Characters of representations for molecular motions
| Motion |
E |
C2.(z) |
C2.(y) |
C2.(x) |
i |
σ.(xy) |
σ.(xy) |
σ.(xy) |
| Cartesian 3N |
72 |
0 |
0 |
-4 |
0 |
24 |
4 |
0 |
| Translation (x,y,z) |
3 |
-1 |
-1 |
-1 |
-3 |
1 |
1 |
1 |
| Rotation (Rx,Ry,Rz) |
3 |
-1 |
-1 |
-1 |
3 |
-1 |
-1 |
-1 |
| Vibration |
66 |
2 |
2 |
-2 |
0 |
24 |
4 |
0 |
Decomposition to irreducible representations
| Motion |
A1g |
B1g |
B2g |
B3g |
A1u |
B1u |
B2u |
B3u |
Total |
| Cartesian 3N |
12 |
12 |
7 |
5 |
5 |
7 |
12 |
12 |
72 |
| Translation (x,y,z) |
0 |
0 |
0 |
0 |
0 |
1 |
1 |
1 |
3 |
| Rotation (Rx,Ry,Rz) |
0 |
1 |
1 |
1 |
0 |
0 |
0 |
0 |
3 |
| Vibration |
12 |
11 |
6 |
4 |
5 |
6 |
11 |
11 |
66 |
Molecular parameter
| Number of Atoms (N) |
24
|
| Number of internal coordinates |
66
|
| Number of independant internal coordinates |
12
|
| Number of vibrational modes |
66
|
Force field analysis
Allowed / forbidden vibronational transitions
| Operator |
A1g |
B1g |
B2g |
B3g |
A1u |
B1u |
B2u |
B3u |
Total |
| Linear (IR) |
12 |
11 |
6 |
4 |
5 |
6 |
11 |
11 |
28 / 38 |
| Quadratic (Raman) |
12 |
11 |
6 |
4 |
5 |
6 |
11 |
11 |
33 / 33 |
| IR + Raman |
- - - - |
- - - - |
- - - - |
- - - - |
5 |
- - - - |
- - - - |
- - - - |
0* / 5 |
* Parity Mutual Exclusion Principle
Characters of force fields
(Symmetric powers of vibration representation)
| Force field |
E |
C2.(z) |
C2.(y) |
C2.(x) |
i |
σ.(xy) |
σ.(xy) |
σ.(xy) |
| linear |
66 |
2 |
2 |
-2 |
0 |
24 |
4 |
0 |
| quadratic |
2.211 |
35 |
35 |
35 |
33 |
321 |
41 |
33 |
| cubic |
50.116 |
68 |
68 |
-68 |
0 |
3.104 |
144 |
0 |
| quartic |
864.501 |
629 |
629 |
629 |
561 |
24.081 |
841 |
561 |
| quintic |
12.103.014 |
1.190 |
1.190 |
-1.190 |
0 |
158.424 |
2.660 |
0 |
| sextic |
143.218.999 |
7.735 |
7.735 |
7.735 |
6.545 |
914.641 |
11.585 |
6.545 |
Decomposition to irreducible representations
Column with number of nonvanshing force constants highlighted
| Force field |
A1g |
B1g |
B2g |
B3g |
A1u |
B1u |
B2u |
B3u |
| linear |
12 |
11 |
6 |
4 |
5 |
6 |
11 |
11 |
| quadratic |
343 |
307 |
237 |
235 |
236 |
237 |
307 |
309 |
| cubic |
6.679 |
6.643 |
5.903 |
5.833 |
5.867 |
5.903 |
6.643 |
6.645 |
| quartic |
111.554 |
110.889 |
105.079 |
105.009 |
105.043 |
105.079 |
110.889 |
110.959 |
| quintic |
1.533.161 |
1.532.496 |
1.493.555 |
1.492.295 |
1.492.890 |
1.493.555 |
1.532.496 |
1.532.566 |
| sextic |
18.022.690 |
18.014.290 |
17.788.526 |
17.787.266 |
17.787.861 |
17.788.526 |
18.014.290 |
18.015.550 |
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
2h
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(A1g) ≤ i ≤ pos(B3u) |
|---|
| ..78. |
A1gA1g. | ..66. |
B1gB1g. | ..21. |
B2gB2g. | ..10. |
B3gB3g. | ..15. |
A1uA1u. | ..21. |
B1uB1u. | ..66. |
B2uB2u. | ..66. |
B3uB3u. | | |
| |
| Subtotal: 343 / 8 / 8 |
| Irrep combinations (i,j) with indices: pos(A1g) ≤ i ≤ j ≤ pos(B3u) |
|---|
| Subtotal: 0 / 0 / 28 |
| Total: 343 / 8 / 36 |
Contributions to nonvanishing cubic force field constants
| Irrep combinations (i,i,i) with indices: pos(A1g) ≤ i ≤ pos(B3u) |
|---|
| ..364. |
A1gA1gA1g. | | |
| |
| |
| |
| |
| |
| |
| |
| |
| Subtotal: 364 / 1 / 8 |
| Irrep combinations (i,i,j) (i,j,j) with indices: pos(A1g) ≤ i ≤ j ≤ pos(B3u) |
|---|
| ..792. |
A1gB1gB1g. | ..252. |
A1gB2gB2g. | ..120. |
A1gB3gB3g. | ..180. |
A1gA1uA1u. | ..252. |
A1gB1uB1u. | ..792. |
A1gB2uB2u. | ..792. |
A1gB3uB3u. | | |
| |
| |
| Subtotal: 3.180 / 7 / 56 |
| Irrep combinations (i,j,k) with indices: pos(A1g) ≤ i ≤ j ≤ k ≤ pos(B3u) |
|---|
| ..264. |
B1gB2gB3g. | ..330. |
B1gA1uB1u. | ..1.331. |
B1gB2uB3u. | ..330. |
B2gA1uB2u. | ..396. |
B2gB1uB3u. | ..220. |
B3gA1uB3u. | ..264. |
B3gB1uB2u. | | |
| |
| |
| Subtotal: 3.135 / 7 / 56 |
| Total: 6.679 / 15 / 120 |
Contributions to nonvanishing quartic force field constants
| Irrep combinations (i,i,i,i) with indices: pos(A1g) ≤ i ≤ pos(B3u) |
|---|
| ..1.365. |
A1gA1gA1gA1g. | ..1.001. |
B1gB1gB1gB1g. | ..126. |
B2gB2gB2gB2g. | ..35. |
B3gB3gB3gB3g. | ..70. |
A1uA1uA1uA1u. | ..126. |
B1uB1uB1uB1u. | ..1.001. |
B2uB2uB2uB2u. | ..1.001. |
B3uB3uB3uB3u. | | |
| |
| Subtotal: 4.725 / 8 / 8 |
| Irrep combinations (i,i,i,j) (i,j,j,j) with indices: pos(A1g) ≤ i ≤ j ≤ pos(B3u) |
|---|
| Subtotal: 0 / 0 / 56 |
| Irrep combinations (i,i,j,j) with indices: pos(A1g) ≤ i ≤ j ≤ pos(B3u) |
|---|
| ..5.148. |
A1gA1gB1gB1g. | ..1.638. |
A1gA1gB2gB2g. | ..780. |
A1gA1gB3gB3g. | ..1.170. |
A1gA1gA1uA1u. | ..1.638. |
A1gA1gB1uB1u. | ..5.148. |
A1gA1gB2uB2u. | ..5.148. |
A1gA1gB3uB3u. | ..1.386. |
B1gB1gB2gB2g. | ..660. |
B1gB1gB3gB3g. | ..990. |
B1gB1gA1uA1u. |
| ..1.386. |
B1gB1gB1uB1u. | ..4.356. |
B1gB1gB2uB2u. | ..4.356. |
B1gB1gB3uB3u. | ..210. |
B2gB2gB3gB3g. | ..315. |
B2gB2gA1uA1u. | ..441. |
B2gB2gB1uB1u. | ..1.386. |
B2gB2gB2uB2u. | ..1.386. |
B2gB2gB3uB3u. | ..150. |
B3gB3gA1uA1u. | ..210. |
B3gB3gB1uB1u. |
| ..660. |
B3gB3gB2uB2u. | ..660. |
B3gB3gB3uB3u. | ..315. |
A1uA1uB1uB1u. | ..990. |
A1uA1uB2uB2u. | ..990. |
A1uA1uB3uB3u. | ..1.386. |
B1uB1uB2uB2u. | ..1.386. |
B1uB1uB3uB3u. | ..4.356. |
B2uB2uB3uB3u. | | |
| |
| Subtotal: 48.645 / 28 / 28 |
| Irrep combinations (i,i,j,k) (i,j,j,k) (i,j,k,k) with indices: pos(A1g) ≤ i ≤ j ≤ k ≤ pos(B3u) |
|---|
| Subtotal: 0 / 0 / 168 |
| Irrep combinations (i,j,k,l) with indices: pos(A1g) ≤ i ≤ j ≤ k ≤ l ≤ pos(B3u) |
|---|
| ..3.168. |
A1gB1gB2gB3g. | ..3.960. |
A1gB1gA1uB1u. | ..15.972. |
A1gB1gB2uB3u. | ..3.960. |
A1gB2gA1uB2u. | ..4.752. |
A1gB2gB1uB3u. | ..2.640. |
A1gB3gA1uB3u. | ..3.168. |
A1gB3gB1uB2u. | ..3.630. |
B1gB2gA1uB3u. | ..4.356. |
B1gB2gB1uB2u. | ..2.420. |
B1gB3gA1uB2u. |
| ..2.904. |
B1gB3gB1uB3u. | ..720. |
B2gB3gA1uB1u. | ..2.904. |
B2gB3gB2uB3u. | ..3.630. |
A1uB1uB2uB3u. | | |
| |
| |
| |
| |
| |
| Subtotal: 58.184 / 14 / 70 |
| Total: 111.554 / 50 / 330 |
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