Characters of representations for molecular motions
| Motion |
E |
C4 |
C2 |
(C4)3 |
i |
(S4)3 |
σh |
S4 |
| Cartesian 3N |
36 |
4 |
-4 |
4 |
0 |
0 |
8 |
0 |
| Translation (x,y,z) |
3 |
1 |
-1 |
1 |
-3 |
-1 |
1 |
-1 |
| Rotation (Rx,Ry,Rz) |
3 |
1 |
-1 |
1 |
3 |
1 |
-1 |
1 |
| Vibration |
30 |
2 |
-2 |
2 |
0 |
0 |
8 |
0 |
Decomposition to irreducible representations
| Motion |
Ag |
Bg |
Eg*
|
Au |
Bu |
Eu*
|
Total |
| Cartesian 3N |
6 |
4 |
4 |
4 |
2 |
6 |
26 |
| Translation (x,y,z) |
0 |
0 |
0 |
1 |
0 |
1 |
2 |
| Rotation (Rx,Ry,Rz) |
1 |
0 |
1 |
0 |
0 |
0 |
2 |
| Vibration |
5 |
4 |
3 |
3 |
2 |
5 |
22 |
Molecular parameter
| Number of Atoms (N) |
12
|
| Number of internal coordinates |
30
|
| Number of independant internal coordinates |
5
|
| Number of vibrational modes |
22
|
Force field analysis
Allowed / forbidden vibronational transitions
| Operator |
Ag |
Bg |
Eg*
|
Au |
Bu |
Eu*
|
Total |
| Linear (IR) |
5 |
4 |
3 |
3 |
2 |
5 |
8 / 14 |
| Quadratic (Raman) |
5 |
4 |
3 |
3 |
2 |
5 |
12 / 10 |
| IR + Raman |
- - - - |
- - - - |
- - - - |
- - - - |
2 |
- - - - |
0* / 2 |
* Parity Mutual Exclusion Principle
Characters of force fields
(Symmetric powers of vibration representation)
| Force field |
E |
C4 |
C2 |
(C4)3 |
i |
(S4)3 |
σh |
S4 |
| linear |
30 |
2 |
-2 |
2 |
0 |
0 |
8 |
0 |
| quadratic |
465 |
1 |
17 |
1 |
15 |
-1 |
47 |
-1 |
| cubic |
4.960 |
0 |
-32 |
0 |
0 |
0 |
208 |
0 |
| quartic |
40.920 |
8 |
152 |
8 |
120 |
8 |
792 |
8 |
| quintic |
278.256 |
16 |
-272 |
16 |
0 |
0 |
2.640 |
0 |
| sextic |
1.623.160 |
8 |
952 |
8 |
680 |
-8 |
8.008 |
-8 |
Decomposition to irreducible representations
Column with number of nonvanshing force constants highlighted
| Force field |
Ag |
Bg |
Eg*
|
Au |
Bu |
Eu*
|
| linear |
5 |
4 |
3 |
3 |
2 |
5 |
| quadratic |
68 |
68 |
52 |
53 |
52 |
60 |
| cubic |
642 |
642 |
598 |
590 |
590 |
650 |
| quartic |
5.252 |
5.244 |
5.012 |
5.020 |
5.020 |
5.180 |
| quintic |
35.082 |
35.074 |
34.486 |
34.422 |
34.414 |
35.146 |
| sextic |
204.100 |
204.100 |
201.860 |
201.932 |
201.924 |
203.692 |
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 C
4h
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(Eu) |
|---|
| ..15. |
AgAg. | ..10. |
BgBg. | ..9. |
EgEg. | ..6. |
AuAu. | ..3. |
BuBu. | ..25. |
EuEu. | | |
| |
| |
| |
| Subtotal: 68 / 6 / 6 |
| Irrep combinations (i,j) with indices: pos(Ag) ≤ i ≤ j ≤ pos(Eu) |
|---|
| 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(Eu) |
|---|
| ..35. |
AgAgAg. | | |
| |
| |
| |
| |
| |
| |
| |
| |
| Subtotal: 35 / 1 / 6 |
| Irrep combinations (i,i,j) (i,j,j) with indices: pos(Ag) ≤ i ≤ j ≤ pos(Eu) |
|---|
| ..50. |
AgBgBg. | ..45. |
AgEgEg. | ..30. |
AgAuAu. | ..15. |
AgBuBu. | ..125. |
AgEuEu. | ..48. |
BgEgEg. | ..120. |
BgEuEu. | | |
| |
| |
| Subtotal: 433 / 7 / 30 |
| Irrep combinations (i,j,k) with indices: pos(Ag) ≤ i ≤ j ≤ k ≤ pos(Eu) |
|---|
| ..24. |
BgAuBu. | ..90. |
EgAuEu. | ..60. |
EgBuEu. | | |
| |
| |
| |
| |
| |
| |
| Subtotal: 174 / 3 / 20 |
| Total: 642 / 11 / 56 |
Contributions to nonvanishing quartic force field constants
| Irrep combinations (i,i,i,i) with indices: pos(Ag) ≤ i ≤ pos(Eu) |
|---|
| ..70. |
AgAgAgAg. | ..35. |
BgBgBgBg. | ..66. |
EgEgEgEg. | ..15. |
AuAuAuAu. | ..5. |
BuBuBuBu. | ..365. |
EuEuEuEu. | | |
| |
| |
| |
| Subtotal: 556 / 6 / 6 |
| Irrep combinations (i,i,i,j) (i,j,j,j) with indices: pos(Ag) ≤ i ≤ j ≤ pos(Eu) |
|---|
| Subtotal: 0 / 0 / 30 |
| Irrep combinations (i,i,j,j) with indices: pos(Ag) ≤ i ≤ j ≤ pos(Eu) |
|---|
| ..150. |
AgAgBgBg. | ..135. |
AgAgEgEg. | ..90. |
AgAgAuAu. | ..45. |
AgAgBuBu. | ..375. |
AgAgEuEu. | ..90. |
BgBgEgEg. | ..60. |
BgBgAuAu. | ..30. |
BgBgBuBu. | ..250. |
BgBgEuEu. | ..54. |
EgEgAuAu. |
| ..27. |
EgEgBuBu. | ..585. |
EgEgEuEu. | ..18. |
AuAuBuBu. | ..150. |
AuAuEuEu. | ..75. |
BuBuEuEu. | | |
| |
| |
| |
| |
| Subtotal: 2.134 / 15 / 15 |
| Irrep combinations (i,i,j,k) (i,j,j,k) (i,j,k,k) with indices: pos(Ag) ≤ i ≤ j ≤ k ≤ pos(Eu) |
|---|
| ..72. |
EgEgAuBu. | ..240. |
AgBgEgEg. | ..600. |
AgBgEuEu. | ..180. |
AuBuEuEu. | | |
| |
| |
| |
| |
| |
| Subtotal: 1.092 / 4 / 60 |
| Irrep combinations (i,j,k,l) with indices: pos(Ag) ≤ i ≤ j ≤ k ≤ l ≤ pos(Eu) |
|---|
| ..120. |
AgBgAuBu. | ..450. |
AgEgAuEu. | ..300. |
AgEgBuEu. | ..360. |
BgEgAuEu. | ..240. |
BgEgBuEu. | | |
| |
| |
| |
| |
| Subtotal: 1.470 / 5 / 15 |
| Total: 5.252 / 30 / 126 |
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Last update November, 13th 2023 by A. Gelessus, Impressum, Datenschutzerklärung/DataPrivacyStatement