Characters of representations for molecular motions
| Motion |
E |
8C3 |
3C2 |
6S4 |
6σd |
| Cartesian 3N |
78 |
0 |
-2 |
0 |
8 |
| Translation (x,y,z) |
3 |
0 |
-1 |
-1 |
1 |
| Rotation (Rx,Ry,Rz) |
3 |
0 |
-1 |
1 |
-1 |
| Vibration |
72 |
0 |
0 |
0 |
8 |
Decomposition to irreducible representations
| Motion |
A1 |
A2 |
E |
T1 |
T2 |
Total |
| Cartesian 3N |
5 |
1 |
6 |
8 |
12 |
32 |
| Translation (x,y,z) |
0 |
0 |
0 |
0 |
1 |
1 |
| Rotation (Rx,Ry,Rz) |
0 |
0 |
0 |
1 |
0 |
1 |
| Vibration |
5 |
1 |
6 |
7 |
11 |
30 |
Molecular parameter
| Number of Atoms (N) |
26
|
| Number of internal coordinates |
72
|
| Number of independant internal coordinates |
5
|
| Number of vibrational modes |
30
|
Force field analysis
Allowed / forbidden vibronational transitions
| Operator |
A1 |
A2 |
E |
T1 |
T2 |
Total |
| Linear (IR) |
5 |
1 |
6 |
7 |
11 |
11 / 19 |
| Quadratic (Raman) |
5 |
1 |
6 |
7 |
11 |
22 / 8 |
| IR + Raman |
- - - - |
1 |
- - - - |
7 |
11 |
11 / 8 |
Characters of force fields
(Symmetric powers of vibration representation)
| Force field |
E |
8C3 |
3C2 |
6S4 |
6σd |
| linear |
72 |
0 |
0 |
0 |
8 |
| quadratic |
2.628 |
0 |
36 |
0 |
68 |
| cubic |
64.824 |
24 |
0 |
0 |
376 |
| quartic |
1.215.450 |
0 |
666 |
18 |
2.010 |
| quintic |
18.474.840 |
0 |
0 |
0 |
8.856 |
| sextic |
237.093.780 |
300 |
8.436 |
0 |
37.268 |
Decomposition to irreducible representations
Column with number of nonvanshing force constants highlighted
| Force field |
A1 |
A2 |
E |
T1 |
T2 |
| linear |
5 |
1 |
6 |
7 |
11 |
| quadratic |
131 |
97 |
228 |
307 |
341 |
| cubic |
2.803 |
2.615 |
5.394 |
8.009 |
8.197 |
| quartic |
51.234 |
50.220 |
101.454 |
151.350 |
152.346 |
| quintic |
771.999 |
767.571 |
1.539.570 |
2.307.141 |
2.311.569 |
| sextic |
9.889.379 |
9.870.745 |
19.759.824 |
29.626.351 |
29.644.985 |
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. | ..1. |
A2A2. | ..21. |
EE. | ..28. |
T1T1. | ..66. |
T2T2. | | |
| |
| |
| |
| |
| Subtotal: 131 / 5 / 5 |
| Irrep combinations (i,j) with indices: pos(A1) ≤ i ≤ j ≤ pos(T2) |
|---|
| Subtotal: 0 / 0 / 10 |
| Total: 131 / 5 / 15 |
Contributions to nonvanishing cubic force field constants
| Irrep combinations (i,i,i) with indices: pos(A1) ≤ i ≤ pos(T2) |
|---|
| ..35. |
A1A1A1. | ..56. |
EEE. | ..35. |
T1T1T1. | ..286. |
T2T2T2. | | |
| |
| |
| |
| |
| |
| Subtotal: 412 / 4 / 5 |
| Irrep combinations (i,i,j) (i,j,j) with indices: pos(A1) ≤ i ≤ j ≤ pos(T2) |
|---|
| ..308. |
T1T1T2. | ..5. |
A1A2A2. | ..105. |
A1EE. | ..140. |
A1T1T1. | ..330. |
A1T2T2. | ..15. |
A2EE. | ..168. |
ET1T1. | ..396. |
ET2T2. | ..385. |
T1T2T2. | | |
| Subtotal: 1.852 / 9 / 20 |
| Irrep combinations (i,j,k) with indices: pos(A1) ≤ i ≤ j ≤ k ≤ pos(T2) |
|---|
| ..77. |
A2T1T2. | ..462. |
ET1T2. | | |
| |
| |
| |
| |
| |
| |
| |
| Subtotal: 539 / 2 / 10 |
| Total: 2.803 / 15 / 35 |
Contributions to nonvanishing quartic force field constants
| Irrep combinations (i,i,i,i) with indices: pos(A1) ≤ i ≤ pos(T2) |
|---|
| ..70. |
A1A1A1A1. | ..1. |
A2A2A2A2. | ..231. |
EEEE. | ..616. |
T1T1T1T1. | ..3.212. |
T2T2T2T2. | | |
| |
| |
| |
| |
| Subtotal: 4.130 / 5 / 5 |
| Irrep combinations (i,i,i,j) (i,j,j,j) with indices: pos(A1) ≤ i ≤ j ≤ pos(T2) |
|---|
| ..2.156. |
T1T1T1T2. | ..280. |
A1EEE. | ..175. |
A1T1T1T1. | ..1.430. |
A1T2T2T2. | ..56. |
A2EEE. | ..84. |
A2T1T1T1. | ..165. |
A2T2T2T2. | ..672. |
ET1T1T1. | ..2.640. |
ET2T2T2. | ..5.082. |
T1T2T2T2. |
| Subtotal: 12.740 / 10 / 20 |
| Irrep combinations (i,i,j,j) with indices: pos(A1) ≤ i ≤ j ≤ pos(T2) |
|---|
| ..15. |
A1A1A2A2. | ..315. |
A1A1EE. | ..420. |
A1A1T1T1. | ..990. |
A1A1T2T2. | ..21. |
A2A2EE. | ..28. |
A2A2T1T1. | ..66. |
A2A2T2T2. | ..1.176. |
EET1T1. | ..2.772. |
EET2T2. | ..6.699. |
T1T1T2T2. |
| Subtotal: 12.502 / 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) |
|---|
| ..2.772. |
EET1T2. | ..1.540. |
A1T1T1T2. | ..231. |
A2T1T1T2. | ..3.234. |
ET1T1T2. | ..75. |
A1A2EE. | ..840. |
A1ET1T1. | ..1.980. |
A1ET2T2. | ..1.925. |
A1T1T2T2. | ..168. |
A2ET1T1. | ..396. |
A2ET2T2. |
| ..462. |
A2T1T2T2. | ..5.082. |
ET1T2T2. | | |
| |
| |
| |
| |
| |
| |
| |
| Subtotal: 18.705 / 12 / 30 |
| Irrep combinations (i,j,k,l) with indices: pos(A1) ≤ i ≤ j ≤ k ≤ l ≤ pos(T2) |
|---|
| ..385. |
A1A2T1T2. | ..2.310. |
A1ET1T2. | ..462. |
A2ET1T2. | | |
| |
| |
| |
| |
| |
| |
| Subtotal: 3.157 / 3 / 5 |
| Total: 51.234 / 40 / 70 |
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