Characters of representations for molecular motions
Motion |
E |
2S4 |
C2 |
2C'2 |
2σd |
Cartesian 3N |
36 |
0 |
-4 |
-4 |
4 |
Translation (x,y,z) |
3 |
-1 |
-1 |
-1 |
1 |
Rotation (Rx,Ry,Rz) |
3 |
1 |
-1 |
-1 |
-1 |
Vibration |
30 |
0 |
-2 |
-2 |
4 |
Decomposition to irreducible representations
Motion |
A1 |
A2 |
B1 |
B2 |
E |
Total |
Cartesian 3N |
4 |
4 |
2 |
6 |
10 |
26 |
Translation (x,y,z) |
0 |
0 |
0 |
1 |
1 |
2 |
Rotation (Rx,Ry,Rz) |
0 |
1 |
0 |
0 |
1 |
2 |
Vibration |
4 |
3 |
2 |
5 |
8 |
22 |
Molecular parameter
Number of Atoms (N) |
12
|
Number of internal coordinates |
30
|
Number of independant internal coordinates |
4
|
Number of vibrational modes |
22
|
Force field analysis
Allowed / forbidden vibronational transitions
Operator |
A1 |
A2 |
B1 |
B2 |
E |
Total |
Linear (IR) |
4 |
3 |
2 |
5 |
8 |
13 / 9 |
Quadratic (Raman) |
4 |
3 |
2 |
5 |
8 |
19 / 3 |
IR + Raman |
- - - - |
3 |
- - - - |
5 |
8 |
13 / 3 |
Characters of force fields
(Symmetric powers of vibration representation)
Force field |
E |
2S4 |
C2 |
2C'2 |
2σd |
linear |
30 |
0 |
-2 |
-2 |
4 |
quadratic |
465 |
-1 |
17 |
17 |
23 |
cubic |
4.960 |
0 |
-32 |
-32 |
72 |
quartic |
40.920 |
8 |
152 |
152 |
256 |
quintic |
278.256 |
0 |
-272 |
-272 |
680 |
sextic |
1.623.160 |
-8 |
952 |
952 |
1.904 |
Decomposition to irreducible representations
Column with number of nonvanshing force constants highlighted
Force field |
A1 |
A2 |
B1 |
B2 |
E |
linear |
4 |
3 |
2 |
5 |
8 |
quadratic |
70 |
50 |
59 |
62 |
112 |
cubic |
626 |
606 |
590 |
642 |
1.248 |
quartic |
5.238 |
5.034 |
5.106 |
5.158 |
10.192 |
quintic |
34.850 |
34.646 |
34.510 |
34.986 |
69.632 |
sextic |
203.726 |
202.298 |
202.778 |
203.254 |
405.552 |
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
2d
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) |
..10. |
A1A1. | ..6. |
A2A2. | ..3. |
B1B1. | ..15. |
B2B2. | ..36. |
EE. | | |
| |
| |
| |
| |
Subtotal: 70 / 5 / 5 |
Irrep combinations (i,j) with indices: pos(A1) ≤ i ≤ j ≤ pos(E) |
Subtotal: 0 / 0 / 10 |
Total: 70 / 5 / 15 |
Contributions to nonvanishing cubic force field constants
Irrep combinations (i,i,i) with indices: pos(A1) ≤ i ≤ pos(E) |
..20. |
A1A1A1. | | |
| |
| |
| |
| |
| |
| |
| |
| |
Subtotal: 20 / 1 / 5 |
Irrep combinations (i,i,j) (i,j,j) with indices: pos(A1) ≤ i ≤ j ≤ pos(E) |
..24. |
A1A2A2. | ..12. |
A1B1B1. | ..60. |
A1B2B2. | ..144. |
A1EE. | ..84. |
A2EE. | ..72. |
B1EE. | ..180. |
B2EE. | | |
| |
| |
Subtotal: 576 / 7 / 20 |
Irrep combinations (i,j,k) with indices: pos(A1) ≤ i ≤ j ≤ k ≤ pos(E) |
..30. |
A2B1B2. | | |
| |
| |
| |
| |
| |
| |
| |
| |
Subtotal: 30 / 1 / 10 |
Total: 626 / 9 / 35 |
Contributions to nonvanishing quartic force field constants
Irrep combinations (i,i,i,i) with indices: pos(A1) ≤ i ≤ pos(E) |
..35. |
A1A1A1A1. | ..15. |
A2A2A2A2. | ..5. |
B1B1B1B1. | ..70. |
B2B2B2B2. | ..996. |
EEEE. | | |
| |
| |
| |
| |
Subtotal: 1.121 / 5 / 5 |
Irrep combinations (i,i,i,j) (i,j,j,j) with indices: pos(A1) ≤ i ≤ j ≤ pos(E) |
Subtotal: 0 / 0 / 20 |
Irrep combinations (i,i,j,j) with indices: pos(A1) ≤ i ≤ j ≤ pos(E) |
..60. |
A1A1A2A2. | ..30. |
A1A1B1B1. | ..150. |
A1A1B2B2. | ..360. |
A1A1EE. | ..18. |
A2A2B1B1. | ..90. |
A2A2B2B2. | ..216. |
A2A2EE. | ..45. |
B1B1B2B2. | ..108. |
B1B1EE. | ..540. |
B2B2EE. |
Subtotal: 1.617 / 10 / 10 |
Irrep combinations (i,i,j,k) (i,j,j,k) (i,j,k,k) with indices: pos(A1) ≤ i ≤ j ≤ k ≤ pos(E) |
..336. |
A1A2EE. | ..288. |
A1B1EE. | ..720. |
A1B2EE. | ..216. |
A2B1EE. | ..540. |
A2B2EE. | ..280. |
B1B2EE. | | |
| |
| |
| |
Subtotal: 2.380 / 6 / 30 |
Irrep combinations (i,j,k,l) with indices: pos(A1) ≤ i ≤ j ≤ k ≤ l ≤ pos(E) |
..120. |
A1A2B1B2. | | |
| |
| |
| |
| |
| |
| |
| |
| |
Subtotal: 120 / 1 / 5 |
Total: 5.238 / 22 / 70 |
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Last update November, 13th 2023 by A. Gelessus, Impressum, Datenschutzerklärung/DataPrivacyStatement