Characters of representations for molecular motions
| Motion |
E |
C4 |
C2 |
(C4)3 |
i |
(S4)3 |
σh |
S4 |
| Cartesian 3N |
21 |
3 |
-3 |
3 |
-3 |
-1 |
5 |
-1 |
| 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 |
-1 |
5 |
-1 |
Decomposition to irreducible representations
| Motion |
Ag |
Bg |
Eg*
|
Au |
Bu |
Eu*
|
Total |
| Cartesian 3N |
3 |
2 |
2 |
3 |
1 |
4 |
15 |
| Translation (x,y,z) |
0 |
0 |
0 |
1 |
0 |
1 |
2 |
| Rotation (Rx,Ry,Rz) |
1 |
0 |
1 |
0 |
0 |
0 |
2 |
| Vibration |
2 |
2 |
1 |
2 |
1 |
3 |
11 |
Molecular parameter
| Number of Atoms (N) |
7
|
| Number of internal coordinates |
15
|
| Number of independant internal coordinates |
2
|
| Number of vibrational modes |
11
|
Force field analysis
Allowed / forbidden vibronational transitions
| Operator |
Ag |
Bg |
Eg*
|
Au |
Bu |
Eu*
|
Total |
| Linear (IR) |
2 |
2 |
1 |
2 |
1 |
3 |
5 / 6 |
| Quadratic (Raman) |
2 |
2 |
1 |
2 |
1 |
3 |
5 / 6 |
| IR + Raman |
- - - - |
- - - - |
- - - - |
- - - - |
1 |
- - - - |
0* / 1 |
* Parity Mutual Exclusion Principle
Characters of force fields
(Symmetric powers of vibration representation)
| Force field |
E |
C4 |
C2 |
(C4)3 |
i |
(S4)3 |
σh |
S4 |
| linear |
15 |
1 |
-1 |
1 |
-3 |
-1 |
5 |
-1 |
| quadratic |
120 |
0 |
8 |
0 |
12 |
0 |
20 |
0 |
| cubic |
680 |
0 |
-8 |
0 |
-28 |
0 |
60 |
0 |
| quartic |
3.060 |
4 |
36 |
4 |
72 |
4 |
160 |
4 |
| quintic |
11.628 |
4 |
-36 |
4 |
-144 |
-4 |
376 |
-4 |
| sextic |
38.760 |
0 |
120 |
0 |
300 |
0 |
820 |
0 |
Decomposition to irreducible representations
Column with number of nonvanshing force constants highlighted
| Force field |
Ag |
Bg |
Eg*
|
Au |
Bu |
Eu*
|
| linear |
2 |
2 |
1 |
2 |
1 |
3 |
| quadratic |
20 |
20 |
13 |
12 |
12 |
15 |
| cubic |
88 |
88 |
75 |
80 |
80 |
97 |
| quartic |
418 |
414 |
367 |
358 |
358 |
389 |
| quintic |
1.478 |
1.478 |
1.393 |
1.422 |
1.418 |
1.523 |
| sextic |
5.000 |
5.000 |
4.765 |
4.720 |
4.720 |
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 C
4h
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(Ag) ≤ i ≤ pos(Eu) |
|---|
| ..3. |
AgAg. | ..3. |
BgBg. | ..1. |
EgEg. | ..3. |
AuAu. | ..1. |
BuBu. | ..9. |
EuEu. | | |
| |
| |
| |
| Subtotal: 20 / 6 / 6 |
| Irrep combinations (i,j) with indices: pos(Ag) ≤ i ≤ j ≤ pos(Eu) |
|---|
| Subtotal: 0 / 0 / 15 |
| Total: 20 / 6 / 21 |
Contributions to nonvanishing cubic force field constants
| Irrep combinations (i,i,i) with indices: pos(Ag) ≤ i ≤ pos(Eu) |
|---|
| ..4. |
AgAgAg. | | |
| |
| |
| |
| |
| |
| |
| |
| |
| Subtotal: 4 / 1 / 6 |
| Irrep combinations (i,i,j) (i,j,j) with indices: pos(Ag) ≤ i ≤ j ≤ pos(Eu) |
|---|
| ..6. |
AgBgBg. | ..2. |
AgEgEg. | ..6. |
AgAuAu. | ..2. |
AgBuBu. | ..18. |
AgEuEu. | ..4. |
BgEgEg. | ..24. |
BgEuEu. | | |
| |
| |
| Subtotal: 62 / 7 / 30 |
| Irrep combinations (i,j,k) with indices: pos(Ag) ≤ i ≤ j ≤ k ≤ pos(Eu) |
|---|
| ..4. |
BgAuBu. | ..12. |
EgAuEu. | ..6. |
EgBuEu. | | |
| |
| |
| |
| |
| |
| |
| Subtotal: 22 / 3 / 20 |
| Total: 88 / 11 / 56 |
Contributions to nonvanishing quartic force field constants
| Irrep combinations (i,i,i,i) with indices: pos(Ag) ≤ i ≤ pos(Eu) |
|---|
| ..5. |
AgAgAgAg. | ..5. |
BgBgBgBg. | ..3. |
EgEgEgEg. | ..5. |
AuAuAuAu. | ..1. |
BuBuBuBu. | ..66. |
EuEuEuEu. | | |
| |
| |
| |
| Subtotal: 85 / 6 / 6 |
| Irrep combinations (i,i,i,j) (i,j,j,j) with indices: pos(Ag) ≤ i ≤ j ≤ pos(Eu) |
|---|
| Subtotal: 0 / 0 / 30 |
| Irrep combinations (i,i,j,j) with indices: pos(Ag) ≤ i ≤ j ≤ pos(Eu) |
|---|
| ..9. |
AgAgBgBg. | ..3. |
AgAgEgEg. | ..9. |
AgAgAuAu. | ..3. |
AgAgBuBu. | ..27. |
AgAgEuEu. | ..3. |
BgBgEgEg. | ..9. |
BgBgAuAu. | ..3. |
BgBgBuBu. | ..27. |
BgBgEuEu. | ..3. |
EgEgAuAu. |
| ..1. |
EgEgBuBu. | ..33. |
EgEgEuEu. | ..3. |
AuAuBuBu. | ..27. |
AuAuEuEu. | ..9. |
BuBuEuEu. | | |
| |
| |
| |
| |
| Subtotal: 169 / 15 / 15 |
| Irrep combinations (i,i,j,k) (i,j,j,k) (i,j,k,k) with indices: pos(Ag) ≤ i ≤ j ≤ k ≤ pos(Eu) |
|---|
| ..4. |
EgEgAuBu. | ..8. |
AgBgEgEg. | ..48. |
AgBgEuEu. | ..24. |
AuBuEuEu. | | |
| |
| |
| |
| |
| |
| Subtotal: 84 / 4 / 60 |
| Irrep combinations (i,j,k,l) with indices: pos(Ag) ≤ i ≤ j ≤ k ≤ l ≤ pos(Eu) |
|---|
| ..8. |
AgBgAuBu. | ..24. |
AgEgAuEu. | ..12. |
AgEgBuEu. | ..24. |
BgEgAuEu. | ..12. |
BgEgBuEu. | | |
| |
| |
| |
| |
| Subtotal: 80 / 5 / 15 |
| Total: 418 / 30 / 126 |
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Last update November, 13th 2023 by A. Gelessus, Impressum, Datenschutzerklärung/DataPrivacyStatement