Characters of representations for molecular motions
| Motion |
E |
8C3 |
6C2 |
6C4 |
3C2 |
| Cartesian 3N |
48 |
0 |
0 |
0 |
0 |
| Translation (x,y,z) |
3 |
0 |
-1 |
1 |
-1 |
| Rotation (Rx,Ry,Rz) |
3 |
0 |
-1 |
1 |
-1 |
| Vibration |
42 |
0 |
2 |
-2 |
2 |
Decomposition to irreducible representations
| Motion |
A1 |
A2 |
E |
T1 |
T2 |
Total |
| Cartesian 3N |
2 |
2 |
4 |
6 |
6 |
20 |
| Translation (x,y,z) |
0 |
0 |
0 |
1 |
0 |
1 |
| Rotation (Rx,Ry,Rz) |
0 |
0 |
0 |
1 |
0 |
1 |
| Vibration |
2 |
2 |
4 |
4 |
6 |
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 |
A1 |
A2 |
E |
T1 |
T2 |
Total |
| Linear (IR) |
2 |
2 |
4 |
4 |
6 |
4 / 14 |
| Quadratic (Raman) |
2 |
2 |
4 |
4 |
6 |
12 / 6 |
| IR + Raman |
- - - - |
2 |
- - - - |
- - - - |
- - - - |
0 / 2 |
Characters of force fields
(Symmetric powers of vibration representation)
| Force field |
E |
8C3 |
6C2 |
6C4 |
3C2 |
| linear |
42 |
0 |
2 |
-2 |
2 |
| quadratic |
903 |
0 |
23 |
3 |
23 |
| cubic |
13.244 |
14 |
44 |
-4 |
44 |
| quartic |
148.995 |
0 |
275 |
15 |
275 |
| quintic |
1.370.754 |
0 |
506 |
-26 |
506 |
| sextic |
10.737.573 |
105 |
2.277 |
37 |
2.277 |
Decomposition to irreducible representations
Column with number of nonvanshing force constants highlighted
| Force field |
A1 |
A2 |
E |
T1 |
T2 |
| linear |
2 |
2 |
4 |
4 |
6 |
| quadratic |
47 |
34 |
81 |
105 |
115 |
| cubic |
572 |
552 |
1.110 |
1.638 |
1.662 |
| quartic |
6.315 |
6.170 |
12.485 |
18.525 |
18.655 |
| quintic |
57.298 |
57.058 |
114.356 |
171.148 |
171.414 |
| sextic |
448.297 |
447.140 |
895.332 |
1.341.352 |
1.342.472 |
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
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(A1) ≤ i ≤ pos(T2) |
|---|
| ..3. |
A1A1. | ..3. |
A2A2. | ..10. |
EE. | ..10. |
T1T1. | ..21. |
T2T2. | | |
| |
| |
| |
| |
| Subtotal: 47 / 5 / 5 |
| Irrep combinations (i,j) with indices: pos(A1) ≤ i ≤ j ≤ pos(T2) |
|---|
| Subtotal: 0 / 0 / 10 |
| Total: 47 / 5 / 15 |
Contributions to nonvanishing cubic force field constants
| Irrep combinations (i,i,i) with indices: pos(A1) ≤ i ≤ pos(T2) |
|---|
| ..4. |
A1A1A1. | ..20. |
EEE. | ..4. |
T1T1T1. | ..56. |
T2T2T2. | | |
| |
| |
| |
| |
| |
| Subtotal: 84 / 4 / 5 |
| Irrep combinations (i,i,j) (i,j,j) with indices: pos(A1) ≤ i ≤ j ≤ pos(T2) |
|---|
| ..60. |
T1T1T2. | ..6. |
A1A2A2. | ..20. |
A1EE. | ..20. |
A1T1T1. | ..42. |
A1T2T2. | ..12. |
A2EE. | ..40. |
ET1T1. | ..84. |
ET2T2. | ..60. |
T1T2T2. | | |
| Subtotal: 344 / 9 / 20 |
| Irrep combinations (i,j,k) with indices: pos(A1) ≤ i ≤ j ≤ k ≤ pos(T2) |
|---|
| ..48. |
A2T1T2. | ..96. |
ET1T2. | | |
| |
| |
| |
| |
| |
| |
| |
| Subtotal: 144 / 2 / 10 |
| Total: 572 / 15 / 35 |
Contributions to nonvanishing quartic force field constants
| Irrep combinations (i,i,i,i) with indices: pos(A1) ≤ i ≤ pos(T2) |
|---|
| ..5. |
A1A1A1A1. | ..5. |
A2A2A2A2. | ..55. |
EEEE. | ..90. |
T1T1T1T1. | ..357. |
T2T2T2T2. | | |
| |
| |
| |
| |
| Subtotal: 512 / 5 / 5 |
| Irrep combinations (i,i,i,j) (i,j,j,j) with indices: pos(A1) ≤ i ≤ j ≤ pos(T2) |
|---|
| ..240. |
T1T1T1T2. | ..40. |
A1EEE. | ..8. |
A1T1T1T1. | ..112. |
A1T2T2T2. | ..40. |
A2EEE. | ..40. |
A2T1T1T1. | ..40. |
A2T2T2T2. | ..80. |
ET1T1T1. | ..280. |
ET2T2T2. | ..504. |
T1T2T2T2. |
| Subtotal: 1.384 / 10 / 20 |
| Irrep combinations (i,i,j,j) with indices: pos(A1) ≤ i ≤ j ≤ pos(T2) |
|---|
| ..9. |
A1A1A2A2. | ..30. |
A1A1EE. | ..30. |
A1A1T1T1. | ..63. |
A1A1T2T2. | ..30. |
A2A2EE. | ..30. |
A2A2T1T1. | ..63. |
A2A2T2T2. | ..200. |
EET1T1. | ..420. |
EET2T2. | ..720. |
T1T1T2T2. |
| Subtotal: 1.595 / 10 / 10 |
| Irrep combinations (i,i,j,k) (i,j,j,k) (i,j,k,k) with indices: pos(A1) ≤ i ≤ j ≤ k ≤ pos(T2) |
|---|
| ..384. |
EET1T2. | ..120. |
A1T1T1T2. | ..72. |
A2T1T1T2. | ..384. |
ET1T1T2. | ..24. |
A1A2EE. | ..80. |
A1ET1T1. | ..168. |
A1ET2T2. | ..120. |
A1T1T2T2. | ..80. |
A2ET1T1. | ..168. |
A2ET2T2. |
| ..168. |
A2T1T2T2. | ..576. |
ET1T2T2. | | |
| |
| |
| |
| |
| |
| |
| |
| Subtotal: 2.344 / 12 / 30 |
| Irrep combinations (i,j,k,l) with indices: pos(A1) ≤ i ≤ j ≤ k ≤ l ≤ pos(T2) |
|---|
| ..96. |
A1A2T1T2. | ..192. |
A1ET1T2. | ..192. |
A2ET1T2. | | |
| |
| |
| |
| |
| |
| |
| Subtotal: 480 / 3 / 5 |
| Total: 6.315 / 40 / 70 |
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