Characters of representations for molecular motions
| Motion |
E |
C3 |
(C3)2 |
i |
(S6)5 |
S6 |
| Cartesian 3N |
156 |
0 |
0 |
0 |
0 |
0 |
| Translation (x,y,z) |
3 |
0 |
0 |
-3 |
0 |
0 |
| Rotation (Rx,Ry,Rz) |
3 |
0 |
0 |
3 |
0 |
0 |
| Vibration |
150 |
0 |
0 |
0 |
0 |
0 |
Decomposition to irreducible representations
| Motion |
Ag |
Eg*
|
Au |
Eu*
|
Total |
| Cartesian 3N |
26 |
26 |
26 |
26 |
104 |
| Translation (x,y,z) |
0 |
0 |
1 |
1 |
2 |
| Rotation (Rx,Ry,Rz) |
1 |
1 |
0 |
0 |
2 |
| Vibration |
25 |
25 |
25 |
25 |
100 |
Molecular parameter
| Number of Atoms (N) |
52
|
| Number of internal coordinates |
150
|
| Number of independant internal coordinates |
25
|
| Number of vibrational modes |
100
|
Force field analysis
Allowed / forbidden vibronational transitions
| Operator |
Ag |
Eg*
|
Au |
Eu*
|
Total |
| Linear (IR) |
25 |
25 |
25 |
25 |
50 / 50 |
| Quadratic (Raman) |
25 |
25 |
25 |
25 |
50 / 50 |
| IR + Raman |
- - - - |
- - - - |
- - - - |
- - - - |
0* / 0 |
* Parity Mutual Exclusion Principle
Characters of force fields
(Symmetric powers of vibration representation)
| Force field |
E |
C3 |
(C3)2 |
i |
(S6)5 |
S6 |
| linear |
150 |
0 |
0 |
0 |
0 |
0 |
| quadratic |
11.325 |
0 |
0 |
75 |
0 |
0 |
| cubic |
573.800 |
50 |
50 |
0 |
0 |
0 |
| quartic |
21.947.850 |
0 |
0 |
2.850 |
0 |
0 |
| quintic |
675.993.780 |
0 |
0 |
0 |
0 |
0 |
| sextic |
17.463.172.650 |
1.275 |
1.275 |
73.150 |
25 |
25 |
Decomposition to irreducible representations
Column with number of nonvanshing force constants highlighted
| Force field |
Ag |
Eg*
|
Au |
Eu*
|
| linear |
25 |
25 |
25 |
25 |
| quadratic |
1.900 |
1.900 |
1.875 |
1.875 |
| cubic |
95.650 |
95.625 |
95.650 |
95.625 |
| quartic |
3.658.450 |
3.658.450 |
3.657.500 |
3.657.500 |
| quintic |
112.665.630 |
112.665.630 |
112.665.630 |
112.665.630 |
| sextic |
2.910.541.400 |
2.910.540.750 |
2.910.517.000 |
2.910.516.375 |
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 S
6
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) |
|---|
| ..325. |
AgAg. | ..625. |
EgEg. | ..325. |
AuAu. | ..625. |
EuEu. | | |
| |
| |
| |
| |
| |
| Subtotal: 1.900 / 4 / 4 |
| Irrep combinations (i,j) with indices: pos(Ag) ≤ i ≤ j ≤ pos(Eu) |
|---|
| Subtotal: 0 / 0 / 6 |
| Total: 1.900 / 4 / 10 |
Contributions to nonvanishing cubic force field constants
| Irrep combinations (i,i,i) with indices: pos(Ag) ≤ i ≤ pos(Eu) |
|---|
| ..2.925. |
AgAgAg. | ..5.850. |
EgEgEg. | | |
| |
| |
| |
| |
| |
| |
| |
| Subtotal: 8.775 / 2 / 4 |
| Irrep combinations (i,i,j) (i,j,j) with indices: pos(Ag) ≤ i ≤ j ≤ pos(Eu) |
|---|
| ..15.625. |
AgEgEg. | ..8.125. |
AgAuAu. | ..15.625. |
AgEuEu. | ..16.250. |
EgEuEu. | | |
| |
| |
| |
| |
| |
| Subtotal: 55.625 / 4 / 12 |
| Irrep combinations (i,j,k) with indices: pos(Ag) ≤ i ≤ j ≤ k ≤ pos(Eu) |
|---|
| ..31.250. |
EgAuEu. | | |
| |
| |
| |
| |
| |
| |
| |
| |
| Subtotal: 31.250 / 1 / 4 |
| Total: 95.650 / 7 / 20 |
Contributions to nonvanishing quartic force field constants
| Irrep combinations (i,i,i,i) with indices: pos(Ag) ≤ i ≤ pos(Eu) |
|---|
| ..20.475. |
AgAgAgAg. | ..105.625. |
EgEgEgEg. | ..20.475. |
AuAuAuAu. | ..105.625. |
EuEuEuEu. | | |
| |
| |
| |
| |
| |
| Subtotal: 252.200 / 4 / 4 |
| Irrep combinations (i,i,i,j) (i,j,j,j) with indices: pos(Ag) ≤ i ≤ j ≤ pos(Eu) |
|---|
| ..146.250. |
AgEgEgEg. | ..146.250. |
AuEuEuEu. | | |
| |
| |
| |
| |
| |
| |
| |
| Subtotal: 292.500 / 2 / 12 |
| Irrep combinations (i,i,j,j) with indices: pos(Ag) ≤ i ≤ j ≤ pos(Eu) |
|---|
| ..203.125. |
AgAgEgEg. | ..105.625. |
AgAgAuAu. | ..203.125. |
AgAgEuEu. | ..203.125. |
EgEgAuAu. | ..601.875. |
EgEgEuEu. | ..203.125. |
AuAuEuEu. | | |
| |
| |
| |
| Subtotal: 1.520.000 / 6 / 6 |
| Irrep combinations (i,i,j,k) (i,j,j,k) (i,j,k,k) with indices: pos(Ag) ≤ i ≤ j ≤ k ≤ pos(Eu) |
|---|
| ..406.250. |
EgEgAuEu. | ..406.250. |
AgEgEuEu. | | |
| |
| |
| |
| |
| |
| |
| |
| Subtotal: 812.500 / 2 / 12 |
| Irrep combinations (i,j,k,l) with indices: pos(Ag) ≤ i ≤ j ≤ k ≤ l ≤ pos(Eu) |
|---|
| ..781.250. |
AgEgAuEu. | | |
| |
| |
| |
| |
| |
| |
| |
| |
| Subtotal: 781.250 / 1 / 1 |
| Total: 3.658.450 / 15 / 35 |
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Last update November, 13th 2023 by A. Gelessus, Impressum, Datenschutzerklärung/DataPrivacyStatement