Characters of representations for molecular motions
| Motion |
E |
8C3 |
3C2 |
6S4 |
6σd |
| Cartesian 3N |
51 |
0 |
-1 |
-1 |
5 |
| Translation (x,y,z) |
3 |
0 |
-1 |
-1 |
1 |
| Rotation (Rx,Ry,Rz) |
3 |
0 |
-1 |
1 |
-1 |
| Vibration |
45 |
0 |
1 |
-1 |
5 |
Decomposition to irreducible representations
| Motion |
A1 |
A2 |
E |
T1 |
T2 |
Total |
| Cartesian 3N |
3 |
1 |
4 |
5 |
8 |
21 |
| Translation (x,y,z) |
0 |
0 |
0 |
0 |
1 |
1 |
| Rotation (Rx,Ry,Rz) |
0 |
0 |
0 |
1 |
0 |
1 |
| Vibration |
3 |
1 |
4 |
4 |
7 |
19 |
Molecular parameter
| Number of Atoms (N) |
17
|
| Number of internal coordinates |
45
|
| Number of independant internal coordinates |
3
|
| Number of vibrational modes |
19
|
Force field analysis
Allowed / forbidden vibronational transitions
| Operator |
A1 |
A2 |
E |
T1 |
T2 |
Total |
| Linear (IR) |
3 |
1 |
4 |
4 |
7 |
7 / 12 |
| Quadratic (Raman) |
3 |
1 |
4 |
4 |
7 |
14 / 5 |
| IR + Raman |
- - - - |
1 |
- - - - |
4 |
7 |
7 / 5 |
Characters of force fields
(Symmetric powers of vibration representation)
| Force field |
E |
8C3 |
3C2 |
6S4 |
6σd |
| linear |
45 |
0 |
1 |
-1 |
5 |
| quadratic |
1.035 |
0 |
23 |
1 |
35 |
| cubic |
16.215 |
15 |
23 |
-1 |
135 |
| quartic |
194.580 |
0 |
276 |
12 |
580 |
| quintic |
1.906.884 |
0 |
276 |
-12 |
1.876 |
| sextic |
15.890.700 |
120 |
2.300 |
12 |
6.300 |
Decomposition to irreducible representations
Column with number of nonvanshing force constants highlighted
| Force field |
A1 |
A2 |
E |
T1 |
T2 |
| linear |
3 |
1 |
4 |
4 |
7 |
| quadratic |
55 |
37 |
92 |
118 |
135 |
| cubic |
717 |
650 |
1.352 |
1.990 |
2.058 |
| quartic |
8.290 |
7.994 |
16.284 |
24.146 |
24.430 |
| quintic |
79.954 |
79.022 |
158.976 |
237.854 |
238.798 |
| sextic |
664.018 |
660.862 |
1.324.760 |
1.984.478 |
1.987.622 |
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
d
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) |
|---|
| ..6. |
A1A1. | ..1. |
A2A2. | ..10. |
EE. | ..10. |
T1T1. | ..28. |
T2T2. | | |
| |
| |
| |
| |
| Subtotal: 55 / 5 / 5 |
| Irrep combinations (i,j) with indices: pos(A1) ≤ i ≤ j ≤ pos(T2) |
|---|
| Subtotal: 0 / 0 / 10 |
| Total: 55 / 5 / 15 |
Contributions to nonvanishing cubic force field constants
| Irrep combinations (i,i,i) with indices: pos(A1) ≤ i ≤ pos(T2) |
|---|
| ..10. |
A1A1A1. | ..20. |
EEE. | ..4. |
T1T1T1. | ..84. |
T2T2T2. | | |
| |
| |
| |
| |
| |
| Subtotal: 118 / 4 / 5 |
| Irrep combinations (i,i,j) (i,j,j) with indices: pos(A1) ≤ i ≤ j ≤ pos(T2) |
|---|
| ..70. |
T1T1T2. | ..3. |
A1A2A2. | ..30. |
A1EE. | ..30. |
A1T1T1. | ..84. |
A1T2T2. | ..6. |
A2EE. | ..40. |
ET1T1. | ..112. |
ET2T2. | ..84. |
T1T2T2. | | |
| Subtotal: 459 / 9 / 20 |
| Irrep combinations (i,j,k) with indices: pos(A1) ≤ i ≤ j ≤ k ≤ pos(T2) |
|---|
| ..28. |
A2T1T2. | ..112. |
ET1T2. | | |
| |
| |
| |
| |
| |
| |
| |
| Subtotal: 140 / 2 / 10 |
| Total: 717 / 15 / 35 |
Contributions to nonvanishing quartic force field constants
| Irrep combinations (i,i,i,i) with indices: pos(A1) ≤ i ≤ pos(T2) |
|---|
| ..15. |
A1A1A1A1. | ..1. |
A2A2A2A2. | ..55. |
EEEE. | ..90. |
T1T1T1T1. | ..616. |
T2T2T2T2. | | |
| |
| |
| |
| |
| Subtotal: 777 / 5 / 5 |
| Irrep combinations (i,i,i,j) (i,j,j,j) with indices: pos(A1) ≤ i ≤ j ≤ pos(T2) |
|---|
| ..280. |
T1T1T1T2. | ..60. |
A1EEE. | ..12. |
A1T1T1T1. | ..252. |
A1T2T2T2. | ..20. |
A2EEE. | ..20. |
A2T1T1T1. | ..35. |
A2T2T2T2. | ..80. |
ET1T1T1. | ..448. |
ET2T2T2. | ..784. |
T1T2T2T2. |
| Subtotal: 1.991 / 10 / 20 |
| Irrep combinations (i,i,j,j) with indices: pos(A1) ≤ i ≤ j ≤ pos(T2) |
|---|
| ..6. |
A1A1A2A2. | ..60. |
A1A1EE. | ..60. |
A1A1T1T1. | ..168. |
A1A1T2T2. | ..10. |
A2A2EE. | ..10. |
A2A2T1T1. | ..28. |
A2A2T2T2. | ..200. |
EET1T1. | ..560. |
EET2T2. | ..966. |
T1T1T2T2. |
| Subtotal: 2.068 / 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) |
|---|
| ..448. |
EET1T2. | ..210. |
A1T1T1T2. | ..42. |
A2T1T1T2. | ..448. |
ET1T1T2. | ..18. |
A1A2EE. | ..120. |
A1ET1T1. | ..336. |
A1ET2T2. | ..252. |
A1T1T2T2. | ..40. |
A2ET1T1. | ..112. |
A2ET2T2. |
| ..112. |
A2T1T2T2. | ..784. |
ET1T2T2. | | |
| |
| |
| |
| |
| |
| |
| |
| Subtotal: 2.922 / 12 / 30 |
| Irrep combinations (i,j,k,l) with indices: pos(A1) ≤ i ≤ j ≤ k ≤ l ≤ pos(T2) |
|---|
| ..84. |
A1A2T1T2. | ..336. |
A1ET1T2. | ..112. |
A2ET1T2. | | |
| |
| |
| |
| |
| |
| |
| Subtotal: 532 / 3 / 5 |
| Total: 8.290 / 40 / 70 |
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