Characters of representations for molecular motions
Motion |
E |
2C3 |
3C'2 |
i |
2S6 |
3σd |
Cartesian 3N |
48 |
0 |
0 |
0 |
0 |
8 |
Translation (x,y,z) |
3 |
0 |
-1 |
-3 |
0 |
1 |
Rotation (Rx,Ry,Rz) |
3 |
0 |
-1 |
3 |
0 |
-1 |
Vibration |
42 |
0 |
2 |
0 |
0 |
8 |
Decomposition to irreducible representations
Motion |
A1g |
A2g |
Eg |
A1u |
A2u |
Eu |
Total |
Cartesian 3N |
6 |
2 |
8 |
2 |
6 |
8 |
32 |
Translation (x,y,z) |
0 |
0 |
0 |
0 |
1 |
1 |
2 |
Rotation (Rx,Ry,Rz) |
0 |
1 |
1 |
0 |
0 |
0 |
2 |
Vibration |
6 |
1 |
7 |
2 |
5 |
7 |
28 |
Molecular parameter
Number of Atoms (N) |
16
|
Number of internal coordinates |
42
|
Number of independant internal coordinates |
6
|
Number of vibrational modes |
28
|
Force field analysis
Allowed / forbidden vibronational transitions
Operator |
A1g |
A2g |
Eg |
A1u |
A2u |
Eu |
Total |
Linear (IR) |
6 |
1 |
7 |
2 |
5 |
7 |
12 / 16 |
Quadratic (Raman) |
6 |
1 |
7 |
2 |
5 |
7 |
13 / 15 |
IR + Raman |
- - - - |
1 |
- - - - |
2 |
- - - - |
- - - - |
0* / 3 |
* Parity Mutual Exclusion Principle
Characters of force fields
(Symmetric powers of vibration representation)
Force field |
E |
2C3 |
3C'2 |
i |
2S6 |
3σd |
linear |
42 |
0 |
2 |
0 |
0 |
8 |
quadratic |
903 |
0 |
23 |
21 |
0 |
53 |
cubic |
13.244 |
14 |
44 |
0 |
0 |
256 |
quartic |
148.995 |
0 |
275 |
231 |
0 |
1.095 |
quintic |
1.370.754 |
0 |
506 |
0 |
0 |
4.056 |
sextic |
10.737.573 |
105 |
2.277 |
1.771 |
7 |
13.803 |
Decomposition to irreducible representations
Column with number of nonvanshing force constants highlighted
Force field |
A1g |
A2g |
Eg |
A1u |
A2u |
Eu |
linear |
6 |
1 |
7 |
2 |
5 |
7 |
quadratic |
96 |
58 |
154 |
66 |
81 |
147 |
cubic |
1.181 |
1.031 |
2.205 |
1.053 |
1.159 |
2.205 |
quartic |
12.778 |
12.093 |
24.871 |
12.192 |
12.602 |
24.794 |
quintic |
115.370 |
113.089 |
228.459 |
113.342 |
115.117 |
228.459 |
sextic |
898.984 |
890.944 |
1.789.872 |
891.785 |
897.548 |
1.789.284 |
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. | ..1. |
A2gA2g. | ..28. |
EgEg. | ..3. |
A1uA1u. | ..15. |
A2uA2u. | ..28. |
EuEu. | | |
| |
| |
| |
Subtotal: 96 / 6 / 6 |
Irrep combinations (i,j) with indices: pos(A1g) ≤ i ≤ j ≤ pos(Eu) |
Subtotal: 0 / 0 / 15 |
Total: 96 / 6 / 21 |
Contributions to nonvanishing cubic force field constants
Irrep combinations (i,i,i) with indices: pos(A1g) ≤ i ≤ pos(Eu) |
..56. |
A1gA1gA1g. | ..84. |
EgEgEg. | | |
| |
| |
| |
| |
| |
| |
| |
Subtotal: 140 / 2 / 6 |
Irrep combinations (i,i,j) (i,j,j) with indices: pos(A1g) ≤ i ≤ j ≤ pos(Eu) |
..6. |
A1gA2gA2g. | ..168. |
A1gEgEg. | ..18. |
A1gA1uA1u. | ..90. |
A1gA2uA2u. | ..168. |
A1gEuEu. | ..21. |
A2gEgEg. | ..21. |
A2gEuEu. | ..196. |
EgEuEu. | | |
| |
Subtotal: 688 / 8 / 30 |
Irrep combinations (i,j,k) with indices: pos(A1g) ≤ i ≤ j ≤ k ≤ pos(Eu) |
..10. |
A2gA1uA2u. | ..98. |
EgA1uEu. | ..245. |
EgA2uEu. | | |
| |
| |
| |
| |
| |
| |
Subtotal: 353 / 3 / 20 |
Total: 1.181 / 13 / 56 |
Contributions to nonvanishing quartic force field constants
Irrep combinations (i,i,i,i) with indices: pos(A1g) ≤ i ≤ pos(Eu) |
..126. |
A1gA1gA1gA1g. | ..1. |
A2gA2gA2gA2g. | ..406. |
EgEgEgEg. | ..5. |
A1uA1uA1uA1u. | ..70. |
A2uA2uA2uA2u. | ..406. |
EuEuEuEu. | | |
| |
| |
| |
Subtotal: 1.014 / 6 / 6 |
Irrep combinations (i,i,i,j) (i,j,j,j) with indices: pos(A1g) ≤ i ≤ j ≤ pos(Eu) |
..504. |
A1gEgEgEg. | ..84. |
A2gEgEgEg. | ..168. |
A1uEuEuEu. | ..420. |
A2uEuEuEu. | | |
| |
| |
| |
| |
| |
Subtotal: 1.176 / 4 / 30 |
Irrep combinations (i,i,j,j) with indices: pos(A1g) ≤ i ≤ j ≤ pos(Eu) |
..21. |
A1gA1gA2gA2g. | ..588. |
A1gA1gEgEg. | ..63. |
A1gA1gA1uA1u. | ..315. |
A1gA1gA2uA2u. | ..588. |
A1gA1gEuEu. | ..28. |
A2gA2gEgEg. | ..3. |
A2gA2gA1uA1u. | ..15. |
A2gA2gA2uA2u. | ..28. |
A2gA2gEuEu. | ..84. |
EgEgA1uA1u. |
..420. |
EgEgA2uA2u. | ..2.009. |
EgEgEuEu. | ..45. |
A1uA1uA2uA2u. | ..84. |
A1uA1uEuEu. | ..420. |
A2uA2uEuEu. | | |
| |
| |
| |
| |
Subtotal: 4.711 / 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) |
..210. |
EgEgA1uA2u. | ..392. |
EgEgA1uEu. | ..980. |
EgEgA2uEu. | ..126. |
A1gA2gEgEg. | ..126. |
A1gA2gEuEu. | ..1.176. |
A1gEgEuEu. | ..196. |
A2gEgEuEu. | ..210. |
A1uA2uEuEu. | | |
| |
Subtotal: 3.416 / 8 / 60 |
Irrep combinations (i,j,k,l) with indices: pos(A1g) ≤ i ≤ j ≤ k ≤ l ≤ pos(Eu) |
..60. |
A1gA2gA1uA2u. | ..588. |
A1gEgA1uEu. | ..1.470. |
A1gEgA2uEu. | ..98. |
A2gEgA1uEu. | ..245. |
A2gEgA2uEu. | | |
| |
| |
| |
| |
Subtotal: 2.461 / 5 / 15 |
Total: 12.778 / 38 / 126 |
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