Characters of representations for molecular motions
| Motion |
E |
2C8 |
2C4 |
2(C8)3 |
C2 |
4C'2 |
4C''2 |
| Cartesian 3N |
99 |
2.414 |
1 |
-0.414 |
-1 |
-1 |
-1 |
| Translation (x,y,z) |
3 |
2.414 |
1 |
-0.414 |
-1 |
-1 |
-1 |
| Rotation (Rx,Ry,Rz) |
3 |
2.414 |
1 |
-0.414 |
-1 |
-1 |
-1 |
| Vibration |
93 |
-2.414 |
-1 |
0.414 |
1 |
1 |
1 |
Decomposition to irreducible representations
| Motion |
A1 |
A2 |
B1 |
B2 |
E1 |
E2 |
E3 |
Total |
| Cartesian 3N |
6 |
7 |
6 |
6 |
13 |
12 |
12 |
62 |
| Translation (x,y,z) |
0 |
1 |
0 |
0 |
1 |
0 |
0 |
2 |
| Rotation (Rx,Ry,Rz) |
0 |
1 |
0 |
0 |
1 |
0 |
0 |
2 |
| Vibration |
6 |
5 |
6 |
6 |
11 |
12 |
12 |
58 |
Molecular parameter
| Number of Atoms (N) |
33
|
| Number of internal coordinates |
93
|
| Number of independant internal coordinates |
6
|
| Number of vibrational modes |
58
|
Force field analysis
Allowed / forbidden vibronational transitions
| Operator |
A1 |
A2 |
B1 |
B2 |
E1 |
E2 |
E3 |
Total |
| Linear (IR) |
6 |
5 |
6 |
6 |
11 |
12 |
12 |
16 / 42 |
| Quadratic (Raman) |
6 |
5 |
6 |
6 |
11 |
12 |
12 |
29 / 29 |
| IR + Raman |
- - - - |
- - - - |
6 |
6 |
11 |
- - - - |
12 |
11 / 24 |
Characters of force fields
(Symmetric powers of vibration representation)
| Force field |
E |
2C8 |
2C4 |
2(C8)3 |
C2 |
4C'2 |
4C''2 |
| linear |
93 |
-2.414 |
-1 |
0.414 |
1 |
1 |
1 |
| quadratic |
4.371 |
2.414 |
1 |
-0.414 |
47 |
47 |
47 |
| cubic |
138.415 |
-1.000 |
-1 |
-1.000 |
47 |
47 |
47 |
| quartic |
3.321.960 |
-0.000 |
24 |
-0.000 |
1.128 |
1.128 |
1.128 |
| quintic |
64.446.024 |
0.000 |
-24 |
0.000 |
1.128 |
1.128 |
1.128 |
| sextic |
1.052.618.392 |
-0.000 |
24 |
-0.000 |
18.424 |
18.424 |
18.424 |
Decomposition to irreducible representations
Column with number of nonvanshing force constants highlighted
| Force field |
A1 |
A2 |
B1 |
B2 |
E1 |
E2 |
E3 |
| linear |
6 |
5 |
6 |
6 |
11 |
12 |
12 |
| quadratic |
300 |
253 |
276 |
276 |
541 |
552 |
540 |
| cubic |
8.677 |
8.630 |
8.654 |
8.654 |
17.296 |
17.308 |
17.296 |
| quartic |
208.260 |
207.132 |
207.696 |
207.696 |
415.104 |
415.380 |
415.104 |
| quintic |
4.028.508 |
4.027.380 |
4.027.944 |
4.027.944 |
8.055.612 |
8.055.900 |
8.055.612 |
| sextic |
65.799.016 |
65.780.592 |
65.789.804 |
65.789.804 |
131.574.996 |
131.579.596 |
131.574.996 |
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
8
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(E3) |
|---|
| ..21. |
A1A1. | ..15. |
A2A2. | ..21. |
B1B1. | ..21. |
B2B2. | ..66. |
E1E1. | ..78. |
E2E2. | ..78. |
E3E3. | | |
| |
| |
| Subtotal: 300 / 7 / 7 |
| Irrep combinations (i,j) with indices: pos(A1) ≤ i ≤ j ≤ pos(E3) |
|---|
| Subtotal: 0 / 0 / 21 |
| Total: 300 / 7 / 28 |
Contributions to nonvanishing cubic force field constants
| Irrep combinations (i,i,i) with indices: pos(A1) ≤ i ≤ pos(E3) |
|---|
| ..56. |
A1A1A1. | | |
| |
| |
| |
| |
| |
| |
| |
| |
| Subtotal: 56 / 1 / 7 |
| Irrep combinations (i,i,j) (i,j,j) with indices: pos(A1) ≤ i ≤ j ≤ pos(E3) |
|---|
| ..792. |
E1E1E2. | ..90. |
A1A2A2. | ..126. |
A1B1B1. | ..126. |
A1B2B2. | ..396. |
A1E1E1. | ..468. |
A1E2E2. | ..468. |
A1E3E3. | ..275. |
A2E1E1. | ..330. |
A2E2E2. | ..330. |
A2E3E3. |
| ..468. |
B1E2E2. | ..468. |
B2E2E2. | ..936. |
E2E3E3. | | |
| |
| |
| |
| |
| |
| |
| Subtotal: 5.273 / 13 / 42 |
| Irrep combinations (i,j,k) with indices: pos(A1) ≤ i ≤ j ≤ k ≤ pos(E3) |
|---|
| ..180. |
A2B1B2. | ..792. |
B1E1E3. | ..792. |
B2E1E3. | ..1.584. |
E1E2E3. | | |
| |
| |
| |
| |
| |
| Subtotal: 3.348 / 4 / 35 |
| Total: 8.677 / 18 / 84 |
Contributions to nonvanishing quartic force field constants
| Irrep combinations (i,i,i,i) with indices: pos(A1) ≤ i ≤ pos(E3) |
|---|
| ..126. |
A1A1A1A1. | ..70. |
A2A2A2A2. | ..126. |
B1B1B1B1. | ..126. |
B2B2B2B2. | ..2.211. |
E1E1E1E1. | ..4.446. |
E2E2E2E2. | ..3.081. |
E3E3E3E3. | | |
| |
| |
| Subtotal: 10.186 / 7 / 7 |
| Irrep combinations (i,i,i,j) (i,j,j,j) with indices: pos(A1) ≤ i ≤ j ≤ pos(E3) |
|---|
| ..3.432. |
E1E1E1E3. | ..4.004. |
E1E3E3E3. | | |
| |
| |
| |
| |
| |
| |
| |
| Subtotal: 7.436 / 2 / 42 |
| Irrep combinations (i,i,j,j) with indices: pos(A1) ≤ i ≤ j ≤ pos(E3) |
|---|
| ..315. |
A1A1A2A2. | ..441. |
A1A1B1B1. | ..441. |
A1A1B2B2. | ..1.386. |
A1A1E1E1. | ..1.638. |
A1A1E2E2. | ..1.638. |
A1A1E3E3. | ..315. |
A2A2B1B1. | ..315. |
A2A2B2B2. | ..990. |
A2A2E1E1. | ..1.170. |
A2A2E2E2. |
| ..1.170. |
A2A2E3E3. | ..441. |
B1B1B2B2. | ..1.386. |
B1B1E1E1. | ..1.638. |
B1B1E2E2. | ..1.638. |
B1B1E3E3. | ..1.386. |
B2B2E1E1. | ..1.638. |
B2B2E2E2. | ..1.638. |
B2B2E3E3. | ..8.778. |
E1E1E2E2. | ..13.926. |
E1E1E3E3. |
| ..10.440. |
E2E2E3E3. | | |
| |
| |
| |
| |
| |
| |
| |
| |
| Subtotal: 52.728 / 21 / 21 |
| Irrep combinations (i,i,j,k) (i,j,j,k) (i,j,k,k) with indices: pos(A1) ≤ i ≤ j ≤ k ≤ pos(E3) |
|---|
| ..4.752. |
A1E1E1E2. | ..3.960. |
A2E1E1E2. | ..4.752. |
B1E1E1E2. | ..4.752. |
B2E1E1E2. | ..20.592. |
E1E2E2E3. | ..1.650. |
A1A2E1E1. | ..1.980. |
A1A2E2E2. | ..1.980. |
A1A2E3E3. | ..2.808. |
A1B1E2E2. | ..2.808. |
A1B2E2E2. |
| ..5.616. |
A1E2E3E3. | ..2.340. |
A2B1E2E2. | ..2.340. |
A2B2E2E2. | ..4.680. |
A2E2E3E3. | ..1.980. |
B1B2E1E1. | ..2.376. |
B1B2E2E2. | ..2.376. |
B1B2E3E3. | ..5.616. |
B1E2E3E3. | ..5.616. |
B2E2E3E3. | | |
| Subtotal: 82.974 / 19 / 105 |
| Irrep combinations (i,j,k,l) with indices: pos(A1) ≤ i ≤ j ≤ k ≤ l ≤ pos(E3) |
|---|
| ..1.080. |
A1A2B1B2. | ..4.752. |
A1B1E1E3. | ..4.752. |
A1B2E1E3. | ..9.504. |
A1E1E2E3. | ..3.960. |
A2B1E1E3. | ..3.960. |
A2B2E1E3. | ..7.920. |
A2E1E2E3. | ..9.504. |
B1E1E2E3. | ..9.504. |
B2E1E2E3. | | |
| Subtotal: 54.936 / 9 / 35 |
| Total: 208.260 / 58 / 210 |
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