Characters of representations for molecular motions
| Motion |
E |
8C3 |
6C2 |
6C4 |
3C2 |
| Cartesian 3N |
156 |
0 |
-2 |
0 |
0 |
| Translation (x,y,z) |
3 |
0 |
-1 |
1 |
-1 |
| Rotation (Rx,Ry,Rz) |
3 |
0 |
-1 |
1 |
-1 |
| Vibration |
150 |
0 |
0 |
-2 |
2 |
Decomposition to irreducible representations
| Motion |
A1 |
A2 |
E |
T1 |
T2 |
Total |
| Cartesian 3N |
6 |
7 |
13 |
20 |
19 |
65 |
| Translation (x,y,z) |
0 |
0 |
0 |
1 |
0 |
1 |
| Rotation (Rx,Ry,Rz) |
0 |
0 |
0 |
1 |
0 |
1 |
| Vibration |
6 |
7 |
13 |
18 |
19 |
63 |
Molecular parameter
| Number of Atoms (N) |
52
|
| Number of internal coordinates |
150
|
| Number of independant internal coordinates |
6
|
| Number of vibrational modes |
63
|
Force field analysis
Allowed / forbidden vibronational transitions
| Operator |
A1 |
A2 |
E |
T1 |
T2 |
Total |
| Linear (IR) |
6 |
7 |
13 |
18 |
19 |
18 / 45 |
| Quadratic (Raman) |
6 |
7 |
13 |
18 |
19 |
38 / 25 |
| IR + Raman |
- - - - |
7 |
- - - - |
- - - - |
- - - - |
0 / 7 |
Characters of force fields
(Symmetric powers of vibration representation)
| Force field |
E |
8C3 |
6C2 |
6C4 |
3C2 |
| linear |
150 |
0 |
0 |
-2 |
2 |
| quadratic |
11.325 |
0 |
75 |
3 |
77 |
| cubic |
573.800 |
50 |
0 |
-4 |
152 |
| quartic |
21.947.850 |
0 |
2.850 |
42 |
3.002 |
| quintic |
675.993.780 |
0 |
0 |
-80 |
5.852 |
| sextic |
17.463.172.650 |
1.275 |
73.150 |
118 |
79.002 |
Decomposition to irreducible representations
Column with number of nonvanshing force constants highlighted
| Force field |
A1 |
A2 |
E |
T1 |
T2 |
| linear |
6 |
7 |
13 |
18 |
19 |
| quadratic |
501 |
462 |
963 |
1.388 |
1.424 |
| cubic |
23.943 |
23.945 |
47.838 |
71.705 |
71.707 |
| quartic |
915.592 |
914.146 |
1.829.738 |
2.742.404 |
2.743.808 |
| quintic |
28.167.119 |
28.167.159 |
56.334.278 |
84.498.471 |
84.498.511 |
| sextic |
727.660.811 |
727.624.177 |
1.455.283.713 |
2.182.868.448 |
2.182.904.964 |
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 O
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(A1) ≤ i ≤ pos(T2) |
|---|
| ..21. |
A1A1. | ..28. |
A2A2. | ..91. |
EE. | ..171. |
T1T1. | ..190. |
T2T2. | | |
| |
| |
| |
| |
| Subtotal: 501 / 5 / 5 |
| Irrep combinations (i,j) with indices: pos(A1) ≤ i ≤ j ≤ pos(T2) |
|---|
| Subtotal: 0 / 0 / 10 |
| Total: 501 / 5 / 15 |
Contributions to nonvanishing cubic force field constants
| Irrep combinations (i,i,i) with indices: pos(A1) ≤ i ≤ pos(T2) |
|---|
| ..56. |
A1A1A1. | ..455. |
EEE. | ..816. |
T1T1T1. | ..1.330. |
T2T2T2. | | |
| |
| |
| |
| |
| |
| Subtotal: 2.657 / 4 / 5 |
| Irrep combinations (i,i,j) (i,j,j) with indices: pos(A1) ≤ i ≤ j ≤ pos(T2) |
|---|
| ..3.249. |
T1T1T2. | ..168. |
A1A2A2. | ..546. |
A1EE. | ..1.026. |
A1T1T1. | ..1.140. |
A1T2T2. | ..546. |
A2EE. | ..2.223. |
ET1T1. | ..2.470. |
ET2T2. | ..3.078. |
T1T2T2. | | |
| Subtotal: 14.446 / 9 / 20 |
| Irrep combinations (i,j,k) with indices: pos(A1) ≤ i ≤ j ≤ k ≤ pos(T2) |
|---|
| ..2.394. |
A2T1T2. | ..4.446. |
ET1T2. | | |
| |
| |
| |
| |
| |
| |
| |
| Subtotal: 6.840 / 2 / 10 |
| Total: 23.943 / 15 / 35 |
Contributions to nonvanishing quartic force field constants
| Irrep combinations (i,i,i,i) with indices: pos(A1) ≤ i ≤ pos(T2) |
|---|
| ..126. |
A1A1A1A1. | ..210. |
A2A2A2A2. | ..4.186. |
EEEE. | ..20.691. |
T1T1T1T1. | ..25.460. |
T2T2T2T2. | | |
| |
| |
| |
| |
| Subtotal: 50.673 / 5 / 5 |
| Irrep combinations (i,i,i,j) (i,j,j,j) with indices: pos(A1) ≤ i ≤ j ≤ pos(T2) |
|---|
| ..58.482. |
T1T1T1T2. | ..2.730. |
A1EEE. | ..4.896. |
A1T1T1T1. | ..7.980. |
A1T2T2T2. | ..3.185. |
A2EEE. | ..7.980. |
A2T1T1T1. | ..6.783. |
A2T2T2T2. | ..25.194. |
ET1T1T1. | ..29.640. |
ET2T2T2. | ..64.980. |
T1T2T2T2. |
| Subtotal: 211.850 / 10 / 20 |
| Irrep combinations (i,i,j,j) with indices: pos(A1) ≤ i ≤ j ≤ pos(T2) |
|---|
| ..588. |
A1A1A2A2. | ..1.911. |
A1A1EE. | ..3.591. |
A1A1T1T1. | ..3.990. |
A1A1T2T2. | ..2.548. |
A2A2EE. | ..4.788. |
A2A2T1T1. | ..5.320. |
A2A2T2T2. | ..31.122. |
EET1T1. | ..34.580. |
EET2T2. | ..123.633. |
T1T1T2T2. |
| Subtotal: 212.071 / 10 / 10 |
| Irrep combinations (i,i,j,k) (i,j,j,k) (i,j,k,k) with indices: pos(A1) ≤ i ≤ j ≤ k ≤ pos(T2) |
|---|
| ..57.798. |
EET1T2. | ..19.494. |
A1T1T1T2. | ..20.349. |
A2T1T1T2. | ..80.028. |
ET1T1T2. | ..3.276. |
A1A2EE. | ..13.338. |
A1ET1T1. | ..14.820. |
A1ET2T2. | ..18.468. |
A1T1T2T2. | ..15.561. |
A2ET1T1. | ..17.290. |
A2ET2T2. |
| ..23.940. |
A2T1T2T2. | ..84.474. |
ET1T2T2. | | |
| |
| |
| |
| |
| |
| |
| |
| Subtotal: 368.836 / 12 / 30 |
| Irrep combinations (i,j,k,l) with indices: pos(A1) ≤ i ≤ j ≤ k ≤ l ≤ pos(T2) |
|---|
| ..14.364. |
A1A2T1T2. | ..26.676. |
A1ET1T2. | ..31.122. |
A2ET1T2. | | |
| |
| |
| |
| |
| |
| |
| Subtotal: 72.162 / 3 / 5 |
| Total: 915.592 / 40 / 70 |
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Last update November, 13th 2023 by A. Gelessus, Impressum, Datenschutzerklärung/DataPrivacyStatement