Applications of SymmetryMatricesWhy Matrices? The matrix representations of the point groups operations will generate a character table. We can use this table to predict properties.

Definitions and RulesMatrix = ordered array of numbers

Multiplying MatricesThe number of columns of matrix #1 must = number of rows of matrix #2Fill in answer matrix from left to right and top to bottomThe first answer number comes from the sum of [(row 1 elements of matrix #1) X (column 1 elements of matrix #2)]The answer matrix has same number of rows as matrix #1

The answer matrix has same number of columns as matrix #2

Relevant example:

Exercise 4-4

Representations of Point GroupsMatrix Representations of C2v Choose set of x,y,z axesz is usually the Cn axisxz plane is usually the plane of the moleculeExamine what happens after the molecule undergoes each symmetry operation in the point group (E, C2, 2s)

Transformation Matrix = matrix expressing the effect of a symmetry operation on the x,y,z axes

E Transformation Matrixx,y,z x,y,zWhat matrix times x,y,z doesnt change anything?

transformationmatrixE Transformation Matrix

C2 Transformation Matrixx,y,z -x, -y, zCorrect matrix is:

sv(xz) Transformation Matrixx,y,z x,-y,zCorrect matrix is:

sv(yz) Transformation Matrixx,y,z -x,y,zCorrect matrix is:

These 4 matrices are the Matrix Representation of the C2v point groupAll point group properties transfer to the matrices as wellExample: Esv(xz) = sv(xz)

B.Reducible and Irreducible RepresentationsCharacter = sum of diagonal from upper left to lower right (only defined for square matrices)

The set of characters = a reducible representation (G) or shorthand version of the matrix representation

For C2v Point Group:

EC2sv(xz)sv(yz)3-111

Reducible and Irreducible RepresentationsEach matrix in the C2v matrix representation can be block diagonalizedTo block diagonalize, make each nonzero element into a 1x1 matrix

When you do this, the x,y, and z axes can be treated independentlyPositions 1,1 always describe x-axisPositions 2,2 always describe y-axisPositions 3,3 always describe z-axisGenerate a partial character table from this treatment

EC2sv(xz)sv(yz)IrreducibleRepresentationsReducible Repr.

Axis usedEC2sv(xz)sv(yz)x1-11-1y1-1-11z1111G3-111

Character TablesThe C2v Character TableWe have found three of the irreducible representations of the character table through matrix mathOne more (A2) irreducible representation is derived from the first three due to the properties of character tables (below)

Rx, Ry, Rz stand for rotation about the x, y, z axes respectively

xs are p and d orbitals

4)Other symbols we need to knowR = any symmetry operation c = character (#)i,j = different representations (A1, B2, etc)h = order of the group (4 total operations in the C2v case)

C2vEC2sv(xz)sv(yz)A11111zx2, y2, z2A211-1-1RzxyB11-11-1x, RyxzB21-1-11y, Rxyz

The C3v Character Table for NH3

1)The threefold symmetry of NH3 makes for complex transformation matrices

Though more complex, the C3v Character Table can be generated similarly to that of the C2v group

Notes on Character TablesMultiple operations in the same class are listed togetherDifferent C2 axes are listed separately with primes ()Those through outer atoms are Those not through outer atoms are Symmetry of orbitals are listed except for s orbitals, which are always in the first listed A irreducible representationIrreducible Representation LabelsDegeneracy (dimension) is determined by the character of E operationA if E = 1 and c of Cn = 1B if E = 1 and c of Cn = -1E if E = 2 (doubly degenerate)T if E = 3 (triply degenerate)

Subscripts1 if symmetric to perpendicular C2 axis (or sv)2 if antisymmetric to perpendicular C2 axis (or sv)g if symmetric to iu if antisymmetric to iPrimes if symmetric to sh if antisymmetric to sh

Applications of SymmetryChiral MoleculesMolecules not superimposable with their mirror images are called chiral or dissymetricThey may still have some symmetry operations: E, Cn Chiral molecules cannot have i, s, or Sn symmetry operations

Molecular VibrationsTo use symmetry, we must assign axes

to each atom of the moleculeThe z-axis is usually the Cn axisThe x-axis is in the molecular planeThe y-axis is perpendicular to the molecular plane

Degrees of Freedom = possible atomic movements in the molecule3N degrees of freedom for a molecule of N atomsNonlinear molecules3 translations (along x, y, z)3 rotations (around x, y, z)3N 6 vibrations

c.Linear moleculesOnly 2 rotations change the molecule3N 5 vibrations

3.We will use group theory to determine the symmetry of all nine motions and then assign them to translation, rotation, and vibration

Look at the C2v character table

Add up how many vectors stay the same after an operationIf the atom moves, none of its vectors stay the sameIf the atom stays and the vector is unchanged = +1If the atom stays and the vector is reversed = -1

Reduce the reducible representation to its irreducible components

C2vEC2sv(xz)sv(yz)A11111zx2, y2, z2A211-1-1RzxyB11-11-1x, RyxzB21-1-11y, Rxyz

C2vEC2sv(xz)sv(yz)G9-131

nA1 = [(1x9x1)+(1x-1x1)+(1x3x1)+(1x1x1)] = 3 A1 nA2 = [(1x9x1)+(1x-1x1)+(1x3x-1)+(1x1x-1)] = 1 A2 nB1 = [(1x9x1)+(1x-1x-1)+(1x3x1)+(1x-1x1)] = 3 B1 nB2 = [(1x9x1)+(1x-1x-1)+(1x3x-1)+(1x1x1)] = 2 B2 All motions of water match 3A1 + A2 + 3B1 + 2B2

Use the character table to remove translations

x, y, z = A1 + B1 + B2

Use the character table to remove rotationsRx, Ry, Rz = A2 + B1 + B2

The motions remaining are the vibrations = 2A1 + B1 A1 = totally symmetricB1 = antisymmetric to C2 and to reflection in yz plane

Symmetry and IRIR only sees a vibration if the vibration changes the molecules dipoleMotion along the x, y, z axes creates a changed dipoleInfrared Active vibrations match up with x, y, z on character tableInfrared Inactive vibrations dontFor water, all three vibrations are infrared active

Examples and Exercises 4-7 and 4-8

Molecular Vibrations of ML2(CO)2 complexesThe symmetry of cis- ML2(CO)2 complexes is C2v The C=O stretch has only one possible direction of motionInstead of using xyz vectors at each atom, we can use a single vector

c)Reducible representation from the 2 vectors

d)2 possible vibrations from reduction formula: A1 + B1 (see both)

C2vEC2sv(xz)sv(yz)G2020

The symmetry of trans- ML2(CO)2 complexes is D2h Symmetry operations on the vectors generate a reducible representationReduction formula give 2 irreducible representations

Only the B3u representation is IR Active

We can tell cis from trans by the number of C=O IR bands