Characters of representations for molecular motions
| Motion |
E |
8C3 |
3C2 |
6S4 |
6σd |
| Cartesian 3N |
66 |
0 |
-2 |
0 |
6 |
| Translation (x,y,z) |
3 |
0 |
-1 |
-1 |
1 |
| Rotation (Rx,Ry,Rz) |
3 |
0 |
-1 |
1 |
-1 |
| Vibration |
60 |
0 |
0 |
0 |
6 |
Decomposition to irreducible representations
| Motion |
A1 |
A2 |
E |
T1 |
T2 |
Total |
| Cartesian 3N |
4 |
1 |
5 |
7 |
10 |
27 |
| Translation (x,y,z) |
0 |
0 |
0 |
0 |
1 |
1 |
| Rotation (Rx,Ry,Rz) |
0 |
0 |
0 |
1 |
0 |
1 |
| Vibration |
4 |
1 |
5 |
6 |
9 |
25 |
Molecular parameter
| Number of Atoms (N) |
22
|
| Number of internal coordinates |
60
|
| Number of independant internal coordinates |
4
|
| Number of vibrational modes |
25
|
Force field analysis
Allowed / forbidden vibronational transitions
| Operator |
A1 |
A2 |
E |
T1 |
T2 |
Total |
| Linear (IR) |
4 |
1 |
5 |
6 |
9 |
9 / 16 |
| Quadratic (Raman) |
4 |
1 |
5 |
6 |
9 |
18 / 7 |
| IR + Raman |
- - - - |
1 |
- - - - |
6 |
9 |
9 / 7 |
Characters of force fields
(Symmetric powers of vibration representation)
| Force field |
E |
8C3 |
3C2 |
6S4 |
6σd |
| linear |
60 |
0 |
0 |
0 |
6 |
| quadratic |
1.830 |
0 |
30 |
0 |
48 |
| cubic |
37.820 |
20 |
0 |
0 |
218 |
| quartic |
595.665 |
0 |
465 |
15 |
1.071 |
| quintic |
7.624.512 |
0 |
0 |
0 |
4.032 |
| sextic |
82.598.880 |
210 |
4.960 |
0 |
15.456 |
Decomposition to irreducible representations
Column with number of nonvanshing force constants highlighted
| Force field |
A1 |
A2 |
E |
T1 |
T2 |
| linear |
4 |
1 |
5 |
6 |
9 |
| quadratic |
92 |
68 |
160 |
213 |
237 |
| cubic |
1.637 |
1.528 |
3.145 |
4.673 |
4.782 |
| quartic |
25.149 |
24.606 |
49.755 |
74.136 |
74.664 |
| quintic |
318.696 |
316.680 |
635.376 |
952.056 |
954.072 |
| sextic |
3.446.174 |
3.438.446 |
6.884.410 |
10.320.376 |
10.328.104 |
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) |
|---|
| ..10. |
A1A1. | ..1. |
A2A2. | ..15. |
EE. | ..21. |
T1T1. | ..45. |
T2T2. | | |
| |
| |
| |
| |
| Subtotal: 92 / 5 / 5 |
| Irrep combinations (i,j) with indices: pos(A1) ≤ i ≤ j ≤ pos(T2) |
|---|
| Subtotal: 0 / 0 / 10 |
| Total: 92 / 5 / 15 |
Contributions to nonvanishing cubic force field constants
| Irrep combinations (i,i,i) with indices: pos(A1) ≤ i ≤ pos(T2) |
|---|
| ..20. |
A1A1A1. | ..35. |
EEE. | ..20. |
T1T1T1. | ..165. |
T2T2T2. | | |
| |
| |
| |
| |
| |
| Subtotal: 240 / 4 / 5 |
| Irrep combinations (i,i,j) (i,j,j) with indices: pos(A1) ≤ i ≤ j ≤ pos(T2) |
|---|
| ..189. |
T1T1T2. | ..4. |
A1A2A2. | ..60. |
A1EE. | ..84. |
A1T1T1. | ..180. |
A1T2T2. | ..10. |
A2EE. | ..105. |
ET1T1. | ..225. |
ET2T2. | ..216. |
T1T2T2. | | |
| Subtotal: 1.073 / 9 / 20 |
| Irrep combinations (i,j,k) with indices: pos(A1) ≤ i ≤ j ≤ k ≤ pos(T2) |
|---|
| ..54. |
A2T1T2. | ..270. |
ET1T2. | | |
| |
| |
| |
| |
| |
| |
| |
| Subtotal: 324 / 2 / 10 |
| Total: 1.637 / 15 / 35 |
Contributions to nonvanishing quartic force field constants
| Irrep combinations (i,i,i,i) with indices: pos(A1) ≤ i ≤ pos(T2) |
|---|
| ..35. |
A1A1A1A1. | ..1. |
A2A2A2A2. | ..120. |
EEEE. | ..357. |
T1T1T1T1. | ..1.530. |
T2T2T2T2. | | |
| |
| |
| |
| |
| Subtotal: 2.043 / 5 / 5 |
| Irrep combinations (i,i,i,j) (i,j,j,j) with indices: pos(A1) ≤ i ≤ j ≤ pos(T2) |
|---|
| ..1.134. |
T1T1T1T2. | ..140. |
A1EEE. | ..80. |
A1T1T1T1. | ..660. |
A1T2T2T2. | ..35. |
A2EEE. | ..56. |
A2T1T1T1. | ..84. |
A2T2T2T2. | ..350. |
ET1T1T1. | ..1.200. |
ET2T2T2. | ..2.430. |
T1T2T2T2. |
| Subtotal: 6.169 / 10 / 20 |
| Irrep combinations (i,i,j,j) with indices: pos(A1) ≤ i ≤ j ≤ pos(T2) |
|---|
| ..10. |
A1A1A2A2. | ..150. |
A1A1EE. | ..210. |
A1A1T1T1. | ..450. |
A1A1T2T2. | ..15. |
A2A2EE. | ..21. |
A2A2T1T1. | ..45. |
A2A2T2T2. | ..630. |
EET1T1. | ..1.350. |
EET2T2. | ..3.375. |
T1T1T2T2. |
| Subtotal: 6.256 / 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) |
|---|
| ..1.350. |
EET1T2. | ..756. |
A1T1T1T2. | ..135. |
A2T1T1T2. | ..1.620. |
ET1T1T2. | ..40. |
A1A2EE. | ..420. |
A1ET1T1. | ..900. |
A1ET2T2. | ..864. |
A1T1T2T2. | ..105. |
A2ET1T1. | ..225. |
A2ET2T2. |
| ..270. |
A2T1T2T2. | ..2.430. |
ET1T2T2. | | |
| |
| |
| |
| |
| |
| |
| |
| Subtotal: 9.115 / 12 / 30 |
| Irrep combinations (i,j,k,l) with indices: pos(A1) ≤ i ≤ j ≤ k ≤ l ≤ pos(T2) |
|---|
| ..216. |
A1A2T1T2. | ..1.080. |
A1ET1T2. | ..270. |
A2ET1T2. | | |
| |
| |
| |
| |
| |
| |
| Subtotal: 1.566 / 3 / 5 |
| Total: 25.149 / 40 / 70 |
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