Characters of representations for molecular motions
| Motion |
E |
4C3 |
4(C3)2 |
3C2 |
i |
4(S6)5 |
4S6 |
3σh |
| Cartesian 3N |
57 |
0 |
0 |
-3 |
-3 |
0 |
0 |
9 |
| Translation (x,y,z) |
3 |
0 |
0 |
-1 |
-3 |
0 |
0 |
1 |
| Rotation (Rx,Ry,Rz) |
3 |
0 |
0 |
-1 |
3 |
0 |
0 |
-1 |
| Vibration |
51 |
0 |
0 |
-1 |
-3 |
0 |
0 |
9 |
Decomposition to irreducible representations
| Motion |
Ag |
Eg*
|
Tg |
Au |
Eu*
|
Tu |
Total |
| Cartesian 3N |
3 |
3 |
6 |
1 |
1 |
9 |
23 |
| Translation (x,y,z) |
0 |
0 |
0 |
0 |
0 |
1 |
1 |
| Rotation (Rx,Ry,Rz) |
0 |
0 |
1 |
0 |
0 |
0 |
1 |
| Vibration |
3 |
3 |
5 |
1 |
1 |
8 |
21 |
Molecular parameter
| Number of Atoms (N) |
19
|
| Number of internal coordinates |
51
|
| Number of independant internal coordinates |
3
|
| Number of vibrational modes |
21
|
Force field analysis
Allowed / forbidden vibronational transitions
| Operator |
Ag |
Eg*
|
Tg |
Au |
Eu*
|
Tu |
Total |
| Linear (IR) |
3 |
3 |
5 |
1 |
1 |
8 |
8 / 13 |
| Quadratic (Raman) |
3 |
3 |
5 |
1 |
1 |
8 |
11 / 10 |
| IR + Raman |
- - - - |
- - - - |
- - - - |
1 |
1 |
- - - - |
0* / 2 |
* Parity Mutual Exclusion Principle
Characters of force fields
(Symmetric powers of vibration representation)
| Force field |
E |
4C3 |
4(C3)2 |
3C2 |
i |
4(S6)5 |
4S6 |
3σh |
| linear |
51 |
0 |
0 |
-1 |
-3 |
0 |
0 |
9 |
| quadratic |
1.326 |
0 |
0 |
26 |
30 |
0 |
0 |
66 |
| cubic |
23.426 |
17 |
17 |
-26 |
-82 |
-1 |
-1 |
354 |
| quartic |
316.251 |
0 |
0 |
351 |
459 |
0 |
0 |
1.671 |
| quintic |
3.478.761 |
0 |
0 |
-351 |
-1.161 |
0 |
0 |
6.831 |
| sextic |
32.468.436 |
153 |
153 |
3.276 |
4.788 |
9 |
9 |
25.564 |
Decomposition to irreducible representations
Column with number of nonvanshing force constants highlighted
| Force field |
Ag |
Eg*
|
Tg |
Au |
Eu*
|
Tu |
| linear |
3 |
3 |
5 |
1 |
1 |
8 |
| quadratic |
68 |
68 |
158 |
49 |
49 |
167 |
| cubic |
1.019 |
1.011 |
2.877 |
938 |
929 |
2.986 |
| quartic |
13.449 |
13.449 |
39.336 |
12.993 |
12.993 |
39.639 |
| quintic |
145.710 |
145.710 |
433.890 |
144.099 |
144.099 |
435.888 |
| sextic |
1.356.710 |
1.356.629 |
4.055.548 |
1.349.914 |
1.349.842 |
4.060.742 |
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 T
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(Ag) ≤ i ≤ pos(Tu) |
|---|
| ..6. |
AgAg. | ..9. |
EgEg. | ..15. |
TgTg. | ..1. |
AuAu. | ..1. |
EuEu. | ..36. |
TuTu. | | |
| |
| |
| |
| Subtotal: 68 / 6 / 6 |
| Irrep combinations (i,j) with indices: pos(Ag) ≤ i ≤ j ≤ pos(Tu) |
|---|
| Subtotal: 0 / 0 / 15 |
| Total: 68 / 6 / 21 |
Contributions to nonvanishing cubic force field constants
| Irrep combinations (i,i,i) with indices: pos(Ag) ≤ i ≤ pos(Tu) |
|---|
| ..10. |
AgAgAg. | ..20. |
EgEgEg. | ..45. |
TgTgTg. | | |
| |
| |
| |
| |
| |
| |
| Subtotal: 75 / 3 / 6 |
| Irrep combinations (i,i,j) (i,j,j) with indices: pos(Ag) ≤ i ≤ j ≤ pos(Tu) |
|---|
| ..27. |
AgEgEg. | ..45. |
AgTgTg. | ..3. |
AgAuAu. | ..3. |
AgEuEu. | ..108. |
AgTuTu. | ..90. |
EgTgTg. | ..6. |
EgEuEu. | ..216. |
EgTuTu. | ..320. |
TgTuTu. | | |
| Subtotal: 818 / 9 / 30 |
| Irrep combinations (i,j,k) with indices: pos(Ag) ≤ i ≤ j ≤ k ≤ pos(Tu) |
|---|
| ..6. |
EgAuEu. | ..40. |
TgAuTu. | ..80. |
TgEuTu. | | |
| |
| |
| |
| |
| |
| |
| Subtotal: 126 / 3 / 20 |
| Total: 1.019 / 15 / 56 |
Contributions to nonvanishing quartic force field constants
| Irrep combinations (i,i,i,i) with indices: pos(Ag) ≤ i ≤ pos(Tu) |
|---|
| ..15. |
AgAgAgAg. | ..36. |
EgEgEgEg. | ..295. |
TgTgTgTg. | ..1. |
AuAuAuAu. | ..1. |
EuEuEuEu. | ..1.626. |
TuTuTuTu. | | |
| |
| |
| |
| Subtotal: 1.974 / 6 / 6 |
| Irrep combinations (i,i,i,j) (i,j,j,j) with indices: pos(Ag) ≤ i ≤ j ≤ pos(Tu) |
|---|
| ..60. |
AgEgEgEg. | ..135. |
AgTgTgTg. | ..240. |
EgTgTgTg. | ..2. |
AuEuEuEu. | ..176. |
AuTuTuTu. | ..336. |
EuTuTuTu. | | |
| |
| |
| |
| Subtotal: 949 / 6 / 30 |
| Irrep combinations (i,i,j,j) with indices: pos(Ag) ≤ i ≤ j ≤ pos(Tu) |
|---|
| ..54. |
AgAgEgEg. | ..90. |
AgAgTgTg. | ..6. |
AgAgAuAu. | ..6. |
AgAgEuEu. | ..216. |
AgAgTuTu. | ..315. |
EgEgTgTg. | ..9. |
EgEgAuAu. | ..21. |
EgEgEuEu. | ..756. |
EgEgTuTu. | ..15. |
TgTgAuAu. |
| ..45. |
TgTgEuEu. | ..3.220. |
TgTgTuTu. | ..1. |
AuAuEuEu. | ..36. |
AuAuTuTu. | ..108. |
EuEuTuTu. | | |
| |
| |
| |
| |
| Subtotal: 4.898 / 15 / 15 |
| Irrep combinations (i,i,j,k) (i,j,j,k) (i,j,k,k) with indices: pos(Ag) ≤ i ≤ j ≤ k ≤ pos(Tu) |
|---|
| ..12. |
EgEgAuEu. | ..30. |
TgTgAuEu. | ..200. |
TgTgAuTu. | ..400. |
TgTgEuTu. | ..270. |
AgEgTgTg. | ..18. |
AgEgEuEu. | ..648. |
AgEgTuTu. | ..960. |
AgTgTuTu. | ..1.920. |
EgTgTuTu. | ..72. |
AuEuTuTu. |
| Subtotal: 4.530 / 10 / 60 |
| Irrep combinations (i,j,k,l) with indices: pos(Ag) ≤ i ≤ j ≤ k ≤ l ≤ pos(Tu) |
|---|
| ..18. |
AgEgAuEu. | ..120. |
AgTgAuTu. | ..240. |
AgTgEuTu. | ..240. |
EgTgAuTu. | ..480. |
EgTgEuTu. | | |
| |
| |
| |
| |
| Subtotal: 1.098 / 5 / 15 |
| Total: 13.449 / 42 / 126 |
Calculate contributions to
Last update November, 13th 2023 by A. Gelessus, Impressum, Datenschutzerklärung/DataPrivacyStatement