|
C3v |
E |
2C3 |
3sigmav |
|
|
|
A1 |
1 |
1 |
1 |
Z |
x2+y2,z2 |
|
aA1 |
MA1(E) |
MA1(C3) |
MA1(sv) |
|
|
|
A2 |
1 |
1 |
-1 |
Rz |
|
|
aA2 |
MA2(E) |
MA2(C3) |
MA2(sv) |
|
|
|
E |
2 |
-1 |
0 |
x,y Rx,Ry |
x2-y2,xy xz,yz |
|
aE |
ME(E) |
ME(C3) |
ME(sv) |
|
|
chi(R) = nu(R)*f(R) (1)
where
chi(R) => reducible representation for that symmetry operation
nu(R) => the number of atoms which remain unshifted after each symmetry operation, R, is performed.
f(R) => a value for character contributions for the symmetry operation
Xi(R) = d(R)*chi(R) (2)
where
Xi(R) => the value for each symmetry element
d(R) => the number of operations of the class of symmetry operation from the character table
Mi(R) = chi(R)*Xi(R) (3)
where
Mi(R) => molecular specific element
ai = 1/h*sumRMi(R) (4)
ai = 1/h*sumRchi(R)*Xi(R) (5)
where
ai => multiplier of the irreducible representation
h => the inverse of the order of the symmetry elements
|
E |
C3 |
Sigmav |
|||
|
# unshifted atoms |
nu(E) |
nu(C3) |
nu(sv) |
nu(R) |
|
|
times the character contribution |
f(E) |
f(C3) |
f(sv) |
f(R) |
|
|
chii(E) |
chii(C3) |
chii(sv) |
Chii(R) |
||
|
times the number of operations of class |
1 |
2 |
3 |
d(R) |
|
|
Xi(E) |
XI(C3) |
Xi(sv) |
Xi(R) |
||
|
A1 |
A2 |
E |
||
|
aA1 |
aA2 |
aE |
Total spectral terms |
|
|
- |
1 |
1 |
2 |
less rotational and translational |
|
vA1 |
vA2 |
vE |
vibrational terms |
|
|
IR-Raman |
IR-Raman |
|||
|
vA1 + vE |
IR active |
vA1 + vE |
Raman active |
vA2 inactive vibration bands
showing the Irreducible Representations
for NH3
|
C3v |
E |
2C3 |
3Fv |
|
|
|
|
|
A1 |
1 |
1 |
1 |
z |
x2+y2,z2 |
|
NH3 |
3 |
12 |
0 |
6 |
|
|
|
|
A2 |
1 |
1 |
-1 |
Rz |
|
|
NH3 |
1 |
12 |
0 |
-6 |
|
|
|
|
E |
2 |
-1 |
0 |
x,y |
x2-y2,xy |
|
NH3 |
4 |
24 |
0 |
0 |
Rx,Ry |
xz,yz |
|
E |
C3 |
Sigmav |
|||
|
# unshifted atoms |
4 |
1 |
2 |
nu(R) |
|
|
times the character contribution |
3 |
0 |
1 |
f(R) |
|
|
|
|
12 |
0 |
2 |
Pi(R) |
|
times the number of operations of the class |
1 |
2 |
3 |
d(R) |
|
|
|
|
12 |
0 |
6 |
Oi(R) |
|
A1 |
A2 |
E |
|
|
|
|
3 |
1 |
4 |
Total spectral terms |
|
- |
1 |
1 |
2 |
Less rotational and translational |
|
|
2 |
0 |
2 |
Vibrational terms |
|
|
IR-Raman |
|
IR-Raman |
|
|
|
4 |
IR active |
4 |
Raman active |
0 inactive bands
Table 3. Molecular Vibration Calculations
for SF4 with Td Symmetry
|
Td |
E |
8C3 |
3C2 |
6S4 |
6Fd |
|
|
|
|
|
A1 |
1 |
1 |
1 |
1 |
1 |
|
x2+y3+z2 |
|
SF4 |
1 |
15 |
0 |
-3 |
-6 |
18 |
|
|
|
|
A2 |
1 |
1 |
1 |
-1 |
-1 |
|
|
|
SF4 |
0 |
15 |
0 |
-3 |
6 |
-18 |
|
|
|
|
E |
2 |
-1 |
2 |
0 |
0 |
|
2z2-x2-y2 |
|
SF4 |
1 |
30 |
0 |
-6 |
0 |
0 |
|
x2-y2 |
|
|
T1 |
3 |
0 |
-1 |
1 |
-1 |
Rx,Ry,Rz |
|
|
SF4 |
1 |
45 |
0 |
3 |
-6 |
-18 |
|
|
|
|
T2 |
3 |
0 |
-1 |
-1 |
1 |
x,y,z |
xy,xz,yz |
|
SF4 |
3 |
45 |
0 |
3 |
6 |
18 |
|
|
|
|
|
|
|
|
|
|
|
|
|
5 |
2 |
1 |
1 |
3 |
# unshifted atoms |
|||
|
x |
3 |
0 |
-1 |
-1 |
1 |
times the character contribution |
||
|
|
|
15 |
0 |
-1 |
-1 |
3 |
|
|
|
x |
1 |
0 |
3 |
6 |
6 |
times the number of operations of class |
||
|
|
|
15 |
0 |
-3 |
-6 |
18 |
Xi(R) |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
A1 |
A2 |
E |
T1 |
T2 |
|
|
|
SF4 |
1 |
0 |
1 |
1 |
3 |
Total Spectral Terms |
||
|
|
- |
|
|
|
1 |
1 |
less rotational and translational |
|
|
|
|
1 |
0 |
1 |
0 |
2 |
Vibration |
terms |
|
|
|
|
|
|
|
IR |
|
|
|
|
|
Raman |
|
Raman |
|
Raman |
|
|
2 IR active bands 4 Raman active bands
0 inactive vibration bands
Table 4. Molecular Vibration Calculations
for SF4 with C2v Symmetry
|
C2v |
E |
C2 |
Fv |
Fv' |
|
|
|
|
|
A1 |
1 |
1 |
1 |
1 |
z |
x2,y2,z2 |
|
SF4 |
5 |
15 |
-1 |
3 |
3 |
|
|
|
|
A2 |
1 |
1 |
-1 |
-1 |
Rz |
xy |
|
SF4 |
2 |
15 |
-1 |
-3 |
-3 |
|
|
|
|
B1 |
1 |
-1 |
1 |
-1 |
x,Ry |
xz |
|
SF4 |
4 |
15 |
1 |
3 |
-3 |
|
|
|
|
B2 |
1 |
-1 |
-1 |
1 |
y,Rx |
yz |
|
SF4 |
4 |
15 |
1 |
-3 |
3 |
|
|
|
|
|
|
|
|
|
|
|
|
|
5 |
1 |
3 |
3 |
# unshifted atoms |
||
|
x |
3 |
-1 |
1 |
1 |
times the character contribution |
||
|
|
|
15 |
-1 |
3 |
3 |
|
|
|
|
x |
1 |
1 |
1 |
1 |
times the number of operations of class |
|
|
|
|
15 |
-1 |
3 |
3 |
Xi(R) |
|
|
|
|
|
|
|
|
|
|
|
|
|
A1 |
A2 |
B1 |
B2 |
|
|
|
SF4 |
|
5 |
2 |
4 |
4 |
Total Spectral Terms |
|
|
- |
1 |
1 |
2 |
2 |
less rotational and translational |
||
|
|
|
4 |
1 |
2 |
2 |
vibrational terms |
|
|
|
|
IR |
|
IR |
IR |
|
|
|
|
|
Raman |
Raman |
Raman |
Raman |
|
|
|
|
|
8 IR active |
9 Raman active |
|
|
||
The Questions for this Problem Set can be found at this link.
In answering the Questions, it is best to build a model of the suggested structure. You will then carry out the symmetry operations and observe how many atoms of the molecule move during a symmetry operation. These numbers are then typed into the spreadsheet. The spreadsheet then makes the group theory molecular vibration calculations and predicts the number IR and Raman active fundamental bands that your model should possess. Check those numbers against what you are told in the Problem Set. If they agree, then you observed the correct number of atoms moving for each symmetry operation. If they do not agree, recheck your model with each symmetry operation until you do get agreement.