Characters of representations for molecular motions
| Motion |
E |
2C3 |
3σv |
| Cartesian 3N |
15 |
0 |
3 |
| Translation (x,y,z) |
3 |
0 |
1 |
| Rotation (Rx,Ry,Rz) |
3 |
0 |
-1 |
| Vibration |
9 |
0 |
3 |
Decomposition to irreducible representations
| Motion |
A1 |
A2 |
E |
Total |
| Cartesian 3N |
4 |
1 |
5 |
10 |
| Translation (x,y,z) |
1 |
0 |
1 |
2 |
| Rotation (Rx,Ry,Rz) |
0 |
1 |
1 |
2 |
| Vibration |
3 |
0 |
3 |
6 |
Molecular parameter
| Number of Atoms (N) |
5
|
| Number of internal coordinates |
9
|
| Number of independant internal coordinates |
3
|
| Number of vibrational modes |
6
|
Force field analysis
Allowed / forbidden vibronational transitions
| Operator |
A1 |
A2 |
E |
Total |
| Linear (IR) |
3 |
0 |
3 |
6 / 0 |
| Quadratic (Raman) |
3 |
0 |
3 |
6 / 0 |
| IR + Raman |
3 |
0 |
3 |
6 / 0 |
Characters of force fields
(Symmetric powers of vibration representation)
| Force field |
E |
2C3 |
3σv |
| linear |
9 |
0 |
3 |
| quadratic |
45 |
0 |
9 |
| cubic |
165 |
3 |
19 |
| quartic |
495 |
0 |
39 |
| quintic |
1.287 |
0 |
69 |
| sextic |
3.003 |
6 |
119 |
Decomposition to irreducible representations
Column with number of nonvanshing force constants highlighted
| Force field |
A1 |
A2 |
E |
| linear |
3 |
0 |
3 |
| quadratic |
12 |
3 |
15 |
| cubic |
38 |
19 |
54 |
| quartic |
102 |
63 |
165 |
| quintic |
249 |
180 |
429 |
| sextic |
562 |
443 |
999 |
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 C
3v
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. | ..6. |
EE. | | |
| |
| |
| |
| |
| |
| |
| |
| Subtotal: 12 / 2 / 3 |
| Irrep combinations (i,j) with indices: pos(A1) ≤ i ≤ j ≤ pos(E) |
|---|
| Subtotal: 0 / 0 / 3 |
| Total: 12 / 2 / 6 |
Contributions to nonvanishing cubic force field constants
| Irrep combinations (i,i,i) with indices: pos(A1) ≤ i ≤ pos(E) |
|---|
| ..10. |
A1A1A1. | ..10. |
EEE. | | |
| |
| |
| |
| |
| |
| |
| |
| Subtotal: 20 / 2 / 3 |
| Irrep combinations (i,i,j) (i,j,j) with indices: pos(A1) ≤ i ≤ j ≤ pos(E) |
|---|
| ..18. |
A1EE. | | |
| |
| |
| |
| |
| |
| |
| |
| |
| Subtotal: 18 / 1 / 6 |
| Irrep combinations (i,j,k) with indices: pos(A1) ≤ i ≤ j ≤ k ≤ pos(E) |
|---|
| Subtotal: 0 / 0 / 1 |
| Total: 38 / 3 / 10 |
Contributions to nonvanishing quartic force field constants
| Irrep combinations (i,i,i,i) with indices: pos(A1) ≤ i ≤ pos(E) |
|---|
| ..15. |
A1A1A1A1. | ..21. |
EEEE. | | |
| |
| |
| |
| |
| |
| |
| |
| Subtotal: 36 / 2 / 3 |
| Irrep combinations (i,i,i,j) (i,j,j,j) with indices: pos(A1) ≤ i ≤ j ≤ pos(E) |
|---|
| ..30. |
A1EEE. | | |
| |
| |
| |
| |
| |
| |
| |
| |
| Subtotal: 30 / 1 / 6 |
| Irrep combinations (i,i,j,j) with indices: pos(A1) ≤ i ≤ j ≤ pos(E) |
|---|
| ..36. |
A1A1EE. | | |
| |
| |
| |
| |
| |
| |
| |
| |
| Subtotal: 36 / 1 / 3 |
| Irrep combinations (i,i,j,k) (i,j,j,k) (i,j,k,k) with indices: pos(A1) ≤ i ≤ j ≤ k ≤ pos(E) |
|---|
| Subtotal: 0 / 0 / 3 |
| Irrep combinations (i,j,k,l) with indices: pos(A1) ≤ i ≤ j ≤ k ≤ l ≤ pos(E) |
|---|
| Subtotal: 0 / 0 / 0 |
| Total: 102 / 4 / 15 |
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