Characters of representations for molecular motions
Motion |
E |
C3 |
(C3)2 |
i |
(S6)5 |
S6 |
Cartesian 3N |
21 |
0 |
0 |
-3 |
0 |
0 |
Translation (x,y,z) |
3 |
0 |
0 |
-3 |
0 |
0 |
Rotation (Rx,Ry,Rz) |
3 |
0 |
0 |
3 |
0 |
0 |
Vibration |
15 |
0 |
0 |
-3 |
0 |
0 |
Decomposition to irreducible representations
Motion |
Ag |
Eg*
|
Au |
Eu*
|
Total |
Cartesian 3N |
3 |
3 |
4 |
4 |
14 |
Translation (x,y,z) |
0 |
0 |
1 |
1 |
2 |
Rotation (Rx,Ry,Rz) |
1 |
1 |
0 |
0 |
2 |
Vibration |
2 |
2 |
3 |
3 |
10 |
Molecular parameter
Number of Atoms (N) |
7
|
Number of internal coordinates |
15
|
Number of independant internal coordinates |
2
|
Number of vibrational modes |
10
|
Force field analysis
Allowed / forbidden vibronational transitions
Operator |
Ag |
Eg*
|
Au |
Eu*
|
Total |
Linear (IR) |
2 |
2 |
3 |
3 |
6 / 4 |
Quadratic (Raman) |
2 |
2 |
3 |
3 |
4 / 6 |
IR + Raman |
- - - - |
- - - - |
- - - - |
- - - - |
0* / 0 |
* Parity Mutual Exclusion Principle
Characters of force fields
(Symmetric powers of vibration representation)
Force field |
E |
C3 |
(C3)2 |
i |
(S6)5 |
S6 |
linear |
15 |
0 |
0 |
-3 |
0 |
0 |
quadratic |
120 |
0 |
0 |
12 |
0 |
0 |
cubic |
680 |
5 |
5 |
-28 |
-1 |
-1 |
quartic |
3.060 |
0 |
0 |
72 |
0 |
0 |
quintic |
11.628 |
0 |
0 |
-144 |
0 |
0 |
sextic |
38.760 |
15 |
15 |
300 |
3 |
3 |
Decomposition to irreducible representations
Column with number of nonvanshing force constants highlighted
Force field |
Ag |
Eg*
|
Au |
Eu*
|
linear |
2 |
2 |
3 |
3 |
quadratic |
22 |
22 |
18 |
18 |
cubic |
110 |
108 |
120 |
117 |
quartic |
522 |
522 |
498 |
498 |
quintic |
1.914 |
1.914 |
1.962 |
1.962 |
sextic |
6.516 |
6.507 |
6.414 |
6.408 |
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 S
6
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) |
..3. |
AgAg. | ..4. |
EgEg. | ..6. |
AuAu. | ..9. |
EuEu. | | |
| |
| |
| |
| |
| |
Subtotal: 22 / 4 / 4 |
Irrep combinations (i,j) with indices: pos(Ag) ≤ i ≤ j ≤ pos(Eu) |
Subtotal: 0 / 0 / 6 |
Total: 22 / 4 / 10 |
Contributions to nonvanishing cubic force field constants
Irrep combinations (i,i,i) with indices: pos(Ag) ≤ i ≤ pos(Eu) |
..4. |
AgAgAg. | ..8. |
EgEgEg. | | |
| |
| |
| |
| |
| |
| |
| |
Subtotal: 12 / 2 / 4 |
Irrep combinations (i,i,j) (i,j,j) with indices: pos(Ag) ≤ i ≤ j ≤ pos(Eu) |
..8. |
AgEgEg. | ..12. |
AgAuAu. | ..18. |
AgEuEu. | ..24. |
EgEuEu. | | |
| |
| |
| |
| |
| |
Subtotal: 62 / 4 / 12 |
Irrep combinations (i,j,k) with indices: pos(Ag) ≤ i ≤ j ≤ k ≤ pos(Eu) |
..36. |
EgAuEu. | | |
| |
| |
| |
| |
| |
| |
| |
| |
Subtotal: 36 / 1 / 4 |
Total: 110 / 7 / 20 |
Contributions to nonvanishing quartic force field constants
Irrep combinations (i,i,i,i) with indices: pos(Ag) ≤ i ≤ pos(Eu) |
..5. |
AgAgAgAg. | ..9. |
EgEgEgEg. | ..15. |
AuAuAuAu. | ..36. |
EuEuEuEu. | | |
| |
| |
| |
| |
| |
Subtotal: 65 / 4 / 4 |
Irrep combinations (i,i,i,j) (i,j,j,j) with indices: pos(Ag) ≤ i ≤ j ≤ pos(Eu) |
..16. |
AgEgEgEg. | ..60. |
AuEuEuEu. | | |
| |
| |
| |
| |
| |
| |
| |
Subtotal: 76 / 2 / 12 |
Irrep combinations (i,i,j,j) with indices: pos(Ag) ≤ i ≤ j ≤ pos(Eu) |
..12. |
AgAgEgEg. | ..18. |
AgAgAuAu. | ..27. |
AgAgEuEu. | ..24. |
EgEgAuAu. | ..72. |
EgEgEuEu. | ..54. |
AuAuEuEu. | | |
| |
| |
| |
Subtotal: 207 / 6 / 6 |
Irrep combinations (i,i,j,k) (i,j,j,k) (i,j,k,k) with indices: pos(Ag) ≤ i ≤ j ≤ k ≤ pos(Eu) |
..54. |
EgEgAuEu. | ..48. |
AgEgEuEu. | | |
| |
| |
| |
| |
| |
| |
| |
Subtotal: 102 / 2 / 12 |
Irrep combinations (i,j,k,l) with indices: pos(Ag) ≤ i ≤ j ≤ k ≤ l ≤ pos(Eu) |
..72. |
AgEgAuEu. | | |
| |
| |
| |
| |
| |
| |
| |
| |
Subtotal: 72 / 1 / 1 |
Total: 522 / 15 / 35 |
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Last update November, 13th 2023 by A. Gelessus, Impressum, Datenschutzerklärung/DataPrivacyStatement