Characters of representations for molecular motions
| Motion |
E |
2C3 |
3C'2 |
i |
2S6 |
3σd |
| Cartesian 3N |
90 |
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 |
84 |
0 |
2 |
0 |
0 |
6 |
Decomposition to irreducible representations
| Motion |
A1g |
A2g |
Eg |
A1u |
A2u |
Eu |
Total |
| Cartesian 3N |
9 |
6 |
15 |
6 |
9 |
15 |
60 |
| Translation (x,y,z) |
0 |
0 |
0 |
0 |
1 |
1 |
2 |
| Rotation (Rx,Ry,Rz) |
0 |
1 |
1 |
0 |
0 |
0 |
2 |
| Vibration |
9 |
5 |
14 |
6 |
8 |
14 |
56 |
Molecular parameter
| Number of Atoms (N) |
30
|
| Number of internal coordinates |
84
|
| Number of independant internal coordinates |
9
|
| Number of vibrational modes |
56
|
Force field analysis
Allowed / forbidden vibronational transitions
| Operator |
A1g |
A2g |
Eg |
A1u |
A2u |
Eu |
Total |
| Linear (IR) |
9 |
5 |
14 |
6 |
8 |
14 |
22 / 34 |
| Quadratic (Raman) |
9 |
5 |
14 |
6 |
8 |
14 |
23 / 33 |
| IR + Raman |
- - - - |
5 |
- - - - |
6 |
- - - - |
- - - - |
0* / 11 |
* Parity Mutual Exclusion Principle
Characters of force fields
(Symmetric powers of vibration representation)
| Force field |
E |
2C3 |
3C'2 |
i |
2S6 |
3σd |
| linear |
84 |
0 |
2 |
0 |
0 |
6 |
| quadratic |
3.570 |
0 |
44 |
42 |
0 |
60 |
| cubic |
102.340 |
28 |
86 |
0 |
0 |
290 |
| quartic |
2.225.895 |
0 |
989 |
903 |
0 |
1.725 |
| quintic |
39.175.752 |
0 |
1.892 |
0 |
0 |
7.116 |
| sextic |
581.106.988 |
406 |
15.136 |
13.244 |
14 |
32.416 |
Decomposition to irreducible representations
Column with number of nonvanshing force constants highlighted
| Force field |
A1g |
A2g |
Eg |
A1u |
A2u |
Eu |
| linear |
9 |
5 |
14 |
6 |
8 |
14 |
| quadratic |
327 |
275 |
602 |
290 |
298 |
588 |
| cubic |
8.627 |
8.439 |
17.052 |
8.482 |
8.584 |
17.052 |
| quartic |
186.245 |
184.888 |
371.133 |
185.232 |
185.600 |
370.832 |
| quintic |
3.266.898 |
3.262.394 |
6.529.292 |
3.263.340 |
3.265.952 |
6.529.292 |
| sextic |
48.438.644 |
48.414.868 |
96.853.302 |
48.420.224 |
48.428.864 |
96.848.892 |
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) |
|---|
| ..45. |
A1gA1g. | ..15. |
A2gA2g. | ..105. |
EgEg. | ..21. |
A1uA1u. | ..36. |
A2uA2u. | ..105. |
EuEu. | | |
| |
| |
| |
| Subtotal: 327 / 6 / 6 |
| Irrep combinations (i,j) with indices: pos(A1g) ≤ i ≤ j ≤ pos(Eu) |
|---|
| Subtotal: 0 / 0 / 15 |
| Total: 327 / 6 / 21 |
Contributions to nonvanishing cubic force field constants
| Irrep combinations (i,i,i) with indices: pos(A1g) ≤ i ≤ pos(Eu) |
|---|
| ..165. |
A1gA1gA1g. | ..560. |
EgEgEg. | | |
| |
| |
| |
| |
| |
| |
| |
| Subtotal: 725 / 2 / 6 |
| Irrep combinations (i,i,j) (i,j,j) with indices: pos(A1g) ≤ i ≤ j ≤ pos(Eu) |
|---|
| ..135. |
A1gA2gA2g. | ..945. |
A1gEgEg. | ..189. |
A1gA1uA1u. | ..324. |
A1gA2uA2u. | ..945. |
A1gEuEu. | ..455. |
A2gEgEg. | ..455. |
A2gEuEu. | ..1.470. |
EgEuEu. | | |
| |
| Subtotal: 4.918 / 8 / 30 |
| Irrep combinations (i,j,k) with indices: pos(A1g) ≤ i ≤ j ≤ k ≤ pos(Eu) |
|---|
| ..240. |
A2gA1uA2u. | ..1.176. |
EgA1uEu. | ..1.568. |
EgA2uEu. | | |
| |
| |
| |
| |
| |
| |
| Subtotal: 2.984 / 3 / 20 |
| Total: 8.627 / 13 / 56 |
Contributions to nonvanishing quartic force field constants
| Irrep combinations (i,i,i,i) with indices: pos(A1g) ≤ i ≤ pos(Eu) |
|---|
| ..495. |
A1gA1gA1gA1g. | ..70. |
A2gA2gA2gA2g. | ..5.565. |
EgEgEgEg. | ..126. |
A1uA1uA1uA1u. | ..330. |
A2uA2uA2uA2u. | ..5.565. |
EuEuEuEu. | | |
| |
| |
| |
| Subtotal: 12.151 / 6 / 6 |
| Irrep combinations (i,i,i,j) (i,j,j,j) with indices: pos(A1g) ≤ i ≤ j ≤ pos(Eu) |
|---|
| ..5.040. |
A1gEgEgEg. | ..2.800. |
A2gEgEgEg. | ..3.360. |
A1uEuEuEu. | ..4.480. |
A2uEuEuEu. | | |
| |
| |
| |
| |
| |
| Subtotal: 15.680 / 4 / 30 |
| Irrep combinations (i,i,j,j) with indices: pos(A1g) ≤ i ≤ j ≤ pos(Eu) |
|---|
| ..675. |
A1gA1gA2gA2g. | ..4.725. |
A1gA1gEgEg. | ..945. |
A1gA1gA1uA1u. | ..1.620. |
A1gA1gA2uA2u. | ..4.725. |
A1gA1gEuEu. | ..1.575. |
A2gA2gEgEg. | ..315. |
A2gA2gA1uA1u. | ..540. |
A2gA2gA2uA2u. | ..1.575. |
A2gA2gEuEu. | ..2.205. |
EgEgA1uA1u. |
| ..3.780. |
EgEgA2uA2u. | ..30.331. |
EgEgEuEu. | ..756. |
A1uA1uA2uA2u. | ..2.205. |
A1uA1uEuEu. | ..3.780. |
A2uA2uEuEu. | | |
| |
| |
| |
| |
| Subtotal: 59.752 / 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) |
|---|
| ..4.368. |
EgEgA1uA2u. | ..8.820. |
EgEgA1uEu. | ..11.760. |
EgEgA2uEu. | ..4.095. |
A1gA2gEgEg. | ..4.095. |
A1gA2gEuEu. | ..13.230. |
A1gEgEuEu. | ..7.350. |
A2gEgEuEu. | ..4.368. |
A1uA2uEuEu. | | |
| |
| Subtotal: 58.086 / 8 / 60 |
| Irrep combinations (i,j,k,l) with indices: pos(A1g) ≤ i ≤ j ≤ k ≤ l ≤ pos(Eu) |
|---|
| ..2.160. |
A1gA2gA1uA2u. | ..10.584. |
A1gEgA1uEu. | ..14.112. |
A1gEgA2uEu. | ..5.880. |
A2gEgA1uEu. | ..7.840. |
A2gEgA2uEu. | | |
| |
| |
| |
| |
| Subtotal: 40.576 / 5 / 15 |
| Total: 186.245 / 38 / 126 |
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