C566 Master Lecture Notes
Weeks 5 - 8
Vibrational wavefunction now can be written out (to a reasonable approximation, at low
vibrational energies) in terms of the separated coordinates,
vib
(Q
i
) =
1
(Q
1
)
2
(Q
2
)
…
3N-6
(Q
3N-6
)
E
vib
= E
1
+ E
2
+ …+ E
3N-6
= h
1
(
1
+ 1/2) + h
2
(
2
+ 1/2)+ … + h
3N
6
(
3N
6
+ 1/2)
Need to now use this, and what we know about molecular symmetry, to devise our
spectroscopic selection rules.
Can now characterize the species of various molecular vibrational coordinates.
C
2h
I
C
2
i
h
A
g
1
1
1
1
R
z
xx
,
yy
,
zz
,
xy
B
g
1
-1
1
-1
R
x
, R
y
xz
,
yz
A
u
1
1
-1
-1
T
z
B
u
1
-1
-1
1
T
x
, T
y
The C
2
axis corresponds to the z axis in our coordinate system, and for glyoxal (below),
the C-C axis is the x-axis.
For fun, consider rotation about the x-axis
(R
x
) = B
g
C
C
O
O
H
H
R
x
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(
n
) = A
u
(
m
) = B
u
Several Observations:
(a)
Symmetry species (A
g
, A
u
, B
g
, B
u
) are vectors in symmetry operation space
e.g.,
A
u
= (1, 1,
1,
1)
B
g
= (1,
1, 1,
1)
(b)
All are mutually orthogonal;
A
g
B
g
= 1
1 + 1
1 + 1
1 + 1
1 = 0
A
g
A
g
= 1
1+1
1+1
1+1
1 = 4 = the order of the point group
(c)
Direct product:
A
g
B
g
= (1,1,1,1)
(1,-1,1,-1)=(1,-1,1,-1) = B
g
B
g
B
u
= (1,-1,1,-1)
(1,-1,-1,1) = (1,1, -1,-1) = A
u
B
u
B
u
= A
g
Direct products will be used to determine the species of combinations of properties, such
as combinations of vibrational modes, electronic-vibration combinations, as well as the
overall electronic state of a molecule built up from the occupation of MO’s.
C
C
O
O
H
H
C
C
O
O
H
H
_
_