Characters of representations for molecular motions
Motion |
E |
C2.(z) |
C2.(y) |
C2.(x) |
i |
σ.(xy) |
σ.(xy) |
σ.(xy) |
Cartesian 3N |
36 |
-4 |
0 |
0 |
0 |
8 |
4 |
4 |
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 |
30 |
-2 |
2 |
2 |
0 |
8 |
4 |
4 |
Decomposition to irreducible representations
Motion |
A1g |
B1g |
B2g |
B3g |
A1u |
B1u |
B2u |
B3u |
Total |
Cartesian 3N |
6 |
4 |
4 |
4 |
2 |
4 |
6 |
6 |
36 |
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 |
6 |
3 |
3 |
3 |
2 |
3 |
5 |
5 |
30 |
Molecular parameter
Number of Atoms (N) |
12
|
Number of internal coordinates |
30
|
Number of independant internal coordinates |
6
|
Number of vibrational modes |
30
|
Force field analysis
Allowed / forbidden vibronational transitions
Operator |
A1g |
B1g |
B2g |
B3g |
A1u |
B1u |
B2u |
B3u |
Total |
Linear (IR) |
6 |
3 |
3 |
3 |
2 |
3 |
5 |
5 |
13 / 17 |
Quadratic (Raman) |
6 |
3 |
3 |
3 |
2 |
3 |
5 |
5 |
15 / 15 |
IR + Raman |
- - - - |
- - - - |
- - - - |
- - - - |
2 |
- - - - |
- - - - |
- - - - |
0* / 2 |
* 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 |
30 |
-2 |
2 |
2 |
0 |
8 |
4 |
4 |
quadratic |
465 |
17 |
17 |
17 |
15 |
47 |
23 |
23 |
cubic |
4.960 |
-32 |
32 |
32 |
0 |
208 |
72 |
72 |
quartic |
40.920 |
152 |
152 |
152 |
120 |
792 |
256 |
256 |
quintic |
278.256 |
-272 |
272 |
272 |
0 |
2.640 |
680 |
680 |
sextic |
1.623.160 |
952 |
952 |
952 |
680 |
8.008 |
1.904 |
1.904 |
Decomposition to irreducible representations
Column with number of nonvanshing force constants highlighted
Force field |
A1g |
B1g |
B2g |
B3g |
A1u |
B1u |
B2u |
B3u |
linear |
6 |
3 |
3 |
3 |
2 |
3 |
5 |
5 |
quadratic |
78 |
58 |
52 |
52 |
51 |
54 |
60 |
60 |
cubic |
668 |
616 |
598 |
598 |
580 |
600 |
650 |
650 |
quartic |
5.350 |
5.146 |
5.012 |
5.012 |
4.994 |
5.046 |
5.180 |
5.180 |
quintic |
35.316 |
34.840 |
34.486 |
34.486 |
34.316 |
34.520 |
35.146 |
35.146 |
sextic |
204.814 |
203.386 |
201.860 |
201.860 |
201.690 |
202.166 |
203.692 |
203.692 |
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) |
..21. |
A1gA1g. | ..6. |
B1gB1g. | ..6. |
B2gB2g. | ..6. |
B3gB3g. | ..3. |
A1uA1u. | ..6. |
B1uB1u. | ..15. |
B2uB2u. | ..15. |
B3uB3u. | | |
| |
Subtotal: 78 / 8 / 8 |
Irrep combinations (i,j) with indices: pos(A1g) ≤ i ≤ j ≤ pos(B3u) |
Subtotal: 0 / 0 / 28 |
Total: 78 / 8 / 36 |
Contributions to nonvanishing cubic force field constants
Irrep combinations (i,i,i) with indices: pos(A1g) ≤ i ≤ pos(B3u) |
..56. |
A1gA1gA1g. | | |
| |
| |
| |
| |
| |
| |
| |
| |
Subtotal: 56 / 1 / 8 |
Irrep combinations (i,i,j) (i,j,j) with indices: pos(A1g) ≤ i ≤ j ≤ pos(B3u) |
..36. |
A1gB1gB1g. | ..36. |
A1gB2gB2g. | ..36. |
A1gB3gB3g. | ..18. |
A1gA1uA1u. | ..36. |
A1gB1uB1u. | ..90. |
A1gB2uB2u. | ..90. |
A1gB3uB3u. | | |
| |
| |
Subtotal: 342 / 7 / 56 |
Irrep combinations (i,j,k) with indices: pos(A1g) ≤ i ≤ j ≤ k ≤ pos(B3u) |
..27. |
B1gB2gB3g. | ..18. |
B1gA1uB1u. | ..75. |
B1gB2uB3u. | ..30. |
B2gA1uB2u. | ..45. |
B2gB1uB3u. | ..30. |
B3gA1uB3u. | ..45. |
B3gB1uB2u. | | |
| |
| |
Subtotal: 270 / 7 / 56 |
Total: 668 / 15 / 120 |
Contributions to nonvanishing quartic force field constants
Irrep combinations (i,i,i,i) with indices: pos(A1g) ≤ i ≤ pos(B3u) |
..126. |
A1gA1gA1gA1g. | ..15. |
B1gB1gB1gB1g. | ..15. |
B2gB2gB2gB2g. | ..15. |
B3gB3gB3gB3g. | ..5. |
A1uA1uA1uA1u. | ..15. |
B1uB1uB1uB1u. | ..70. |
B2uB2uB2uB2u. | ..70. |
B3uB3uB3uB3u. | | |
| |
Subtotal: 331 / 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) |
..126. |
A1gA1gB1gB1g. | ..126. |
A1gA1gB2gB2g. | ..126. |
A1gA1gB3gB3g. | ..63. |
A1gA1gA1uA1u. | ..126. |
A1gA1gB1uB1u. | ..315. |
A1gA1gB2uB2u. | ..315. |
A1gA1gB3uB3u. | ..36. |
B1gB1gB2gB2g. | ..36. |
B1gB1gB3gB3g. | ..18. |
B1gB1gA1uA1u. |
..36. |
B1gB1gB1uB1u. | ..90. |
B1gB1gB2uB2u. | ..90. |
B1gB1gB3uB3u. | ..36. |
B2gB2gB3gB3g. | ..18. |
B2gB2gA1uA1u. | ..36. |
B2gB2gB1uB1u. | ..90. |
B2gB2gB2uB2u. | ..90. |
B2gB2gB3uB3u. | ..18. |
B3gB3gA1uA1u. | ..36. |
B3gB3gB1uB1u. |
..90. |
B3gB3gB2uB2u. | ..90. |
B3gB3gB3uB3u. | ..18. |
A1uA1uB1uB1u. | ..45. |
A1uA1uB2uB2u. | ..45. |
A1uA1uB3uB3u. | ..90. |
B1uB1uB2uB2u. | ..90. |
B1uB1uB3uB3u. | ..225. |
B2uB2uB3uB3u. | | |
| |
Subtotal: 2.520 / 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) |
..162. |
A1gB1gB2gB3g. | ..108. |
A1gB1gA1uB1u. | ..450. |
A1gB1gB2uB3u. | ..180. |
A1gB2gA1uB2u. | ..270. |
A1gB2gB1uB3u. | ..180. |
A1gB3gA1uB3u. | ..270. |
A1gB3gB1uB2u. | ..90. |
B1gB2gA1uB3u. | ..135. |
B1gB2gB1uB2u. | ..90. |
B1gB3gA1uB2u. |
..135. |
B1gB3gB1uB3u. | ..54. |
B2gB3gA1uB1u. | ..225. |
B2gB3gB2uB3u. | ..150. |
A1uB1uB2uB3u. | | |
| |
| |
| |
| |
| |
Subtotal: 2.499 / 14 / 70 |
Total: 5.350 / 50 / 330 |
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