Lecture 8
Symmetry: Character Tables
All symmetry operations associated with a point group are given in a character table having the point group on top left and the symmetry operations in the first row.
H2O,
SO2
The z axis coincides with the principal symmetry axis (C2 above) and the molecule (H2O, SO2) lies in the yz plane (σv’ above).
NH3,
CH3Cl
CHEM 331 – 2014
Dr. Roland Roesler
8-1
Symmetry: Character Tables
BF3,
CO32ˉ
2C3 stands for C31 and C32. C33 = E.
The first column provides a list of symmetry labels corresponding to irreducible representations:
A and B indicate non-degenerate
A indicates symmetric with respect to rotation about the principal axis
B indicates anti-symmetric with respect to rotation about the principal axis CHEM 331 – 2014
Dr. Roland Roesler
8-2
Symmetry: Character Tables
BF3,
CO32ˉ
E indicates doubly-degenerate and T indicates triply-degenerate
’ indicates symmetric with respect to σh
” indicates anti-symmetric with respect to σh g and u indicate symmetric and anti-symmetric with respect to inversion
Subscript 1 and 2 indicate symmetric and anti-symmetric with respect to rotation about the C2 secondary axis, or σv if no C2 present
The numbers in the body of the character table are the characters.
CHEM 331 – 2014
Dr. Roland Roesler
8-3
Symmetry: Character Tables
The reduction formula: ai: 1/g × Σ(nR × χ(R) × χ(IR)) ai: number of times an irreducible representation contributes to the reducible representation g: order of the group, i.e. the total number of symmetry operations in the group nR: the number of operations in a particular class of operation χ(R): the character in the reducible representation corresponding to the class of operation χ(IR): the corresponding character in the irreducible representation ai
CHEM 331 – 2014
Dr. Roland Roesler
8-4
MO Theory: Polyatomic Molecules
H2 O
The LCAO (Linear Combination of Atomic Orbitals) for the 2 H atoms under
C2v symmetry is assessed (two 1s orbitals).
Γ2H(1s)
2
0
0
2
This
All symmetry operations associated with a point group are given in a character table having the point group on top left and the symmetry operations in the first row.
H2O,
SO2
The z axis coincides with the principal symmetry axis (C2 above) and the molecule (H2O, SO2) lies in the yz plane (σv’ above).
NH3,
CH3Cl
CHEM 331 – 2014
Dr. Roland Roesler
8-1
Symmetry: Character Tables
BF3,
CO32ˉ
2C3 stands for C31 and C32. C33 = E.
The first column provides a list of symmetry labels corresponding to irreducible representations:
A and B indicate non-degenerate
A indicates symmetric with respect to rotation about the principal axis
B indicates anti-symmetric with respect to rotation about the principal axis CHEM 331 – 2014
Dr. Roland Roesler
8-2
Symmetry: Character Tables
BF3,
CO32ˉ
E indicates doubly-degenerate and T indicates triply-degenerate
’ indicates symmetric with respect to σh
” indicates anti-symmetric with respect to σh g and u indicate symmetric and anti-symmetric with respect to inversion
Subscript 1 and 2 indicate symmetric and anti-symmetric with respect to rotation about the C2 secondary axis, or σv if no C2 present
The numbers in the body of the character table are the characters.
CHEM 331 – 2014
Dr. Roland Roesler
8-3
Symmetry: Character Tables
The reduction formula: ai: 1/g × Σ(nR × χ(R) × χ(IR)) ai: number of times an irreducible representation contributes to the reducible representation g: order of the group, i.e. the total number of symmetry operations in the group nR: the number of operations in a particular class of operation χ(R): the character in the reducible representation corresponding to the class of operation χ(IR): the corresponding character in the irreducible representation ai
CHEM 331 – 2014
Dr. Roland Roesler
8-4
MO Theory: Polyatomic Molecules
H2 O
The LCAO (Linear Combination of Atomic Orbitals) for the 2 H atoms under
C2v symmetry is assessed (two 1s orbitals).
Γ2H(1s)
2
0
0
2
This