# Chapter 4 Chapter 4 Symmetry and its Applications Symmetry = do something to a molecule

and have it look the same Reducible vs. Irreducible Representations of Groups:

Example: effect of operations on x, y, z axes alone Examine the effect of the operation on the x, y, and z axes.

If unchanged, character = 1; if reversed, character = -1. Representations of Groups:

Example: effect of operations on x, y, z axes together Gxyz

Gx Gy Gz

Irreducibile represenations

Mullikan symbols Character Tables

Definitions by Example:

Order = # symmetry operations = 6 (sum of numbers on top) Classes = grouping of similar operations = 3 (number of groups on top) Dimensions = # under E operation

Order of group = (characters under E)2 Symmetry Applications: Vibrations

1. Examine x, y, z changes for each atom. 2. An atom that moves gets zeros for each. 3. Those not moving get 1 or -1 for each axis.

4. Add them up to get reducible representation for atomic motions. 5. Break reducible representation up to get list of irreducible representation. # irrrep= 1/order (#operations in class)(character of redrep)(character of irrerep)

6. All motions = degrees of freedom = 3n Translations = 3 and go by x, y, z functions. Rotations = 3 and go by Rx. Ry, and Rz functions.

7. Vibrations are whats left over. IR active translate as x, y, z functions. Raman active translate as xy, xz, yz, z2, etc. functions.

Vibrational Analysis for Water

Consider result of operation on each axis, accounting for change in sign, and then add

that column. C2: z stays the same = 1 x rotates by 180o and

sign changes, so = -1 G

all motions

Deconvoluting reducible representation to irreducible representations.

# A1 = [(1)(1)(9) + (1)(1)(-1) + (1)(1)(3) + (1)(1)(1) =

Deconvoluting reducible representation to irreducible representations.

G

Identify translational and rotational modes. Identify Vibrational Modes as IR or Raman Active.

= 3 A

+ 1 A

+ 3 B

+ 2 B

1 2 1

2 all motions

Vibrational Analysis for Ammonia

G all motions

Vibrational Analysis of Carbonyl Vibrations: count number of unmoved CO groups

Vibrational Analysis of Carbonyl Vibrations

G CO groups

Vibrational Analysis of Carbonyl Vibrations G

CO groups