Characters of representations for molecular motions
| Motion |
E |
C2.(z) |
C2.(y) |
C2.(x) |
i |
σ.(xy) |
σ.(xy) |
σ.(xy) |
| Cartesian 3N |
144 |
0 |
0 |
0 |
0 |
8 |
0 |
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 |
138 |
2 |
2 |
2 |
0 |
8 |
0 |
0 |
Decomposition to irreducible representations
| Motion |
A1g |
B1g |
B2g |
B3g |
A1u |
B1u |
B2u |
B3u |
Total |
| Cartesian 3N |
19 |
19 |
17 |
17 |
17 |
17 |
19 |
19 |
144 |
| 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 |
19 |
18 |
16 |
16 |
17 |
16 |
18 |
18 |
138 |
Molecular parameter
| Number of Atoms (N) |
48
|
| Number of internal coordinates |
138
|
| Number of independant internal coordinates |
19
|
| Number of vibrational modes |
138
|
Force field analysis
Allowed / forbidden vibronational transitions
| Operator |
A1g |
B1g |
B2g |
B3g |
A1u |
B1u |
B2u |
B3u |
Total |
| Linear (IR) |
19 |
18 |
16 |
16 |
17 |
16 |
18 |
18 |
52 / 86 |
| Quadratic (Raman) |
19 |
18 |
16 |
16 |
17 |
16 |
18 |
18 |
69 / 69 |
| IR + Raman |
- - - - |
- - - - |
- - - - |
- - - - |
17 |
- - - - |
- - - - |
- - - - |
0* / 17 |
* 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 |
138 |
2 |
2 |
2 |
0 |
8 |
0 |
0 |
| quadratic |
9.591 |
71 |
71 |
71 |
69 |
101 |
69 |
69 |
| cubic |
447.580 |
140 |
140 |
140 |
0 |
640 |
0 |
0 |
| quartic |
15.777.195 |
2.555 |
2.555 |
2.555 |
2.415 |
4.815 |
2.415 |
2.415 |
| quintic |
448.072.338 |
4.970 |
4.970 |
4.970 |
0 |
25.752 |
0 |
0 |
| sextic |
10.679.057.389 |
62.125 |
62.125 |
62.125 |
57.155 |
148.291 |
57.155 |
57.155 |
Decomposition to irreducible representations
Column with number of nonvanshing force constants highlighted
| Force field |
A1g |
B1g |
B2g |
B3g |
A1u |
B1u |
B2u |
B3u |
| linear |
19 |
18 |
16 |
16 |
17 |
16 |
18 |
18 |
| quadratic |
1.264 |
1.194 |
1.186 |
1.186 |
1.187 |
1.186 |
1.194 |
1.194 |
| cubic |
56.080 |
56.010 |
55.850 |
55.850 |
55.920 |
55.850 |
56.010 |
56.010 |
| quartic |
1.974.615 |
1.972.130 |
1.971.530 |
1.971.530 |
1.971.600 |
1.971.530 |
1.972.130 |
1.972.130 |
| quintic |
56.014.125 |
56.011.640 |
56.005.202 |
56.005.202 |
56.007.687 |
56.005.202 |
56.011.640 |
56.011.640 |
| sextic |
1.334.945.440 |
1.334.885.800 |
1.334.863.016 |
1.334.863.016 |
1.334.865.501 |
1.334.863.016 |
1.334.885.800 |
1.334.885.800 |
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) |
|---|
| ..190. |
A1gA1g. | ..171. |
B1gB1g. | ..136. |
B2gB2g. | ..136. |
B3gB3g. | ..153. |
A1uA1u. | ..136. |
B1uB1u. | ..171. |
B2uB2u. | ..171. |
B3uB3u. | | |
| |
| Subtotal: 1.264 / 8 / 8 |
| Irrep combinations (i,j) with indices: pos(A1g) ≤ i ≤ j ≤ pos(B3u) |
|---|
| Subtotal: 0 / 0 / 28 |
| Total: 1.264 / 8 / 36 |
Contributions to nonvanishing cubic force field constants
| Irrep combinations (i,i,i) with indices: pos(A1g) ≤ i ≤ pos(B3u) |
|---|
| ..1.330. |
A1gA1gA1g. | | |
| |
| |
| |
| |
| |
| |
| |
| |
| Subtotal: 1.330 / 1 / 8 |
| Irrep combinations (i,i,j) (i,j,j) with indices: pos(A1g) ≤ i ≤ j ≤ pos(B3u) |
|---|
| ..3.249. |
A1gB1gB1g. | ..2.584. |
A1gB2gB2g. | ..2.584. |
A1gB3gB3g. | ..2.907. |
A1gA1uA1u. | ..2.584. |
A1gB1uB1u. | ..3.249. |
A1gB2uB2u. | ..3.249. |
A1gB3uB3u. | | |
| |
| |
| Subtotal: 20.406 / 7 / 56 |
| Irrep combinations (i,j,k) with indices: pos(A1g) ≤ i ≤ j ≤ k ≤ pos(B3u) |
|---|
| ..4.608. |
B1gB2gB3g. | ..4.896. |
B1gA1uB1u. | ..5.832. |
B1gB2uB3u. | ..4.896. |
B2gA1uB2u. | ..4.608. |
B2gB1uB3u. | ..4.896. |
B3gA1uB3u. | ..4.608. |
B3gB1uB2u. | | |
| |
| |
| Subtotal: 34.344 / 7 / 56 |
| Total: 56.080 / 15 / 120 |
Contributions to nonvanishing quartic force field constants
| Irrep combinations (i,i,i,i) with indices: pos(A1g) ≤ i ≤ pos(B3u) |
|---|
| ..7.315. |
A1gA1gA1gA1g. | ..5.985. |
B1gB1gB1gB1g. | ..3.876. |
B2gB2gB2gB2g. | ..3.876. |
B3gB3gB3gB3g. | ..4.845. |
A1uA1uA1uA1u. | ..3.876. |
B1uB1uB1uB1u. | ..5.985. |
B2uB2uB2uB2u. | ..5.985. |
B3uB3uB3uB3u. | | |
| |
| Subtotal: 41.743 / 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) |
|---|
| ..32.490. |
A1gA1gB1gB1g. | ..25.840. |
A1gA1gB2gB2g. | ..25.840. |
A1gA1gB3gB3g. | ..29.070. |
A1gA1gA1uA1u. | ..25.840. |
A1gA1gB1uB1u. | ..32.490. |
A1gA1gB2uB2u. | ..32.490. |
A1gA1gB3uB3u. | ..23.256. |
B1gB1gB2gB2g. | ..23.256. |
B1gB1gB3gB3g. | ..26.163. |
B1gB1gA1uA1u. |
| ..23.256. |
B1gB1gB1uB1u. | ..29.241. |
B1gB1gB2uB2u. | ..29.241. |
B1gB1gB3uB3u. | ..18.496. |
B2gB2gB3gB3g. | ..20.808. |
B2gB2gA1uA1u. | ..18.496. |
B2gB2gB1uB1u. | ..23.256. |
B2gB2gB2uB2u. | ..23.256. |
B2gB2gB3uB3u. | ..20.808. |
B3gB3gA1uA1u. | ..18.496. |
B3gB3gB1uB1u. |
| ..23.256. |
B3gB3gB2uB2u. | ..23.256. |
B3gB3gB3uB3u. | ..20.808. |
A1uA1uB1uB1u. | ..26.163. |
A1uA1uB2uB2u. | ..26.163. |
A1uA1uB3uB3u. | ..23.256. |
B1uB1uB2uB2u. | ..23.256. |
B1uB1uB3uB3u. | ..29.241. |
B2uB2uB3uB3u. | | |
| |
| Subtotal: 697.488 / 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) |
|---|
| ..87.552. |
A1gB1gB2gB3g. | ..93.024. |
A1gB1gA1uB1u. | ..110.808. |
A1gB1gB2uB3u. | ..93.024. |
A1gB2gA1uB2u. | ..87.552. |
A1gB2gB1uB3u. | ..93.024. |
A1gB3gA1uB3u. | ..87.552. |
A1gB3gB1uB2u. | ..88.128. |
B1gB2gA1uB3u. | ..82.944. |
B1gB2gB1uB2u. | ..88.128. |
B1gB3gA1uB2u. |
| ..82.944. |
B1gB3gB1uB3u. | ..69.632. |
B2gB3gA1uB1u. | ..82.944. |
B2gB3gB2uB3u. | ..88.128. |
A1uB1uB2uB3u. | | |
| |
| |
| |
| |
| |
| Subtotal: 1.235.384 / 14 / 70 |
| Total: 1.974.615 / 50 / 330 |
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