Characters of representations for molecular motions
Motion |
E |
2C3 |
3C'2 |
Cartesian 3N |
156 |
0 |
-2 |
Translation (x,y,z) |
3 |
0 |
-1 |
Rotation (Rx,Ry,Rz) |
3 |
0 |
-1 |
Vibration |
150 |
0 |
0 |
Decomposition to irreducible representations
Motion |
A1 |
A2 |
E |
Total |
Cartesian 3N |
25 |
27 |
52 |
104 |
Translation (x,y,z) |
0 |
1 |
1 |
2 |
Rotation (Rx,Ry,Rz) |
0 |
1 |
1 |
2 |
Vibration |
25 |
25 |
50 |
100 |
Molecular parameter
Number of Atoms (N) |
52
|
Number of internal coordinates |
150
|
Number of independant internal coordinates |
25
|
Number of vibrational modes |
100
|
Force field analysis
Allowed / forbidden vibronational transitions
Operator |
A1 |
A2 |
E |
Total |
Linear (IR) |
25 |
25 |
50 |
75 / 25 |
Quadratic (Raman) |
25 |
25 |
50 |
75 / 25 |
IR + Raman |
- - - - |
- - - - |
50 |
50 / 0 |
Characters of force fields
(Symmetric powers of vibration representation)
Force field |
E |
2C3 |
3C'2 |
linear |
150 |
0 |
0 |
quadratic |
11.325 |
0 |
75 |
cubic |
573.800 |
50 |
0 |
quartic |
21.947.850 |
0 |
2.850 |
quintic |
675.993.780 |
0 |
0 |
sextic |
17.463.172.650 |
1.275 |
73.150 |
Decomposition to irreducible representations
Column with number of nonvanshing force constants highlighted
Force field |
A1 |
A2 |
E |
linear |
25 |
25 |
50 |
quadratic |
1.925 |
1.850 |
3.775 |
cubic |
95.650 |
95.650 |
191.250 |
quartic |
3.659.400 |
3.656.550 |
7.315.950 |
quintic |
112.665.630 |
112.665.630 |
225.331.260 |
sextic |
2.910.565.775 |
2.910.492.625 |
5.821.057.125 |
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) |
..325. |
A1A1. | ..325. |
A2A2. | ..1.275. |
EE. | | |
| |
| |
| |
| |
| |
| |
Subtotal: 1.925 / 3 / 3 |
Irrep combinations (i,j) with indices: pos(A1) ≤ i ≤ j ≤ pos(E) |
Subtotal: 0 / 0 / 3 |
Total: 1.925 / 3 / 6 |
Contributions to nonvanishing cubic force field constants
Irrep combinations (i,i,i) with indices: pos(A1) ≤ i ≤ pos(E) |
..2.925. |
A1A1A1. | ..22.100. |
EEE. | | |
| |
| |
| |
| |
| |
| |
| |
Subtotal: 25.025 / 2 / 3 |
Irrep combinations (i,i,j) (i,j,j) with indices: pos(A1) ≤ i ≤ j ≤ pos(E) |
..8.125. |
A1A2A2. | ..31.875. |
A1EE. | ..30.625. |
A2EE. | | |
| |
| |
| |
| |
| |
| |
Subtotal: 70.625 / 3 / 6 |
Irrep combinations (i,j,k) with indices: pos(A1) ≤ i ≤ j ≤ k ≤ pos(E) |
Subtotal: 0 / 0 / 1 |
Total: 95.650 / 5 / 10 |
Contributions to nonvanishing quartic force field constants
Irrep combinations (i,i,i,i) with indices: pos(A1) ≤ i ≤ pos(E) |
..20.475. |
A1A1A1A1. | ..20.475. |
A2A2A2A2. | ..813.450. |
EEEE. | | |
| |
| |
| |
| |
| |
| |
Subtotal: 854.400 / 3 / 3 |
Irrep combinations (i,i,i,j) (i,j,j,j) with indices: pos(A1) ≤ i ≤ j ≤ pos(E) |
..552.500. |
A1EEE. | ..552.500. |
A2EEE. | | |
| |
| |
| |
| |
| |
| |
| |
Subtotal: 1.105.000 / 2 / 6 |
Irrep combinations (i,i,j,j) with indices: pos(A1) ≤ i ≤ j ≤ pos(E) |
..105.625. |
A1A1A2A2. | ..414.375. |
A1A1EE. | ..414.375. |
A2A2EE. | | |
| |
| |
| |
| |
| |
| |
Subtotal: 934.375 / 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) |
..765.625. |
A1A2EE. | | |
| |
| |
| |
| |
| |
| |
| |
| |
Subtotal: 765.625 / 1 / 3 |
Irrep combinations (i,j,k,l) with indices: pos(A1) ≤ i ≤ j ≤ k ≤ l ≤ pos(E) |
Subtotal: 0 / 0 / 0 |
Total: 3.659.400 / 9 / 15 |
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Last update November, 13th 2023 by A. Gelessus, Impressum, Datenschutzerklärung/DataPrivacyStatement