Characters of representations for molecular motions
| Motion |
E |
C2.(z) |
C2.(y) |
C2.(x) |
i |
σ.(xy) |
σ.(xy) |
σ.(xy) |
| Cartesian 3N |
90 |
0 |
0 |
-2 |
0 |
30 |
2 |
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 |
84 |
2 |
2 |
0 |
0 |
30 |
2 |
0 |
Decomposition to irreducible representations
| Motion |
A1g |
B1g |
B2g |
B3g |
A1u |
B1u |
B2u |
B3u |
Total |
| Cartesian 3N |
15 |
15 |
8 |
7 |
7 |
8 |
15 |
15 |
90 |
| 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 |
15 |
14 |
7 |
6 |
7 |
7 |
14 |
14 |
84 |
Molecular parameter
| Number of Atoms (N) |
30
|
| Number of internal coordinates |
84
|
| Number of independant internal coordinates |
15
|
| Number of vibrational modes |
84
|
Force field analysis
Allowed / forbidden vibronational transitions
| Operator |
A1g |
B1g |
B2g |
B3g |
A1u |
B1u |
B2u |
B3u |
Total |
| Linear (IR) |
15 |
14 |
7 |
6 |
7 |
7 |
14 |
14 |
35 / 49 |
| Quadratic (Raman) |
15 |
14 |
7 |
6 |
7 |
7 |
14 |
14 |
42 / 42 |
| IR + Raman |
- - - - |
- - - - |
- - - - |
- - - - |
7 |
- - - - |
- - - - |
- - - - |
0* / 7 |
* 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 |
84 |
2 |
2 |
0 |
0 |
30 |
2 |
0 |
| quadratic |
3.570 |
44 |
44 |
42 |
42 |
492 |
44 |
42 |
| cubic |
102.340 |
86 |
86 |
0 |
0 |
5.770 |
86 |
0 |
| quartic |
2.225.895 |
989 |
989 |
903 |
903 |
53.853 |
989 |
903 |
| quintic |
39.175.752 |
1.892 |
1.892 |
0 |
0 |
423.516 |
1.892 |
0 |
| sextic |
581.106.988 |
15.136 |
15.136 |
13.244 |
13.244 |
2.907.424 |
15.136 |
13.244 |
Decomposition to irreducible representations
Column with number of nonvanshing force constants highlighted
| Force field |
A1g |
B1g |
B2g |
B3g |
A1u |
B1u |
B2u |
B3u |
| linear |
15 |
14 |
7 |
6 |
7 |
7 |
14 |
14 |
| quadratic |
540 |
497 |
385 |
384 |
385 |
385 |
497 |
497 |
| cubic |
13.546 |
13.503 |
12.082 |
12.039 |
12.082 |
12.082 |
13.503 |
13.503 |
| quartic |
285.678 |
284.732 |
271.516 |
271.473 |
271.516 |
271.516 |
284.732 |
284.732 |
| quintic |
4.950.618 |
4.949.672 |
4.844.266 |
4.843.320 |
4.844.266 |
4.844.266 |
4.949.672 |
4.949.672 |
| sextic |
73.012.444 |
72.998.254 |
72.275.182 |
72.274.236 |
72.275.182 |
72.275.182 |
72.998.254 |
72.998.254 |
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) |
|---|
| ..120. |
A1gA1g. | ..105. |
B1gB1g. | ..28. |
B2gB2g. | ..21. |
B3gB3g. | ..28. |
A1uA1u. | ..28. |
B1uB1u. | ..105. |
B2uB2u. | ..105. |
B3uB3u. | | |
| |
| Subtotal: 540 / 8 / 8 |
| Irrep combinations (i,j) with indices: pos(A1g) ≤ i ≤ j ≤ pos(B3u) |
|---|
| Subtotal: 0 / 0 / 28 |
| Total: 540 / 8 / 36 |
Contributions to nonvanishing cubic force field constants
| Irrep combinations (i,i,i) with indices: pos(A1g) ≤ i ≤ pos(B3u) |
|---|
| ..680. |
A1gA1gA1g. | | |
| |
| |
| |
| |
| |
| |
| |
| |
| Subtotal: 680 / 1 / 8 |
| Irrep combinations (i,i,j) (i,j,j) with indices: pos(A1g) ≤ i ≤ j ≤ pos(B3u) |
|---|
| ..1.575. |
A1gB1gB1g. | ..420. |
A1gB2gB2g. | ..315. |
A1gB3gB3g. | ..420. |
A1gA1uA1u. | ..420. |
A1gB1uB1u. | ..1.575. |
A1gB2uB2u. | ..1.575. |
A1gB3uB3u. | | |
| |
| |
| Subtotal: 6.300 / 7 / 56 |
| Irrep combinations (i,j,k) with indices: pos(A1g) ≤ i ≤ j ≤ k ≤ pos(B3u) |
|---|
| ..588. |
B1gB2gB3g. | ..686. |
B1gA1uB1u. | ..2.744. |
B1gB2uB3u. | ..686. |
B2gA1uB2u. | ..686. |
B2gB1uB3u. | ..588. |
B3gA1uB3u. | ..588. |
B3gB1uB2u. | | |
| |
| |
| Subtotal: 6.566 / 7 / 56 |
| Total: 13.546 / 15 / 120 |
Contributions to nonvanishing quartic force field constants
| Irrep combinations (i,i,i,i) with indices: pos(A1g) ≤ i ≤ pos(B3u) |
|---|
| ..3.060. |
A1gA1gA1gA1g. | ..2.380. |
B1gB1gB1gB1g. | ..210. |
B2gB2gB2gB2g. | ..126. |
B3gB3gB3gB3g. | ..210. |
A1uA1uA1uA1u. | ..210. |
B1uB1uB1uB1u. | ..2.380. |
B2uB2uB2uB2u. | ..2.380. |
B3uB3uB3uB3u. | | |
| |
| Subtotal: 10.956 / 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) |
|---|
| ..12.600. |
A1gA1gB1gB1g. | ..3.360. |
A1gA1gB2gB2g. | ..2.520. |
A1gA1gB3gB3g. | ..3.360. |
A1gA1gA1uA1u. | ..3.360. |
A1gA1gB1uB1u. | ..12.600. |
A1gA1gB2uB2u. | ..12.600. |
A1gA1gB3uB3u. | ..2.940. |
B1gB1gB2gB2g. | ..2.205. |
B1gB1gB3gB3g. | ..2.940. |
B1gB1gA1uA1u. |
| ..2.940. |
B1gB1gB1uB1u. | ..11.025. |
B1gB1gB2uB2u. | ..11.025. |
B1gB1gB3uB3u. | ..588. |
B2gB2gB3gB3g. | ..784. |
B2gB2gA1uA1u. | ..784. |
B2gB2gB1uB1u. | ..2.940. |
B2gB2gB2uB2u. | ..2.940. |
B2gB2gB3uB3u. | ..588. |
B3gB3gA1uA1u. | ..588. |
B3gB3gB1uB1u. |
| ..2.205. |
B3gB3gB2uB2u. | ..2.205. |
B3gB3gB3uB3u. | ..784. |
A1uA1uB1uB1u. | ..2.940. |
A1uA1uB2uB2u. | ..2.940. |
A1uA1uB3uB3u. | ..2.940. |
B1uB1uB2uB2u. | ..2.940. |
B1uB1uB3uB3u. | ..11.025. |
B2uB2uB3uB3u. | | |
| |
| Subtotal: 120.666 / 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) |
|---|
| ..8.820. |
A1gB1gB2gB3g. | ..10.290. |
A1gB1gA1uB1u. | ..41.160. |
A1gB1gB2uB3u. | ..10.290. |
A1gB2gA1uB2u. | ..10.290. |
A1gB2gB1uB3u. | ..8.820. |
A1gB3gA1uB3u. | ..8.820. |
A1gB3gB1uB2u. | ..9.604. |
B1gB2gA1uB3u. | ..9.604. |
B1gB2gB1uB2u. | ..8.232. |
B1gB3gA1uB2u. |
| ..8.232. |
B1gB3gB1uB3u. | ..2.058. |
B2gB3gA1uB1u. | ..8.232. |
B2gB3gB2uB3u. | ..9.604. |
A1uB1uB2uB3u. | | |
| |
| |
| |
| |
| |
| Subtotal: 154.056 / 14 / 70 |
| Total: 285.678 / 50 / 330 |
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