Characters of representations for molecular motions
| Motion |
E |
2S4 |
C2 |
2C'2 |
2σd |
| Cartesian 3N |
156 |
0 |
0 |
0 |
14 |
| Translation (x,y,z) |
3 |
-1 |
-1 |
-1 |
1 |
| Rotation (Rx,Ry,Rz) |
3 |
1 |
-1 |
-1 |
-1 |
| Vibration |
150 |
0 |
2 |
2 |
14 |
Decomposition to irreducible representations
| Motion |
A1 |
A2 |
B1 |
B2 |
E |
Total |
| Cartesian 3N |
23 |
16 |
16 |
23 |
39 |
117 |
| Translation (x,y,z) |
0 |
0 |
0 |
1 |
1 |
2 |
| Rotation (Rx,Ry,Rz) |
0 |
1 |
0 |
0 |
1 |
2 |
| Vibration |
23 |
15 |
16 |
22 |
37 |
113 |
Molecular parameter
| Number of Atoms (N) |
52
|
| Number of internal coordinates |
150
|
| Number of independant internal coordinates |
23
|
| Number of vibrational modes |
113
|
Force field analysis
Allowed / forbidden vibronational transitions
| Operator |
A1 |
A2 |
B1 |
B2 |
E |
Total |
| Linear (IR) |
23 |
15 |
16 |
22 |
37 |
59 / 54 |
| Quadratic (Raman) |
23 |
15 |
16 |
22 |
37 |
98 / 15 |
| IR + Raman |
- - - - |
15 |
- - - - |
22 |
37 |
59 / 15 |
Characters of force fields
(Symmetric powers of vibration representation)
| Force field |
E |
2S4 |
C2 |
2C'2 |
2σd |
| linear |
150 |
0 |
2 |
2 |
14 |
| quadratic |
11.325 |
1 |
77 |
77 |
173 |
| cubic |
573.800 |
0 |
152 |
152 |
1.512 |
| quartic |
21.947.850 |
38 |
3.002 |
3.002 |
11.866 |
| quintic |
675.993.780 |
0 |
5.852 |
5.852 |
79.492 |
| sextic |
17.463.172.650 |
38 |
79.002 |
79.002 |
490.042 |
Decomposition to irreducible representations
Column with number of nonvanshing force constants highlighted
| Force field |
A1 |
A2 |
B1 |
B2 |
E |
| linear |
23 |
15 |
16 |
22 |
37 |
| quadratic |
1.488 |
1.363 |
1.401 |
1.449 |
2.812 |
| cubic |
72.160 |
71.328 |
71.404 |
72.084 |
143.412 |
| quartic |
2.747.583 |
2.740.149 |
2.741.631 |
2.746.063 |
5.486.212 |
| quintic |
84.521.290 |
84.478.618 |
84.481.544 |
84.518.364 |
168.996.982 |
| sextic |
2.183.048.727 |
2.182.764.205 |
2.182.803.687 |
2.183.009.207 |
4.365.773.412 |
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 D
2d
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) |
|---|
| ..276. |
A1A1. | ..120. |
A2A2. | ..136. |
B1B1. | ..253. |
B2B2. | ..703. |
EE. | | |
| |
| |
| |
| |
| Subtotal: 1.488 / 5 / 5 |
| Irrep combinations (i,j) with indices: pos(A1) ≤ i ≤ j ≤ pos(E) |
|---|
| Subtotal: 0 / 0 / 10 |
| Total: 1.488 / 5 / 15 |
Contributions to nonvanishing cubic force field constants
| Irrep combinations (i,i,i) with indices: pos(A1) ≤ i ≤ pos(E) |
|---|
| ..2.300. |
A1A1A1. | | |
| |
| |
| |
| |
| |
| |
| |
| |
| Subtotal: 2.300 / 1 / 5 |
| Irrep combinations (i,i,j) (i,j,j) with indices: pos(A1) ≤ i ≤ j ≤ pos(E) |
|---|
| ..2.760. |
A1A2A2. | ..3.128. |
A1B1B1. | ..5.819. |
A1B2B2. | ..16.169. |
A1EE. | ..9.990. |
A2EE. | ..11.248. |
B1EE. | ..15.466. |
B2EE. | | |
| |
| |
| Subtotal: 64.580 / 7 / 20 |
| Irrep combinations (i,j,k) with indices: pos(A1) ≤ i ≤ j ≤ k ≤ pos(E) |
|---|
| ..5.280. |
A2B1B2. | | |
| |
| |
| |
| |
| |
| |
| |
| |
| Subtotal: 5.280 / 1 / 10 |
| Total: 72.160 / 9 / 35 |
Contributions to nonvanishing quartic force field constants
| Irrep combinations (i,i,i,i) with indices: pos(A1) ≤ i ≤ pos(E) |
|---|
| ..14.950. |
A1A1A1A1. | ..3.060. |
A2A2A2A2. | ..3.876. |
B1B1B1B1. | ..12.650. |
B2B2B2B2. | ..338.846. |
EEEE. | | |
| |
| |
| |
| |
| Subtotal: 373.382 / 5 / 5 |
| Irrep combinations (i,i,i,j) (i,j,j,j) with indices: pos(A1) ≤ i ≤ j ≤ pos(E) |
|---|
| Subtotal: 0 / 0 / 20 |
| Irrep combinations (i,i,j,j) with indices: pos(A1) ≤ i ≤ j ≤ pos(E) |
|---|
| ..33.120. |
A1A1A2A2. | ..37.536. |
A1A1B1B1. | ..69.828. |
A1A1B2B2. | ..194.028. |
A1A1EE. | ..16.320. |
A2A2B1B1. | ..30.360. |
A2A2B2B2. | ..84.360. |
A2A2EE. | ..34.408. |
B1B1B2B2. | ..95.608. |
B1B1EE. | ..177.859. |
B2B2EE. |
| Subtotal: 773.427 / 10 / 10 |
| Irrep combinations (i,i,j,k) (i,j,j,k) (i,j,k,k) with indices: pos(A1) ≤ i ≤ j ≤ k ≤ pos(E) |
|---|
| ..229.770. |
A1A2EE. | ..258.704. |
A1B1EE. | ..355.718. |
A1B2EE. | ..168.720. |
A2B1EE. | ..231.990. |
A2B2EE. | ..234.432. |
B1B2EE. | | |
| |
| |
| |
| Subtotal: 1.479.334 / 6 / 30 |
| Irrep combinations (i,j,k,l) with indices: pos(A1) ≤ i ≤ j ≤ k ≤ l ≤ pos(E) |
|---|
| ..121.440. |
A1A2B1B2. | | |
| |
| |
| |
| |
| |
| |
| |
| |
| Subtotal: 121.440 / 1 / 5 |
| Total: 2.747.583 / 22 / 70 |
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