Characters of representations for molecular motions
| Motion |
E |
C2.(z) |
C2.(y) |
C2.(x) |
i |
σ.(xy) |
σ.(xy) |
σ.(xy) |
| Cartesian 3N |
18 |
0 |
0 |
-2 |
0 |
6 |
2 |
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 |
12 |
2 |
2 |
0 |
0 |
6 |
2 |
0 |
Decomposition to irreducible representations
| Motion |
A1g |
B1g |
B2g |
B3g |
A1u |
B1u |
B2u |
B3u |
Total |
| Cartesian 3N |
3 |
3 |
2 |
1 |
1 |
2 |
3 |
3 |
18 |
| 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 |
2 |
1 |
0 |
1 |
1 |
2 |
2 |
12 |
Molecular parameter
| Number of Atoms (N) |
6
|
| Number of internal coordinates |
12
|
| Number of independant internal coordinates |
3
|
| Number of vibrational modes |
12
|
Force field analysis
Allowed / forbidden vibronational transitions
| Operator |
A1g |
B1g |
B2g |
B3g |
A1u |
B1u |
B2u |
B3u |
Total |
| Linear (IR) |
3 |
2 |
1 |
0 |
1 |
1 |
2 |
2 |
5 / 7 |
| Quadratic (Raman) |
3 |
2 |
1 |
0 |
1 |
1 |
2 |
2 |
6 / 6 |
| IR + Raman |
- - - - |
- - - - |
- - - - |
- - - - |
1 |
- - - - |
- - - - |
- - - - |
0* / 1 |
* 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 |
12 |
2 |
2 |
0 |
0 |
6 |
2 |
0 |
| quadratic |
78 |
8 |
8 |
6 |
6 |
24 |
8 |
6 |
| cubic |
364 |
14 |
14 |
0 |
0 |
74 |
14 |
0 |
| quartic |
1.365 |
35 |
35 |
21 |
21 |
195 |
35 |
21 |
| quintic |
4.368 |
56 |
56 |
0 |
0 |
456 |
56 |
0 |
| sextic |
12.376 |
112 |
112 |
56 |
56 |
976 |
112 |
56 |
Decomposition to irreducible representations
Column with number of nonvanshing force constants highlighted
| Force field |
A1g |
B1g |
B2g |
B3g |
A1u |
B1u |
B2u |
B3u |
| linear |
3 |
2 |
1 |
0 |
1 |
1 |
2 |
2 |
| quadratic |
18 |
11 |
7 |
6 |
7 |
7 |
11 |
11 |
| cubic |
60 |
53 |
38 |
31 |
38 |
38 |
53 |
53 |
| quartic |
216 |
188 |
148 |
141 |
148 |
148 |
188 |
188 |
| quintic |
624 |
596 |
496 |
468 |
496 |
496 |
596 |
596 |
| sextic |
1.732 |
1.648 |
1.432 |
1.404 |
1.432 |
1.432 |
1.648 |
1.648 |
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. | ..3. |
B1gB1g. | ..1. |
B2gB2g. | ..1. |
A1uA1u. | ..1. |
B1uB1u. | ..3. |
B2uB2u. | ..3. |
B3uB3u. | | |
| |
| |
| Subtotal: 18 / 7 / 8 |
| Irrep combinations (i,j) with indices: pos(A1g) ≤ i ≤ j ≤ pos(B3u) |
|---|
| Subtotal: 0 / 0 / 28 |
| Total: 18 / 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) |
|---|
| ..9. |
A1gB1gB1g. | ..3. |
A1gB2gB2g. | ..3. |
A1gA1uA1u. | ..3. |
A1gB1uB1u. | ..9. |
A1gB2uB2u. | ..9. |
A1gB3uB3u. | | |
| |
| |
| |
| Subtotal: 36 / 6 / 56 |
| Irrep combinations (i,j,k) with indices: pos(A1g) ≤ i ≤ j ≤ k ≤ pos(B3u) |
|---|
| ..2. |
B1gA1uB1u. | ..8. |
B1gB2uB3u. | ..2. |
B2gA1uB2u. | ..2. |
B2gB1uB3u. | | |
| |
| |
| |
| |
| |
| Subtotal: 14 / 4 / 56 |
| Total: 60 / 11 / 120 |
Contributions to nonvanishing quartic force field constants
| Irrep combinations (i,i,i,i) with indices: pos(A1g) ≤ i ≤ pos(B3u) |
|---|
| ..15. |
A1gA1gA1gA1g. | ..5. |
B1gB1gB1gB1g. | ..1. |
B2gB2gB2gB2g. | ..1. |
A1uA1uA1uA1u. | ..1. |
B1uB1uB1uB1u. | ..5. |
B2uB2uB2uB2u. | ..5. |
B3uB3uB3uB3u. | | |
| |
| |
| Subtotal: 33 / 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) |
|---|
| ..18. |
A1gA1gB1gB1g. | ..6. |
A1gA1gB2gB2g. | ..6. |
A1gA1gA1uA1u. | ..6. |
A1gA1gB1uB1u. | ..18. |
A1gA1gB2uB2u. | ..18. |
A1gA1gB3uB3u. | ..3. |
B1gB1gB2gB2g. | ..3. |
B1gB1gA1uA1u. | ..3. |
B1gB1gB1uB1u. | ..9. |
B1gB1gB2uB2u. |
| ..9. |
B1gB1gB3uB3u. | ..1. |
B2gB2gA1uA1u. | ..1. |
B2gB2gB1uB1u. | ..3. |
B2gB2gB2uB2u. | ..3. |
B2gB2gB3uB3u. | ..1. |
A1uA1uB1uB1u. | ..3. |
A1uA1uB2uB2u. | ..3. |
A1uA1uB3uB3u. | ..3. |
B1uB1uB2uB2u. | ..3. |
B1uB1uB3uB3u. |
| ..9. |
B2uB2uB3uB3u. | | |
| |
| |
| |
| |
| |
| |
| |
| |
| Subtotal: 129 / 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) |
|---|
| ..6. |
A1gB1gA1uB1u. | ..24. |
A1gB1gB2uB3u. | ..6. |
A1gB2gA1uB2u. | ..6. |
A1gB2gB1uB3u. | ..4. |
B1gB2gA1uB3u. | ..4. |
B1gB2gB1uB2u. | ..4. |
A1uB1uB2uB3u. | | |
| |
| |
| Subtotal: 54 / 7 / 70 |
| Total: 216 / 35 / 330 |
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Last update November, 13th 2023 by A. Gelessus, Impressum, Datenschutzerklärung/DataPrivacyStatement