Characters of representations for molecular motions
Motion |
E |
4C3 |
4(C3)2 |
3C2 |
Cartesian 3N |
36 |
0 |
0 |
-4 |
Translation (x,y,z) |
3 |
0 |
0 |
-1 |
Rotation (Rx,Ry,Rz) |
3 |
0 |
0 |
-1 |
Vibration |
30 |
0 |
0 |
-2 |
Decomposition to irreducible representations
Motion |
A |
E*
|
T |
Total |
Cartesian 3N |
2 |
2 |
10 |
14 |
Translation (x,y,z) |
0 |
0 |
1 |
1 |
Rotation (Rx,Ry,Rz) |
0 |
0 |
1 |
1 |
Vibration |
2 |
2 |
8 |
12 |
Molecular parameter
Number of Atoms (N) |
12
|
Number of internal coordinates |
30
|
Number of independant internal coordinates |
2
|
Number of vibrational modes |
12
|
Force field analysis
Allowed / forbidden vibronational transitions
Operator |
A |
E*
|
T |
Total |
Linear (IR) |
2 |
2 |
8 |
8 / 4 |
Quadratic (Raman) |
2 |
2 |
8 |
12 / 0 |
IR + Raman |
- - - - |
- - - - |
8 |
8 / 0 |
Characters of force fields
(Symmetric powers of vibration representation)
Force field |
E |
4C3 |
4(C3)2 |
3C2 |
linear |
30 |
0 |
0 |
-2 |
quadratic |
465 |
0 |
0 |
17 |
cubic |
4.960 |
10 |
10 |
-32 |
quartic |
40.920 |
0 |
0 |
152 |
quintic |
278.256 |
0 |
0 |
-272 |
sextic |
1.623.160 |
55 |
55 |
952 |
Decomposition to irreducible representations
Column with number of nonvanshing force constants highlighted
Force field |
A |
E*
|
T |
linear |
2 |
2 |
8 |
quadratic |
43 |
43 |
112 |
cubic |
412 |
402 |
1.248 |
quartic |
3.448 |
3.448 |
10.192 |
quintic |
23.120 |
23.120 |
69.632 |
sextic |
135.538 |
135.483 |
405.552 |
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) |
..3. |
AA. | ..4. |
EE. | ..36. |
TT. | | |
| |
| |
| |
| |
| |
| |
Subtotal: 43 / 3 / 3 |
Irrep combinations (i,j) with indices: pos(A) ≤ i ≤ j ≤ pos(T) |
Subtotal: 0 / 0 / 3 |
Total: 43 / 3 / 6 |
Contributions to nonvanishing cubic force field constants
Irrep combinations (i,i,i) with indices: pos(A) ≤ i ≤ pos(T) |
..4. |
AAA. | ..8. |
EEE. | ..176. |
TTT. | | |
| |
| |
| |
| |
| |
| |
Subtotal: 188 / 3 / 3 |
Irrep combinations (i,i,j) (i,j,j) with indices: pos(A) ≤ i ≤ j ≤ pos(T) |
..8. |
AEE. | ..72. |
ATT. | ..144. |
ETT. | | |
| |
| |
| |
| |
| |
| |
Subtotal: 224 / 3 / 6 |
Irrep combinations (i,j,k) with indices: pos(A) ≤ i ≤ j ≤ k ≤ pos(T) |
Subtotal: 0 / 0 / 1 |
Total: 412 / 6 / 10 |
Contributions to nonvanishing quartic force field constants
Irrep combinations (i,i,i,i) with indices: pos(A) ≤ i ≤ pos(T) |
..5. |
AAAA. | ..9. |
EEEE. | ..1.626. |
TTTT. | | |
| |
| |
| |
| |
| |
| |
Subtotal: 1.640 / 3 / 3 |
Irrep combinations (i,i,i,j) (i,j,j,j) with indices: pos(A) ≤ i ≤ j ≤ pos(T) |
..16. |
AEEE. | ..352. |
ATTT. | ..672. |
ETTT. | | |
| |
| |
| |
| |
| |
| |
Subtotal: 1.040 / 3 / 6 |
Irrep combinations (i,i,j,j) with indices: pos(A) ≤ i ≤ j ≤ pos(T) |
..12. |
AAEE. | ..108. |
AATT. | ..360. |
EETT. | | |
| |
| |
| |
| |
| |
| |
Subtotal: 480 / 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) |
..288. |
AETT. | | |
| |
| |
| |
| |
| |
| |
| |
| |
Subtotal: 288 / 1 / 3 |
Irrep combinations (i,j,k,l) with indices: pos(A) ≤ i ≤ j ≤ k ≤ l ≤ pos(T) |
Subtotal: 0 / 0 / 0 |
Total: 3.448 / 10 / 15 |
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Last update November, 13th 2023 by A. Gelessus, Impressum, Datenschutzerklärung/DataPrivacyStatement