Characters of representations for molecular motions
| Motion |
E |
C2.(z) |
σv.(xz) |
σd.(yz) |
| Cartesian 3N |
156 |
-2 |
4 |
14 |
| Translation (x,y,z) |
3 |
-1 |
1 |
1 |
| Rotation (Rx,Ry,Rz) |
3 |
-1 |
-1 |
-1 |
| Vibration |
150 |
0 |
4 |
14 |
Decomposition to irreducible representations
| Motion |
A1 |
A2 |
B1 |
B2 |
Total |
| Cartesian 3N |
43 |
34 |
37 |
42 |
156 |
| Translation (x,y,z) |
1 |
0 |
1 |
1 |
3 |
| Rotation (Rx,Ry,Rz) |
0 |
1 |
1 |
1 |
3 |
| Vibration |
42 |
33 |
35 |
40 |
150 |
Molecular parameter
| Number of Atoms (N) |
52
|
| Number of internal coordinates |
150
|
| Number of independant internal coordinates |
42
|
| Number of vibrational modes |
150
|
Force field analysis
Allowed / forbidden vibronational transitions
| Operator |
A1 |
A2 |
B1 |
B2 |
Total |
| Linear (IR) |
42 |
33 |
35 |
40 |
117 / 33 |
| Quadratic (Raman) |
42 |
33 |
35 |
40 |
150 / 0 |
| IR + Raman |
42 |
- - - - |
35 |
40 |
117 / 0 |
Characters of force fields
(Symmetric powers of vibration representation)
| Force field |
E |
C2.(z) |
σv.(xz) |
σd.(yz) |
| linear |
150 |
0 |
4 |
14 |
| quadratic |
11.325 |
75 |
83 |
173 |
| cubic |
573.800 |
0 |
312 |
1.512 |
| quartic |
21.947.850 |
2.850 |
3.466 |
11.866 |
| quintic |
675.993.780 |
0 |
12.320 |
79.492 |
| sextic |
17.463.172.650 |
73.150 |
97.174 |
490.042 |
Decomposition to irreducible representations
Column with number of nonvanshing force constants highlighted
| Force field |
A1 |
A2 |
B1 |
B2 |
| linear |
42 |
33 |
35 |
40 |
| quadratic |
2.914 |
2.786 |
2.790 |
2.835 |
| cubic |
143.906 |
142.994 |
143.150 |
143.750 |
| quartic |
5.491.508 |
5.483.842 |
5.484.150 |
5.488.350 |
| quintic |
169.021.398 |
168.975.492 |
168.981.652 |
169.015.238 |
| sextic |
4.365.958.254 |
4.365.664.646 |
4.365.676.658 |
4.365.873.092 |
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) |
|---|
| ..903. |
A1A1. | ..561. |
A2A2. | ..630. |
B1B1. | ..820. |
B2B2. | | |
| |
| |
| |
| |
| |
| Subtotal: 2.914 / 4 / 4 |
| Irrep combinations (i,j) with indices: pos(A1) ≤ i ≤ j ≤ pos(B2) |
|---|
| Subtotal: 0 / 0 / 6 |
| Total: 2.914 / 4 / 10 |
Contributions to nonvanishing cubic force field constants
| Irrep combinations (i,i,i) with indices: pos(A1) ≤ i ≤ pos(B2) |
|---|
| ..13.244. |
A1A1A1. | | |
| |
| |
| |
| |
| |
| |
| |
| |
| Subtotal: 13.244 / 1 / 4 |
| Irrep combinations (i,i,j) (i,j,j) with indices: pos(A1) ≤ i ≤ j ≤ pos(B2) |
|---|
| ..23.562. |
A1A2A2. | ..26.460. |
A1B1B1. | ..34.440. |
A1B2B2. | | |
| |
| |
| |
| |
| |
| |
| Subtotal: 84.462 / 3 / 12 |
| Irrep combinations (i,j,k) with indices: pos(A1) ≤ i ≤ j ≤ k ≤ pos(B2) |
|---|
| ..46.200. |
A2B1B2. | | |
| |
| |
| |
| |
| |
| |
| |
| |
| Subtotal: 46.200 / 1 / 4 |
| Total: 143.906 / 5 / 20 |
Contributions to nonvanishing quartic force field constants
| Irrep combinations (i,i,i,i) with indices: pos(A1) ≤ i ≤ pos(B2) |
|---|
| ..148.995. |
A1A1A1A1. | ..58.905. |
A2A2A2A2. | ..73.815. |
B1B1B1B1. | ..123.410. |
B2B2B2B2. | | |
| |
| |
| |
| |
| |
| Subtotal: 405.125 / 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) |
|---|
| ..506.583. |
A1A1A2A2. | ..568.890. |
A1A1B1B1. | ..740.460. |
A1A1B2B2. | ..353.430. |
A2A2B1B1. | ..460.020. |
A2A2B2B2. | ..516.600. |
B1B1B2B2. | | |
| |
| |
| |
| Subtotal: 3.145.983 / 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) |
|---|
| ..1.940.400. |
A1A2B1B2. | | |
| |
| |
| |
| |
| |
| |
| |
| |
| Subtotal: 1.940.400 / 1 / 1 |
| Total: 5.491.508 / 11 / 35 |
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