Characters of representations for molecular motions
| Motion |
E |
C4 |
C2 |
(C4)3 |
i |
(S4)3 |
σh |
S4 |
| Cartesian 3N |
48 |
0 |
0 |
0 |
0 |
0 |
0 |
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 |
42 |
-2 |
2 |
-2 |
0 |
0 |
0 |
0 |
Decomposition to irreducible representations
| Motion |
Ag |
Bg |
Eg*
|
Au |
Bu |
Eu*
|
Total |
| Cartesian 3N |
6 |
6 |
6 |
6 |
6 |
6 |
36 |
| Translation (x,y,z) |
0 |
0 |
0 |
1 |
0 |
1 |
2 |
| Rotation (Rx,Ry,Rz) |
1 |
0 |
1 |
0 |
0 |
0 |
2 |
| Vibration |
5 |
6 |
5 |
5 |
6 |
5 |
32 |
Molecular parameter
| Number of Atoms (N) |
16
|
| Number of internal coordinates |
42
|
| Number of independant internal coordinates |
5
|
| Number of vibrational modes |
32
|
Force field analysis
Allowed / forbidden vibronational transitions
| Operator |
Ag |
Bg |
Eg*
|
Au |
Bu |
Eu*
|
Total |
| Linear (IR) |
5 |
6 |
5 |
5 |
6 |
5 |
10 / 22 |
| Quadratic (Raman) |
5 |
6 |
5 |
5 |
6 |
5 |
16 / 16 |
| IR + Raman |
- - - - |
- - - - |
- - - - |
- - - - |
6 |
- - - - |
0* / 6 |
* 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 |
42 |
-2 |
2 |
-2 |
0 |
0 |
0 |
0 |
| quadratic |
903 |
3 |
23 |
3 |
21 |
1 |
21 |
1 |
| cubic |
13.244 |
-4 |
44 |
-4 |
0 |
0 |
0 |
0 |
| quartic |
148.995 |
15 |
275 |
15 |
231 |
11 |
231 |
11 |
| quintic |
1.370.754 |
-26 |
506 |
-26 |
0 |
0 |
0 |
0 |
| sextic |
10.737.573 |
37 |
2.277 |
37 |
1.771 |
11 |
1.771 |
11 |
Decomposition to irreducible representations
Column with number of nonvanshing force constants highlighted
| Force field |
Ag |
Bg |
Eg*
|
Au |
Bu |
Eu*
|
| linear |
5 |
6 |
5 |
5 |
6 |
5 |
| quadratic |
122 |
120 |
110 |
111 |
110 |
110 |
| cubic |
1.660 |
1.662 |
1.650 |
1.660 |
1.662 |
1.650 |
| quartic |
18.723 |
18.710 |
18.590 |
18.602 |
18.600 |
18.590 |
| quintic |
171.401 |
171.414 |
171.281 |
171.401 |
171.414 |
171.281 |
| sextic |
1.342.936 |
1.342.912 |
1.341.912 |
1.342.045 |
1.342.032 |
1.341.912 |
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) |
|---|
| ..15. |
AgAg. | ..21. |
BgBg. | ..25. |
EgEg. | ..15. |
AuAu. | ..21. |
BuBu. | ..25. |
EuEu. | | |
| |
| |
| |
| Subtotal: 122 / 6 / 6 |
| Irrep combinations (i,j) with indices: pos(Ag) ≤ i ≤ j ≤ pos(Eu) |
|---|
| Subtotal: 0 / 0 / 15 |
| Total: 122 / 6 / 21 |
Contributions to nonvanishing cubic force field constants
| Irrep combinations (i,i,i) with indices: pos(Ag) ≤ i ≤ pos(Eu) |
|---|
| ..35. |
AgAgAg. | | |
| |
| |
| |
| |
| |
| |
| |
| |
| Subtotal: 35 / 1 / 6 |
| Irrep combinations (i,i,j) (i,j,j) with indices: pos(Ag) ≤ i ≤ j ≤ pos(Eu) |
|---|
| ..105. |
AgBgBg. | ..125. |
AgEgEg. | ..75. |
AgAuAu. | ..105. |
AgBuBu. | ..125. |
AgEuEu. | ..180. |
BgEgEg. | ..180. |
BgEuEu. | | |
| |
| |
| Subtotal: 895 / 7 / 30 |
| Irrep combinations (i,j,k) with indices: pos(Ag) ≤ i ≤ j ≤ k ≤ pos(Eu) |
|---|
| ..180. |
BgAuBu. | ..250. |
EgAuEu. | ..300. |
EgBuEu. | | |
| |
| |
| |
| |
| |
| |
| Subtotal: 730 / 3 / 20 |
| Total: 1.660 / 11 / 56 |
Contributions to nonvanishing quartic force field constants
| Irrep combinations (i,i,i,i) with indices: pos(Ag) ≤ i ≤ pos(Eu) |
|---|
| ..70. |
AgAgAgAg. | ..126. |
BgBgBgBg. | ..365. |
EgEgEgEg. | ..70. |
AuAuAuAu. | ..126. |
BuBuBuBu. | ..365. |
EuEuEuEu. | | |
| |
| |
| |
| Subtotal: 1.122 / 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) |
|---|
| ..315. |
AgAgBgBg. | ..375. |
AgAgEgEg. | ..225. |
AgAgAuAu. | ..315. |
AgAgBuBu. | ..375. |
AgAgEuEu. | ..525. |
BgBgEgEg. | ..315. |
BgBgAuAu. | ..441. |
BgBgBuBu. | ..525. |
BgBgEuEu. | ..375. |
EgEgAuAu. |
| ..525. |
EgEgBuBu. | ..1.525. |
EgEgEuEu. | ..315. |
AuAuBuBu. | ..375. |
AuAuEuEu. | ..525. |
BuBuEuEu. | | |
| |
| |
| |
| |
| Subtotal: 7.051 / 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) |
|---|
| ..900. |
EgEgAuBu. | ..900. |
AgBgEgEg. | ..900. |
AgBgEuEu. | ..900. |
AuBuEuEu. | | |
| |
| |
| |
| |
| |
| Subtotal: 3.600 / 4 / 60 |
| Irrep combinations (i,j,k,l) with indices: pos(Ag) ≤ i ≤ j ≤ k ≤ l ≤ pos(Eu) |
|---|
| ..900. |
AgBgAuBu. | ..1.250. |
AgEgAuEu. | ..1.500. |
AgEgBuEu. | ..1.500. |
BgEgAuEu. | ..1.800. |
BgEgBuEu. | | |
| |
| |
| |
| |
| Subtotal: 6.950 / 5 / 15 |
| Total: 18.723 / 30 / 126 |
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Last update November, 13th 2023 by A. Gelessus, Impressum, Datenschutzerklärung/DataPrivacyStatement