Characters of representations for molecular motions
| Motion |
E |
2C3 |
3C'2 |
i |
2S6 |
3σd |
| Cartesian 3N |
156 |
0 |
-2 |
0 |
0 |
14 |
| Translation (x,y,z) |
3 |
0 |
-1 |
-3 |
0 |
1 |
| Rotation (Rx,Ry,Rz) |
3 |
0 |
-1 |
3 |
0 |
-1 |
| Vibration |
150 |
0 |
0 |
0 |
0 |
14 |
Decomposition to irreducible representations
| Motion |
A1g |
A2g |
Eg |
A1u |
A2u |
Eu |
Total |
| Cartesian 3N |
16 |
10 |
26 |
9 |
17 |
26 |
104 |
| Translation (x,y,z) |
0 |
0 |
0 |
0 |
1 |
1 |
2 |
| Rotation (Rx,Ry,Rz) |
0 |
1 |
1 |
0 |
0 |
0 |
2 |
| Vibration |
16 |
9 |
25 |
9 |
16 |
25 |
100 |
Molecular parameter
| Number of Atoms (N) |
52
|
| Number of internal coordinates |
150
|
| Number of independant internal coordinates |
16
|
| Number of vibrational modes |
100
|
Force field analysis
Allowed / forbidden vibronational transitions
| Operator |
A1g |
A2g |
Eg |
A1u |
A2u |
Eu |
Total |
| Linear (IR) |
16 |
9 |
25 |
9 |
16 |
25 |
41 / 59 |
| Quadratic (Raman) |
16 |
9 |
25 |
9 |
16 |
25 |
41 / 59 |
| IR + Raman |
- - - - |
9 |
- - - - |
9 |
- - - - |
- - - - |
0* / 18 |
* Parity Mutual Exclusion Principle
Characters of force fields
(Symmetric powers of vibration representation)
| Force field |
E |
2C3 |
3C'2 |
i |
2S6 |
3σd |
| linear |
150 |
0 |
0 |
0 |
0 |
14 |
| quadratic |
11.325 |
0 |
75 |
75 |
0 |
173 |
| cubic |
573.800 |
50 |
0 |
0 |
0 |
1.512 |
| quartic |
21.947.850 |
0 |
2.850 |
2.850 |
0 |
11.866 |
| quintic |
675.993.780 |
0 |
0 |
0 |
0 |
79.492 |
| sextic |
17.463.172.650 |
1.275 |
73.150 |
73.150 |
25 |
490.042 |
Decomposition to irreducible representations
Column with number of nonvanshing force constants highlighted
| Force field |
A1g |
A2g |
Eg |
A1u |
A2u |
Eu |
| linear |
16 |
9 |
25 |
9 |
16 |
25 |
| quadratic |
1.012 |
888 |
1.900 |
913 |
962 |
1.875 |
| cubic |
48.203 |
47.447 |
95.625 |
47.447 |
48.203 |
95.625 |
| quartic |
1.832.904 |
1.825.546 |
3.658.450 |
1.826.496 |
1.831.004 |
3.657.500 |
| quintic |
56.352.688 |
56.312.942 |
112.665.630 |
56.312.942 |
56.352.688 |
112.665.630 |
| sextic |
1.455.411.498 |
1.455.129.902 |
2.910.540.750 |
1.455.154.277 |
1.455.362.723 |
2.910.516.375 |
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
3d
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(Eu) |
|---|
| ..136. |
A1gA1g. | ..45. |
A2gA2g. | ..325. |
EgEg. | ..45. |
A1uA1u. | ..136. |
A2uA2u. | ..325. |
EuEu. | | |
| |
| |
| |
| Subtotal: 1.012 / 6 / 6 |
| Irrep combinations (i,j) with indices: pos(A1g) ≤ i ≤ j ≤ pos(Eu) |
|---|
| Subtotal: 0 / 0 / 15 |
| Total: 1.012 / 6 / 21 |
Contributions to nonvanishing cubic force field constants
| Irrep combinations (i,i,i) with indices: pos(A1g) ≤ i ≤ pos(Eu) |
|---|
| ..816. |
A1gA1gA1g. | ..2.925. |
EgEgEg. | | |
| |
| |
| |
| |
| |
| |
| |
| Subtotal: 3.741 / 2 / 6 |
| Irrep combinations (i,i,j) (i,j,j) with indices: pos(A1g) ≤ i ≤ j ≤ pos(Eu) |
|---|
| ..720. |
A1gA2gA2g. | ..5.200. |
A1gEgEg. | ..720. |
A1gA1uA1u. | ..2.176. |
A1gA2uA2u. | ..5.200. |
A1gEuEu. | ..2.700. |
A2gEgEg. | ..2.700. |
A2gEuEu. | ..8.125. |
EgEuEu. | | |
| |
| Subtotal: 27.541 / 8 / 30 |
| Irrep combinations (i,j,k) with indices: pos(A1g) ≤ i ≤ j ≤ k ≤ pos(Eu) |
|---|
| ..1.296. |
A2gA1uA2u. | ..5.625. |
EgA1uEu. | ..10.000. |
EgA2uEu. | | |
| |
| |
| |
| |
| |
| |
| Subtotal: 16.921 / 3 / 20 |
| Total: 48.203 / 13 / 56 |
Contributions to nonvanishing quartic force field constants
| Irrep combinations (i,i,i,i) with indices: pos(A1g) ≤ i ≤ pos(Eu) |
|---|
| ..3.876. |
A1gA1gA1gA1g. | ..495. |
A2gA2gA2gA2g. | ..52.975. |
EgEgEgEg. | ..495. |
A1uA1uA1uA1u. | ..3.876. |
A2uA2uA2uA2u. | ..52.975. |
EuEuEuEu. | | |
| |
| |
| |
| Subtotal: 114.692 / 6 / 6 |
| Irrep combinations (i,i,i,j) (i,j,j,j) with indices: pos(A1g) ≤ i ≤ j ≤ pos(Eu) |
|---|
| ..46.800. |
A1gEgEgEg. | ..26.325. |
A2gEgEgEg. | ..26.325. |
A1uEuEuEu. | ..46.800. |
A2uEuEuEu. | | |
| |
| |
| |
| |
| |
| Subtotal: 146.250 / 4 / 30 |
| Irrep combinations (i,i,j,j) with indices: pos(A1g) ≤ i ≤ j ≤ pos(Eu) |
|---|
| ..6.120. |
A1gA1gA2gA2g. | ..44.200. |
A1gA1gEgEg. | ..6.120. |
A1gA1gA1uA1u. | ..18.496. |
A1gA1gA2uA2u. | ..44.200. |
A1gA1gEuEu. | ..14.625. |
A2gA2gEgEg. | ..2.025. |
A2gA2gA1uA1u. | ..6.120. |
A2gA2gA2uA2u. | ..14.625. |
A2gA2gEuEu. | ..14.625. |
EgEgA1uA1u. |
| ..44.200. |
EgEgA2uA2u. | ..301.250. |
EgEgEuEu. | ..6.120. |
A1uA1uA2uA2u. | ..14.625. |
A1uA1uEuEu. | ..44.200. |
A2uA2uEuEu. | | |
| |
| |
| |
| |
| Subtotal: 581.551 / 15 / 15 |
| Irrep combinations (i,i,j,k) (i,j,j,k) (i,j,k,k) with indices: pos(A1g) ≤ i ≤ j ≤ k ≤ pos(Eu) |
|---|
| ..43.200. |
EgEgA1uA2u. | ..73.125. |
EgEgA1uEu. | ..130.000. |
EgEgA2uEu. | ..43.200. |
A1gA2gEgEg. | ..43.200. |
A1gA2gEuEu. | ..130.000. |
A1gEgEuEu. | ..73.125. |
A2gEgEuEu. | ..43.200. |
A1uA2uEuEu. | | |
| |
| Subtotal: 579.050 / 8 / 60 |
| Irrep combinations (i,j,k,l) with indices: pos(A1g) ≤ i ≤ j ≤ k ≤ l ≤ pos(Eu) |
|---|
| ..20.736. |
A1gA2gA1uA2u. | ..90.000. |
A1gEgA1uEu. | ..160.000. |
A1gEgA2uEu. | ..50.625. |
A2gEgA1uEu. | ..90.000. |
A2gEgA2uEu. | | |
| |
| |
| |
| |
| Subtotal: 411.361 / 5 / 15 |
| Total: 1.832.904 / 38 / 126 |
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