Mode Analysis
- Page ID
- 1855
Γ_{Total} = Γ_{Stretch} + Γ_{Bend} + Γ_{Translation} + Γ_{Rotation}
and
Γ_{Vibration }= Γ_{Stretch }+ Γ_{Bend}
For easier understanding, there are steps immediately followed by examples using the D3h.
Finding Γ Total
First, add all the x, y and z rows on the Character Tables of Symmetry groups. If x, y or z are in () on the far right then only count them once, otherwise count the row a second time (Keep the column separated). This is called Γ_{x,y,z}. Next, move the molecule with the designated column symbols and if an atom does not move then it is counted. Finally, multiply Γ_{x,y,z} and UMA and it will equal Γtotal.
D3h | E | 2C_{2} | 3C_{2} | σ_{h} | 2S_{3} | 3σ_{v} | IR | Raman |
---|---|---|---|---|---|---|---|---|
A_{1}' | 1 | 1 | 1 | 1 | 1 | 1 | x^{2}+y^{2}, z^{2} | |
A_{2}' | 1 | 1 | -1 | 1 | 1 | -1 | Rz | |
E' | 2 | -1 | 0 | 2 | -1 | 0 | (x,y) | (xy, x^{2}-y^{2}) |
A_{1}" | 1 | 1 | 1 | -1 | -1 | -1 | ||
A_{2}" | 1 | 1 | -1 | -1 | -1 | 1 | z | |
E" | 2 | -1 | 0 | -2 | 1 | 0 | (Rx,Ry) | (xz, yz) |
Γ_{x,y,z} | 2+1=3 | -1+1=0 | 0-1=-1 | 2-1=1 | -1-1=-2 | 0+1=1 | ||
UMA (unmoved atoms) |
6 |
3 |
2 |
4 |
1 |
4 |
||
Γ_{total} | (3)(6)=18 | (0)(3)= 0 | (-1)(2)=-2 | (1)(4)=4 | (-2)(1)=-2 | (1)(4)=4 |
D3h molecule
-need to find the irreducible representation of Gamma total.
Take Γtotal multiply by the number in front of the symbols (the order) and multiply by each number inside of the character table. Add up each row, and divide each row by the total order. For the D_{3}h the order is 12.
D3h | 1E | 2C_{2} | 3C_{2} | σh | 2S_{3} | 3σ_{v} | add the up and divide by 12 |
---|---|---|---|---|---|---|---|
A_{1}' | (1x1x18)=18 | (2x1x0)=0 | (3x1x-2)=-6 | (1x1x4)=4 | (2x1x-2)=-4 | (3x1x4)=12 | (24/12)=2 |
A_{2}' | (1x1x18)=18 | (2x1x0)=0 | (3x-1x-2)=-6 | (1x1x4)=4 | (2x1x-2)=-4 | (3x-1x4)=-12 | (0/12)=0 |
E' | (1x2x18)=36 | (2x-1x0)=0 | (3x0x-2)=0 | (1x2x4)=8 | (2x-1x-2)=4 | (3x0x4)=0 | (48/12)=4 |
A_{1}" | (1x1x18)=18 | (2x1x0)=0 | (3x1x-2)=-6 | (1x-1x4)=-4 | (2x-1x-2)=4 | (3x-1x4)=-12 | (0/12)=0 |
A_{2}" | (1x1x18)=18 | (2x1x0)=0 | (3x-1x-2)=6 | (1x-1x4)=-4 | (2x-1x-2)=4 | (3x1x4)=12 | (36/12)=3 |
E" | (1x2x18)=36 | (2x-1x0)=0 | (3x0x-2)=0 | (1x-2x4)=-8 | (2x1x-2)=-4 | (3x0x4)=0 | (24/12)=2 |
Γ_{total}= 2A1' + A2' + 4E' + 3A2" + 2E"
Γ Translation
The irreducible form of Γtrans, one needs to look at the second to last column. look at the row that the x, y and z and take the irreducible representation. For instance the D3h would be Γ_{trans}= E'+A2"
Γ Rotation
One can find Γ_{rot} the same way as Γ_{trans}. Instead of looking at x,y,z, one would look at the Rx, Ry, Rz. For the D3h. The Γ_{rot} = A2'+E"
Γ Vibration
To find Γ_{Vibration}, just take Γ_{tot} -Γ_{trans}-Γ_{rot}= Γ_{vibration}.
D3h example
Γ_{tot} | 2A1' + A2' + 4E' + 3A2" + 2E" |
Γ_{trans} | - E' - A2" |
Γ_{rot} | - A2' -E" |
Γ_{vibration} | 2A1' + 3E' + 2A2" + E" |
Number of Vibrational Active IR Bands
Only R_{x}, R_{y}, R_{z}, x, y, and z can be ir active. which means only A2', E', A2", and E" can be IR active bands for the D_{3}h. Next add up the number in front of the irreducible representation and that is how many IR active bonds. For instance for the same problem there are 3E'+2A2". There are 5 bands, three of them (meaning the E) are two fold degenerate.
Number of Vibrational Active Raman bands
Only x^{2}+y^{2}, z^{2}, xy, xz, yz, x2-y2 can be Raman active. which means only A1', E', and E" can be raman active for the D3h. Next add up the number in front of the irreducible representation, and that is how many Raman active bonds there are. For instance for the same problem there are 3E' + E". There are four bands, 4 of which are two fold degenerate.
Finding ΓStretch
Looking at the molecules point group, do each of the symmetry representation and count the number of unmoved bonds. Γ_{stretch} = Γ_{σ} =Γ_{rad} . next multiply unmoved bonds by the symmetry operations and then the numbers inside the character tables. Then add the rows up and divide by the order of the point group.
D_{3}h | E | 2C_{2} | 3C_{2} | σh | 2S_{3} | 3σ_{v} | IR | Raman |
---|---|---|---|---|---|---|---|---|
A_{1}' | 1 | 1 | 1 | 1 | 1 | 1 | x^{2}+y^{2}, z^{2} | |
A_{2}' | 1 | 1 | -1 | 1 | 1 | -1 | Rz | |
E' | 2 | -1 | 0 | 2 | -1 | 0 | (x,y) | (xy, x^{2}-y^{2}) |
A_{1}" | 1 | 1 | 1 | -1 | -1 | -1 | ||
A_{2}" | 1 | 1 | -1 | -1 | -1 | 1 | z | |
E" | 2 | -1 | 0 | -2 | 1 | 0 | (Rx,Ry) | (xz, yz) |
Γ_{x,y,z} | 2+1=3 | -1+1=0 | 0-1=-1 | 2-1=1 | -1-1=-2 | 0+1=1 | ||
UMB (unmoved bonds) |
5 |
2 |
1 |
3 |
0 |
3 |
D_{3}h | 1E | 2C_{2} | 3C_{2} | σh | 2S_{3} | 3σ_{v} | add the up and divide by 12 |
---|---|---|---|---|---|---|---|
A_{1}' | (1x1x5)=5 | (2x1x2)=4 | (3x1x1)=3 | (1x1x3)=3 | (2x1x0)=0 | (3x1x3)=9 | (24/12)=2 |
A_{2}' | (1x1x5)=5 | (2x1x2)=4 | (3x-1x1)=-3 | (1x1x3)=3 | (2x1x0)=0 | (3x-1x3)=-9 | (0/12)=0 |
E' | (1x2x5)=10 | (2x-1x2)=-4 | (3x0x1)=0 | (1x2x3)=6 | (2x-1x0)=0 | (3x0x3)=0 | (12/12)=1 |
A_{1}" | (1x1x5)=5 | (2x1x2)=4 | (3x1x1)=3 | (1x-1x3)=-3 | (2x-1x0=0 | (3x-1x3)=-9 | (0/12)=0 |
A_{2}" | (1x1x5)=5 | (2x1x2)=4 | (3x-1x1)=-3 | (1x-1x3)=-3 | (2x-1x0)=0 | (3x1x3)=9 | (12/12)=1 |
E" | (1x2x5)=10 | (2x-1x2)=-4 | (3x0x1)=0 | (1x-2x3)=-6 | (2x1x0)=0 | (3x0x3)=0 | (24/12)=0 |
Γ_{stretch} = Γ_{σ} = Γ_{rad} = 2A1' + 1E'+ 1A2"
Γ_{π}= Γ_{tan }=