to the totally symmetric species A]. The carbon 2p 2py, and 2p AOs are given by x, y, or z times a radial function. Their symmetry behavior is the same as that of the functions x, y, and z, respectively. From the formulas for rotation of coordinates [Eq. (15.52)], we see that any proper rotation sends each of the functions x, y, and z into some linear combination of x, y, and z. Any improper rotation is the product of some proper rotation and an inversion (Problem 12.15) the inversion simply converts each coordinate to its negative. Hence the three carbon 2p orbitals are sent into linear combinations of one another by each symmetry operation. They must therefore transform according to one of the triply degenerate symmetry species. Further investigation (which is omitted) shows the symmetry species of the 2p AOs to be T2. 

Since each methane symmetry operator permutes the hydrogen Is orbitals among themselves, (15.42) is sent into itself by each symmetry operation and belongs to the totally symmetric species Aj. We need three more symmetry functions. The construction of these is not obvious without the use of group theory, and we shall simply write down the results. The remaining three orthogonal (unnormalized) symmetry-adapted basis functions can be taken as... 

The vibrational modes of the Raman lines used in this experiment belong to the totally symmetric species, so that the rotational diffusion constant obtained from their Raman linewidths is for the rotational motion (tumbling motion) of the solute molecule around the x axis perpendicular to the molecular z axis. Figs. 1-3 show the rotational diffusion constants obtained for benzene, cyclohexane and chloroform-d in aqueous solution and reference solution plotted against the reciprocal viscosity of the solvent. 

More generally stated, the rule for the existence (nonvanishing) of the integrals j, ix " dr is that the triple direct product of the species of M, and must contain the totally symmetric species. Thus if is the species of l/v, that of product ... 

---