Characters of representations for molecular motions
Motion |
E |
8C3 |
3C2 |
6S4 |
6σd |
Cartesian 3N |
156 |
0 |
0 |
0 |
14 |
Translation (x,y,z) |
3 |
0 |
-1 |
-1 |
1 |
Rotation (Rx,Ry,Rz) |
3 |
0 |
-1 |
1 |
-1 |
Vibration |
150 |
0 |
2 |
0 |
14 |
Decomposition to irreducible representations
Motion |
A1 |
A2 |
E |
T1 |
T2 |
Total |
Cartesian 3N |
10 |
3 |
13 |
16 |
23 |
65 |
Translation (x,y,z) |
0 |
0 |
0 |
0 |
1 |
1 |
Rotation (Rx,Ry,Rz) |
0 |
0 |
0 |
1 |
0 |
1 |
Vibration |
10 |
3 |
13 |
15 |
22 |
63 |
Molecular parameter
Number of Atoms (N) |
52
|
Number of internal coordinates |
150
|
Number of independant internal coordinates |
10
|
Number of vibrational modes |
63
|
Force field analysis
Allowed / forbidden vibronational transitions
Operator |
A1 |
A2 |
E |
T1 |
T2 |
Total |
Linear (IR) |
10 |
3 |
13 |
15 |
22 |
22 / 41 |
Quadratic (Raman) |
10 |
3 |
13 |
15 |
22 |
45 / 18 |
IR + Raman |
- - - - |
3 |
- - - - |
15 |
22 |
22 / 18 |
Characters of force fields
(Symmetric powers of vibration representation)
Force field |
E |
8C3 |
3C2 |
6S4 |
6σd |
linear |
150 |
0 |
2 |
0 |
14 |
quadratic |
11.325 |
0 |
77 |
1 |
173 |
cubic |
573.800 |
50 |
152 |
0 |
1.512 |
quartic |
21.947.850 |
0 |
3.002 |
38 |
11.866 |
quintic |
675.993.780 |
0 |
5.852 |
0 |
79.492 |
sextic |
17.463.172.650 |
1.275 |
79.002 |
38 |
490.042 |
Decomposition to irreducible representations
Column with number of nonvanshing force constants highlighted
Force field |
A1 |
A2 |
E |
T1 |
T2 |
linear |
10 |
3 |
13 |
15 |
22 |
quadratic |
525 |
438 |
963 |
1.363 |
1.449 |
cubic |
24.322 |
23.566 |
47.838 |
71.328 |
72.084 |
quartic |
917.845 |
911.893 |
1.829.738 |
2.740.149 |
2.746.063 |
quintic |
28.187.012 |
28.147.266 |
56.334.278 |
84.478.618 |
84.518.364 |
sextic |
727.765.014 |
727.519.974 |
1.455.283.713 |
2.182.764.205 |
2.183.009.207 |
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) |
..55. |
A1A1. | ..6. |
A2A2. | ..91. |
EE. | ..120. |
T1T1. | ..253. |
T2T2. | | |
| |
| |
| |
| |
Subtotal: 525 / 5 / 5 |
Irrep combinations (i,j) with indices: pos(A1) ≤ i ≤ j ≤ pos(T2) |
Subtotal: 0 / 0 / 10 |
Total: 525 / 5 / 15 |
Contributions to nonvanishing cubic force field constants
Irrep combinations (i,i,i) with indices: pos(A1) ≤ i ≤ pos(T2) |
..220. |
A1A1A1. | ..455. |
EEE. | ..455. |
T1T1T1. | ..2.024. |
T2T2T2. | | |
| |
| |
| |
| |
| |
Subtotal: 3.154 / 4 / 5 |
Irrep combinations (i,i,j) (i,j,j) with indices: pos(A1) ≤ i ≤ j ≤ pos(T2) |
..2.640. |
T1T1T2. | ..60. |
A1A2A2. | ..910. |
A1EE. | ..1.200. |
A1T1T1. | ..2.530. |
A1T2T2. | ..234. |
A2EE. | ..1.560. |
ET1T1. | ..3.289. |
ET2T2. | ..3.465. |
T1T2T2. | | |
Subtotal: 15.888 / 9 / 20 |
Irrep combinations (i,j,k) with indices: pos(A1) ≤ i ≤ j ≤ k ≤ pos(T2) |
..990. |
A2T1T2. | ..4.290. |
ET1T2. | | |
| |
| |
| |
| |
| |
| |
| |
Subtotal: 5.280 / 2 / 10 |
Total: 24.322 / 15 / 35 |
Contributions to nonvanishing quartic force field constants
Irrep combinations (i,i,i,i) with indices: pos(A1) ≤ i ≤ pos(T2) |
..715. |
A1A1A1A1. | ..15. |
A2A2A2A2. | ..4.186. |
EEEE. | ..10.320. |
T1T1T1T1. | ..44.781. |
T2T2T2T2. | | |
| |
| |
| |
| |
Subtotal: 60.017 / 5 / 5 |
Irrep combinations (i,i,i,j) (i,j,j,j) with indices: pos(A1) ≤ i ≤ j ≤ pos(T2) |
..39.600. |
T1T1T1T2. | ..4.550. |
A1EEE. | ..4.550. |
A1T1T1T1. | ..20.240. |
A1T2T2T2. | ..1.365. |
A2EEE. | ..2.040. |
A2T1T1T1. | ..4.620. |
A2T2T2T2. | ..14.560. |
ET1T1T1. | ..46.046. |
ET2T2T2. | ..83.490. |
T1T2T2T2. |
Subtotal: 221.061 / 10 / 20 |
Irrep combinations (i,i,j,j) with indices: pos(A1) ≤ i ≤ j ≤ pos(T2) |
..330. |
A1A1A2A2. | ..5.005. |
A1A1EE. | ..6.600. |
A1A1T1T1. | ..13.915. |
A1A1T2T2. | ..546. |
A2A2EE. | ..720. |
A2A2T1T1. | ..1.518. |
A2A2T2T2. | ..21.840. |
EET1T1. | ..46.046. |
EET2T2. | ..115.335. |
T1T1T2T2. |
Subtotal: 211.855 / 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) |
..55.770. |
EET1T2. | ..26.400. |
A1T1T1T2. | ..6.930. |
A2T1T1T2. | ..64.350. |
ET1T1T2. | ..2.340. |
A1A2EE. | ..15.600. |
A1ET1T1. | ..32.890. |
A1ET2T2. | ..34.650. |
A1T1T2T2. | ..4.680. |
A2ET1T1. | ..9.867. |
A2ET2T2. |
..11.385. |
A2T1T2T2. | ..94.380. |
ET1T2T2. | | |
| |
| |
| |
| |
| |
| |
| |
Subtotal: 359.242 / 12 / 30 |
Irrep combinations (i,j,k,l) with indices: pos(A1) ≤ i ≤ j ≤ k ≤ l ≤ pos(T2) |
..9.900. |
A1A2T1T2. | ..42.900. |
A1ET1T2. | ..12.870. |
A2ET1T2. | | |
| |
| |
| |
| |
| |
| |
Subtotal: 65.670 / 3 / 5 |
Total: 917.845 / 40 / 70 |
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