Characters of representations for molecular motions
| Motion |
E |
C2 |
| Cartesian 3N |
12 |
0 |
| Translation (x,y,z) |
3 |
-1 |
| Rotation (Rx,Ry,Rz) |
3 |
-1 |
| Vibration |
6 |
2 |
Decomposition to irreducible representations
| Motion |
A |
B |
Total |
| Cartesian 3N |
6 |
6 |
12 |
| Translation (x,y,z) |
1 |
2 |
3 |
| Rotation (Rx,Ry,Rz) |
1 |
2 |
3 |
| Vibration |
4 |
2 |
6 |
Molecular parameter
| Number of Atoms (N) |
4
|
| Number of internal coordinates |
6
|
| Number of independant internal coordinates |
4
|
| Number of vibrational modes |
6
|
Force field analysis
Allowed / forbidden vibronational transitions
| Operator |
A |
B |
Total |
| Linear (IR) |
4 |
2 |
6 / 0 |
| Quadratic (Raman) |
4 |
2 |
6 / 0 |
| IR + Raman |
4 |
2 |
6 / 0 |
Characters of force fields
(Symmetric powers of vibration representation)
| Force field |
E |
C2 |
| linear |
6 |
2 |
| quadratic |
21 |
5 |
| cubic |
56 |
8 |
| quartic |
126 |
14 |
| quintic |
252 |
20 |
| sextic |
462 |
30 |
Decomposition to irreducible representations
Column with number of nonvanshing force constants highlighted
| Force field |
A |
B |
| linear |
4 |
2 |
| quadratic |
13 |
8 |
| cubic |
32 |
24 |
| quartic |
70 |
56 |
| quintic |
136 |
116 |
| sextic |
246 |
216 |
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
2
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(A) ≤ i ≤ pos(B) |
|---|
| ..10. |
AA. | ..3. |
BB. | | |
| |
| |
| |
| |
| |
| |
| |
| Subtotal: 13 / 2 / 2 |
| Irrep combinations (i,j) with indices: pos(A) ≤ i ≤ j ≤ pos(B) |
|---|
| Subtotal: 0 / 0 / 1 |
| Total: 13 / 2 / 3 |
Contributions to nonvanishing cubic force field constants
| Irrep combinations (i,i,i) with indices: pos(A) ≤ i ≤ pos(B) |
|---|
| ..20. |
AAA. | | |
| |
| |
| |
| |
| |
| |
| |
| |
| Subtotal: 20 / 1 / 2 |
| Irrep combinations (i,i,j) (i,j,j) with indices: pos(A) ≤ i ≤ j ≤ pos(B) |
|---|
| ..12. |
ABB. | | |
| |
| |
| |
| |
| |
| |
| |
| |
| Subtotal: 12 / 1 / 2 |
| Irrep combinations (i,j,k) with indices: pos(A) ≤ i ≤ j ≤ k ≤ pos(B) |
|---|
| Subtotal: 0 / 0 / 0 |
| Total: 32 / 2 / 4 |
Contributions to nonvanishing quartic force field constants
| Irrep combinations (i,i,i,i) with indices: pos(A) ≤ i ≤ pos(B) |
|---|
| ..35. |
AAAA. | ..5. |
BBBB. | | |
| |
| |
| |
| |
| |
| |
| |
| Subtotal: 40 / 2 / 2 |
| Irrep combinations (i,i,i,j) (i,j,j,j) with indices: pos(A) ≤ i ≤ j ≤ pos(B) |
|---|
| Subtotal: 0 / 0 / 2 |
| Irrep combinations (i,i,j,j) with indices: pos(A) ≤ i ≤ j ≤ pos(B) |
|---|
| ..30. |
AABB. | | |
| |
| |
| |
| |
| |
| |
| |
| |
| Subtotal: 30 / 1 / 1 |
| Irrep combinations (i,i,j,k) (i,j,j,k) (i,j,k,k) with indices: pos(A) ≤ i ≤ j ≤ k ≤ pos(B) |
|---|
| Subtotal: 0 / 0 / 0 |
| Irrep combinations (i,j,k,l) with indices: pos(A) ≤ i ≤ j ≤ k ≤ l ≤ pos(B) |
|---|
| Subtotal: 0 / 0 / 0 |
| Total: 70 / 3 / 5 |
Calculate contributions to
Last update November, 13th 2023 by A. Gelessus, Impressum, Datenschutzerklärung/DataPrivacyStatement