Characters of representations for molecular motions
| Motion |
E |
2C6 |
2C3 |
C2 |
3C'2 |
3C''2 |
| Cartesian 3N |
75 |
2 |
0 |
-1 |
-1 |
-1 |
| Translation (x,y,z) |
3 |
2 |
0 |
-1 |
-1 |
-1 |
| Rotation (Rx,Ry,Rz) |
3 |
2 |
0 |
-1 |
-1 |
-1 |
| Vibration |
69 |
-2 |
0 |
1 |
1 |
1 |
Decomposition to irreducible representations
| Motion |
A1 |
A2 |
B1 |
B2 |
E1 |
E2 |
Total |
| Cartesian 3N |
6 |
7 |
6 |
6 |
13 |
12 |
50 |
| Translation (x,y,z) |
0 |
1 |
0 |
0 |
1 |
0 |
2 |
| Rotation (Rx,Ry,Rz) |
0 |
1 |
0 |
0 |
1 |
0 |
2 |
| Vibration |
6 |
5 |
6 |
6 |
11 |
12 |
46 |
Molecular parameter
| Number of Atoms (N) |
25
|
| Number of internal coordinates |
69
|
| Number of independant internal coordinates |
6
|
| Number of vibrational modes |
46
|
Force field analysis
Allowed / forbidden vibronational transitions
| Operator |
A1 |
A2 |
B1 |
B2 |
E1 |
E2 |
Total |
| Linear (IR) |
6 |
5 |
6 |
6 |
11 |
12 |
16 / 30 |
| Quadratic (Raman) |
6 |
5 |
6 |
6 |
11 |
12 |
29 / 17 |
| IR + Raman |
- - - - |
- - - - |
6 |
6 |
11 |
- - - - |
11 / 12 |
Characters of force fields
(Symmetric powers of vibration representation)
| Force field |
E |
2C6 |
2C3 |
C2 |
3C'2 |
3C''2 |
| linear |
69 |
-2 |
0 |
1 |
1 |
1 |
| quadratic |
2.415 |
2 |
0 |
35 |
35 |
35 |
| cubic |
57.155 |
-1 |
23 |
35 |
35 |
35 |
| quartic |
1.028.790 |
0 |
0 |
630 |
630 |
630 |
| quintic |
15.020.334 |
0 |
0 |
630 |
630 |
630 |
| sextic |
185.250.786 |
12 |
276 |
7.770 |
7.770 |
7.770 |
Decomposition to irreducible representations
Column with number of nonvanshing force constants highlighted
| Force field |
A1 |
A2 |
B1 |
B2 |
E1 |
E2 |
| linear |
6 |
5 |
6 |
6 |
11 |
12 |
| quadratic |
222 |
187 |
198 |
198 |
397 |
408 |
| cubic |
4.787 |
4.752 |
4.764 |
4.764 |
9.516 |
9.528 |
| quartic |
86.100 |
85.470 |
85.680 |
85.680 |
171.360 |
171.570 |
| quintic |
1.252.062 |
1.251.432 |
1.251.642 |
1.251.642 |
2.503.284 |
2.503.494 |
| sextic |
15.442.146 |
15.434.376 |
15.436.962 |
15.436.962 |
30.873.792 |
30.876.378 |
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
6
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(E2) |
|---|
| ..21. |
A1A1. | ..15. |
A2A2. | ..21. |
B1B1. | ..21. |
B2B2. | ..66. |
E1E1. | ..78. |
E2E2. | | |
| |
| |
| |
| Subtotal: 222 / 6 / 6 |
| Irrep combinations (i,j) with indices: pos(A1) ≤ i ≤ j ≤ pos(E2) |
|---|
| Subtotal: 0 / 0 / 15 |
| Total: 222 / 6 / 21 |
Contributions to nonvanishing cubic force field constants
| Irrep combinations (i,i,i) with indices: pos(A1) ≤ i ≤ pos(E2) |
|---|
| ..56. |
A1A1A1. | ..364. |
E2E2E2. | | |
| |
| |
| |
| |
| |
| |
| |
| Subtotal: 420 / 2 / 6 |
| Irrep combinations (i,i,j) (i,j,j) with indices: pos(A1) ≤ i ≤ j ≤ pos(E2) |
|---|
| ..792. |
E1E1E2. | ..90. |
A1A2A2. | ..126. |
A1B1B1. | ..126. |
A1B2B2. | ..396. |
A1E1E1. | ..468. |
A1E2E2. | ..275. |
A2E1E1. | ..330. |
A2E2E2. | | |
| |
| Subtotal: 2.603 / 8 / 30 |
| Irrep combinations (i,j,k) with indices: pos(A1) ≤ i ≤ j ≤ k ≤ pos(E2) |
|---|
| ..180. |
A2B1B2. | ..792. |
B1E1E2. | ..792. |
B2E1E2. | | |
| |
| |
| |
| |
| |
| |
| Subtotal: 1.764 / 3 / 20 |
| Total: 4.787 / 13 / 56 |
Contributions to nonvanishing quartic force field constants
| Irrep combinations (i,i,i,i) with indices: pos(A1) ≤ i ≤ pos(E2) |
|---|
| ..126. |
A1A1A1A1. | ..70. |
A2A2A2A2. | ..126. |
B1B1B1B1. | ..126. |
B2B2B2B2. | ..2.211. |
E1E1E1E1. | ..3.081. |
E2E2E2E2. | | |
| |
| |
| |
| Subtotal: 5.740 / 6 / 6 |
| Irrep combinations (i,i,i,j) (i,j,j,j) with indices: pos(A1) ≤ i ≤ j ≤ pos(E2) |
|---|
| ..2.184. |
A1E2E2E2. | ..1.820. |
A2E2E2E2. | ..1.716. |
B1E1E1E1. | ..1.716. |
B2E1E1E1. | | |
| |
| |
| |
| |
| |
| Subtotal: 7.436 / 4 / 30 |
| Irrep combinations (i,i,j,j) with indices: pos(A1) ≤ i ≤ j ≤ pos(E2) |
|---|
| ..315. |
A1A1A2A2. | ..441. |
A1A1B1B1. | ..441. |
A1A1B2B2. | ..1.386. |
A1A1E1E1. | ..1.638. |
A1A1E2E2. | ..315. |
A2A2B1B1. | ..315. |
A2A2B2B2. | ..990. |
A2A2E1E1. | ..1.170. |
A2A2E2E2. | ..441. |
B1B1B2B2. |
| ..1.386. |
B1B1E1E1. | ..1.638. |
B1B1E2E2. | ..1.386. |
B2B2E1E1. | ..1.638. |
B2B2E2E2. | ..13.926. |
E1E1E2E2. | | |
| |
| |
| |
| |
| Subtotal: 27.426 / 15 / 15 |
| Irrep combinations (i,i,j,k) (i,j,j,k) (i,j,k,k) with indices: pos(A1) ≤ i ≤ j ≤ k ≤ pos(E2) |
|---|
| ..4.752. |
A1E1E1E2. | ..3.960. |
A2E1E1E2. | ..1.650. |
A1A2E1E1. | ..1.980. |
A1A2E2E2. | ..1.980. |
B1B2E1E1. | ..2.376. |
B1B2E2E2. | ..5.148. |
B1E1E2E2. | ..5.148. |
B2E1E2E2. | | |
| |
| Subtotal: 26.994 / 8 / 60 |
| Irrep combinations (i,j,k,l) with indices: pos(A1) ≤ i ≤ j ≤ k ≤ l ≤ pos(E2) |
|---|
| ..1.080. |
A1A2B1B2. | ..4.752. |
A1B1E1E2. | ..4.752. |
A1B2E1E2. | ..3.960. |
A2B1E1E2. | ..3.960. |
A2B2E1E2. | | |
| |
| |
| |
| |
| Subtotal: 18.504 / 5 / 15 |
| Total: 86.100 / 38 / 126 |
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