Characters of representations for molecular motions
| Motion |
E |
2C3 |
3C'2 |
i |
2S6 |
3σd |
| Cartesian 3N |
54 |
0 |
0 |
0 |
0 |
6 |
| Translation (x,y,z) |
3 |
0 |
-1 |
-3 |
0 |
1 |
| Rotation (Rx,Ry,Rz) |
3 |
0 |
-1 |
3 |
0 |
-1 |
| Vibration |
48 |
0 |
2 |
0 |
0 |
6 |
Decomposition to irreducible representations
| Motion |
A1g |
A2g |
Eg |
A1u |
A2u |
Eu |
Total |
| Cartesian 3N |
6 |
3 |
9 |
3 |
6 |
9 |
36 |
| Translation (x,y,z) |
0 |
0 |
0 |
0 |
1 |
1 |
2 |
| Rotation (Rx,Ry,Rz) |
0 |
1 |
1 |
0 |
0 |
0 |
2 |
| Vibration |
6 |
2 |
8 |
3 |
5 |
8 |
32 |
Molecular parameter
| Number of Atoms (N) |
18
|
| Number of internal coordinates |
48
|
| Number of independant internal coordinates |
6
|
| Number of vibrational modes |
32
|
Force field analysis
Allowed / forbidden vibronational transitions
| Operator |
A1g |
A2g |
Eg |
A1u |
A2u |
Eu |
Total |
| Linear (IR) |
6 |
2 |
8 |
3 |
5 |
8 |
13 / 19 |
| Quadratic (Raman) |
6 |
2 |
8 |
3 |
5 |
8 |
14 / 18 |
| IR + Raman |
- - - - |
2 |
- - - - |
3 |
- - - - |
- - - - |
0* / 5 |
* Parity Mutual Exclusion Principle
Characters of force fields
(Symmetric powers of vibration representation)
| Force field |
E |
2C3 |
3C'2 |
i |
2S6 |
3σd |
| linear |
48 |
0 |
2 |
0 |
0 |
6 |
| quadratic |
1.176 |
0 |
26 |
24 |
0 |
42 |
| cubic |
19.600 |
16 |
50 |
0 |
0 |
182 |
| quartic |
249.900 |
0 |
350 |
300 |
0 |
798 |
| quintic |
2.598.960 |
0 |
650 |
0 |
0 |
2.814 |
| sextic |
22.957.480 |
136 |
3.250 |
2.600 |
8 |
9.730 |
Decomposition to irreducible representations
Column with number of nonvanshing force constants highlighted
| Force field |
A1g |
A2g |
Eg |
A1u |
A2u |
Eu |
| linear |
6 |
2 |
8 |
3 |
5 |
8 |
| quadratic |
117 |
83 |
200 |
92 |
100 |
192 |
| cubic |
1.694 |
1.578 |
3.264 |
1.603 |
1.669 |
3.264 |
| quartic |
21.137 |
20.563 |
41.700 |
20.688 |
20.912 |
41.600 |
| quintic |
217.446 |
215.714 |
433.160 |
216.039 |
217.121 |
433.160 |
| sextic |
1.916.609 |
1.910.119 |
3.826.656 |
1.911.308 |
1.914.548 |
3.825.792 |
Further Reading
- J.K.G. Watson, J. Mol. Spec. 41 229 (1972)
The Numbers of Structural Parameters and Potential Constants of Molecules
- X.F. Zhou, P. Pulay. J. Comp. Chem. 10 No. 7, 935-938 (1989)
Characters for Symmetric and Antisymmetric Higher Powers of Representations:
Application to the Number of Anharmonic Force Constants in Symmetrical Molecules
- F. Varga, L. Nemes, J.K.G. Watson. J. Phys. B: At. Mol. Opt. Phys. 10 No. 7, 5043-5048 (1996)
The number of anharmonic potential constants of the fullerenes C60 and C70
Contributions to nonvanishing force field constants
pos(X) : Position of irreducible representation (irrep) X in character table of D
3d
Subtotal: <Number of nonvanishing force constants in subsection> / <number of nonzero irrep combinations in subsection> / <number of irrep combinations in subsection>
Total: <Number of nonvanishing force constants in force field> / <number of nonzero irrep combinations in force field> / <number of irrep combinations in force field>
Contributions to nonvanishing quadratic force field constants
| Irrep combinations (i,i) with indices: pos(A1g) ≤ i ≤ pos(Eu) |
|---|
| ..21. |
A1gA1g. | ..3. |
A2gA2g. | ..36. |
EgEg. | ..6. |
A1uA1u. | ..15. |
A2uA2u. | ..36. |
EuEu. | | |
| |
| |
| |
| Subtotal: 117 / 6 / 6 |
| Irrep combinations (i,j) with indices: pos(A1g) ≤ i ≤ j ≤ pos(Eu) |
|---|
| Subtotal: 0 / 0 / 15 |
| Total: 117 / 6 / 21 |
Contributions to nonvanishing cubic force field constants
| Irrep combinations (i,i,i) with indices: pos(A1g) ≤ i ≤ pos(Eu) |
|---|
| ..56. |
A1gA1gA1g. | ..120. |
EgEgEg. | | |
| |
| |
| |
| |
| |
| |
| |
| Subtotal: 176 / 2 / 6 |
| Irrep combinations (i,i,j) (i,j,j) with indices: pos(A1g) ≤ i ≤ j ≤ pos(Eu) |
|---|
| ..18. |
A1gA2gA2g. | ..216. |
A1gEgEg. | ..36. |
A1gA1uA1u. | ..90. |
A1gA2uA2u. | ..216. |
A1gEuEu. | ..56. |
A2gEgEg. | ..56. |
A2gEuEu. | ..288. |
EgEuEu. | | |
| |
| Subtotal: 976 / 8 / 30 |
| Irrep combinations (i,j,k) with indices: pos(A1g) ≤ i ≤ j ≤ k ≤ pos(Eu) |
|---|
| ..30. |
A2gA1uA2u. | ..192. |
EgA1uEu. | ..320. |
EgA2uEu. | | |
| |
| |
| |
| |
| |
| |
| Subtotal: 542 / 3 / 20 |
| Total: 1.694 / 13 / 56 |
Contributions to nonvanishing quartic force field constants
| Irrep combinations (i,i,i,i) with indices: pos(A1g) ≤ i ≤ pos(Eu) |
|---|
| ..126. |
A1gA1gA1gA1g. | ..5. |
A2gA2gA2gA2g. | ..666. |
EgEgEgEg. | ..15. |
A1uA1uA1uA1u. | ..70. |
A2uA2uA2uA2u. | ..666. |
EuEuEuEu. | | |
| |
| |
| |
| Subtotal: 1.548 / 6 / 6 |
| Irrep combinations (i,i,i,j) (i,j,j,j) with indices: pos(A1g) ≤ i ≤ j ≤ pos(Eu) |
|---|
| ..720. |
A1gEgEgEg. | ..240. |
A2gEgEgEg. | ..360. |
A1uEuEuEu. | ..600. |
A2uEuEuEu. | | |
| |
| |
| |
| |
| |
| Subtotal: 1.920 / 4 / 30 |
| Irrep combinations (i,i,j,j) with indices: pos(A1g) ≤ i ≤ j ≤ pos(Eu) |
|---|
| ..63. |
A1gA1gA2gA2g. | ..756. |
A1gA1gEgEg. | ..126. |
A1gA1gA1uA1u. | ..315. |
A1gA1gA2uA2u. | ..756. |
A1gA1gEuEu. | ..108. |
A2gA2gEgEg. | ..18. |
A2gA2gA1uA1u. | ..45. |
A2gA2gA2uA2u. | ..108. |
A2gA2gEuEu. | ..216. |
EgEgA1uA1u. |
| ..540. |
EgEgA2uA2u. | ..3.376. |
EgEgEuEu. | ..90. |
A1uA1uA2uA2u. | ..216. |
A1uA1uEuEu. | ..540. |
A2uA2uEuEu. | | |
| |
| |
| |
| |
| Subtotal: 7.273 / 15 / 15 |
| Irrep combinations (i,i,j,k) (i,j,j,k) (i,j,k,k) with indices: pos(A1g) ≤ i ≤ j ≤ k ≤ pos(Eu) |
|---|
| ..420. |
EgEgA1uA2u. | ..864. |
EgEgA1uEu. | ..1.440. |
EgEgA2uEu. | ..336. |
A1gA2gEgEg. | ..336. |
A1gA2gEuEu. | ..1.728. |
A1gEgEuEu. | ..576. |
A2gEgEuEu. | ..420. |
A1uA2uEuEu. | | |
| |
| Subtotal: 6.120 / 8 / 60 |
| Irrep combinations (i,j,k,l) with indices: pos(A1g) ≤ i ≤ j ≤ k ≤ l ≤ pos(Eu) |
|---|
| ..180. |
A1gA2gA1uA2u. | ..1.152. |
A1gEgA1uEu. | ..1.920. |
A1gEgA2uEu. | ..384. |
A2gEgA1uEu. | ..640. |
A2gEgA2uEu. | | |
| |
| |
| |
| |
| Subtotal: 4.276 / 5 / 15 |
| Total: 21.137 / 38 / 126 |
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