Characters of representations for molecular motions
| Motion |
E |
C2.(z) |
σv.(xz) |
σd.(yz) |
| Cartesian 3N |
72 |
0 |
24 |
0 |
| Translation (x,y,z) |
3 |
-1 |
1 |
1 |
| Rotation (Rx,Ry,Rz) |
3 |
-1 |
-1 |
-1 |
| Vibration |
66 |
2 |
24 |
0 |
Decomposition to irreducible representations
| Motion |
A1 |
A2 |
B1 |
B2 |
Total |
| Cartesian 3N |
24 |
12 |
24 |
12 |
72 |
| Translation (x,y,z) |
1 |
0 |
1 |
1 |
3 |
| Rotation (Rx,Ry,Rz) |
0 |
1 |
1 |
1 |
3 |
| Vibration |
23 |
11 |
22 |
10 |
66 |
Molecular parameter
| Number of Atoms (N) |
24
|
| Number of internal coordinates |
66
|
| Number of independant internal coordinates |
23
|
| Number of vibrational modes |
66
|
Force field analysis
Allowed / forbidden vibronational transitions
| Operator |
A1 |
A2 |
B1 |
B2 |
Total |
| Linear (IR) |
23 |
11 |
22 |
10 |
55 / 11 |
| Quadratic (Raman) |
23 |
11 |
22 |
10 |
66 / 0 |
| IR + Raman |
23 |
- - - - |
22 |
10 |
55 / 0 |
Characters of force fields
(Symmetric powers of vibration representation)
| Force field |
E |
C2.(z) |
σv.(xz) |
σd.(yz) |
| linear |
66 |
2 |
24 |
0 |
| quadratic |
2.211 |
35 |
321 |
33 |
| cubic |
50.116 |
68 |
3.104 |
0 |
| quartic |
864.501 |
629 |
24.081 |
561 |
| quintic |
12.103.014 |
1.190 |
158.424 |
0 |
| sextic |
143.218.999 |
7.735 |
914.641 |
6.545 |
Decomposition to irreducible representations
Column with number of nonvanshing force constants highlighted
| Force field |
A1 |
A2 |
B1 |
B2 |
| linear |
23 |
11 |
22 |
10 |
| quadratic |
650 |
473 |
616 |
472 |
| cubic |
13.322 |
11.770 |
13.288 |
11.736 |
| quartic |
222.443 |
210.122 |
221.848 |
210.088 |
| quintic |
3.065.657 |
2.986.445 |
3.065.062 |
2.985.850 |
| sextic |
36.036.980 |
35.576.387 |
36.029.840 |
35.575.792 |
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) |
|---|
| ..276. |
A1A1. | ..66. |
A2A2. | ..253. |
B1B1. | ..55. |
B2B2. | | |
| |
| |
| |
| |
| |
| Subtotal: 650 / 4 / 4 |
| Irrep combinations (i,j) with indices: pos(A1) ≤ i ≤ j ≤ pos(B2) |
|---|
| Subtotal: 0 / 0 / 6 |
| Total: 650 / 4 / 10 |
Contributions to nonvanishing cubic force field constants
| Irrep combinations (i,i,i) with indices: pos(A1) ≤ i ≤ pos(B2) |
|---|
| ..2.300. |
A1A1A1. | | |
| |
| |
| |
| |
| |
| |
| |
| |
| Subtotal: 2.300 / 1 / 4 |
| Irrep combinations (i,i,j) (i,j,j) with indices: pos(A1) ≤ i ≤ j ≤ pos(B2) |
|---|
| ..1.518. |
A1A2A2. | ..5.819. |
A1B1B1. | ..1.265. |
A1B2B2. | | |
| |
| |
| |
| |
| |
| |
| Subtotal: 8.602 / 3 / 12 |
| Irrep combinations (i,j,k) with indices: pos(A1) ≤ i ≤ j ≤ k ≤ pos(B2) |
|---|
| ..2.420. |
A2B1B2. | | |
| |
| |
| |
| |
| |
| |
| |
| |
| Subtotal: 2.420 / 1 / 4 |
| Total: 13.322 / 5 / 20 |
Contributions to nonvanishing quartic force field constants
| Irrep combinations (i,i,i,i) with indices: pos(A1) ≤ i ≤ pos(B2) |
|---|
| ..14.950. |
A1A1A1A1. | ..1.001. |
A2A2A2A2. | ..12.650. |
B1B1B1B1. | ..715. |
B2B2B2B2. | | |
| |
| |
| |
| |
| |
| Subtotal: 29.316 / 4 / 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) |
|---|
| ..18.216. |
A1A1A2A2. | ..69.828. |
A1A1B1B1. | ..15.180. |
A1A1B2B2. | ..16.698. |
A2A2B1B1. | ..3.630. |
A2A2B2B2. | ..13.915. |
B1B1B2B2. | | |
| |
| |
| |
| Subtotal: 137.467 / 6 / 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) |
|---|
| ..55.660. |
A1A2B1B2. | | |
| |
| |
| |
| |
| |
| |
| |
| |
| Subtotal: 55.660 / 1 / 1 |
| Total: 222.443 / 11 / 35 |
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