Characters of representations for molecular motions
Motion |
E |
2C4 |
C2 |
2σv |
2σd |
Cartesian 3N |
18 |
2 |
-2 |
4 |
2 |
Translation (x,y,z) |
3 |
1 |
-1 |
1 |
1 |
Rotation (Rx,Ry,Rz) |
3 |
1 |
-1 |
-1 |
-1 |
Vibration |
12 |
0 |
0 |
4 |
2 |
Decomposition to irreducible representations
Motion |
A1 |
A2 |
B1 |
B2 |
E |
Total |
Cartesian 3N |
4 |
1 |
2 |
1 |
5 |
13 |
Translation (x,y,z) |
1 |
0 |
0 |
0 |
1 |
2 |
Rotation (Rx,Ry,Rz) |
0 |
1 |
0 |
0 |
1 |
2 |
Vibration |
3 |
0 |
2 |
1 |
3 |
9 |
Molecular parameter
Number of Atoms (N) |
6
|
Number of internal coordinates |
12
|
Number of independant internal coordinates |
3
|
Number of vibrational modes |
9
|
Force field analysis
Allowed / forbidden vibronational transitions
Operator |
A1 |
A2 |
B1 |
B2 |
E |
Total |
Linear (IR) |
3 |
0 |
2 |
1 |
3 |
6 / 3 |
Quadratic (Raman) |
3 |
0 |
2 |
1 |
3 |
9 / 0 |
IR + Raman |
3 |
0 |
- - - - |
- - - - |
3 |
6 / 0 |
Characters of force fields
(Symmetric powers of vibration representation)
Force field |
E |
2C4 |
C2 |
2σv |
2σd |
linear |
12 |
0 |
0 |
4 |
2 |
quadratic |
78 |
0 |
6 |
14 |
8 |
cubic |
364 |
0 |
0 |
36 |
14 |
quartic |
1.365 |
3 |
21 |
85 |
35 |
quintic |
4.368 |
0 |
0 |
176 |
56 |
sextic |
12.376 |
0 |
56 |
344 |
112 |
Decomposition to irreducible representations
Column with number of nonvanshing force constants highlighted
Force field |
A1 |
A2 |
B1 |
B2 |
E |
linear |
3 |
0 |
2 |
1 |
3 |
quadratic |
16 |
5 |
12 |
9 |
18 |
cubic |
58 |
33 |
51 |
40 |
91 |
quartic |
204 |
144 |
185 |
160 |
336 |
quintic |
604 |
488 |
576 |
516 |
1.092 |
sextic |
1.668 |
1.440 |
1.612 |
1.496 |
3.080 |
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
4v
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(E) |
..6. |
A1A1. | ..3. |
B1B1. | ..1. |
B2B2. | ..6. |
EE. | | |
| |
| |
| |
| |
| |
Subtotal: 16 / 4 / 5 |
Irrep combinations (i,j) with indices: pos(A1) ≤ i ≤ j ≤ pos(E) |
Subtotal: 0 / 0 / 10 |
Total: 16 / 4 / 15 |
Contributions to nonvanishing cubic force field constants
Irrep combinations (i,i,i) with indices: pos(A1) ≤ i ≤ pos(E) |
..10. |
A1A1A1. | | |
| |
| |
| |
| |
| |
| |
| |
| |
Subtotal: 10 / 1 / 5 |
Irrep combinations (i,i,j) (i,j,j) with indices: pos(A1) ≤ i ≤ j ≤ pos(E) |
..9. |
A1B1B1. | ..3. |
A1B2B2. | ..18. |
A1EE. | ..12. |
B1EE. | ..6. |
B2EE. | | |
| |
| |
| |
| |
Subtotal: 48 / 5 / 20 |
Irrep combinations (i,j,k) with indices: pos(A1) ≤ i ≤ j ≤ k ≤ pos(E) |
Subtotal: 0 / 0 / 10 |
Total: 58 / 6 / 35 |
Contributions to nonvanishing quartic force field constants
Irrep combinations (i,i,i,i) with indices: pos(A1) ≤ i ≤ pos(E) |
..15. |
A1A1A1A1. | ..5. |
B1B1B1B1. | ..1. |
B2B2B2B2. | ..36. |
EEEE. | | |
| |
| |
| |
| |
| |
Subtotal: 57 / 4 / 5 |
Irrep combinations (i,i,i,j) (i,j,j,j) with indices: pos(A1) ≤ i ≤ j ≤ pos(E) |
Subtotal: 0 / 0 / 20 |
Irrep combinations (i,i,j,j) with indices: pos(A1) ≤ i ≤ j ≤ pos(E) |
..18. |
A1A1B1B1. | ..6. |
A1A1B2B2. | ..36. |
A1A1EE. | ..3. |
B1B1B2B2. | ..18. |
B1B1EE. | ..6. |
B2B2EE. | | |
| |
| |
| |
Subtotal: 87 / 6 / 10 |
Irrep combinations (i,i,j,k) (i,j,j,k) (i,j,k,k) with indices: pos(A1) ≤ i ≤ j ≤ k ≤ pos(E) |
..36. |
A1B1EE. | ..18. |
A1B2EE. | ..6. |
B1B2EE. | | |
| |
| |
| |
| |
| |
| |
Subtotal: 60 / 3 / 30 |
Irrep combinations (i,j,k,l) with indices: pos(A1) ≤ i ≤ j ≤ k ≤ l ≤ pos(E) |
Subtotal: 0 / 0 / 5 |
Total: 204 / 13 / 70 |
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Last update November, 13th 2023 by A. Gelessus, Impressum, Datenschutzerklärung/DataPrivacyStatement