Characters of representations for molecular motions
| Motion |
E |
C2.(z) |
C2.(y) |
C2.(x) |
i |
σ.(xy) |
σ.(xy) |
σ.(xy) |
| Cartesian 3N |
78 |
0 |
0 |
-6 |
0 |
26 |
6 |
0 |
| Translation (x,y,z) |
3 |
-1 |
-1 |
-1 |
-3 |
1 |
1 |
1 |
| Rotation (Rx,Ry,Rz) |
3 |
-1 |
-1 |
-1 |
3 |
-1 |
-1 |
-1 |
| Vibration |
72 |
2 |
2 |
-4 |
0 |
26 |
6 |
0 |
Decomposition to irreducible representations
| Motion |
A1g |
B1g |
B2g |
B3g |
A1u |
B1u |
B2u |
B3u |
Total |
| Cartesian 3N |
13 |
13 |
8 |
5 |
5 |
8 |
13 |
13 |
78 |
| Translation (x,y,z) |
0 |
0 |
0 |
0 |
0 |
1 |
1 |
1 |
3 |
| Rotation (Rx,Ry,Rz) |
0 |
1 |
1 |
1 |
0 |
0 |
0 |
0 |
3 |
| Vibration |
13 |
12 |
7 |
4 |
5 |
7 |
12 |
12 |
72 |
Molecular parameter
| Number of Atoms (N) |
26
|
| Number of internal coordinates |
72
|
| Number of independant internal coordinates |
13
|
| Number of vibrational modes |
72
|
Force field analysis
Allowed / forbidden vibronational transitions
| Operator |
A1g |
B1g |
B2g |
B3g |
A1u |
B1u |
B2u |
B3u |
Total |
| Linear (IR) |
13 |
12 |
7 |
4 |
5 |
7 |
12 |
12 |
31 / 41 |
| Quadratic (Raman) |
13 |
12 |
7 |
4 |
5 |
7 |
12 |
12 |
36 / 36 |
| IR + Raman |
- - - - |
- - - - |
- - - - |
- - - - |
5 |
- - - - |
- - - - |
- - - - |
0* / 5 |
* Parity Mutual Exclusion Principle
Characters of force fields
(Symmetric powers of vibration representation)
| Force field |
E |
C2.(z) |
C2.(y) |
C2.(x) |
i |
σ.(xy) |
σ.(xy) |
σ.(xy) |
| linear |
72 |
2 |
2 |
-4 |
0 |
26 |
6 |
0 |
| quadratic |
2.628 |
38 |
38 |
44 |
36 |
374 |
54 |
36 |
| cubic |
64.824 |
74 |
74 |
-156 |
0 |
3.874 |
254 |
0 |
| quartic |
1.215.450 |
740 |
740 |
970 |
666 |
32.100 |
1.380 |
666 |
| quintic |
18.474.840 |
1.406 |
1.406 |
-3.116 |
0 |
225.030 |
5.466 |
0 |
| sextic |
237.093.780 |
9.842 |
9.842 |
14.364 |
8.436 |
1.381.730 |
22.946 |
8.436 |
Decomposition to irreducible representations
Column with number of nonvanshing force constants highlighted
| Force field |
A1g |
B1g |
B2g |
B3g |
A1u |
B1u |
B2u |
B3u |
| linear |
13 |
12 |
7 |
4 |
5 |
7 |
12 |
12 |
| quadratic |
406 |
363 |
283 |
280 |
281 |
283 |
363 |
369 |
| cubic |
8.618 |
8.575 |
7.670 |
7.549 |
7.586 |
7.670 |
8.575 |
8.581 |
| quartic |
156.589 |
155.650 |
147.970 |
147.849 |
147.886 |
147.970 |
155.650 |
155.886 |
| quintic |
2.338.129 |
2.337.190 |
2.282.299 |
2.279.802 |
2.280.505 |
2.282.299 |
2.337.190 |
2.337.426 |
| sextic |
29.818.672 |
29.804.775 |
29.465.079 |
29.462.582 |
29.463.285 |
29.465.079 |
29.804.775 |
29.809.533 |
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
2h
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(B3u) |
|---|
| ..91. |
A1gA1g. | ..78. |
B1gB1g. | ..28. |
B2gB2g. | ..10. |
B3gB3g. | ..15. |
A1uA1u. | ..28. |
B1uB1u. | ..78. |
B2uB2u. | ..78. |
B3uB3u. | | |
| |
| Subtotal: 406 / 8 / 8 |
| Irrep combinations (i,j) with indices: pos(A1g) ≤ i ≤ j ≤ pos(B3u) |
|---|
| Subtotal: 0 / 0 / 28 |
| Total: 406 / 8 / 36 |
Contributions to nonvanishing cubic force field constants
| Irrep combinations (i,i,i) with indices: pos(A1g) ≤ i ≤ pos(B3u) |
|---|
| ..455. |
A1gA1gA1g. | | |
| |
| |
| |
| |
| |
| |
| |
| |
| Subtotal: 455 / 1 / 8 |
| Irrep combinations (i,i,j) (i,j,j) with indices: pos(A1g) ≤ i ≤ j ≤ pos(B3u) |
|---|
| ..1.014. |
A1gB1gB1g. | ..364. |
A1gB2gB2g. | ..130. |
A1gB3gB3g. | ..195. |
A1gA1uA1u. | ..364. |
A1gB1uB1u. | ..1.014. |
A1gB2uB2u. | ..1.014. |
A1gB3uB3u. | | |
| |
| |
| Subtotal: 4.095 / 7 / 56 |
| Irrep combinations (i,j,k) with indices: pos(A1g) ≤ i ≤ j ≤ k ≤ pos(B3u) |
|---|
| ..336. |
B1gB2gB3g. | ..420. |
B1gA1uB1u. | ..1.728. |
B1gB2uB3u. | ..420. |
B2gA1uB2u. | ..588. |
B2gB1uB3u. | ..240. |
B3gA1uB3u. | ..336. |
B3gB1uB2u. | | |
| |
| |
| Subtotal: 4.068 / 7 / 56 |
| Total: 8.618 / 15 / 120 |
Contributions to nonvanishing quartic force field constants
| Irrep combinations (i,i,i,i) with indices: pos(A1g) ≤ i ≤ pos(B3u) |
|---|
| ..1.820. |
A1gA1gA1gA1g. | ..1.365. |
B1gB1gB1gB1g. | ..210. |
B2gB2gB2gB2g. | ..35. |
B3gB3gB3gB3g. | ..70. |
A1uA1uA1uA1u. | ..210. |
B1uB1uB1uB1u. | ..1.365. |
B2uB2uB2uB2u. | ..1.365. |
B3uB3uB3uB3u. | | |
| |
| Subtotal: 6.440 / 8 / 8 |
| Irrep combinations (i,i,i,j) (i,j,j,j) with indices: pos(A1g) ≤ i ≤ j ≤ pos(B3u) |
|---|
| Subtotal: 0 / 0 / 56 |
| Irrep combinations (i,i,j,j) with indices: pos(A1g) ≤ i ≤ j ≤ pos(B3u) |
|---|
| ..7.098. |
A1gA1gB1gB1g. | ..2.548. |
A1gA1gB2gB2g. | ..910. |
A1gA1gB3gB3g. | ..1.365. |
A1gA1gA1uA1u. | ..2.548. |
A1gA1gB1uB1u. | ..7.098. |
A1gA1gB2uB2u. | ..7.098. |
A1gA1gB3uB3u. | ..2.184. |
B1gB1gB2gB2g. | ..780. |
B1gB1gB3gB3g. | ..1.170. |
B1gB1gA1uA1u. |
| ..2.184. |
B1gB1gB1uB1u. | ..6.084. |
B1gB1gB2uB2u. | ..6.084. |
B1gB1gB3uB3u. | ..280. |
B2gB2gB3gB3g. | ..420. |
B2gB2gA1uA1u. | ..784. |
B2gB2gB1uB1u. | ..2.184. |
B2gB2gB2uB2u. | ..2.184. |
B2gB2gB3uB3u. | ..150. |
B3gB3gA1uA1u. | ..280. |
B3gB3gB1uB1u. |
| ..780. |
B3gB3gB2uB2u. | ..780. |
B3gB3gB3uB3u. | ..420. |
A1uA1uB1uB1u. | ..1.170. |
A1uA1uB2uB2u. | ..1.170. |
A1uA1uB3uB3u. | ..2.184. |
B1uB1uB2uB2u. | ..2.184. |
B1uB1uB3uB3u. | ..6.084. |
B2uB2uB3uB3u. | | |
| |
| Subtotal: 68.205 / 28 / 28 |
| Irrep combinations (i,i,j,k) (i,j,j,k) (i,j,k,k) with indices: pos(A1g) ≤ i ≤ j ≤ k ≤ pos(B3u) |
|---|
| Subtotal: 0 / 0 / 168 |
| Irrep combinations (i,j,k,l) with indices: pos(A1g) ≤ i ≤ j ≤ k ≤ l ≤ pos(B3u) |
|---|
| ..4.368. |
A1gB1gB2gB3g. | ..5.460. |
A1gB1gA1uB1u. | ..22.464. |
A1gB1gB2uB3u. | ..5.460. |
A1gB2gA1uB2u. | ..7.644. |
A1gB2gB1uB3u. | ..3.120. |
A1gB3gA1uB3u. | ..4.368. |
A1gB3gB1uB2u. | ..5.040. |
B1gB2gA1uB3u. | ..7.056. |
B1gB2gB1uB2u. | ..2.880. |
B1gB3gA1uB2u. |
| ..4.032. |
B1gB3gB1uB3u. | ..980. |
B2gB3gA1uB1u. | ..4.032. |
B2gB3gB2uB3u. | ..5.040. |
A1uB1uB2uB3u. | | |
| |
| |
| |
| |
| |
| Subtotal: 81.944 / 14 / 70 |
| Total: 156.589 / 50 / 330 |
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