Characters of representations for molecular motions
| Motion |
E |
C2.(z) |
σv.(xz) |
σd.(yz) |
| Cartesian 3N |
15 |
-3 |
5 |
3 |
| Translation (x,y,z) |
3 |
-1 |
1 |
1 |
| Rotation (Rx,Ry,Rz) |
3 |
-1 |
-1 |
-1 |
| Vibration |
9 |
-1 |
5 |
3 |
Decomposition to irreducible representations
| Motion |
A1 |
A2 |
B1 |
B2 |
Total |
| Cartesian 3N |
5 |
1 |
5 |
4 |
15 |
| Translation (x,y,z) |
1 |
0 |
1 |
1 |
3 |
| Rotation (Rx,Ry,Rz) |
0 |
1 |
1 |
1 |
3 |
| Vibration |
4 |
0 |
3 |
2 |
9 |
Molecular parameter
| Number of Atoms (N) |
5
|
| Number of internal coordinates |
9
|
| Number of independant internal coordinates |
4
|
| Number of vibrational modes |
9
|
Force field analysis
Allowed / forbidden vibronational transitions
| Operator |
A1 |
A2 |
B1 |
B2 |
Total |
| Linear (IR) |
4 |
0 |
3 |
2 |
9 / 0 |
| Quadratic (Raman) |
4 |
0 |
3 |
2 |
9 / 0 |
| IR + Raman |
4 |
- - - - |
3 |
2 |
9 / 0 |
Characters of force fields
(Symmetric powers of vibration representation)
| Force field |
E |
C2.(z) |
σv.(xz) |
σd.(yz) |
| linear |
9 |
-1 |
5 |
3 |
| quadratic |
45 |
5 |
17 |
9 |
| cubic |
165 |
-5 |
45 |
19 |
| quartic |
495 |
15 |
103 |
39 |
| quintic |
1.287 |
-15 |
211 |
69 |
| sextic |
3.003 |
35 |
399 |
119 |
Decomposition to irreducible representations
Column with number of nonvanshing force constants highlighted
| Force field |
A1 |
A2 |
B1 |
B2 |
| linear |
4 |
0 |
3 |
2 |
| quadratic |
19 |
6 |
12 |
8 |
| cubic |
56 |
24 |
49 |
36 |
| quartic |
163 |
92 |
136 |
104 |
| quintic |
388 |
248 |
361 |
290 |
| sextic |
889 |
630 |
812 |
672 |
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
2v
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(B2) |
|---|
| ..10. |
A1A1. | ..6. |
B1B1. | ..3. |
B2B2. | | |
| |
| |
| |
| |
| |
| |
| Subtotal: 19 / 3 / 4 |
| Irrep combinations (i,j) with indices: pos(A1) ≤ i ≤ j ≤ pos(B2) |
|---|
| Subtotal: 0 / 0 / 6 |
| Total: 19 / 3 / 10 |
Contributions to nonvanishing cubic force field constants
| Irrep combinations (i,i,i) with indices: pos(A1) ≤ i ≤ pos(B2) |
|---|
| ..20. |
A1A1A1. | | |
| |
| |
| |
| |
| |
| |
| |
| |
| Subtotal: 20 / 1 / 4 |
| Irrep combinations (i,i,j) (i,j,j) with indices: pos(A1) ≤ i ≤ j ≤ pos(B2) |
|---|
| ..24. |
A1B1B1. | ..12. |
A1B2B2. | | |
| |
| |
| |
| |
| |
| |
| |
| Subtotal: 36 / 2 / 12 |
| Irrep combinations (i,j,k) with indices: pos(A1) ≤ i ≤ j ≤ k ≤ pos(B2) |
|---|
| Subtotal: 0 / 0 / 4 |
| Total: 56 / 3 / 20 |
Contributions to nonvanishing quartic force field constants
| Irrep combinations (i,i,i,i) with indices: pos(A1) ≤ i ≤ pos(B2) |
|---|
| ..35. |
A1A1A1A1. | ..15. |
B1B1B1B1. | ..5. |
B2B2B2B2. | | |
| |
| |
| |
| |
| |
| |
| Subtotal: 55 / 3 / 4 |
| Irrep combinations (i,i,i,j) (i,j,j,j) with indices: pos(A1) ≤ i ≤ j ≤ pos(B2) |
|---|
| Subtotal: 0 / 0 / 12 |
| Irrep combinations (i,i,j,j) with indices: pos(A1) ≤ i ≤ j ≤ pos(B2) |
|---|
| ..60. |
A1A1B1B1. | ..30. |
A1A1B2B2. | ..18. |
B1B1B2B2. | | |
| |
| |
| |
| |
| |
| |
| Subtotal: 108 / 3 / 6 |
| Irrep combinations (i,i,j,k) (i,j,j,k) (i,j,k,k) with indices: pos(A1) ≤ i ≤ j ≤ k ≤ pos(B2) |
|---|
| Subtotal: 0 / 0 / 12 |
| Irrep combinations (i,j,k,l) with indices: pos(A1) ≤ i ≤ j ≤ k ≤ l ≤ pos(B2) |
|---|
| Subtotal: 0 / 0 / 1 |
| Total: 163 / 6 / 35 |
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