Chapter 16 – Symmetry
Symmetry elements
E
identity
Cn
n-fold rotation
σ
mirror plane
i
inversion
Sn
n-fold rotation-reflection
H 2O
E ,σ v , σ v' , C2
NH 3
E , σ v (1), σ v (2), σ v (3)
C3 , C32 = C3−1
Principal rotation
axis – axis of the
highest symmetry
⇒ C2 v
σv ⇒ mirror plane
⇒ C3v
σh ⇒ mirror plane
contains princ.
rotational axis
┴ to the princ. axis
CHFClBr
E
Symmetry operations
⇒ C1
group
H
C=C
Br
H
F
H
C=C
H
E,σ
H
E , C2 , C2' , C2''
H
σ , σ ' , σ '' , i
⇒ Cs
⇒ D2 h
group
Symmetry operators can be represented by matrices
Example:
H
x
O
H y
z
, x ⊥ to the plane
of the molecule
⎛ 1 0 0⎞
⎜
⎟
Eˆ : ⎜ 0 1 0 ⎟
⎜ 0 0 1⎟
⎝
⎠
⎛ −1 0 0 ⎞
⎜
⎟
Cˆ 2 : ⎜ 0 − 1 0 ⎟
⎜ 0 0 1⎟
⎝
⎠
⎛ 1 0
σˆ v : ⎜⎜ 0 − 1
⎜ 0 0
⎝
⎛ −1 0
σˆ v' : ⎜⎜ 0 1
⎜ 0 0
⎝
0⎞
⎟
0⎟
1 ⎟⎠
0⎞
⎟
0⎟
1 ⎟⎠
form a representation
for the C2v group
Cˆ 2 iCˆ 2 = Eˆ
Cˆ 2 iσˆ v = σˆ v'
etc.
In this case, there are simpler representations
C2 v Eˆ Cˆ 2
σˆ 2
σˆ v'
A1
1
1
1
1
z
x2 , y2 , z 2
A2
1
1 −1
−1
Rz
xy
B1
1 −1
B2
1 −1 −1
1 −1
1
x, Rx xz
character
table
y, Rx yz
Ri = rotation
about I axis
irreducible representations
O pz → a1
A1 = totally symmetric
representation
O px → b1
O p y → b2
for C2V all irreducible representations are one-dimensional
⇒ no degeneracies
C3V is an example of a group with a degenerate representation
z
N
x
rotate 120°, “mixes” x and y
E is a two-fold degenerate representation
y
The different representations are orthogonal
A1 xA2 = 1i1 + 2i1i1 + 3i1i(−1) = 0
A2 xE = 1i2 + 2(1)(−1) + 0 = 0
Symmetry labels of MOs of water
Electronic structure
C2v group
∫ψ 1 Hˆ ψ 2 dτ = 0
if ψ1, ψ2 not the
a22 → A1
same symmetry
b1b2 → A2
∫ψ 1 Aˆψ 2 dτ = 0
if ψ 1 Âψ 2 does not
a2b2 → A1
b1a2 → B2
Selection rules
contain totally symmetric
representation
1
1
a1 → a1
1
2
a1 → b1
∫ a za dτ ≠ 0
∫ a zb dτ = 0
∫ a xb dτ ≠ 0
1
1
etc.
allowed
transitions
C3v
e2 → 4 1 0
= c1 (1 1 1) + c2 (1 1 − 1) + c3 (2 − 1 0)
c1 = c2 = c3 = 1 ⇒ e 2 → e, a1 , a2
two electrons in an e orbital → E, A1, A2 states
Symmetries of vibrational normal modes
1 ⎛ ∂ 2V ⎞ 2
V = ∑ ⎜ 2 ⎟Qi ,
2 i ⎝ 2Qi ⎠
Qi are normal
coordinates
Ψ = ψ 1 ( Q1 )ψ 2 ( Q2 )…ψ N ( QN )
1⎞
⎛
E = ∑ ⎜ n j + ⎟hv j
2⎠
j ⎝
Vibrations of the water molecule
a1
a1
b2
Procedure (p. 412) for determining how many normal modes
there are of each symmetry
H 2O = A1 , A1 , B2
ψ 0 ( Q j ) Q jψ m ( Q j )
must belong to the same representation as
x, y, or z to be IR active
C2V: A1 vibrations are IR active
B2 vibration is also IR active
Td: example CH4: A1, E, 2T2 vibrations
IR
forbidden
IR
active
x2, y2, z2, xy, xz, yz
belong to
representations that
are Raman active
An example of a more complicated group – D2h (example: ethylene)