Characters of representations for molecular motions
Motion |
E |
8C3 |
6C2 |
6C4 |
3C2 |
i |
6S4 |
8S6 |
3σh |
6σd |
Cartesian 3N |
48 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
8 |
Translation (x,y,z) |
3 |
0 |
-1 |
1 |
-1 |
-3 |
-1 |
0 |
1 |
1 |
Rotation (Rx,Ry,Rz) |
3 |
0 |
-1 |
1 |
-1 |
3 |
1 |
0 |
-1 |
-1 |
Vibration |
42 |
0 |
2 |
-2 |
2 |
0 |
0 |
0 |
0 |
8 |
Decomposition to irreducible representations
Motion |
A1g |
A2g |
Eg |
T1g |
T2g |
A1u |
A2u |
Eu |
T1u |
T2u |
Total |
Cartesian 3N |
2 |
0 |
2 |
2 |
4 |
0 |
2 |
2 |
4 |
2 |
20 |
Translation (x,y,z) |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
1 |
0 |
1 |
Rotation (Rx,Ry,Rz) |
0 |
0 |
0 |
1 |
0 |
0 |
0 |
0 |
0 |
0 |
1 |
Vibration |
2 |
0 |
2 |
1 |
4 |
0 |
2 |
2 |
3 |
2 |
18 |
Molecular parameter
Number of Atoms (N) |
16
|
Number of internal coordinates |
42
|
Number of independant internal coordinates |
2
|
Number of vibrational modes |
18
|
Force field analysis
Allowed / forbidden vibronational transitions
Operator |
A1g |
A2g |
Eg |
T1g |
T2g |
A1u |
A2u |
Eu |
T1u |
T2u |
Total |
Linear (IR) |
2 |
0 |
2 |
1 |
4 |
0 |
2 |
2 |
3 |
2 |
3 / 15 |
Quadratic (Raman) |
2 |
0 |
2 |
1 |
4 |
0 |
2 |
2 |
3 |
2 |
8 / 10 |
IR + Raman |
- - - - |
0 |
- - - - |
1 |
- - - - |
0 |
2 |
2 |
- - - - |
2 |
0* / 7 |
* Parity Mutual Exclusion Principle
Characters of force fields
(Symmetric powers of vibration representation)
Force field |
E |
8C3 |
6C2 |
6C4 |
3C2 |
i |
6S4 |
8S6 |
3σh |
6σd |
linear |
42 |
0 |
2 |
-2 |
2 |
0 |
0 |
0 |
0 |
8 |
quadratic |
903 |
0 |
23 |
3 |
23 |
21 |
1 |
0 |
21 |
53 |
cubic |
13.244 |
14 |
44 |
-4 |
44 |
0 |
0 |
0 |
0 |
256 |
quartic |
148.995 |
0 |
275 |
15 |
275 |
231 |
11 |
0 |
231 |
1.095 |
quintic |
1.370.754 |
0 |
506 |
-26 |
506 |
0 |
0 |
0 |
0 |
4.056 |
sextic |
10.737.573 |
105 |
2.277 |
37 |
2.277 |
1.771 |
11 |
7 |
1.771 |
13.803 |
Decomposition to irreducible representations
Column with number of nonvanshing force constants highlighted
Force field |
A1g |
A2g |
Eg |
T1g |
T2g |
A1u |
A2u |
Eu |
T1u |
T2u |
linear |
2 |
0 |
2 |
1 |
4 |
0 |
2 |
2 |
3 |
2 |
quadratic |
32 |
12 |
44 |
46 |
64 |
15 |
22 |
37 |
59 |
51 |
cubic |
318 |
244 |
555 |
787 |
863 |
254 |
308 |
555 |
851 |
799 |
quartic |
3.315 |
2.966 |
6.281 |
9.127 |
9.463 |
3.000 |
3.204 |
6.204 |
9.398 |
9.192 |
quintic |
29.156 |
28.022 |
57.178 |
85.067 |
86.214 |
28.142 |
29.036 |
57.178 |
86.081 |
85.200 |
sextic |
226.024 |
221.992 |
447.960 |
668.952 |
672.960 |
222.273 |
225.148 |
447.372 |
672.400 |
669.512 |
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 O
h
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(T2u) |
..3. |
A1gA1g. | ..3. |
EgEg. | ..1. |
T1gT1g. | ..10. |
T2gT2g. | ..3. |
A2uA2u. | ..3. |
EuEu. | ..6. |
T1uT1u. | ..3. |
T2uT2u. | | |
| |
Subtotal: 32 / 8 / 10 |
Irrep combinations (i,j) with indices: pos(A1g) ≤ i ≤ j ≤ pos(T2u) |
Subtotal: 0 / 0 / 45 |
Total: 32 / 8 / 55 |
Contributions to nonvanishing cubic force field constants
Irrep combinations (i,i,i) with indices: pos(A1g) ≤ i ≤ pos(T2u) |
..4. |
A1gA1gA1g. | ..4. |
EgEgEg. | ..20. |
T2gT2gT2g. | | |
| |
| |
| |
| |
| |
| |
Subtotal: 28 / 3 / 10 |
Irrep combinations (i,i,j) (i,j,j) with indices: pos(A1g) ≤ i ≤ j ≤ pos(T2u) |
..4. |
T1gT1gT2g. | ..6. |
A1gEgEg. | ..2. |
A1gT1gT1g. | ..20. |
A1gT2gT2g. | ..6. |
A1gA2uA2u. | ..6. |
A1gEuEu. | ..12. |
A1gT1uT1u. | ..6. |
A1gT2uT2u. | ..2. |
EgT1gT1g. | ..20. |
EgT2gT2g. |
..6. |
EgEuEu. | ..12. |
EgT1uT1u. | ..6. |
EgT2uT2u. | ..6. |
T1gT2gT2g. | ..3. |
T1gT1uT1u. | ..1. |
T1gT2uT2u. | ..24. |
T2gT1uT1u. | ..12. |
T2gT2uT2u. | | |
| |
Subtotal: 154 / 18 / 90 |
Irrep combinations (i,j,k) with indices: pos(A1g) ≤ i ≤ j ≤ k ≤ pos(T2u) |
..8. |
EgT1gT2g. | ..8. |
EgA2uEu. | ..12. |
EgT1uT2u. | ..4. |
T1gA2uT2u. | ..6. |
T1gEuT1u. | ..4. |
T1gEuT2u. | ..6. |
T1gT1uT2u. | ..24. |
T2gA2uT1u. | ..24. |
T2gEuT1u. | ..16. |
T2gEuT2u. |
..24. |
T2gT1uT2u. | | |
| |
| |
| |
| |
| |
| |
| |
| |
Subtotal: 136 / 11 / 120 |
Total: 318 / 32 / 220 |
Contributions to nonvanishing quartic force field constants
Irrep combinations (i,i,i,i) with indices: pos(A1g) ≤ i ≤ pos(T2u) |
..5. |
A1gA1gA1gA1g. | ..6. |
EgEgEgEg. | ..2. |
T1gT1gT1gT1g. | ..90. |
T2gT2gT2gT2g. | ..5. |
A2uA2uA2uA2u. | ..6. |
EuEuEuEu. | ..36. |
T1uT1uT1uT1u. | ..11. |
T2uT2uT2uT2u. | | |
| |
Subtotal: 161 / 8 / 10 |
Irrep combinations (i,i,i,j) (i,j,j,j) with indices: pos(A1g) ≤ i ≤ j ≤ pos(T2u) |
..4. |
T1gT1gT1gT2g. | ..36. |
T1uT1uT1uT2u. | ..8. |
A1gEgEgEg. | ..40. |
A1gT2gT2gT2g. | ..40. |
EgT2gT2gT2g. | ..40. |
T1gT2gT2gT2g. | ..8. |
A2uEuEuEu. | ..20. |
A2uT1uT1uT1u. | ..16. |
EuT1uT1uT1u. | ..4. |
EuT2uT2uT2u. |
..18. |
T1uT2uT2uT2u. | | |
| |
| |
| |
| |
| |
| |
| |
| |
Subtotal: 234 / 11 / 90 |
Irrep combinations (i,i,j,j) with indices: pos(A1g) ≤ i ≤ j ≤ pos(T2u) |
..9. |
A1gA1gEgEg. | ..3. |
A1gA1gT1gT1g. | ..30. |
A1gA1gT2gT2g. | ..9. |
A1gA1gA2uA2u. | ..9. |
A1gA1gEuEu. | ..18. |
A1gA1gT1uT1u. | ..9. |
A1gA1gT2uT2u. | ..6. |
EgEgT1gT1g. | ..60. |
EgEgT2gT2g. | ..9. |
EgEgA2uA2u. |
..19. |
EgEgEuEu. | ..36. |
EgEgT1uT1u. | ..18. |
EgEgT2uT2u. | ..30. |
T1gT1gT2gT2g. | ..3. |
T1gT1gA2uA2u. | ..6. |
T1gT1gEuEu. | ..18. |
T1gT1gT1uT1u. | ..9. |
T1gT1gT2uT2u. | ..30. |
T2gT2gA2uA2u. | ..60. |
T2gT2gEuEu. |
..198. |
T2gT2gT1uT1u. | ..96. |
T2gT2gT2uT2u. | ..9. |
A2uA2uEuEu. | ..18. |
A2uA2uT1uT1u. | ..9. |
A2uA2uT2uT2u. | ..36. |
EuEuT1uT1u. | ..18. |
EuEuT2uT2u. | ..57. |
T1uT1uT2uT2u. | | |
| |
Subtotal: 832 / 28 / 45 |
Irrep combinations (i,i,j,k) (i,j,j,k) (i,j,k,k) with indices: pos(A1g) ≤ i ≤ j ≤ k ≤ pos(T2u) |
..16. |
EgEgT1gT2g. | ..12. |
EgEgA2uEu. | ..24. |
EgEgT1uT2u. | ..4. |
T1gT1gA2uEu. | ..6. |
T1gT1gA2uT1u. | ..6. |
T1gT1gEuT1u. | ..4. |
T1gT1gEuT2u. | ..12. |
T1gT1gT1uT2u. | ..40. |
T2gT2gA2uEu. | ..60. |
T2gT2gA2uT1u. |
..24. |
T2gT2gA2uT2u. | ..96. |
T2gT2gEuT1u. | ..64. |
T2gT2gEuT2u. | ..156. |
T2gT2gT1uT2u. | ..24. |
EuEuT1uT2u. | ..8. |
A1gT1gT1gT2g. | ..8. |
EgT1gT1gT2g. | ..12. |
A2uT1uT1uT2u. | ..36. |
EuT1uT1uT2u. | ..4. |
A1gEgT1gT1g. |
..40. |
A1gEgT2gT2g. | ..12. |
A1gEgEuEu. | ..24. |
A1gEgT1uT1u. | ..12. |
A1gEgT2uT2u. | ..12. |
A1gT1gT2gT2g. | ..6. |
A1gT1gT1uT1u. | ..2. |
A1gT1gT2uT2u. | ..48. |
A1gT2gT1uT1u. | ..24. |
A1gT2gT2uT2u. | ..32. |
EgT1gT2gT2g. |
..18. |
EgT1gT1uT1u. | ..8. |
EgT1gT2uT2u. | ..72. |
EgT2gT1uT1u. | ..32. |
EgT2gT2uT2u. | ..16. |
T1gT2gEuEu. | ..60. |
T1gT2gT1uT1u. | ..28. |
T1gT2gT2uT2u. | ..24. |
A2uEuT1uT1u. | ..12. |
A2uEuT2uT2u. | ..18. |
A2uT1uT2uT2u. |
..24. |
EuT1uT2uT2u. | | |
| |
| |
| |
| |
| |
| |
| |
| |
Subtotal: 1.140 / 41 / 360 |
Irrep combinations (i,j,k,l) with indices: pos(A1g) ≤ i ≤ j ≤ k ≤ l ≤ pos(T2u) |
..16. |
A1gEgT1gT2g. | ..16. |
A1gEgA2uEu. | ..24. |
A1gEgT1uT2u. | ..8. |
A1gT1gA2uT2u. | ..12. |
A1gT1gEuT1u. | ..8. |
A1gT1gEuT2u. | ..12. |
A1gT1gT1uT2u. | ..48. |
A1gT2gA2uT1u. | ..48. |
A1gT2gEuT1u. | ..32. |
A1gT2gEuT2u. |
..48. |
A1gT2gT1uT2u. | ..12. |
EgT1gA2uT1u. | ..8. |
EgT1gA2uT2u. | ..24. |
EgT1gEuT1u. | ..16. |
EgT1gEuT2u. | ..24. |
EgT1gT1uT2u. | ..48. |
EgT2gA2uT1u. | ..32. |
EgT2gA2uT2u. | ..96. |
EgT2gEuT1u. | ..64. |
EgT2gEuT2u. |
..96. |
EgT2gT1uT2u. | ..16. |
T1gT2gA2uEu. | ..24. |
T1gT2gA2uT1u. | ..16. |
T1gT2gA2uT2u. | ..48. |
T1gT2gEuT1u. | ..32. |
T1gT2gEuT2u. | ..96. |
T1gT2gT1uT2u. | ..24. |
A2uEuT1uT2u. | | |
| |
Subtotal: 948 / 28 / 210 |
Total: 3.315 / 116 / 715 |
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Last update November, 13th 2023 by A. Gelessus, Impressum, Datenschutzerklärung/DataPrivacyStatement