Characters of representations for molecular motions
Motion |
E |
4C3 |
4(C3)2 |
3C2 |
Cartesian 3N |
156 |
0 |
0 |
0 |
Translation (x,y,z) |
3 |
0 |
0 |
-1 |
Rotation (Rx,Ry,Rz) |
3 |
0 |
0 |
-1 |
Vibration |
150 |
0 |
0 |
2 |
Decomposition to irreducible representations
Motion |
A |
E*
|
T |
Total |
Cartesian 3N |
13 |
13 |
39 |
65 |
Translation (x,y,z) |
0 |
0 |
1 |
1 |
Rotation (Rx,Ry,Rz) |
0 |
0 |
1 |
1 |
Vibration |
13 |
13 |
37 |
63 |
Molecular parameter
Number of Atoms (N) |
52
|
Number of internal coordinates |
150
|
Number of independant internal coordinates |
13
|
Number of vibrational modes |
63
|
Force field analysis
Allowed / forbidden vibronational transitions
Operator |
A |
E*
|
T |
Total |
Linear (IR) |
13 |
13 |
37 |
37 / 26 |
Quadratic (Raman) |
13 |
13 |
37 |
63 / 0 |
IR + Raman |
- - - - |
- - - - |
37 |
37 / 0 |
Characters of force fields
(Symmetric powers of vibration representation)
Force field |
E |
4C3 |
4(C3)2 |
3C2 |
linear |
150 |
0 |
0 |
2 |
quadratic |
11.325 |
0 |
0 |
77 |
cubic |
573.800 |
50 |
50 |
152 |
quartic |
21.947.850 |
0 |
0 |
3.002 |
quintic |
675.993.780 |
0 |
0 |
5.852 |
sextic |
17.463.172.650 |
1.275 |
1.275 |
79.002 |
Decomposition to irreducible representations
Column with number of nonvanshing force constants highlighted
Force field |
A |
E*
|
T |
linear |
13 |
13 |
37 |
quadratic |
963 |
963 |
2.812 |
cubic |
47.888 |
47.838 |
143.412 |
quartic |
1.829.738 |
1.829.738 |
5.486.212 |
quintic |
56.334.278 |
56.334.278 |
168.996.982 |
sextic |
1.455.284.988 |
1.455.283.713 |
4.365.773.412 |
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
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(A) ≤ i ≤ pos(T) |
..91. |
AA. | ..169. |
EE. | ..703. |
TT. | | |
| |
| |
| |
| |
| |
| |
Subtotal: 963 / 3 / 3 |
Irrep combinations (i,j) with indices: pos(A) ≤ i ≤ j ≤ pos(T) |
Subtotal: 0 / 0 / 3 |
Total: 963 / 3 / 6 |
Contributions to nonvanishing cubic force field constants
Irrep combinations (i,i,i) with indices: pos(A) ≤ i ≤ pos(T) |
..455. |
AAA. | ..910. |
EEE. | ..16.909. |
TTT. | | |
| |
| |
| |
| |
| |
| |
Subtotal: 18.274 / 3 / 3 |
Irrep combinations (i,i,j) (i,j,j) with indices: pos(A) ≤ i ≤ j ≤ pos(T) |
..2.197. |
AEE. | ..9.139. |
ATT. | ..18.278. |
ETT. | | |
| |
| |
| |
| |
| |
| |
Subtotal: 29.614 / 3 / 6 |
Irrep combinations (i,j,k) with indices: pos(A) ≤ i ≤ j ≤ k ≤ pos(T) |
Subtotal: 0 / 0 / 1 |
Total: 47.888 / 6 / 10 |
Contributions to nonvanishing quartic force field constants
Irrep combinations (i,i,i,i) with indices: pos(A) ≤ i ≤ pos(T) |
..1.820. |
AAAA. | ..8.281. |
EEEE. | ..585.599. |
TTTT. | | |
| |
| |
| |
| |
| |
| |
Subtotal: 595.700 / 3 / 3 |
Irrep combinations (i,i,i,j) (i,j,j,j) with indices: pos(A) ≤ i ≤ j ≤ pos(T) |
..11.830. |
AEEE. | ..219.817. |
ATTT. | ..438.672. |
ETTT. | | |
| |
| |
| |
| |
| |
| |
Subtotal: 670.319 / 3 / 6 |
Irrep combinations (i,i,j,j) with indices: pos(A) ≤ i ≤ j ≤ pos(T) |
..15.379. |
AAEE. | ..63.973. |
AATT. | ..246.753. |
EETT. | | |
| |
| |
| |
| |
| |
| |
Subtotal: 326.105 / 3 / 3 |
Irrep combinations (i,i,j,k) (i,j,j,k) (i,j,k,k) with indices: pos(A) ≤ i ≤ j ≤ k ≤ pos(T) |
..237.614. |
AETT. | | |
| |
| |
| |
| |
| |
| |
| |
| |
Subtotal: 237.614 / 1 / 3 |
Irrep combinations (i,j,k,l) with indices: pos(A) ≤ i ≤ j ≤ k ≤ l ≤ pos(T) |
Subtotal: 0 / 0 / 0 |
Total: 1.829.738 / 10 / 15 |
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Last update November, 13th 2023 by A. Gelessus, Impressum, Datenschutzerklärung/DataPrivacyStatement