Characters of representations for molecular motions
Motion |
E |
C2.(z) |
σv.(xz) |
σd.(yz) |
Cartesian 3N |
21 |
-1 |
5 |
3 |
Translation (x,y,z) |
3 |
-1 |
1 |
1 |
Rotation (Rx,Ry,Rz) |
3 |
-1 |
-1 |
-1 |
Vibration |
15 |
1 |
5 |
3 |
Decomposition to irreducible representations
Motion |
A1 |
A2 |
B1 |
B2 |
Total |
Cartesian 3N |
7 |
3 |
6 |
5 |
21 |
Translation (x,y,z) |
1 |
0 |
1 |
1 |
3 |
Rotation (Rx,Ry,Rz) |
0 |
1 |
1 |
1 |
3 |
Vibration |
6 |
2 |
4 |
3 |
15 |
Molecular parameter
Number of Atoms (N) |
7
|
Number of internal coordinates |
15
|
Number of independant internal coordinates |
6
|
Number of vibrational modes |
15
|
Force field analysis
Allowed / forbidden vibronational transitions
Operator |
A1 |
A2 |
B1 |
B2 |
Total |
Linear (IR) |
6 |
2 |
4 |
3 |
13 / 2 |
Quadratic (Raman) |
6 |
2 |
4 |
3 |
15 / 0 |
IR + Raman |
6 |
- - - - |
4 |
3 |
13 / 0 |
Characters of force fields
(Symmetric powers of vibration representation)
Force field |
E |
C2.(z) |
σv.(xz) |
σd.(yz) |
linear |
15 |
1 |
5 |
3 |
quadratic |
120 |
8 |
20 |
12 |
cubic |
680 |
8 |
60 |
28 |
quartic |
3.060 |
36 |
160 |
72 |
quintic |
11.628 |
36 |
376 |
144 |
sextic |
38.760 |
120 |
820 |
300 |
Decomposition to irreducible representations
Column with number of nonvanshing force constants highlighted
Force field |
A1 |
A2 |
B1 |
B2 |
linear |
6 |
2 |
4 |
3 |
quadratic |
40 |
24 |
30 |
26 |
cubic |
194 |
150 |
176 |
160 |
quartic |
832 |
716 |
778 |
734 |
quintic |
3.046 |
2.786 |
2.956 |
2.840 |
sextic |
10.000 |
9.440 |
9.790 |
9.530 |
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
2v
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(B2) |
..21. |
A1A1. | ..3. |
A2A2. | ..10. |
B1B1. | ..6. |
B2B2. | | |
| |
| |
| |
| |
| |
Subtotal: 40 / 4 / 4 |
Irrep combinations (i,j) with indices: pos(A1) ≤ i ≤ j ≤ pos(B2) |
Subtotal: 0 / 0 / 6 |
Total: 40 / 4 / 10 |
Contributions to nonvanishing cubic force field constants
Irrep combinations (i,i,i) with indices: pos(A1) ≤ i ≤ pos(B2) |
..56. |
A1A1A1. | | |
| |
| |
| |
| |
| |
| |
| |
| |
Subtotal: 56 / 1 / 4 |
Irrep combinations (i,i,j) (i,j,j) with indices: pos(A1) ≤ i ≤ j ≤ pos(B2) |
..18. |
A1A2A2. | ..60. |
A1B1B1. | ..36. |
A1B2B2. | | |
| |
| |
| |
| |
| |
| |
Subtotal: 114 / 3 / 12 |
Irrep combinations (i,j,k) with indices: pos(A1) ≤ i ≤ j ≤ k ≤ pos(B2) |
..24. |
A2B1B2. | | |
| |
| |
| |
| |
| |
| |
| |
| |
Subtotal: 24 / 1 / 4 |
Total: 194 / 5 / 20 |
Contributions to nonvanishing quartic force field constants
Irrep combinations (i,i,i,i) with indices: pos(A1) ≤ i ≤ pos(B2) |
..126. |
A1A1A1A1. | ..5. |
A2A2A2A2. | ..35. |
B1B1B1B1. | ..15. |
B2B2B2B2. | | |
| |
| |
| |
| |
| |
Subtotal: 181 / 4 / 4 |
Irrep combinations (i,i,i,j) (i,j,j,j) with indices: pos(A1) ≤ i ≤ j ≤ pos(B2) |
Subtotal: 0 / 0 / 12 |
Irrep combinations (i,i,j,j) with indices: pos(A1) ≤ i ≤ j ≤ pos(B2) |
..63. |
A1A1A2A2. | ..210. |
A1A1B1B1. | ..126. |
A1A1B2B2. | ..30. |
A2A2B1B1. | ..18. |
A2A2B2B2. | ..60. |
B1B1B2B2. | | |
| |
| |
| |
Subtotal: 507 / 6 / 6 |
Irrep combinations (i,i,j,k) (i,j,j,k) (i,j,k,k) with indices: pos(A1) ≤ i ≤ j ≤ k ≤ pos(B2) |
Subtotal: 0 / 0 / 12 |
Irrep combinations (i,j,k,l) with indices: pos(A1) ≤ i ≤ j ≤ k ≤ l ≤ pos(B2) |
..144. |
A1A2B1B2. | | |
| |
| |
| |
| |
| |
| |
| |
| |
Subtotal: 144 / 1 / 1 |
Total: 832 / 11 / 35 |
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Last update November, 13th 2023 by A. Gelessus, Impressum, Datenschutzerklärung/DataPrivacyStatement