Characters of representations for molecular motions
Motion |
E |
2C4 |
C2 |
2C'2 |
2C''2 |
i |
2S4 |
σh |
2σv |
2σd |
Cartesian 3N |
48 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
8 |
Translation (x,y,z) |
3 |
1 |
-1 |
-1 |
-1 |
-3 |
-1 |
1 |
1 |
1 |
Rotation (Rx,Ry,Rz) |
3 |
1 |
-1 |
-1 |
-1 |
3 |
1 |
-1 |
-1 |
-1 |
Vibration |
42 |
-2 |
2 |
2 |
2 |
0 |
0 |
0 |
0 |
8 |
Decomposition to irreducible representations
Motion |
A1g |
A2g |
B1g |
B2g |
Eg |
A1u |
A2u |
B1u |
B2u |
Eu |
Total |
Cartesian 3N |
4 |
2 |
2 |
4 |
6 |
2 |
4 |
4 |
2 |
6 |
36 |
Translation (x,y,z) |
0 |
0 |
0 |
0 |
0 |
0 |
1 |
0 |
0 |
1 |
2 |
Rotation (Rx,Ry,Rz) |
0 |
1 |
0 |
0 |
1 |
0 |
0 |
0 |
0 |
0 |
2 |
Vibration |
4 |
1 |
2 |
4 |
5 |
2 |
3 |
4 |
2 |
5 |
32 |
Molecular parameter
Number of Atoms (N) |
16
|
Number of internal coordinates |
42
|
Number of independant internal coordinates |
4
|
Number of vibrational modes |
32
|
Force field analysis
Allowed / forbidden vibronational transitions
Operator |
A1g |
A2g |
B1g |
B2g |
Eg |
A1u |
A2u |
B1u |
B2u |
Eu |
Total |
Linear (IR) |
4 |
1 |
2 |
4 |
5 |
2 |
3 |
4 |
2 |
5 |
8 / 24 |
Quadratic (Raman) |
4 |
1 |
2 |
4 |
5 |
2 |
3 |
4 |
2 |
5 |
15 / 17 |
IR + Raman |
- - - - |
1 |
- - - - |
- - - - |
- - - - |
2 |
- - - - |
4 |
2 |
- - - - |
0* / 9 |
* Parity Mutual Exclusion Principle
Characters of force fields
(Symmetric powers of vibration representation)
Force field |
E |
2C4 |
C2 |
2C'2 |
2C''2 |
i |
2S4 |
σh |
2σv |
2σd |
linear |
42 |
-2 |
2 |
2 |
2 |
0 |
0 |
0 |
0 |
8 |
quadratic |
903 |
3 |
23 |
23 |
23 |
21 |
1 |
21 |
21 |
53 |
cubic |
13.244 |
-4 |
44 |
44 |
44 |
0 |
0 |
0 |
0 |
256 |
quartic |
148.995 |
15 |
275 |
275 |
275 |
231 |
11 |
231 |
231 |
1.095 |
quintic |
1.370.754 |
-26 |
506 |
506 |
506 |
0 |
0 |
0 |
0 |
4.056 |
sextic |
10.737.573 |
37 |
2.277 |
2.277 |
2.277 |
1.771 |
11 |
1.771 |
1.771 |
13.803 |
Decomposition to irreducible representations
Column with number of nonvanshing force constants highlighted
Force field |
A1g |
A2g |
B1g |
B2g |
Eg |
A1u |
A2u |
B1u |
B2u |
Eu |
linear |
4 |
1 |
2 |
4 |
5 |
2 |
3 |
4 |
2 |
5 |
quadratic |
76 |
46 |
56 |
64 |
110 |
52 |
59 |
59 |
51 |
110 |
cubic |
873 |
787 |
799 |
863 |
1.650 |
809 |
851 |
863 |
799 |
1.650 |
quartic |
9.596 |
9.127 |
9.247 |
9.463 |
18.590 |
9.204 |
9.398 |
9.408 |
9.192 |
18.590 |
quintic |
86.334 |
85.067 |
85.200 |
86.214 |
171.281 |
85.320 |
86.081 |
86.214 |
85.200 |
171.281 |
sextic |
673.984 |
668.952 |
669.952 |
672.960 |
1.341.912 |
669.645 |
672.400 |
672.520 |
669.512 |
1.341.912 |
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
4h
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. | ..3. |
B1gB1g. | ..10. |
B2gB2g. | ..15. |
EgEg. | ..3. |
A1uA1u. | ..6. |
A2uA2u. | ..10. |
B1uB1u. | ..3. |
B2uB2u. | ..15. |
EuEu. |
Subtotal: 76 / 10 / 10 |
Irrep combinations (i,j) with indices: pos(A1g) ≤ i ≤ j ≤ pos(Eu) |
Subtotal: 0 / 0 / 45 |
Total: 76 / 10 / 55 |
Contributions to nonvanishing cubic force field constants
Irrep combinations (i,i,i) with indices: pos(A1g) ≤ i ≤ pos(Eu) |
..20. |
A1gA1gA1g. | | |
| |
| |
| |
| |
| |
| |
| |
| |
Subtotal: 20 / 1 / 10 |
Irrep combinations (i,i,j) (i,j,j) with indices: pos(A1g) ≤ i ≤ j ≤ pos(Eu) |
..4. |
A1gA2gA2g. | ..12. |
A1gB1gB1g. | ..40. |
A1gB2gB2g. | ..60. |
A1gEgEg. | ..12. |
A1gA1uA1u. | ..24. |
A1gA2uA2u. | ..40. |
A1gB1uB1u. | ..12. |
A1gB2uB2u. | ..60. |
A1gEuEu. | ..10. |
A2gEgEg. |
..10. |
A2gEuEu. | ..30. |
B1gEgEg. | ..30. |
B1gEuEu. | ..60. |
B2gEgEg. | ..60. |
B2gEuEu. | | |
| |
| |
| |
| |
Subtotal: 464 / 15 / 90 |
Irrep combinations (i,j,k) with indices: pos(A1g) ≤ i ≤ j ≤ k ≤ pos(Eu) |
..8. |
A2gB1gB2g. | ..6. |
A2gA1uA2u. | ..8. |
A2gB1uB2u. | ..16. |
B1gA1uB1u. | ..12. |
B1gA2uB2u. | ..16. |
B2gA1uB2u. | ..48. |
B2gA2uB1u. | ..50. |
EgA1uEu. | ..75. |
EgA2uEu. | ..100. |
EgB1uEu. |
..50. |
EgB2uEu. | | |
| |
| |
| |
| |
| |
| |
| |
| |
Subtotal: 389 / 11 / 120 |
Total: 873 / 27 / 220 |
Contributions to nonvanishing quartic force field constants
Irrep combinations (i,i,i,i) with indices: pos(A1g) ≤ i ≤ pos(Eu) |
..35. |
A1gA1gA1gA1g. | ..1. |
A2gA2gA2gA2g. | ..5. |
B1gB1gB1gB1g. | ..35. |
B2gB2gB2gB2g. | ..190. |
EgEgEgEg. | ..5. |
A1uA1uA1uA1u. | ..15. |
A2uA2uA2uA2u. | ..35. |
B1uB1uB1uB1u. | ..5. |
B2uB2uB2uB2u. | ..190. |
EuEuEuEu. |
Subtotal: 516 / 10 / 10 |
Irrep combinations (i,i,i,j) (i,j,j,j) with indices: pos(A1g) ≤ i ≤ j ≤ pos(Eu) |
Subtotal: 0 / 0 / 90 |
Irrep combinations (i,i,j,j) with indices: pos(A1g) ≤ i ≤ j ≤ pos(Eu) |
..10. |
A1gA1gA2gA2g. | ..30. |
A1gA1gB1gB1g. | ..100. |
A1gA1gB2gB2g. | ..150. |
A1gA1gEgEg. | ..30. |
A1gA1gA1uA1u. | ..60. |
A1gA1gA2uA2u. | ..100. |
A1gA1gB1uB1u. | ..30. |
A1gA1gB2uB2u. | ..150. |
A1gA1gEuEu. | ..3. |
A2gA2gB1gB1g. |
..10. |
A2gA2gB2gB2g. | ..15. |
A2gA2gEgEg. | ..3. |
A2gA2gA1uA1u. | ..6. |
A2gA2gA2uA2u. | ..10. |
A2gA2gB1uB1u. | ..3. |
A2gA2gB2uB2u. | ..15. |
A2gA2gEuEu. | ..30. |
B1gB1gB2gB2g. | ..45. |
B1gB1gEgEg. | ..9. |
B1gB1gA1uA1u. |
..18. |
B1gB1gA2uA2u. | ..30. |
B1gB1gB1uB1u. | ..9. |
B1gB1gB2uB2u. | ..45. |
B1gB1gEuEu. | ..150. |
B2gB2gEgEg. | ..30. |
B2gB2gA1uA1u. | ..60. |
B2gB2gA2uA2u. | ..100. |
B2gB2gB1uB1u. | ..30. |
B2gB2gB2uB2u. | ..150. |
B2gB2gEuEu. |
..45. |
EgEgA1uA1u. | ..90. |
EgEgA2uA2u. | ..150. |
EgEgB1uB1u. | ..45. |
EgEgB2uB2u. | ..775. |
EgEgEuEu. | ..18. |
A1uA1uA2uA2u. | ..30. |
A1uA1uB1uB1u. | ..9. |
A1uA1uB2uB2u. | ..45. |
A1uA1uEuEu. | ..60. |
A2uA2uB1uB1u. |
..18. |
A2uA2uB2uB2u. | ..90. |
A2uA2uEuEu. | ..30. |
B1uB1uB2uB2u. | ..150. |
B1uB1uEuEu. | ..45. |
B2uB2uEuEu. | | |
| |
| |
| |
| |
Subtotal: 3.031 / 45 / 45 |
Irrep combinations (i,i,j,k) (i,j,j,k) (i,j,k,k) with indices: pos(A1g) ≤ i ≤ j ≤ k ≤ pos(Eu) |
..60. |
EgEgA1uA2u. | ..120. |
EgEgA1uB1u. | ..60. |
EgEgA1uB2u. | ..180. |
EgEgA2uB1u. | ..90. |
EgEgA2uB2u. | ..80. |
EgEgB1uB2u. | ..40. |
A1gA2gEgEg. | ..40. |
A1gA2gEuEu. | ..120. |
A1gB1gEgEg. | ..120. |
A1gB1gEuEu. |
..240. |
A1gB2gEgEg. | ..240. |
A1gB2gEuEu. | ..30. |
A2gB1gEgEg. | ..30. |
A2gB1gEuEu. | ..60. |
A2gB2gEgEg. | ..60. |
A2gB2gEuEu. | ..80. |
B1gB2gEgEg. | ..80. |
B1gB2gEuEu. | ..60. |
A1uA2uEuEu. | ..120. |
A1uB1uEuEu. |
..60. |
A1uB2uEuEu. | ..180. |
A2uB1uEuEu. | ..90. |
A2uB2uEuEu. | ..80. |
B1uB2uEuEu. | | |
| |
| |
| |
| |
| |
Subtotal: 2.320 / 24 / 360 |
Irrep combinations (i,j,k,l) with indices: pos(A1g) ≤ i ≤ j ≤ k ≤ l ≤ pos(Eu) |
..32. |
A1gA2gB1gB2g. | ..24. |
A1gA2gA1uA2u. | ..32. |
A1gA2gB1uB2u. | ..64. |
A1gB1gA1uB1u. | ..48. |
A1gB1gA2uB2u. | ..64. |
A1gB2gA1uB2u. | ..192. |
A1gB2gA2uB1u. | ..200. |
A1gEgA1uEu. | ..300. |
A1gEgA2uEu. | ..400. |
A1gEgB1uEu. |
..200. |
A1gEgB2uEu. | ..8. |
A2gB1gA1uB2u. | ..24. |
A2gB1gA2uB1u. | ..32. |
A2gB2gA1uB1u. | ..24. |
A2gB2gA2uB2u. | ..50. |
A2gEgA1uEu. | ..75. |
A2gEgA2uEu. | ..100. |
A2gEgB1uEu. | ..50. |
A2gEgB2uEu. | ..48. |
B1gB2gA1uA2u. |
..64. |
B1gB2gB1uB2u. | ..100. |
B1gEgA1uEu. | ..150. |
B1gEgA2uEu. | ..200. |
B1gEgB1uEu. | ..100. |
B1gEgB2uEu. | ..200. |
B2gEgA1uEu. | ..300. |
B2gEgA2uEu. | ..400. |
B2gEgB1uEu. | ..200. |
B2gEgB2uEu. | ..48. |
A1uA2uB1uB2u. |
Subtotal: 3.729 / 30 / 210 |
Total: 9.596 / 109 / 715 |
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Last update November, 13th 2023 by A. Gelessus, Impressum, Datenschutzerklärung/DataPrivacyStatement