Characters of representations for molecular motions
Motion |
E |
2C3 |
3C'2 |
Cartesian 3N |
21 |
0 |
-1 |
Translation (x,y,z) |
3 |
0 |
-1 |
Rotation (Rx,Ry,Rz) |
3 |
0 |
-1 |
Vibration |
15 |
0 |
1 |
Decomposition to irreducible representations
Motion |
A1 |
A2 |
E |
Total |
Cartesian 3N |
3 |
4 |
7 |
14 |
Translation (x,y,z) |
0 |
1 |
1 |
2 |
Rotation (Rx,Ry,Rz) |
0 |
1 |
1 |
2 |
Vibration |
3 |
2 |
5 |
10 |
Molecular parameter
Number of Atoms (N) |
7
|
Number of internal coordinates |
15
|
Number of independant internal coordinates |
3
|
Number of vibrational modes |
10
|
Force field analysis
Allowed / forbidden vibronational transitions
Operator |
A1 |
A2 |
E |
Total |
Linear (IR) |
3 |
2 |
5 |
7 / 3 |
Quadratic (Raman) |
3 |
2 |
5 |
8 / 2 |
IR + Raman |
- - - - |
- - - - |
5 |
5 / 0 |
Characters of force fields
(Symmetric powers of vibration representation)
Force field |
E |
2C3 |
3C'2 |
linear |
15 |
0 |
1 |
quadratic |
120 |
0 |
8 |
cubic |
680 |
5 |
8 |
quartic |
3.060 |
0 |
36 |
quintic |
11.628 |
0 |
36 |
sextic |
38.760 |
15 |
120 |
Decomposition to irreducible representations
Column with number of nonvanshing force constants highlighted
Force field |
A1 |
A2 |
E |
linear |
3 |
2 |
5 |
quadratic |
24 |
16 |
40 |
cubic |
119 |
111 |
225 |
quartic |
528 |
492 |
1.020 |
quintic |
1.956 |
1.920 |
3.876 |
sextic |
6.525 |
6.405 |
12.915 |
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) |
..6. |
A1A1. | ..3. |
A2A2. | ..15. |
EE. | | |
| |
| |
| |
| |
| |
| |
Subtotal: 24 / 3 / 3 |
Irrep combinations (i,j) with indices: pos(A1) ≤ i ≤ j ≤ pos(E) |
Subtotal: 0 / 0 / 3 |
Total: 24 / 3 / 6 |
Contributions to nonvanishing cubic force field constants
Irrep combinations (i,i,i) with indices: pos(A1) ≤ i ≤ pos(E) |
..10. |
A1A1A1. | ..35. |
EEE. | | |
| |
| |
| |
| |
| |
| |
| |
Subtotal: 45 / 2 / 3 |
Irrep combinations (i,i,j) (i,j,j) with indices: pos(A1) ≤ i ≤ j ≤ pos(E) |
..9. |
A1A2A2. | ..45. |
A1EE. | ..20. |
A2EE. | | |
| |
| |
| |
| |
| |
| |
Subtotal: 74 / 3 / 6 |
Irrep combinations (i,j,k) with indices: pos(A1) ≤ i ≤ j ≤ k ≤ pos(E) |
Subtotal: 0 / 0 / 1 |
Total: 119 / 5 / 10 |
Contributions to nonvanishing quartic force field constants
Irrep combinations (i,i,i,i) with indices: pos(A1) ≤ i ≤ pos(E) |
..15. |
A1A1A1A1. | ..5. |
A2A2A2A2. | ..120. |
EEEE. | | |
| |
| |
| |
| |
| |
| |
Subtotal: 140 / 3 / 3 |
Irrep combinations (i,i,i,j) (i,j,j,j) with indices: pos(A1) ≤ i ≤ j ≤ pos(E) |
..105. |
A1EEE. | ..70. |
A2EEE. | | |
| |
| |
| |
| |
| |
| |
| |
Subtotal: 175 / 2 / 6 |
Irrep combinations (i,i,j,j) with indices: pos(A1) ≤ i ≤ j ≤ pos(E) |
..18. |
A1A1A2A2. | ..90. |
A1A1EE. | ..45. |
A2A2EE. | | |
| |
| |
| |
| |
| |
| |
Subtotal: 153 / 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) |
..60. |
A1A2EE. | | |
| |
| |
| |
| |
| |
| |
| |
| |
Subtotal: 60 / 1 / 3 |
Irrep combinations (i,j,k,l) with indices: pos(A1) ≤ i ≤ j ≤ k ≤ l ≤ pos(E) |
Subtotal: 0 / 0 / 0 |
Total: 528 / 9 / 15 |
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Last update November, 13th 2023 by A. Gelessus, Impressum, Datenschutzerklärung/DataPrivacyStatement