Characters of representations for molecular motions
| Motion |
E |
8C3 |
3C2 |
6S4 |
6σd |
| Cartesian 3N |
84 |
0 |
0 |
0 |
6 |
| Translation (x,y,z) |
3 |
0 |
-1 |
-1 |
1 |
| Rotation (Rx,Ry,Rz) |
3 |
0 |
-1 |
1 |
-1 |
| Vibration |
78 |
0 |
2 |
0 |
6 |
Decomposition to irreducible representations
| Motion |
A1 |
A2 |
E |
T1 |
T2 |
Total |
| Cartesian 3N |
5 |
2 |
7 |
9 |
12 |
35 |
| Translation (x,y,z) |
0 |
0 |
0 |
0 |
1 |
1 |
| Rotation (Rx,Ry,Rz) |
0 |
0 |
0 |
1 |
0 |
1 |
| Vibration |
5 |
2 |
7 |
8 |
11 |
33 |
Molecular parameter
| Number of Atoms (N) |
28
|
| Number of internal coordinates |
78
|
| Number of independant internal coordinates |
5
|
| Number of vibrational modes |
33
|
Force field analysis
Allowed / forbidden vibronational transitions
| Operator |
A1 |
A2 |
E |
T1 |
T2 |
Total |
| Linear (IR) |
5 |
2 |
7 |
8 |
11 |
11 / 22 |
| Quadratic (Raman) |
5 |
2 |
7 |
8 |
11 |
23 / 10 |
| IR + Raman |
- - - - |
2 |
- - - - |
8 |
11 |
11 / 10 |
Characters of force fields
(Symmetric powers of vibration representation)
| Force field |
E |
8C3 |
3C2 |
6S4 |
6σd |
| linear |
78 |
0 |
2 |
0 |
6 |
| quadratic |
3.081 |
0 |
41 |
1 |
57 |
| cubic |
82.160 |
26 |
80 |
0 |
272 |
| quartic |
1.663.740 |
0 |
860 |
20 |
1.548 |
| quintic |
27.285.336 |
0 |
1.640 |
0 |
6.264 |
| sextic |
377.447.148 |
351 |
12.300 |
20 |
27.420 |
Decomposition to irreducible representations
Column with number of nonvanshing force constants highlighted
| Force field |
A1 |
A2 |
E |
T1 |
T2 |
| linear |
5 |
2 |
7 |
8 |
11 |
| quadratic |
148 |
119 |
267 |
366 |
394 |
| cubic |
3.510 |
3.374 |
6.858 |
10.192 |
10.328 |
| quartic |
69.822 |
69.038 |
138.860 |
207.478 |
208.242 |
| quintic |
1.138.660 |
1.135.528 |
2.274.188 |
3.408.896 |
3.412.028 |
| sextic |
15.735.479 |
15.721.759 |
31.456.887 |
47.172.506 |
47.186.206 |
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) |
|---|
| ..15. |
A1A1. | ..3. |
A2A2. | ..28. |
EE. | ..36. |
T1T1. | ..66. |
T2T2. | | |
| |
| |
| |
| |
| Subtotal: 148 / 5 / 5 |
| Irrep combinations (i,j) with indices: pos(A1) ≤ i ≤ j ≤ pos(T2) |
|---|
| Subtotal: 0 / 0 / 10 |
| Total: 148 / 5 / 15 |
Contributions to nonvanishing cubic force field constants
| Irrep combinations (i,i,i) with indices: pos(A1) ≤ i ≤ pos(T2) |
|---|
| ..35. |
A1A1A1. | ..84. |
EEE. | ..56. |
T1T1T1. | ..286. |
T2T2T2. | | |
| |
| |
| |
| |
| |
| Subtotal: 461 / 4 / 5 |
| Irrep combinations (i,i,j) (i,j,j) with indices: pos(A1) ≤ i ≤ j ≤ pos(T2) |
|---|
| ..396. |
T1T1T2. | ..15. |
A1A2A2. | ..140. |
A1EE. | ..180. |
A1T1T1. | ..330. |
A1T2T2. | ..42. |
A2EE. | ..252. |
ET1T1. | ..462. |
ET2T2. | ..440. |
T1T2T2. | | |
| Subtotal: 2.257 / 9 / 20 |
| Irrep combinations (i,j,k) with indices: pos(A1) ≤ i ≤ j ≤ k ≤ pos(T2) |
|---|
| ..176. |
A2T1T2. | ..616. |
ET1T2. | | |
| |
| |
| |
| |
| |
| |
| |
| Subtotal: 792 / 2 / 10 |
| Total: 3.510 / 15 / 35 |
Contributions to nonvanishing quartic force field constants
| Irrep combinations (i,i,i,i) with indices: pos(A1) ≤ i ≤ pos(T2) |
|---|
| ..70. |
A1A1A1A1. | ..5. |
A2A2A2A2. | ..406. |
EEEE. | ..996. |
T1T1T1T1. | ..3.212. |
T2T2T2T2. | | |
| |
| |
| |
| |
| Subtotal: 4.689 / 5 / 5 |
| Irrep combinations (i,i,i,j) (i,j,j,j) with indices: pos(A1) ≤ i ≤ j ≤ pos(T2) |
|---|
| ..3.168. |
T1T1T1T2. | ..420. |
A1EEE. | ..280. |
A1T1T1T1. | ..1.430. |
A1T2T2T2. | ..168. |
A2EEE. | ..240. |
A2T1T1T1. | ..330. |
A2T2T2T2. | ..1.176. |
ET1T1T1. | ..3.080. |
ET2T2T2. | ..5.808. |
T1T2T2T2. |
| Subtotal: 16.100 / 10 / 20 |
| Irrep combinations (i,i,j,j) with indices: pos(A1) ≤ i ≤ j ≤ pos(T2) |
|---|
| ..45. |
A1A1A2A2. | ..420. |
A1A1EE. | ..540. |
A1A1T1T1. | ..990. |
A1A1T2T2. | ..84. |
A2A2EE. | ..108. |
A2A2T1T1. | ..198. |
A2A2T2T2. | ..2.016. |
EET1T1. | ..3.696. |
EET2T2. | ..8.668. |
T1T1T2T2. |
| Subtotal: 16.765 / 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) |
|---|
| ..4.312. |
EET1T2. | ..1.980. |
A1T1T1T2. | ..616. |
A2T1T1T2. | ..4.928. |
ET1T1T2. | ..210. |
A1A2EE. | ..1.260. |
A1ET1T1. | ..2.310. |
A1ET2T2. | ..2.200. |
A1T1T2T2. | ..504. |
A2ET1T1. | ..924. |
A2ET2T2. |
| ..1.056. |
A2T1T2T2. | ..6.776. |
ET1T2T2. | | |
| |
| |
| |
| |
| |
| |
| |
| Subtotal: 27.076 / 12 / 30 |
| Irrep combinations (i,j,k,l) with indices: pos(A1) ≤ i ≤ j ≤ k ≤ l ≤ pos(T2) |
|---|
| ..880. |
A1A2T1T2. | ..3.080. |
A1ET1T2. | ..1.232. |
A2ET1T2. | | |
| |
| |
| |
| |
| |
| |
| Subtotal: 5.192 / 3 / 5 |
| Total: 69.822 / 40 / 70 |
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