Characters of representations for molecular motions
Motion |
E |
8C3 |
3C2 |
6S4 |
6σd |
Cartesian 3N |
48 |
0 |
0 |
0 |
8 |
Translation (x,y,z) |
3 |
0 |
-1 |
-1 |
1 |
Rotation (Rx,Ry,Rz) |
3 |
0 |
-1 |
1 |
-1 |
Vibration |
42 |
0 |
2 |
0 |
8 |
Decomposition to irreducible representations
Motion |
A1 |
A2 |
E |
T1 |
T2 |
Total |
Cartesian 3N |
4 |
0 |
4 |
4 |
8 |
20 |
Translation (x,y,z) |
0 |
0 |
0 |
0 |
1 |
1 |
Rotation (Rx,Ry,Rz) |
0 |
0 |
0 |
1 |
0 |
1 |
Vibration |
4 |
0 |
4 |
3 |
7 |
18 |
Molecular parameter
Number of Atoms (N) |
16
|
Number of internal coordinates |
42
|
Number of independant internal coordinates |
4
|
Number of vibrational modes |
18
|
Force field analysis
Allowed / forbidden vibronational transitions
Operator |
A1 |
A2 |
E |
T1 |
T2 |
Total |
Linear (IR) |
4 |
0 |
4 |
3 |
7 |
7 / 11 |
Quadratic (Raman) |
4 |
0 |
4 |
3 |
7 |
15 / 3 |
IR + Raman |
- - - - |
0 |
- - - - |
3 |
7 |
7 / 3 |
Characters of force fields
(Symmetric powers of vibration representation)
Force field |
E |
8C3 |
3C2 |
6S4 |
6σd |
linear |
42 |
0 |
2 |
0 |
8 |
quadratic |
903 |
0 |
23 |
1 |
53 |
cubic |
13.244 |
14 |
44 |
0 |
256 |
quartic |
148.995 |
0 |
275 |
11 |
1.095 |
quintic |
1.370.754 |
0 |
506 |
0 |
4.056 |
sextic |
10.737.573 |
105 |
2.277 |
11 |
13.803 |
Decomposition to irreducible representations
Column with number of nonvanshing force constants highlighted
Force field |
A1 |
A2 |
E |
T1 |
T2 |
linear |
4 |
0 |
4 |
3 |
7 |
quadratic |
54 |
27 |
81 |
97 |
123 |
cubic |
626 |
498 |
1.110 |
1.586 |
1.714 |
quartic |
6.519 |
5.966 |
12.485 |
18.319 |
18.861 |
quintic |
58.192 |
56.164 |
114.356 |
170.267 |
172.295 |
sextic |
451.172 |
444.265 |
895.332 |
1.338.464 |
1.345.360 |
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 T
d
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) |
..10. |
A1A1. | ..10. |
EE. | ..6. |
T1T1. | ..28. |
T2T2. | | |
| |
| |
| |
| |
| |
Subtotal: 54 / 4 / 5 |
Irrep combinations (i,j) with indices: pos(A1) ≤ i ≤ j ≤ pos(T2) |
Subtotal: 0 / 0 / 10 |
Total: 54 / 4 / 15 |
Contributions to nonvanishing cubic force field constants
Irrep combinations (i,i,i) with indices: pos(A1) ≤ i ≤ pos(T2) |
..20. |
A1A1A1. | ..20. |
EEE. | ..1. |
T1T1T1. | ..84. |
T2T2T2. | | |
| |
| |
| |
| |
| |
Subtotal: 125 / 4 / 5 |
Irrep combinations (i,i,j) (i,j,j) with indices: pos(A1) ≤ i ≤ j ≤ pos(T2) |
..42. |
T1T1T2. | ..40. |
A1EE. | ..24. |
A1T1T1. | ..112. |
A1T2T2. | ..24. |
ET1T1. | ..112. |
ET2T2. | ..63. |
T1T2T2. | | |
| |
| |
Subtotal: 417 / 7 / 20 |
Irrep combinations (i,j,k) with indices: pos(A1) ≤ i ≤ j ≤ k ≤ pos(T2) |
..84. |
ET1T2. | | |
| |
| |
| |
| |
| |
| |
| |
| |
Subtotal: 84 / 1 / 10 |
Total: 626 / 12 / 35 |
Contributions to nonvanishing quartic force field constants
Irrep combinations (i,i,i,i) with indices: pos(A1) ≤ i ≤ pos(T2) |
..35. |
A1A1A1A1. | ..55. |
EEEE. | ..36. |
T1T1T1T1. | ..616. |
T2T2T2T2. | | |
| |
| |
| |
| |
| |
Subtotal: 742 / 4 / 5 |
Irrep combinations (i,i,i,j) (i,j,j,j) with indices: pos(A1) ≤ i ≤ j ≤ pos(T2) |
..126. |
T1T1T1T2. | ..80. |
A1EEE. | ..4. |
A1T1T1T1. | ..336. |
A1T2T2T2. | ..32. |
ET1T1T1. | ..448. |
ET2T2T2. | ..588. |
T1T2T2T2. | | |
| |
| |
Subtotal: 1.614 / 7 / 20 |
Irrep combinations (i,i,j,j) with indices: pos(A1) ≤ i ≤ j ≤ pos(T2) |
..100. |
A1A1EE. | ..60. |
A1A1T1T1. | ..280. |
A1A1T2T2. | ..120. |
EET1T1. | ..560. |
EET2T2. | ..567. |
T1T1T2T2. | | |
| |
| |
| |
Subtotal: 1.687 / 6 / 10 |
Irrep combinations (i,i,j,k) (i,j,j,k) (i,j,k,k) with indices: pos(A1) ≤ i ≤ j ≤ k ≤ pos(T2) |
..336. |
EET1T2. | ..168. |
A1T1T1T2. | ..252. |
ET1T1T2. | ..96. |
A1ET1T1. | ..448. |
A1ET2T2. | ..252. |
A1T1T2T2. | ..588. |
ET1T2T2. | | |
| |
| |
Subtotal: 2.140 / 7 / 30 |
Irrep combinations (i,j,k,l) with indices: pos(A1) ≤ i ≤ j ≤ k ≤ l ≤ pos(T2) |
..336. |
A1ET1T2. | | |
| |
| |
| |
| |
| |
| |
| |
| |
Subtotal: 336 / 1 / 5 |
Total: 6.519 / 25 / 70 |
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Last update November, 13th 2023 by A. Gelessus, Impressum, Datenschutzerklärung/DataPrivacyStatement