Characters of representations for molecular motions
Motion |
E |
C3 |
(C3)2 |
Cartesian 3N |
36 |
0 |
0 |
Translation (x,y,z) |
3 |
0 |
0 |
Rotation (Rx,Ry,Rz) |
3 |
0 |
0 |
Vibration |
30 |
0 |
0 |
Decomposition to irreducible representations
Motion |
A |
E*
|
Total |
Cartesian 3N |
12 |
12 |
24 |
Translation (x,y,z) |
1 |
1 |
2 |
Rotation (Rx,Ry,Rz) |
1 |
1 |
2 |
Vibration |
10 |
10 |
20 |
Molecular parameter
Number of Atoms (N) |
12
|
Number of internal coordinates |
30
|
Number of independant internal coordinates |
10
|
Number of vibrational modes |
20
|
Force field analysis
Allowed / forbidden vibronational transitions
Operator |
A |
E*
|
Total |
Linear (IR) |
10 |
10 |
20 / 0 |
Quadratic (Raman) |
10 |
10 |
20 / 0 |
IR + Raman |
10 |
10 |
20 / 0 |
Characters of force fields
(Symmetric powers of vibration representation)
Force field |
E |
C3 |
(C3)2 |
linear |
30 |
0 |
0 |
quadratic |
465 |
0 |
0 |
cubic |
4.960 |
10 |
10 |
quartic |
40.920 |
0 |
0 |
quintic |
278.256 |
0 |
0 |
sextic |
1.623.160 |
55 |
55 |
Decomposition to irreducible representations
Column with number of nonvanshing force constants highlighted
Force field |
A |
E*
|
linear |
10 |
10 |
quadratic |
155 |
155 |
cubic |
1.660 |
1.650 |
quartic |
13.640 |
13.640 |
quintic |
92.752 |
92.752 |
sextic |
541.090 |
541.035 |
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
3
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(E) |
..55. |
AA. | ..100. |
EE. | | |
| |
| |
| |
| |
| |
| |
| |
Subtotal: 155 / 2 / 2 |
Irrep combinations (i,j) with indices: pos(A) ≤ i ≤ j ≤ pos(E) |
Subtotal: 0 / 0 / 1 |
Total: 155 / 2 / 3 |
Contributions to nonvanishing cubic force field constants
Irrep combinations (i,i,i) with indices: pos(A) ≤ i ≤ pos(E) |
..220. |
AAA. | ..440. |
EEE. | | |
| |
| |
| |
| |
| |
| |
| |
Subtotal: 660 / 2 / 2 |
Irrep combinations (i,i,j) (i,j,j) with indices: pos(A) ≤ i ≤ j ≤ pos(E) |
..1.000. |
AEE. | | |
| |
| |
| |
| |
| |
| |
| |
| |
Subtotal: 1.000 / 1 / 2 |
Irrep combinations (i,j,k) with indices: pos(A) ≤ i ≤ j ≤ k ≤ pos(E) |
Subtotal: 0 / 0 / 0 |
Total: 1.660 / 3 / 4 |
Contributions to nonvanishing quartic force field constants
Irrep combinations (i,i,i,i) with indices: pos(A) ≤ i ≤ pos(E) |
..715. |
AAAA. | ..3.025. |
EEEE. | | |
| |
| |
| |
| |
| |
| |
| |
Subtotal: 3.740 / 2 / 2 |
Irrep combinations (i,i,i,j) (i,j,j,j) with indices: pos(A) ≤ i ≤ j ≤ pos(E) |
..4.400. |
AEEE. | | |
| |
| |
| |
| |
| |
| |
| |
| |
Subtotal: 4.400 / 1 / 2 |
Irrep combinations (i,i,j,j) with indices: pos(A) ≤ i ≤ j ≤ pos(E) |
..5.500. |
AAEE. | | |
| |
| |
| |
| |
| |
| |
| |
| |
Subtotal: 5.500 / 1 / 1 |
Irrep combinations (i,i,j,k) (i,j,j,k) (i,j,k,k) with indices: pos(A) ≤ i ≤ j ≤ k ≤ pos(E) |
Subtotal: 0 / 0 / 0 |
Irrep combinations (i,j,k,l) with indices: pos(A) ≤ i ≤ j ≤ k ≤ l ≤ pos(E) |
Subtotal: 0 / 0 / 0 |
Total: 13.640 / 4 / 5 |
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Last update November, 13th 2023 by A. Gelessus, Impressum, Datenschutzerklärung/DataPrivacyStatement