Characters of representations for molecular motions
Motion |
E |
2C4 |
C2 |
2C'2 |
2C''2 |
i |
2S4 |
σh |
2σv |
2σd |
Cartesian 3N |
39 |
5 |
-5 |
-5 |
-1 |
-3 |
-1 |
9 |
9 |
5 |
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 |
33 |
3 |
-3 |
-3 |
1 |
-3 |
-1 |
9 |
9 |
5 |
Decomposition to irreducible representations
Motion |
A1g |
A2g |
B1g |
B2g |
Eg |
A1u |
A2u |
B1u |
B2u |
Eu |
Total |
Cartesian 3N |
4 |
2 |
2 |
2 |
4 |
0 |
5 |
0 |
2 |
7 |
28 |
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 |
2 |
3 |
0 |
4 |
0 |
2 |
6 |
24 |
Molecular parameter
Number of Atoms (N) |
13
|
Number of internal coordinates |
33
|
Number of independant internal coordinates |
4
|
Number of vibrational modes |
24
|
Force field analysis
Allowed / forbidden vibronational transitions
Operator |
A1g |
A2g |
B1g |
B2g |
Eg |
A1u |
A2u |
B1u |
B2u |
Eu |
Total |
Linear (IR) |
4 |
1 |
2 |
2 |
3 |
0 |
4 |
0 |
2 |
6 |
10 / 14 |
Quadratic (Raman) |
4 |
1 |
2 |
2 |
3 |
0 |
4 |
0 |
2 |
6 |
11 / 13 |
IR + Raman |
- - - - |
1 |
- - - - |
- - - - |
- - - - |
0 |
- - - - |
0 |
2 |
- - - - |
0* / 3 |
* 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 |
33 |
3 |
-3 |
-3 |
1 |
-3 |
-1 |
9 |
9 |
5 |
quadratic |
561 |
3 |
21 |
21 |
17 |
21 |
-1 |
57 |
57 |
29 |
cubic |
6.545 |
1 |
-55 |
-55 |
17 |
-55 |
1 |
273 |
273 |
105 |
quartic |
58.905 |
9 |
225 |
225 |
153 |
225 |
9 |
1.113 |
1.113 |
385 |
quintic |
435.897 |
27 |
-531 |
-531 |
153 |
-531 |
-9 |
3.969 |
3.969 |
1.141 |
sextic |
2.760.681 |
27 |
1.653 |
1.653 |
969 |
1.653 |
-9 |
12.817 |
12.817 |
3.325 |
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 |
2 |
3 |
0 |
4 |
0 |
2 |
6 |
quadratic |
57 |
26 |
45 |
37 |
63 |
26 |
38 |
28 |
34 |
72 |
cubic |
462 |
377 |
431 |
407 |
784 |
340 |
444 |
362 |
422 |
866 |
quartic |
4.016 |
3.547 |
3.877 |
3.677 |
7.224 |
3.472 |
3.752 |
3.530 |
3.694 |
7.446 |
quintic |
28.019 |
26.836 |
27.691 |
27.155 |
53.991 |
26.314 |
27.686 |
26.552 |
27.430 |
55.116 |
sextic |
175.898 |
171.207 |
174.820 |
172.276 |
343.483 |
170.056 |
173.436 |
170.636 |
172.838 |
346.274 |
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. | ..3. |
B2gB2g. | ..6. |
EgEg. | ..10. |
A2uA2u. | ..3. |
B2uB2u. | ..21. |
EuEu. | | |
| |
Subtotal: 57 / 8 / 10 |
Irrep combinations (i,j) with indices: pos(A1g) ≤ i ≤ j ≤ pos(Eu) |
Subtotal: 0 / 0 / 45 |
Total: 57 / 8 / 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. | ..12. |
A1gB2gB2g. | ..24. |
A1gEgEg. | ..40. |
A1gA2uA2u. | ..12. |
A1gB2uB2u. | ..84. |
A1gEuEu. | ..3. |
A2gEgEg. | ..15. |
A2gEuEu. | ..12. |
B1gEgEg. |
..42. |
B1gEuEu. | ..12. |
B2gEgEg. | ..42. |
B2gEuEu. | | |
| |
| |
| |
| |
| |
| |
Subtotal: 314 / 13 / 90 |
Irrep combinations (i,j,k) with indices: pos(A1g) ≤ i ≤ j ≤ k ≤ pos(Eu) |
..4. |
A2gB1gB2g. | ..16. |
B1gA2uB2u. | ..72. |
EgA2uEu. | ..36. |
EgB2uEu. | | |
| |
| |
| |
| |
| |
Subtotal: 128 / 4 / 120 |
Total: 462 / 18 / 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. | ..5. |
B2gB2gB2gB2g. | ..36. |
EgEgEgEg. | ..35. |
A2uA2uA2uA2u. | ..5. |
B2uB2uB2uB2u. | ..357. |
EuEuEuEu. | | |
| |
Subtotal: 479 / 8 / 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. | ..30. |
A1gA1gB2gB2g. | ..60. |
A1gA1gEgEg. | ..100. |
A1gA1gA2uA2u. | ..30. |
A1gA1gB2uB2u. | ..210. |
A1gA1gEuEu. | ..3. |
A2gA2gB1gB1g. | ..3. |
A2gA2gB2gB2g. | ..6. |
A2gA2gEgEg. |
..10. |
A2gA2gA2uA2u. | ..3. |
A2gA2gB2uB2u. | ..21. |
A2gA2gEuEu. | ..9. |
B1gB1gB2gB2g. | ..18. |
B1gB1gEgEg. | ..30. |
B1gB1gA2uA2u. | ..9. |
B1gB1gB2uB2u. | ..63. |
B1gB1gEuEu. | ..18. |
B2gB2gEgEg. | ..30. |
B2gB2gA2uA2u. |
..9. |
B2gB2gB2uB2u. | ..63. |
B2gB2gEuEu. | ..60. |
EgEgA2uA2u. | ..18. |
EgEgB2uB2u. | ..423. |
EgEgEuEu. | ..30. |
A2uA2uB2uB2u. | ..210. |
A2uA2uEuEu. | ..63. |
B2uB2uEuEu. | | |
| |
Subtotal: 1.569 / 28 / 45 |
Irrep combinations (i,i,j,k) (i,j,j,k) (i,j,k,k) with indices: pos(A1g) ≤ i ≤ j ≤ k ≤ pos(Eu) |
..48. |
EgEgA2uB2u. | ..12. |
A1gA2gEgEg. | ..60. |
A1gA2gEuEu. | ..48. |
A1gB1gEgEg. | ..168. |
A1gB1gEuEu. | ..48. |
A1gB2gEgEg. | ..168. |
A1gB2gEuEu. | ..12. |
A2gB1gEgEg. | ..42. |
A2gB1gEuEu. | ..12. |
A2gB2gEgEg. |
..42. |
A2gB2gEuEu. | ..12. |
B1gB2gEgEg. | ..60. |
B1gB2gEuEu. | ..168. |
A2uB2uEuEu. | | |
| |
| |
| |
| |
| |
Subtotal: 900 / 14 / 360 |
Irrep combinations (i,j,k,l) with indices: pos(A1g) ≤ i ≤ j ≤ k ≤ l ≤ pos(Eu) |
..16. |
A1gA2gB1gB2g. | ..64. |
A1gB1gA2uB2u. | ..288. |
A1gEgA2uEu. | ..144. |
A1gEgB2uEu. | ..16. |
A2gB2gA2uB2u. | ..72. |
A2gEgA2uEu. | ..36. |
A2gEgB2uEu. | ..144. |
B1gEgA2uEu. | ..72. |
B1gEgB2uEu. | ..144. |
B2gEgA2uEu. |
..72. |
B2gEgB2uEu. | | |
| |
| |
| |
| |
| |
| |
| |
| |
Subtotal: 1.068 / 11 / 210 |
Total: 4.016 / 61 / 715 |
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