Characters of representations for molecular motions
Motion |
E |
2C3 |
3C'2 |
Cartesian 3N |
36 |
0 |
0 |
Translation (x,y,z) |
3 |
0 |
-1 |
Rotation (Rx,Ry,Rz) |
3 |
0 |
-1 |
Vibration |
30 |
0 |
2 |
Decomposition to irreducible representations
Motion |
A1 |
A2 |
E |
Total |
Cartesian 3N |
6 |
6 |
12 |
24 |
Translation (x,y,z) |
0 |
1 |
1 |
2 |
Rotation (Rx,Ry,Rz) |
0 |
1 |
1 |
2 |
Vibration |
6 |
4 |
10 |
20 |
Molecular parameter
Number of Atoms (N) |
12
|
Number of internal coordinates |
30
|
Number of independant internal coordinates |
6
|
Number of vibrational modes |
20
|
Force field analysis
Allowed / forbidden vibronational transitions
Operator |
A1 |
A2 |
E |
Total |
Linear (IR) |
6 |
4 |
10 |
14 / 6 |
Quadratic (Raman) |
6 |
4 |
10 |
16 / 4 |
IR + Raman |
- - - - |
- - - - |
10 |
10 / 0 |
Characters of force fields
(Symmetric powers of vibration representation)
Force field |
E |
2C3 |
3C'2 |
linear |
30 |
0 |
2 |
quadratic |
465 |
0 |
17 |
cubic |
4.960 |
10 |
32 |
quartic |
40.920 |
0 |
152 |
quintic |
278.256 |
0 |
272 |
sextic |
1.623.160 |
55 |
952 |
Decomposition to irreducible representations
Column with number of nonvanshing force constants highlighted
Force field |
A1 |
A2 |
E |
linear |
6 |
4 |
10 |
quadratic |
86 |
69 |
155 |
cubic |
846 |
814 |
1.650 |
quartic |
6.896 |
6.744 |
13.640 |
quintic |
46.512 |
46.240 |
92.752 |
sextic |
271.021 |
270.069 |
541.035 |
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 D
3
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(E) |
..21. |
A1A1. | ..10. |
A2A2. | ..55. |
EE. | | |
| |
| |
| |
| |
| |
| |
Subtotal: 86 / 3 / 3 |
Irrep combinations (i,j) with indices: pos(A1) ≤ i ≤ j ≤ pos(E) |
Subtotal: 0 / 0 / 3 |
Total: 86 / 3 / 6 |
Contributions to nonvanishing cubic force field constants
Irrep combinations (i,i,i) with indices: pos(A1) ≤ i ≤ pos(E) |
..56. |
A1A1A1. | ..220. |
EEE. | | |
| |
| |
| |
| |
| |
| |
| |
Subtotal: 276 / 2 / 3 |
Irrep combinations (i,i,j) (i,j,j) with indices: pos(A1) ≤ i ≤ j ≤ pos(E) |
..60. |
A1A2A2. | ..330. |
A1EE. | ..180. |
A2EE. | | |
| |
| |
| |
| |
| |
| |
Subtotal: 570 / 3 / 6 |
Irrep combinations (i,j,k) with indices: pos(A1) ≤ i ≤ j ≤ k ≤ pos(E) |
Subtotal: 0 / 0 / 1 |
Total: 846 / 5 / 10 |
Contributions to nonvanishing quartic force field constants
Irrep combinations (i,i,i,i) with indices: pos(A1) ≤ i ≤ pos(E) |
..126. |
A1A1A1A1. | ..35. |
A2A2A2A2. | ..1.540. |
EEEE. | | |
| |
| |
| |
| |
| |
| |
Subtotal: 1.701 / 3 / 3 |
Irrep combinations (i,i,i,j) (i,j,j,j) with indices: pos(A1) ≤ i ≤ j ≤ pos(E) |
..1.320. |
A1EEE. | ..880. |
A2EEE. | | |
| |
| |
| |
| |
| |
| |
| |
Subtotal: 2.200 / 2 / 6 |
Irrep combinations (i,i,j,j) with indices: pos(A1) ≤ i ≤ j ≤ pos(E) |
..210. |
A1A1A2A2. | ..1.155. |
A1A1EE. | ..550. |
A2A2EE. | | |
| |
| |
| |
| |
| |
| |
Subtotal: 1.915 / 3 / 3 |
Irrep combinations (i,i,j,k) (i,j,j,k) (i,j,k,k) with indices: pos(A1) ≤ i ≤ j ≤ k ≤ pos(E) |
..1.080. |
A1A2EE. | | |
| |
| |
| |
| |
| |
| |
| |
| |
Subtotal: 1.080 / 1 / 3 |
Irrep combinations (i,j,k,l) with indices: pos(A1) ≤ i ≤ j ≤ k ≤ l ≤ pos(E) |
Subtotal: 0 / 0 / 0 |
Total: 6.896 / 9 / 15 |
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Last update November, 13th 2023 by A. Gelessus, Impressum, Datenschutzerklärung/DataPrivacyStatement