Characters of representations for molecular motions
Motion |
E |
2C3 |
3C'2 |
i |
2S6 |
3σd |
Cartesian 3N |
39 |
0 |
-1 |
-3 |
0 |
5 |
Translation (x,y,z) |
3 |
0 |
-1 |
-3 |
0 |
1 |
Rotation (Rx,Ry,Rz) |
3 |
0 |
-1 |
3 |
0 |
-1 |
Vibration |
33 |
0 |
1 |
-3 |
0 |
5 |
Decomposition to irreducible representations
Motion |
A1g |
A2g |
Eg |
A1u |
A2u |
Eu |
Total |
Cartesian 3N |
4 |
2 |
6 |
2 |
5 |
7 |
26 |
Translation (x,y,z) |
0 |
0 |
0 |
0 |
1 |
1 |
2 |
Rotation (Rx,Ry,Rz) |
0 |
1 |
1 |
0 |
0 |
0 |
2 |
Vibration |
4 |
1 |
5 |
2 |
4 |
6 |
22 |
Molecular parameter
Number of Atoms (N) |
13
|
Number of internal coordinates |
33
|
Number of independant internal coordinates |
4
|
Number of vibrational modes |
22
|
Force field analysis
Allowed / forbidden vibronational transitions
Operator |
A1g |
A2g |
Eg |
A1u |
A2u |
Eu |
Total |
Linear (IR) |
4 |
1 |
5 |
2 |
4 |
6 |
10 / 12 |
Quadratic (Raman) |
4 |
1 |
5 |
2 |
4 |
6 |
9 / 13 |
IR + Raman |
- - - - |
1 |
- - - - |
2 |
- - - - |
- - - - |
0* / 3 |
* Parity Mutual Exclusion Principle
Characters of force fields
(Symmetric powers of vibration representation)
Force field |
E |
2C3 |
3C'2 |
i |
2S6 |
3σd |
linear |
33 |
0 |
1 |
-3 |
0 |
5 |
quadratic |
561 |
0 |
17 |
21 |
0 |
29 |
cubic |
6.545 |
11 |
17 |
-55 |
-1 |
105 |
quartic |
58.905 |
0 |
153 |
225 |
0 |
385 |
quintic |
435.897 |
0 |
153 |
-531 |
0 |
1.141 |
sextic |
2.760.681 |
66 |
969 |
1.653 |
6 |
3.325 |
Decomposition to irreducible representations
Column with number of nonvanshing force constants highlighted
Force field |
A1g |
A2g |
Eg |
A1u |
A2u |
Eu |
linear |
4 |
1 |
5 |
2 |
4 |
6 |
quadratic |
60 |
37 |
97 |
42 |
48 |
90 |
cubic |
573 |
512 |
1.080 |
530 |
574 |
1.098 |
quartic |
5.062 |
4.793 |
9.855 |
4.832 |
4.948 |
9.780 |
quintic |
36.604 |
35.957 |
72.561 |
36.122 |
36.616 |
72.738 |
sextic |
231.280 |
229.133 |
460.377 |
229.340 |
230.518 |
459.828 |
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) |
..10. |
A1gA1g. | ..1. |
A2gA2g. | ..15. |
EgEg. | ..3. |
A1uA1u. | ..10. |
A2uA2u. | ..21. |
EuEu. | | |
| |
| |
| |
Subtotal: 60 / 6 / 6 |
Irrep combinations (i,j) with indices: pos(A1g) ≤ i ≤ j ≤ pos(Eu) |
Subtotal: 0 / 0 / 15 |
Total: 60 / 6 / 21 |
Contributions to nonvanishing cubic force field constants
Irrep combinations (i,i,i) with indices: pos(A1g) ≤ i ≤ pos(Eu) |
..20. |
A1gA1gA1g. | ..35. |
EgEgEg. | | |
| |
| |
| |
| |
| |
| |
| |
Subtotal: 55 / 2 / 6 |
Irrep combinations (i,i,j) (i,j,j) with indices: pos(A1g) ≤ i ≤ j ≤ pos(Eu) |
..4. |
A1gA2gA2g. | ..60. |
A1gEgEg. | ..12. |
A1gA1uA1u. | ..40. |
A1gA2uA2u. | ..84. |
A1gEuEu. | ..10. |
A2gEgEg. | ..15. |
A2gEuEu. | ..105. |
EgEuEu. | | |
| |
Subtotal: 330 / 8 / 30 |
Irrep combinations (i,j,k) with indices: pos(A1g) ≤ i ≤ j ≤ k ≤ pos(Eu) |
..8. |
A2gA1uA2u. | ..60. |
EgA1uEu. | ..120. |
EgA2uEu. | | |
| |
| |
| |
| |
| |
| |
Subtotal: 188 / 3 / 20 |
Total: 573 / 13 / 56 |
Contributions to nonvanishing quartic force field constants
Irrep combinations (i,i,i,i) with indices: pos(A1g) ≤ i ≤ pos(Eu) |
..35. |
A1gA1gA1gA1g. | ..1. |
A2gA2gA2gA2g. | ..120. |
EgEgEgEg. | ..5. |
A1uA1uA1uA1u. | ..35. |
A2uA2uA2uA2u. | ..231. |
EuEuEuEu. | | |
| |
| |
| |
Subtotal: 427 / 6 / 6 |
Irrep combinations (i,i,i,j) (i,j,j,j) with indices: pos(A1g) ≤ i ≤ j ≤ pos(Eu) |
..140. |
A1gEgEgEg. | ..35. |
A2gEgEgEg. | ..112. |
A1uEuEuEu. | ..224. |
A2uEuEuEu. | | |
| |
| |
| |
| |
| |
Subtotal: 511 / 4 / 30 |
Irrep combinations (i,i,j,j) with indices: pos(A1g) ≤ i ≤ j ≤ pos(Eu) |
..10. |
A1gA1gA2gA2g. | ..150. |
A1gA1gEgEg. | ..30. |
A1gA1gA1uA1u. | ..100. |
A1gA1gA2uA2u. | ..210. |
A1gA1gEuEu. | ..15. |
A2gA2gEgEg. | ..3. |
A2gA2gA1uA1u. | ..10. |
A2gA2gA2uA2u. | ..21. |
A2gA2gEuEu. | ..45. |
EgEgA1uA1u. |
..150. |
EgEgA2uA2u. | ..780. |
EgEgEuEu. | ..30. |
A1uA1uA2uA2u. | ..63. |
A1uA1uEuEu. | ..210. |
A2uA2uEuEu. | | |
| |
| |
| |
| |
Subtotal: 1.827 / 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) |
..80. |
EgEgA1uA2u. | ..180. |
EgEgA1uEu. | ..360. |
EgEgA2uEu. | ..40. |
A1gA2gEgEg. | ..60. |
A1gA2gEuEu. | ..420. |
A1gEgEuEu. | ..105. |
A2gEgEuEu. | ..120. |
A1uA2uEuEu. | | |
| |
Subtotal: 1.365 / 8 / 60 |
Irrep combinations (i,j,k,l) with indices: pos(A1g) ≤ i ≤ j ≤ k ≤ l ≤ pos(Eu) |
..32. |
A1gA2gA1uA2u. | ..240. |
A1gEgA1uEu. | ..480. |
A1gEgA2uEu. | ..60. |
A2gEgA1uEu. | ..120. |
A2gEgA2uEu. | | |
| |
| |
| |
| |
Subtotal: 932 / 5 / 15 |
Total: 5.062 / 38 / 126 |
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