Characters of representations for molecular motions
Motion |
E |
2S4 |
C2 |
2C'2 |
2σd |
Cartesian 3N |
21 |
-1 |
-3 |
-1 |
5 |
Translation (x,y,z) |
3 |
-1 |
-1 |
-1 |
1 |
Rotation (Rx,Ry,Rz) |
3 |
1 |
-1 |
-1 |
-1 |
Vibration |
15 |
-1 |
-1 |
1 |
5 |
Decomposition to irreducible representations
Motion |
A1 |
A2 |
B1 |
B2 |
E |
Total |
Cartesian 3N |
3 |
1 |
1 |
4 |
6 |
15 |
Translation (x,y,z) |
0 |
0 |
0 |
1 |
1 |
2 |
Rotation (Rx,Ry,Rz) |
0 |
1 |
0 |
0 |
1 |
2 |
Vibration |
3 |
0 |
1 |
3 |
4 |
11 |
Molecular parameter
Number of Atoms (N) |
7
|
Number of internal coordinates |
15
|
Number of independant internal coordinates |
3
|
Number of vibrational modes |
11
|
Force field analysis
Allowed / forbidden vibronational transitions
Operator |
A1 |
A2 |
B1 |
B2 |
E |
Total |
Linear (IR) |
3 |
0 |
1 |
3 |
4 |
7 / 4 |
Quadratic (Raman) |
3 |
0 |
1 |
3 |
4 |
11 / 0 |
IR + Raman |
- - - - |
0 |
- - - - |
3 |
4 |
7 / 0 |
Characters of force fields
(Symmetric powers of vibration representation)
Force field |
E |
2S4 |
C2 |
2C'2 |
2σd |
linear |
15 |
-1 |
-1 |
1 |
5 |
quadratic |
120 |
0 |
8 |
8 |
20 |
cubic |
680 |
0 |
-8 |
8 |
60 |
quartic |
3.060 |
4 |
36 |
36 |
160 |
quintic |
11.628 |
-4 |
-36 |
36 |
376 |
sextic |
38.760 |
0 |
120 |
120 |
820 |
Decomposition to irreducible representations
Column with number of nonvanshing force constants highlighted
Force field |
A1 |
A2 |
B1 |
B2 |
E |
linear |
3 |
0 |
1 |
3 |
4 |
quadratic |
23 |
9 |
13 |
19 |
28 |
cubic |
101 |
67 |
71 |
97 |
172 |
quartic |
437 |
339 |
355 |
417 |
756 |
quintic |
1.551 |
1.345 |
1.365 |
1.535 |
2.916 |
sextic |
5.095 |
4.625 |
4.685 |
5.035 |
9.660 |
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 D
2d
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. | ..1. |
B1B1. | ..6. |
B2B2. | ..10. |
EE. | | |
| |
| |
| |
| |
| |
Subtotal: 23 / 4 / 5 |
Irrep combinations (i,j) with indices: pos(A1) ≤ i ≤ j ≤ pos(E) |
Subtotal: 0 / 0 / 10 |
Total: 23 / 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) |
..3. |
A1B1B1. | ..18. |
A1B2B2. | ..30. |
A1EE. | ..10. |
B1EE. | ..30. |
B2EE. | | |
| |
| |
| |
| |
Subtotal: 91 / 5 / 20 |
Irrep combinations (i,j,k) with indices: pos(A1) ≤ i ≤ j ≤ k ≤ pos(E) |
Subtotal: 0 / 0 / 10 |
Total: 101 / 6 / 35 |
Contributions to nonvanishing quartic force field constants
Irrep combinations (i,i,i,i) with indices: pos(A1) ≤ i ≤ pos(E) |
..15. |
A1A1A1A1. | ..1. |
B1B1B1B1. | ..15. |
B2B2B2B2. | ..90. |
EEEE. | | |
| |
| |
| |
| |
| |
Subtotal: 121 / 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) |
..6. |
A1A1B1B1. | ..36. |
A1A1B2B2. | ..60. |
A1A1EE. | ..6. |
B1B1B2B2. | ..10. |
B1B1EE. | ..60. |
B2B2EE. | | |
| |
| |
| |
Subtotal: 178 / 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) |
..30. |
A1B1EE. | ..90. |
A1B2EE. | ..18. |
B1B2EE. | | |
| |
| |
| |
| |
| |
| |
Subtotal: 138 / 3 / 30 |
Irrep combinations (i,j,k,l) with indices: pos(A1) ≤ i ≤ j ≤ k ≤ l ≤ pos(E) |
Subtotal: 0 / 0 / 5 |
Total: 437 / 13 / 70 |
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Last update November, 13th 2023 by A. Gelessus, Impressum, Datenschutzerklärung/DataPrivacyStatement