Chemical Applications of Group Theory

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Chemical Applicationsof Group Theory

• Edward A. Mottel
• Department of Chemistry
• Rose-Hulman Institute of Technology

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Group Theory

• A group is a mathematically defined collection of
  (symmetry) operations that have a specific set of
  mathematical properties.
• Identity element
• Inverse operation for each member of the group
• Closure
• Associative

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Chemical Applications of Group Theory

• chirality
• vibrational spectroscopy
• ir and Raman
• allowed stretching bending frequencies

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Determination of the Reducible Representation of
a Molecule
Symmetry elements present
C2
E
sv
sv
9
-1
1
3
5
Reducible Representation
Symmetry operations that are applied to the each
atom of the molecule that does not move produce
the reducible representation
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Character Table
A reducible representation can be decomposed into
a set of irreducible representations using a
character table
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Decomposing a Reducible Representation to
Irreducible Representations
ai (1/h) S (CR CiR CR)
aA1 (1/4) (9 1 1) (-1 1 1) ( 1 1
1) (3 1 1) 3
aA2 (1/4) (9 1 1) (-1 1 1) (1 -1
1) (3 -1 1) 1
aB1 (1/4) (9 1 1) (-1 -1 1) (1 1
1) (3 -1 1) 2
aB2 (1/4) (9 1 1) (-1 -1 1) (1 -1
1) (3 1 1) 3
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Decomposing a Reducible Representation to
Irreducible Representations
ai (1/h) S (CR CiR CR)
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Girred 3 A1 A2 2 B1 3 B2
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Degrees of Freedom
Every molecule has 3n energy modes
degrees of freedom
atoms
translation
rotation
vibration
1
3
3
0
0
2
6
3
2
1
3 (linear)
9
3
2
4
3 (nonlinear)
9
3
3
3
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Irreducible Representations 3 A1 A2 2 B1 3
B2
Represents the energy modes of the molecule
every molecule has 3n energy modes
3 translational modes (x, y, z)
non-linear molecule ?
3 rotational modes (Rx, Ry, Rz)
vibrational modes
polyatomic ?
How many vibrational modes does SO2 have?
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Energy Modes of C2v
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Energy Modes of SO2
Girred 3 A1 A2 2 B1 3 B2
Assign the 9 energy modes of SO2
Are all the vibrational modes infrared active?
Are all the vibrational modes Raman active?
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Vibrational Modes of SO2
symmetric stretch A1
symmetric bend A1
asymmetric stretch B2
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