Characters of representations for molecular motions
| Motion |
E |
4C3 |
4(C3)2 |
3C2 |
i |
4(S6)5 |
4S6 |
3σh |
| Cartesian 3N |
48 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
| 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 |
42 |
0 |
0 |
2 |
0 |
0 |
0 |
0 |
Decomposition to irreducible representations
| Motion |
Ag |
Eg*
|
Tg |
Au |
Eu*
|
Tu |
Total |
| Cartesian 3N |
2 |
2 |
6 |
2 |
2 |
6 |
20 |
| Translation (x,y,z) |
0 |
0 |
0 |
0 |
0 |
1 |
1 |
| Rotation (Rx,Ry,Rz) |
0 |
0 |
1 |
0 |
0 |
0 |
1 |
| Vibration |
2 |
2 |
5 |
2 |
2 |
5 |
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 |
Ag |
Eg*
|
Tg |
Au |
Eu*
|
Tu |
Total |
| Linear (IR) |
2 |
2 |
5 |
2 |
2 |
5 |
5 / 13 |
| Quadratic (Raman) |
2 |
2 |
5 |
2 |
2 |
5 |
9 / 9 |
| IR + Raman |
- - - - |
- - - - |
- - - - |
2 |
2 |
- - - - |
0* / 4 |
* 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 |
42 |
0 |
0 |
2 |
0 |
0 |
0 |
0 |
| quadratic |
903 |
0 |
0 |
23 |
21 |
0 |
0 |
21 |
| cubic |
13.244 |
14 |
14 |
44 |
0 |
0 |
0 |
0 |
| quartic |
148.995 |
0 |
0 |
275 |
231 |
0 |
0 |
231 |
| quintic |
1.370.754 |
0 |
0 |
506 |
0 |
0 |
0 |
0 |
| sextic |
10.737.573 |
105 |
105 |
2.277 |
1.771 |
7 |
7 |
1.771 |
Decomposition to irreducible representations
Column with number of nonvanshing force constants highlighted
| Force field |
Ag |
Eg*
|
Tg |
Au |
Eu*
|
Tu |
| linear |
2 |
2 |
5 |
2 |
2 |
5 |
| quadratic |
44 |
44 |
110 |
37 |
37 |
110 |
| cubic |
562 |
555 |
1.650 |
562 |
555 |
1.650 |
| quartic |
6.281 |
6.281 |
18.590 |
6.204 |
6.204 |
18.590 |
| quintic |
57.178 |
57.178 |
171.281 |
57.178 |
57.178 |
171.281 |
| sextic |
448.016 |
447.960 |
1.341.912 |
447.421 |
447.372 |
1.341.912 |
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) |
|---|
| ..3. |
AgAg. | ..4. |
EgEg. | ..15. |
TgTg. | ..3. |
AuAu. | ..4. |
EuEu. | ..15. |
TuTu. | | |
| |
| |
| |
| Subtotal: 44 / 6 / 6 |
| Irrep combinations (i,j) with indices: pos(Ag) ≤ i ≤ j ≤ pos(Tu) |
|---|
| Subtotal: 0 / 0 / 15 |
| Total: 44 / 6 / 21 |
Contributions to nonvanishing cubic force field constants
| Irrep combinations (i,i,i) with indices: pos(Ag) ≤ i ≤ pos(Tu) |
|---|
| ..4. |
AgAgAg. | ..8. |
EgEgEg. | ..45. |
TgTgTg. | | |
| |
| |
| |
| |
| |
| |
| Subtotal: 57 / 3 / 6 |
| Irrep combinations (i,i,j) (i,j,j) with indices: pos(Ag) ≤ i ≤ j ≤ pos(Tu) |
|---|
| ..8. |
AgEgEg. | ..30. |
AgTgTg. | ..6. |
AgAuAu. | ..8. |
AgEuEu. | ..30. |
AgTuTu. | ..60. |
EgTgTg. | ..12. |
EgEuEu. | ..60. |
EgTuTu. | ..125. |
TgTuTu. | | |
| Subtotal: 339 / 9 / 30 |
| Irrep combinations (i,j,k) with indices: pos(Ag) ≤ i ≤ j ≤ k ≤ pos(Tu) |
|---|
| ..16. |
EgAuEu. | ..50. |
TgAuTu. | ..100. |
TgEuTu. | | |
| |
| |
| |
| |
| |
| |
| Subtotal: 166 / 3 / 20 |
| Total: 562 / 15 / 56 |
Contributions to nonvanishing quartic force field constants
| Irrep combinations (i,i,i,i) with indices: pos(Ag) ≤ i ≤ pos(Tu) |
|---|
| ..5. |
AgAgAgAg. | ..9. |
EgEgEgEg. | ..295. |
TgTgTgTg. | ..5. |
AuAuAuAu. | ..9. |
EuEuEuEu. | ..295. |
TuTuTuTu. | | |
| |
| |
| |
| Subtotal: 618 / 6 / 6 |
| Irrep combinations (i,i,i,j) (i,j,j,j) with indices: pos(Ag) ≤ i ≤ j ≤ pos(Tu) |
|---|
| ..16. |
AgEgEgEg. | ..90. |
AgTgTgTg. | ..160. |
EgTgTgTg. | ..16. |
AuEuEuEu. | ..90. |
AuTuTuTu. | ..160. |
EuTuTuTu. | | |
| |
| |
| |
| Subtotal: 532 / 6 / 30 |
| Irrep combinations (i,i,j,j) with indices: pos(Ag) ≤ i ≤ j ≤ pos(Tu) |
|---|
| ..12. |
AgAgEgEg. | ..45. |
AgAgTgTg. | ..9. |
AgAgAuAu. | ..12. |
AgAgEuEu. | ..45. |
AgAgTuTu. | ..150. |
EgEgTgTg. | ..12. |
EgEgAuAu. | ..34. |
EgEgEuEu. | ..150. |
EgEgTuTu. | ..45. |
TgTgAuAu. |
| ..150. |
TgTgEuEu. | ..1.300. |
TgTgTuTu. | ..12. |
AuAuEuEu. | ..45. |
AuAuTuTu. | ..150. |
EuEuTuTu. | | |
| |
| |
| |
| |
| Subtotal: 2.171 / 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) |
|---|
| ..24. |
EgEgAuEu. | ..120. |
TgTgAuEu. | ..250. |
TgTgAuTu. | ..500. |
TgTgEuTu. | ..120. |
AgEgTgTg. | ..24. |
AgEgEuEu. | ..120. |
AgEgTuTu. | ..250. |
AgTgTuTu. | ..500. |
EgTgTuTu. | ..120. |
AuEuTuTu. |
| Subtotal: 2.028 / 10 / 60 |
| Irrep combinations (i,j,k,l) with indices: pos(Ag) ≤ i ≤ j ≤ k ≤ l ≤ pos(Tu) |
|---|
| ..32. |
AgEgAuEu. | ..100. |
AgTgAuTu. | ..200. |
AgTgEuTu. | ..200. |
EgTgAuTu. | ..400. |
EgTgEuTu. | | |
| |
| |
| |
| |
| Subtotal: 932 / 5 / 15 |
| Total: 6.281 / 42 / 126 |
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Last update November, 13th 2023 by A. Gelessus, Impressum, Datenschutzerklärung/DataPrivacyStatement