1. Lecture 12 APPLICATIONS OF GROUP THEORY 1) Chirality
A species is called chiral if its mirror image (enantiomer) is NOT equivalent to the original.
Symmetry criterion of chirality: A species is chiral if there is no improper rotational axes among its symmetry elements (S1 = s, S2 = i, Sn (n > 2)). The following molecules are NOT chiral:
Note that a chiral species may be Cn-symmetric:

2. 2) Molecular vibrations and symmetry
Each n-atomic species has 3n degrees of freedom. Three of them correspond to translation of the species as a whole and three others (two for linear molecules) correspond to rotation of the species as a whole. The remaining 3n-6 (3n-5 for linear species) degrees of freedom correspond to molecular vibrations.
Molecular vibrations appear as normal vibrations (normal modes) and have contributions from all atoms constituting a species. Each normal vibration is characterized by its symmetry, intensity and frequency. The symmetry of normal vibrations is a function of the symmetry of a species.
Group theory can be used to reveal the number and the symmetry of the normal modes.
For this purpose the entire set of 3n Cartesian displacement vectors should be used as a basis for a reducible representation of a molecular symmetry group. This representation can be decomposed into the set of irreducible representations to which the complete set of normal modes belong.

3. 3) Reducible representation Gr for molecular vibrations of H2O
Consider H2O molecule as an example (C2v point group)
Symmetry operations available are E, C2, sv, s’v.
The dimension of the representation Gr is 3n = 9.
Matrices representing the symmetry operations and their characters are given below:
You can note that each of the three atoms contributes 3 to the character of the operation E
What concerns the C2 rotation, only oxygen atom which is unshifted by the C2 contributes -1 to the character of the C2 operation

4. 4) Reducible representation for vibrations of H2O: characters of s’s
Only oxygen atom which is unshifted by the sv contributes 1 to the character of the sv operation.
All three atoms remain unshifted by the s’v. All of them contribute 1 to the character of the s’v operation.

5. 5) Algorithm of vibrational modes search
Count the number of atoms in a species (n) and calculate the number of degrees of freedom (3n). For H2O, 3n=9.
Assign the molecule to the appropriate symmetry point group for (C2v) and find its character table.
Figure out how many atoms remain unshifted when each of the symmetry operations is applied.
Find the reducible representation:
Decompose Gr: Gr = (9)E + (-1)C2 + (1)sv + (3)s’v = 3A1 + 1A2 + 2B1 + 3B2
Remove components corresponding to translation
and rotation of the species. Translations: A1(z) B1(x)+ B2(y)
Rotations: A2(Rz)+ B1(Ry)+B2(Rx)
Remaining are the vibrational modes: 2 A B2
Check: their number should be 3n-6
(for non-linear molecules) or 3n-5 (for linear).
Symmetry operations E C2 sv
s’v
# unshifted atoms 3 1 Gr E 3 C3 C2 -1 s 1 S3 -2
C2v E C2
sxz
syz A1 1 z A2 -1 Rz B1
x, Ry B2
y, Rx