Characters of representations for molecular motions
| Motion |
E |
C2.(z) |
C2.(y) |
C2.(x) |
i |
σ.(xy) |
σ.(xy) |
σ.(xy) |
| Cartesian 3N |
21 |
-3 |
-3 |
-3 |
-3 |
5 |
5 |
5 |
| 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 |
15 |
-1 |
-1 |
-1 |
-3 |
5 |
5 |
5 |
Decomposition to irreducible representations
| Motion |
A1g |
B1g |
B2g |
B3g |
A1u |
B1u |
B2u |
B3u |
Total |
| Cartesian 3N |
3 |
2 |
2 |
2 |
0 |
4 |
4 |
4 |
21 |
| 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 |
3 |
1 |
1 |
1 |
0 |
3 |
3 |
3 |
15 |
Molecular parameter
| Number of Atoms (N) |
7
|
| Number of internal coordinates |
15
|
| Number of independant internal coordinates |
3
|
| Number of vibrational modes |
15
|
Force field analysis
Allowed / forbidden vibronational transitions
| Operator |
A1g |
B1g |
B2g |
B3g |
A1u |
B1u |
B2u |
B3u |
Total |
| Linear (IR) |
3 |
1 |
1 |
1 |
0 |
3 |
3 |
3 |
9 / 6 |
| Quadratic (Raman) |
3 |
1 |
1 |
1 |
0 |
3 |
3 |
3 |
6 / 9 |
| IR + Raman |
- - - - |
- - - - |
- - - - |
- - - - |
0 |
- - - - |
- - - - |
- - - - |
0* / 0 |
* 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 |
15 |
-1 |
-1 |
-1 |
-3 |
5 |
5 |
5 |
| quadratic |
120 |
8 |
8 |
8 |
12 |
20 |
20 |
20 |
| cubic |
680 |
-8 |
-8 |
-8 |
-28 |
60 |
60 |
60 |
| quartic |
3.060 |
36 |
36 |
36 |
72 |
160 |
160 |
160 |
| quintic |
11.628 |
-36 |
-36 |
-36 |
-144 |
376 |
376 |
376 |
| sextic |
38.760 |
120 |
120 |
120 |
300 |
820 |
820 |
820 |
Decomposition to irreducible representations
Column with number of nonvanshing force constants highlighted
| Force field |
A1g |
B1g |
B2g |
B3g |
A1u |
B1u |
B2u |
B3u |
| linear |
3 |
1 |
1 |
1 |
0 |
3 |
3 |
3 |
| quadratic |
27 |
13 |
13 |
13 |
9 |
15 |
15 |
15 |
| cubic |
101 |
75 |
75 |
75 |
63 |
97 |
97 |
97 |
| quartic |
465 |
367 |
367 |
367 |
327 |
389 |
389 |
389 |
| quintic |
1.563 |
1.393 |
1.393 |
1.393 |
1.317 |
1.523 |
1.523 |
1.523 |
| sextic |
5.235 |
4.765 |
4.765 |
4.765 |
4.545 |
4.895 |
4.895 |
4.895 |
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) |
|---|
| ..6. |
A1gA1g. | ..1. |
B1gB1g. | ..1. |
B2gB2g. | ..1. |
B3gB3g. | ..6. |
B1uB1u. | ..6. |
B2uB2u. | ..6. |
B3uB3u. | | |
| |
| |
| Subtotal: 27 / 7 / 8 |
| Irrep combinations (i,j) with indices: pos(A1g) ≤ i ≤ j ≤ pos(B3u) |
|---|
| Subtotal: 0 / 0 / 28 |
| Total: 27 / 7 / 36 |
Contributions to nonvanishing cubic force field constants
| Irrep combinations (i,i,i) with indices: pos(A1g) ≤ i ≤ pos(B3u) |
|---|
| ..10. |
A1gA1gA1g. | | |
| |
| |
| |
| |
| |
| |
| |
| |
| Subtotal: 10 / 1 / 8 |
| Irrep combinations (i,i,j) (i,j,j) with indices: pos(A1g) ≤ i ≤ j ≤ pos(B3u) |
|---|
| ..3. |
A1gB1gB1g. | ..3. |
A1gB2gB2g. | ..3. |
A1gB3gB3g. | ..18. |
A1gB1uB1u. | ..18. |
A1gB2uB2u. | ..18. |
A1gB3uB3u. | | |
| |
| |
| |
| Subtotal: 63 / 6 / 56 |
| Irrep combinations (i,j,k) with indices: pos(A1g) ≤ i ≤ j ≤ k ≤ pos(B3u) |
|---|
| ..1. |
B1gB2gB3g. | ..9. |
B1gB2uB3u. | ..9. |
B2gB1uB3u. | ..9. |
B3gB1uB2u. | | |
| |
| |
| |
| |
| |
| Subtotal: 28 / 4 / 56 |
| Total: 101 / 11 / 120 |
Contributions to nonvanishing quartic force field constants
| Irrep combinations (i,i,i,i) with indices: pos(A1g) ≤ i ≤ pos(B3u) |
|---|
| ..15. |
A1gA1gA1gA1g. | ..1. |
B1gB1gB1gB1g. | ..1. |
B2gB2gB2gB2g. | ..1. |
B3gB3gB3gB3g. | ..15. |
B1uB1uB1uB1u. | ..15. |
B2uB2uB2uB2u. | ..15. |
B3uB3uB3uB3u. | | |
| |
| |
| Subtotal: 63 / 7 / 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) |
|---|
| ..6. |
A1gA1gB1gB1g. | ..6. |
A1gA1gB2gB2g. | ..6. |
A1gA1gB3gB3g. | ..36. |
A1gA1gB1uB1u. | ..36. |
A1gA1gB2uB2u. | ..36. |
A1gA1gB3uB3u. | ..1. |
B1gB1gB2gB2g. | ..1. |
B1gB1gB3gB3g. | ..6. |
B1gB1gB1uB1u. | ..6. |
B1gB1gB2uB2u. |
| ..6. |
B1gB1gB3uB3u. | ..1. |
B2gB2gB3gB3g. | ..6. |
B2gB2gB1uB1u. | ..6. |
B2gB2gB2uB2u. | ..6. |
B2gB2gB3uB3u. | ..6. |
B3gB3gB1uB1u. | ..6. |
B3gB3gB2uB2u. | ..6. |
B3gB3gB3uB3u. | ..36. |
B1uB1uB2uB2u. | ..36. |
B1uB1uB3uB3u. |
| ..36. |
B2uB2uB3uB3u. | | |
| |
| |
| |
| |
| |
| |
| |
| |
| Subtotal: 291 / 21 / 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) |
|---|
| ..3. |
A1gB1gB2gB3g. | ..27. |
A1gB1gB2uB3u. | ..27. |
A1gB2gB1uB3u. | ..27. |
A1gB3gB1uB2u. | ..9. |
B1gB2gB1uB2u. | ..9. |
B1gB3gB1uB3u. | ..9. |
B2gB3gB2uB3u. | | |
| |
| |
| Subtotal: 111 / 7 / 70 |
| Total: 465 / 35 / 330 |
Calculate contributions to
Last update November, 13th 2023 by A. Gelessus, Impressum, Datenschutzerklärung/DataPrivacyStatement