4.3.

3 Character Table

▪ Additional Features of Character Table

1. C3, C23 in the same class → clockwise and anticlockwise direction

2. C2 perpendicular to the principal axis → C2’: pass through several atoms

C2”: pass b/w the atoms

3. horizontal plane: σh

vertical plane: σv, σd

4. in the right side of the column in the character table,

x, y, z px, py, px
Rx, Ry, Rz in the character table
xy, xz, yz dxy, dxz, dyz
totally symmetric s

in C3v (x, y) have the same symm. properties as the E reducible rep.

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 4.3.3 Character Table

▪ Additional Features of Character Table

5. matching the symm. operation w/ list in the top row

confirm any point group

6. labeling of irreducible representation (symm. → 1, antisymm. → -1)

a) letter: dimension of the irreducible representation

Dimension Symm. label

1 A, B
2 E
3 T
b) subscript: 1 → symmetric to a C2 rotation perpendicular to the principal axis

2 → antisymm. to the C2

* if no perpendicular C2,,,

1 – symm. to a vertical plane

2 – antisymm. to a vertical plane

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 4.3.3 Character Table

▪ Additional Features of Character Table

6. c) subscript: g (gerade) → symm. to i

u (ungerade) → antisymm. to i

d) single prime (‘) → symm to σh

double prime (“) → antisymm. to σh

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 4.4 Examples and Application of Symmetry

4.4.1 Chirality
▪ chiral/dissymmetric: molecules that are not superimposable on their mirror image

e.g.) CBrClFI

propeller (C3 axis)

Fig.4.18

condition for chirality: no symm. operation other than E

only proper rotation

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 4.4 Examples and Application of Symmetry

4.4.1 Chirality
▪ optical activity: the ability of chiral molecules to rotate plane-polarzed light

1) clockwise rotation: dextrorotary

2) anticlockwise rotation: levorotary

Fig.4.19

e.g.) [Ru(NH2CH2CH2NH2)3]2+ → D3

Fig.4.20
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 4.4.2 Molecular Vibrations

: symmetry can help to determine the mode of vibration of molecule

▪ water (C2v symm.): x, y, z coordinates should be used for each atom.

x – plane of molecule

y – perpendicular to plane Total 9 transformation

z – C2 axes

Fig.4.21

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 4.4.2 Molecular Vibrations

- to assign translation, rotation, vibration motion → use transformation matrices

for H2O w/ 9 transformation → e.g.) 9 x 9, C2 matrix

[ new axes] = [transformation matrix (9 x 9)] [initial axes]

if position changes (during the operation) → 0

if unchanged changes → 1

if vector direction changes → -1

- use the character of the representation matrice instead of individual matrix

sum of along the diagonal

no-zero entry appears along the diagonal of the matrix only for an atom
that does not change position.

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 4.4.2 Molecular Vibrations

- reducible representation Γ

E: 9 → no change

C2: 2H → 0: change position

O → (-1) + (-1) + 1 = -1: x, y – reversed

z – remains same

σv(xz) (plane of molecule): 3 – 3 + 3 = 3 : x, z – unchanged

y – change the direction

σv’(yz): 2H → 0: changed position

O → x – change direction
4–1+1=1
y, z – unchanged

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 4.4.2 Molecular Vibrations

reducible representation

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 4.4.2 Molecular Vibrations

▪ Reducing Representation to Irreducible Representation

: separate the reducible representation into its component irreducible representations

- reduction formula

# irreducible character of character of

representation of = (1/order)Σ # operations x reducible x irreducible
a given type in the class representation representation

- For H2O,,

nA1 = 1/4[(9)(1) + (-1)(1) + (3)(1) + (1)(1)] = 3

nA2 = 1/4[(9)(1) + (-1)(1) + (3)(-1) + (1)(-1)] = 1

Γ = 3A1 + A2 + 3B1 + +2B2
nB1 = 1/4[(9)(1) + (-1)(-1) + (3)(1) + (1)(-1)] = 3

nB2 = 1/4[(9)(1) + (-1)(-1) + (3)(-1) + (1)(1)] = 2

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 4.4.2 Molecular Vibrations

- according to the character table,,,

translation along x, y, z: A1 + B1 + B2

rotation (Rx, Ry, Rz): A2 + B1 + B2

vibration mode: 2A1 + B1

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 4.4.2 Molecular Vibrations

Example) Using the x, y, z coordinates for each atom in XeF4, determine the reducible
representation for all molecular motions; reduce this representation to its irreducible
components; and classify these representations into translational, rotational, and vibrational
mode.

Fig.4.26

sol) only the coordinates on atoms that do not move when symmetry operations are applied can

give rise to nonzero elements along the diagonals of transformation matrices.

if unchanged → 1

if reverse the direction → -1

if move to another coordinate → 0

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 4.4.2 Molecular Vibrations

There are 15 possible motions to be considered.

If reduced,,
Γ = A1g + A2g + B1g + B2g + Eg + 2A2u + B2u + 3Eu

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 4.4.2 Molecular Vibrations

▪ translational motion: motion through space w/ x, y, z components

z: A2u

x, y: Eu

Fig.4.23

▪ Rotationl motion: rotation about the x, y, z axis (Rx, Ry, Rz)

Rz: A2g

(Rx, Ry): Eg

Fig.4.24

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 4.4.2 Molecular Vibrations

▪ vibrational motion: 15 – 3 – 3 = 9
: change in bond length & angles

motion both within and out of the molecular place

Fig.4.23

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 4.4.2 Molecular Vibrations

Example) Reduce the following representation to their irreducible representation in the point

group indicated (refer to the character table in Appendix C).

C2h E C2 i σh
Γ 4 0 2 2

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 4.4.2 Molecular Vibrations

▪ Infrared Spectra

- infrared active: if there is any change in the dipole moment of the molecule

using group theory: infrared active if it corresponds to an irreducible

representation that has the same symmetry (or transformation) as the

Cartesian coordinates x, y, z

∵ vibrational motion → change the center of charge → change in dipole moment

▪ Selected vibrational modes: for particular type of vibration mode

e.g.) C-O stretching bands cis- and trans-dicarbonyl square planar complex

Fig.4.26

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 4.4.2 Molecular Vibrations

1) cis-ML2(CO2) point group C2v.

Fig.4.27

- either an increase or decrease in the C-O distance

generate the reducible representation usign C-O bond as shown in Fig.4.27

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 4.4.2 Molecular Vibrations

1) cis-ML2(CO2) point group C2v.

reduction
Γ = A1 + B1

A1 → z
→ Both A1, B1 transforms as the Cartesian coordinates z, x
B1 → x

∴ There are two IR active vibrational modes!!

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 4.4.2 Molecular Vibrations

2) trans-ML2(CO2) point group D2h.

reduction
Γ = Ag + B3u

Ag → No → IR-inactive
B3u → x → IR-active

∴ There are one IR active vibrational modes!!

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 4.4.2 Molecular Vibrations

∴ Therefore, to distinguish cis- & trans-ML2(CO)2 by IR.

one C-O stretching band → trans

two C-O stretching band → cis

Example) on p.118

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 4.4.2 Molecular Vibrations

▪ Raman Spectra

- laser → excite molecule to higher electronic states (“virtual” states)

→ decay of excited states to various vibrational states → provide info. about vibrational E

- Raman active if there is a change in polarizability.

xy, yz, xz, x2, y2, z2 functions or linear combination of any of these.

e.g.) XeO4 (Td) → two Raman bands at 778 & 878 cm-1. Confirm these bands!!

reduction
Γ = A1 + T2

∴ Both the A1 & T2 → two Raman active bands!!

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