Characters of representations for molecular motions
| Motion |
E |
2S4 |
C2 |
2C'2 |
2σd |
| Cartesian 3N |
48 |
0 |
0 |
0 |
0 |
| Translation (x,y,z) |
3 |
-1 |
-1 |
-1 |
1 |
| Rotation (Rx,Ry,Rz) |
3 |
1 |
-1 |
-1 |
-1 |
| Vibration |
42 |
0 |
2 |
2 |
0 |
Decomposition to irreducible representations
| Motion |
A1 |
A2 |
B1 |
B2 |
E |
Total |
| Cartesian 3N |
6 |
6 |
6 |
6 |
12 |
36 |
| Translation (x,y,z) |
0 |
0 |
0 |
1 |
1 |
2 |
| Rotation (Rx,Ry,Rz) |
0 |
1 |
0 |
0 |
1 |
2 |
| Vibration |
6 |
5 |
6 |
5 |
10 |
32 |
Molecular parameter
| Number of Atoms (N) |
16
|
| Number of internal coordinates |
42
|
| Number of independant internal coordinates |
6
|
| Number of vibrational modes |
32
|
Force field analysis
Allowed / forbidden vibronational transitions
| Operator |
A1 |
A2 |
B1 |
B2 |
E |
Total |
| Linear (IR) |
6 |
5 |
6 |
5 |
10 |
15 / 17 |
| Quadratic (Raman) |
6 |
5 |
6 |
5 |
10 |
27 / 5 |
| IR + Raman |
- - - - |
5 |
- - - - |
5 |
10 |
15 / 5 |
Characters of force fields
(Symmetric powers of vibration representation)
| Force field |
E |
2S4 |
C2 |
2C'2 |
2σd |
| linear |
42 |
0 |
2 |
2 |
0 |
| quadratic |
903 |
1 |
23 |
23 |
21 |
| cubic |
13.244 |
0 |
44 |
44 |
0 |
| quartic |
148.995 |
11 |
275 |
275 |
231 |
| quintic |
1.370.754 |
0 |
506 |
506 |
0 |
| sextic |
10.737.573 |
11 |
2.277 |
2.277 |
1.771 |
Decomposition to irreducible representations
Column with number of nonvanshing force constants highlighted
| Force field |
A1 |
A2 |
B1 |
B2 |
E |
| linear |
6 |
5 |
6 |
5 |
10 |
| quadratic |
127 |
105 |
116 |
115 |
220 |
| cubic |
1.672 |
1.650 |
1.672 |
1.650 |
3.300 |
| quartic |
18.788 |
18.535 |
18.667 |
18.645 |
37.180 |
| quintic |
171.534 |
171.281 |
171.534 |
171.281 |
342.562 |
| sextic |
1.343.496 |
1.341.472 |
1.342.605 |
1.342.352 |
2.683.824 |
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) |
|---|
| ..21. |
A1A1. | ..15. |
A2A2. | ..21. |
B1B1. | ..15. |
B2B2. | ..55. |
EE. | | |
| |
| |
| |
| |
| Subtotal: 127 / 5 / 5 |
| Irrep combinations (i,j) with indices: pos(A1) ≤ i ≤ j ≤ pos(E) |
|---|
| Subtotal: 0 / 0 / 10 |
| Total: 127 / 5 / 15 |
Contributions to nonvanishing cubic force field constants
| Irrep combinations (i,i,i) with indices: pos(A1) ≤ i ≤ pos(E) |
|---|
| ..56. |
A1A1A1. | | |
| |
| |
| |
| |
| |
| |
| |
| |
| Subtotal: 56 / 1 / 5 |
| Irrep combinations (i,i,j) (i,j,j) with indices: pos(A1) ≤ i ≤ j ≤ pos(E) |
|---|
| ..90. |
A1A2A2. | ..126. |
A1B1B1. | ..90. |
A1B2B2. | ..330. |
A1EE. | ..225. |
A2EE. | ..330. |
B1EE. | ..275. |
B2EE. | | |
| |
| |
| Subtotal: 1.466 / 7 / 20 |
| Irrep combinations (i,j,k) with indices: pos(A1) ≤ i ≤ j ≤ k ≤ pos(E) |
|---|
| ..150. |
A2B1B2. | | |
| |
| |
| |
| |
| |
| |
| |
| |
| Subtotal: 150 / 1 / 10 |
| Total: 1.672 / 9 / 35 |
Contributions to nonvanishing quartic force field constants
| Irrep combinations (i,i,i,i) with indices: pos(A1) ≤ i ≤ pos(E) |
|---|
| ..126. |
A1A1A1A1. | ..70. |
A2A2A2A2. | ..126. |
B1B1B1B1. | ..70. |
B2B2B2B2. | ..2.255. |
EEEE. | | |
| |
| |
| |
| |
| Subtotal: 2.647 / 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) |
|---|
| ..315. |
A1A1A2A2. | ..441. |
A1A1B1B1. | ..315. |
A1A1B2B2. | ..1.155. |
A1A1EE. | ..315. |
A2A2B1B1. | ..225. |
A2A2B2B2. | ..825. |
A2A2EE. | ..315. |
B1B1B2B2. | ..1.155. |
B1B1EE. | ..825. |
B2B2EE. |
| Subtotal: 5.886 / 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) |
|---|
| ..1.350. |
A1A2EE. | ..1.980. |
A1B1EE. | ..1.650. |
A1B2EE. | ..1.650. |
A2B1EE. | ..1.375. |
A2B2EE. | ..1.350. |
B1B2EE. | | |
| |
| |
| |
| Subtotal: 9.355 / 6 / 30 |
| Irrep combinations (i,j,k,l) with indices: pos(A1) ≤ i ≤ j ≤ k ≤ l ≤ pos(E) |
|---|
| ..900. |
A1A2B1B2. | | |
| |
| |
| |
| |
| |
| |
| |
| |
| Subtotal: 900 / 1 / 5 |
| Total: 18.788 / 22 / 70 |
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