Characters of representations for molecular motions
Motion |
E |
8C3 |
6C2 |
6C4 |
3C2 |
i |
6S4 |
8S6 |
3σh |
6σd |
Cartesian 3N |
39 |
0 |
-1 |
5 |
-5 |
-3 |
-1 |
0 |
9 |
5 |
Translation (x,y,z) |
3 |
0 |
-1 |
1 |
-1 |
-3 |
-1 |
0 |
1 |
1 |
Rotation (Rx,Ry,Rz) |
3 |
0 |
-1 |
1 |
-1 |
3 |
1 |
0 |
-1 |
-1 |
Vibration |
33 |
0 |
1 |
3 |
-3 |
-3 |
-1 |
0 |
9 |
5 |
Decomposition to irreducible representations
Motion |
A1g |
A2g |
Eg |
T1g |
T2g |
A1u |
A2u |
Eu |
T1u |
T2u |
Total |
Cartesian 3N |
2 |
0 |
2 |
2 |
2 |
0 |
0 |
0 |
5 |
2 |
15 |
Translation (x,y,z) |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
1 |
0 |
1 |
Rotation (Rx,Ry,Rz) |
0 |
0 |
0 |
1 |
0 |
0 |
0 |
0 |
0 |
0 |
1 |
Vibration |
2 |
0 |
2 |
1 |
2 |
0 |
0 |
0 |
4 |
2 |
13 |
Molecular parameter
Number of Atoms (N) |
13
|
Number of internal coordinates |
33
|
Number of independant internal coordinates |
2
|
Number of vibrational modes |
13
|
Force field analysis
Allowed / forbidden vibronational transitions
Operator |
A1g |
A2g |
Eg |
T1g |
T2g |
A1u |
A2u |
Eu |
T1u |
T2u |
Total |
Linear (IR) |
2 |
0 |
2 |
1 |
2 |
0 |
0 |
0 |
4 |
2 |
4 / 9 |
Quadratic (Raman) |
2 |
0 |
2 |
1 |
2 |
0 |
0 |
0 |
4 |
2 |
6 / 7 |
IR + Raman |
- - - - |
0 |
- - - - |
1 |
- - - - |
0 |
0 |
0 |
- - - - |
2 |
0* / 3 |
* Parity Mutual Exclusion Principle
Characters of force fields
(Symmetric powers of vibration representation)
Force field |
E |
8C3 |
6C2 |
6C4 |
3C2 |
i |
6S4 |
8S6 |
3σh |
6σd |
linear |
33 |
0 |
1 |
3 |
-3 |
-3 |
-1 |
0 |
9 |
5 |
quadratic |
561 |
0 |
17 |
3 |
21 |
21 |
-1 |
0 |
57 |
29 |
cubic |
6.545 |
11 |
17 |
1 |
-55 |
-55 |
1 |
-1 |
273 |
105 |
quartic |
58.905 |
0 |
153 |
9 |
225 |
225 |
9 |
0 |
1.113 |
385 |
quintic |
435.897 |
0 |
153 |
27 |
-531 |
-531 |
-9 |
0 |
3.969 |
1.141 |
sextic |
2.760.681 |
66 |
969 |
27 |
1.653 |
1.653 |
-9 |
6 |
12.817 |
3.325 |
Decomposition to irreducible representations
Column with number of nonvanshing force constants highlighted
Force field |
A1g |
A2g |
Eg |
T1g |
T2g |
A1u |
A2u |
Eu |
T1u |
T2u |
linear |
2 |
0 |
2 |
1 |
2 |
0 |
0 |
0 |
4 |
2 |
quadratic |
23 |
11 |
34 |
26 |
37 |
8 |
10 |
18 |
38 |
34 |
cubic |
166 |
135 |
296 |
377 |
407 |
108 |
130 |
232 |
444 |
422 |
quartic |
1.385 |
1.246 |
2.631 |
3.547 |
3.677 |
1.138 |
1.196 |
2.334 |
3.752 |
3.694 |
quintic |
9.449 |
9.121 |
18.570 |
26.836 |
27.155 |
8.692 |
8.930 |
17.622 |
27.686 |
27.430 |
sextic |
59.004 |
57.926 |
116.894 |
171.207 |
172.276 |
56.502 |
57.082 |
113.554 |
173.436 |
172.838 |
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 O
h
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(T2u) |
..3. |
A1gA1g. | ..3. |
EgEg. | ..1. |
T1gT1g. | ..3. |
T2gT2g. | ..10. |
T1uT1u. | ..3. |
T2uT2u. | | |
| |
| |
| |
Subtotal: 23 / 6 / 10 |
Irrep combinations (i,j) with indices: pos(A1g) ≤ i ≤ j ≤ pos(T2u) |
Subtotal: 0 / 0 / 45 |
Total: 23 / 6 / 55 |
Contributions to nonvanishing cubic force field constants
Irrep combinations (i,i,i) with indices: pos(A1g) ≤ i ≤ pos(T2u) |
..4. |
A1gA1gA1g. | ..4. |
EgEgEg. | ..4. |
T2gT2gT2g. | | |
| |
| |
| |
| |
| |
| |
Subtotal: 12 / 3 / 10 |
Irrep combinations (i,i,j) (i,j,j) with indices: pos(A1g) ≤ i ≤ j ≤ pos(T2u) |
..2. |
T1gT1gT2g. | ..6. |
A1gEgEg. | ..2. |
A1gT1gT1g. | ..6. |
A1gT2gT2g. | ..20. |
A1gT1uT1u. | ..6. |
A1gT2uT2u. | ..2. |
EgT1gT1g. | ..6. |
EgT2gT2g. | ..20. |
EgT1uT1u. | ..6. |
EgT2uT2u. |
..1. |
T1gT2gT2g. | ..6. |
T1gT1uT1u. | ..1. |
T1gT2uT2u. | ..20. |
T2gT1uT1u. | ..6. |
T2gT2uT2u. | | |
| |
| |
| |
| |
Subtotal: 110 / 15 / 90 |
Irrep combinations (i,j,k) with indices: pos(A1g) ≤ i ≤ j ≤ k ≤ pos(T2u) |
..4. |
EgT1gT2g. | ..16. |
EgT1uT2u. | ..8. |
T1gT1uT2u. | ..16. |
T2gT1uT2u. | | |
| |
| |
| |
| |
| |
Subtotal: 44 / 4 / 120 |
Total: 166 / 22 / 220 |
Contributions to nonvanishing quartic force field constants
Irrep combinations (i,i,i,i) with indices: pos(A1g) ≤ i ≤ pos(T2u) |
..5. |
A1gA1gA1gA1g. | ..6. |
EgEgEgEg. | ..2. |
T1gT1gT1gT1g. | ..11. |
T2gT2gT2gT2g. | ..90. |
T1uT1uT1uT1u. | ..11. |
T2uT2uT2uT2u. | | |
| |
| |
| |
Subtotal: 125 / 6 / 10 |
Irrep combinations (i,i,i,j) (i,j,j,j) with indices: pos(A1g) ≤ i ≤ j ≤ pos(T2u) |
..2. |
T1gT1gT1gT2g. | ..80. |
T1uT1uT1uT2u. | ..8. |
A1gEgEgEg. | ..8. |
A1gT2gT2gT2g. | ..4. |
EgT2gT2gT2g. | ..6. |
T1gT2gT2gT2g. | ..24. |
T1uT2uT2uT2u. | | |
| |
| |
Subtotal: 132 / 7 / 90 |
Irrep combinations (i,i,j,j) with indices: pos(A1g) ≤ i ≤ j ≤ pos(T2u) |
..9. |
A1gA1gEgEg. | ..3. |
A1gA1gT1gT1g. | ..9. |
A1gA1gT2gT2g. | ..30. |
A1gA1gT1uT1u. | ..9. |
A1gA1gT2uT2u. | ..6. |
EgEgT1gT1g. | ..18. |
EgEgT2gT2g. | ..60. |
EgEgT1uT1u. | ..18. |
EgEgT2uT2u. | ..9. |
T1gT1gT2gT2g. |
..30. |
T1gT1gT1uT1u. | ..9. |
T1gT1gT2uT2u. | ..96. |
T2gT2gT1uT1u. | ..28. |
T2gT2gT2uT2u. | ..96. |
T1uT1uT2uT2u. | | |
| |
| |
| |
| |
Subtotal: 430 / 15 / 45 |
Irrep combinations (i,i,j,k) (i,j,j,k) (i,j,k,k) with indices: pos(A1g) ≤ i ≤ j ≤ k ≤ pos(T2u) |
..8. |
EgEgT1gT2g. | ..32. |
EgEgT1uT2u. | ..16. |
T1gT1gT1uT2u. | ..56. |
T2gT2gT1uT2u. | ..4. |
A1gT1gT1gT2g. | ..4. |
EgT1gT1gT2g. | ..4. |
A1gEgT1gT1g. | ..12. |
A1gEgT2gT2g. | ..40. |
A1gEgT1uT1u. | ..12. |
A1gEgT2uT2u. |
..2. |
A1gT1gT2gT2g. | ..12. |
A1gT1gT1uT1u. | ..2. |
A1gT1gT2uT2u. | ..40. |
A1gT2gT1uT1u. | ..12. |
A1gT2gT2uT2u. | ..8. |
EgT1gT2gT2g. | ..32. |
EgT1gT1uT1u. | ..8. |
EgT1gT2uT2u. | ..64. |
EgT2gT1uT1u. | ..16. |
EgT2gT2uT2u. |
..52. |
T1gT2gT1uT1u. | ..14. |
T1gT2gT2uT2u. | | |
| |
| |
| |
| |
| |
| |
| |
Subtotal: 450 / 22 / 360 |
Irrep combinations (i,j,k,l) with indices: pos(A1g) ≤ i ≤ j ≤ k ≤ l ≤ pos(T2u) |
..8. |
A1gEgT1gT2g. | ..32. |
A1gEgT1uT2u. | ..16. |
A1gT1gT1uT2u. | ..32. |
A1gT2gT1uT2u. | ..32. |
EgT1gT1uT2u. | ..64. |
EgT2gT1uT2u. | ..64. |
T1gT2gT1uT2u. | | |
| |
| |
Subtotal: 248 / 7 / 210 |
Total: 1.385 / 57 / 715 |
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