Characters of representations for molecular motions
| Motion |
E |
C4 |
C2 |
(C4)3 |
i |
(S4)3 |
σh |
S4 |
| Cartesian 3N |
156 |
0 |
0 |
0 |
0 |
0 |
4 |
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 |
150 |
-2 |
2 |
-2 |
0 |
0 |
4 |
0 |
Decomposition to irreducible representations
| Motion |
Ag |
Bg |
Eg*
|
Au |
Bu |
Eu*
|
Total |
| Cartesian 3N |
20 |
20 |
19 |
19 |
19 |
20 |
117 |
| Translation (x,y,z) |
0 |
0 |
0 |
1 |
0 |
1 |
2 |
| Rotation (Rx,Ry,Rz) |
1 |
0 |
1 |
0 |
0 |
0 |
2 |
| Vibration |
19 |
20 |
18 |
18 |
19 |
19 |
113 |
Molecular parameter
| Number of Atoms (N) |
52
|
| Number of internal coordinates |
150
|
| Number of independant internal coordinates |
19
|
| Number of vibrational modes |
113
|
Force field analysis
Allowed / forbidden vibronational transitions
| Operator |
Ag |
Bg |
Eg*
|
Au |
Bu |
Eu*
|
Total |
| Linear (IR) |
19 |
20 |
18 |
18 |
19 |
19 |
37 / 76 |
| Quadratic (Raman) |
19 |
20 |
18 |
18 |
19 |
19 |
57 / 56 |
| IR + Raman |
- - - - |
- - - - |
- - - - |
- - - - |
19 |
- - - - |
0* / 19 |
* 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 |
150 |
-2 |
2 |
-2 |
0 |
0 |
4 |
0 |
| quadratic |
11.325 |
3 |
77 |
3 |
75 |
1 |
83 |
1 |
| cubic |
573.800 |
-4 |
152 |
-4 |
0 |
0 |
312 |
0 |
| quartic |
21.947.850 |
42 |
3.002 |
42 |
2.850 |
38 |
3.466 |
38 |
| quintic |
675.993.780 |
-80 |
5.852 |
-80 |
0 |
0 |
12.320 |
0 |
| sextic |
17.463.172.650 |
118 |
79.002 |
118 |
73.150 |
38 |
97.174 |
38 |
Decomposition to irreducible representations
Column with number of nonvanshing force constants highlighted
| Force field |
Ag |
Bg |
Eg*
|
Au |
Bu |
Eu*
|
| linear |
19 |
20 |
18 |
18 |
19 |
19 |
| quadratic |
1.446 |
1.444 |
1.405 |
1.406 |
1.405 |
1.407 |
| cubic |
71.782 |
71.784 |
71.667 |
71.704 |
71.706 |
71.745 |
| quartic |
2.744.666 |
2.744.626 |
2.743.029 |
2.743.068 |
2.743.066 |
2.743.183 |
| quintic |
84.501.474 |
84.501.514 |
84.496.951 |
84.498.394 |
84.498.434 |
84.500.031 |
| sextic |
2.182.927.786 |
2.182.927.708 |
2.182.883.703 |
2.182.885.186 |
2.182.885.146 |
2.182.889.709 |
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) |
|---|
| ..190. |
AgAg. | ..210. |
BgBg. | ..324. |
EgEg. | ..171. |
AuAu. | ..190. |
BuBu. | ..361. |
EuEu. | | |
| |
| |
| |
| Subtotal: 1.446 / 6 / 6 |
| Irrep combinations (i,j) with indices: pos(Ag) ≤ i ≤ j ≤ pos(Eu) |
|---|
| Subtotal: 0 / 0 / 15 |
| Total: 1.446 / 6 / 21 |
Contributions to nonvanishing cubic force field constants
| Irrep combinations (i,i,i) with indices: pos(Ag) ≤ i ≤ pos(Eu) |
|---|
| ..1.330. |
AgAgAg. | | |
| |
| |
| |
| |
| |
| |
| |
| |
| Subtotal: 1.330 / 1 / 6 |
| Irrep combinations (i,i,j) (i,j,j) with indices: pos(Ag) ≤ i ≤ j ≤ pos(Eu) |
|---|
| ..3.990. |
AgBgBg. | ..6.156. |
AgEgEg. | ..3.249. |
AgAuAu. | ..3.610. |
AgBuBu. | ..6.859. |
AgEuEu. | ..6.840. |
BgEgEg. | ..7.600. |
BgEuEu. | | |
| |
| |
| Subtotal: 38.304 / 7 / 30 |
| Irrep combinations (i,j,k) with indices: pos(Ag) ≤ i ≤ j ≤ k ≤ pos(Eu) |
|---|
| ..6.840. |
BgAuBu. | ..12.312. |
EgAuEu. | ..12.996. |
EgBuEu. | | |
| |
| |
| |
| |
| |
| |
| Subtotal: 32.148 / 3 / 20 |
| Total: 71.782 / 11 / 56 |
Contributions to nonvanishing quartic force field constants
| Irrep combinations (i,i,i,i) with indices: pos(Ag) ≤ i ≤ pos(Eu) |
|---|
| ..7.315. |
AgAgAgAg. | ..8.855. |
BgBgBgBg. | ..41.211. |
EgEgEgEg. | ..5.985. |
AuAuAuAu. | ..7.315. |
BuBuBuBu. | ..50.730. |
EuEuEuEu. | | |
| |
| |
| |
| Subtotal: 121.411 / 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) |
|---|
| ..39.900. |
AgAgBgBg. | ..61.560. |
AgAgEgEg. | ..32.490. |
AgAgAuAu. | ..36.100. |
AgAgBuBu. | ..68.590. |
AgAgEuEu. | ..68.040. |
BgBgEgEg. | ..35.910. |
BgBgAuAu. | ..39.900. |
BgBgBuBu. | ..75.810. |
BgBgEuEu. | ..55.404. |
EgEgAuAu. |
| ..61.560. |
EgEgBuBu. | ..246.924. |
EgEgEuEu. | ..32.490. |
AuAuBuBu. | ..61.731. |
AuAuEuEu. | ..68.590. |
BuBuEuEu. | | |
| |
| |
| |
| |
| Subtotal: 984.999 / 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) |
|---|
| ..116.964. |
EgEgAuBu. | ..129.960. |
AgBgEgEg. | ..144.400. |
AgBgEuEu. | ..129.960. |
AuBuEuEu. | | |
| |
| |
| |
| |
| |
| Subtotal: 521.284 / 4 / 60 |
| Irrep combinations (i,j,k,l) with indices: pos(Ag) ≤ i ≤ j ≤ k ≤ l ≤ pos(Eu) |
|---|
| ..129.960. |
AgBgAuBu. | ..233.928. |
AgEgAuEu. | ..246.924. |
AgEgBuEu. | ..246.240. |
BgEgAuEu. | ..259.920. |
BgEgBuEu. | | |
| |
| |
| |
| |
| Subtotal: 1.116.972 / 5 / 15 |
| Total: 2.744.666 / 30 / 126 |
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Last update November, 13th 2023 by A. Gelessus, Impressum, Datenschutzerklärung/DataPrivacyStatement