Characters of representations for molecular motions
Motion |
E |
2C28 |
2C14 |
2(C28)3 |
2C7 |
2(C28)5 |
2(C14)3 |
2C4 |
2(C7)2 |
2(C28)9 |
2(C14)5 |
2(C28)11 |
2(C7)3 |
2(C28)13 |
C2 |
14C'2 |
14C''2 |
Cartesian 3N |
0 |
0.000 |
0.000 |
0.000 |
0.000 |
0.000 |
0.000 |
0 |
0.000 |
0.000 |
0.000 |
0.000 |
0.000 |
0.000 |
0 |
0 |
0 |
Translation (x,y,z) |
3 |
2.950 |
2.802 |
2.564 |
2.247 |
1.868 |
1.445 |
1 |
0.555 |
0.132 |
-0.247 |
-0.564 |
-0.802 |
-0.950 |
-1 |
-1 |
-1 |
Rotation (Rx,Ry,Rz) |
3 |
2.950 |
2.802 |
2.564 |
2.247 |
1.868 |
1.445 |
1 |
0.555 |
0.132 |
-0.247 |
-0.564 |
-0.802 |
-0.950 |
-1 |
-1 |
-1 |
Vibration |
-6 |
-5.900 |
-5.604 |
-5.127 |
-4.494 |
-3.736 |
-2.890 |
-2 |
-1.110 |
-0.264 |
0.494 |
1.127 |
1.604 |
1.900 |
2 |
2 |
2 |
Decomposition to irreducible representations
Motion |
A1 |
A2 |
B1 |
B2 |
E1 |
E2 |
E3 |
E4 |
E5 |
E6 |
E7 |
E8 |
E9 |
E10 |
E11 |
E12 |
E13 |
Total |
Cartesian 3N |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
Translation (x,y,z) |
0 |
1 |
0 |
0 |
1 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
2 |
Rotation (Rx,Ry,Rz) |
0 |
1 |
0 |
0 |
1 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
2 |
Vibration |
0 |
-2 |
0 |
0 |
-2 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
-4 |
Molecular parameter
Number of Atoms (N) |
0
|
Number of internal coordinates |
-6
|
Number of independant internal coordinates |
0
|
Number of vibrational modes |
-4
|
Force field analysis
Allowed / forbidden vibronational transitions
Operator |
A1 |
A2 |
B1 |
B2 |
E1 |
E2 |
E3 |
E4 |
E5 |
E6 |
E7 |
E8 |
E9 |
E10 |
E11 |
E12 |
E13 |
Total |
Linear (IR) |
0 |
-2 |
0 |
0 |
-2 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
-4 / 0 |
Quadratic (Raman) |
0 |
-2 |
0 |
0 |
-2 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
-2 / -2 |
IR + Raman |
- - - - |
- - - - |
0 |
0 |
-2 |
- - - - |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
-2 / 0 |
Characters of force fields
(Symmetric powers of vibration representation)
Force field |
E |
2C28 |
2C14 |
2(C28)3 |
2C7 |
2(C28)5 |
2(C14)3 |
2C4 |
2(C7)2 |
2(C28)9 |
2(C14)5 |
2(C28)11 |
2(C7)3 |
2(C28)13 |
C2 |
14C'2 |
14C''2 |
linear |
-6 |
-5.900 |
-5.604 |
-5.127 |
-4.494 |
-3.736 |
-2.890 |
-2 |
-1.110 |
-0.264 |
0.494 |
1.127 |
1.604 |
1.900 |
2 |
2 |
2 |
quadratic |
15 |
14.601 |
13.455 |
11.700 |
9.543 |
7.224 |
4.978 |
3 |
1.418 |
0.282 |
-0.433 |
-0.810 |
-0.961 |
-0.997 |
-1 |
-1 |
-1 |
cubic |
-20 |
-19.403 |
-17.702 |
-15.145 |
-12.098 |
-8.977 |
-6.176 |
-4 |
-2.616 |
-2.035 |
-2.122 |
-2.635 |
-3.286 |
-3.804 |
-4 |
-4 |
-4 |
quartic |
15 |
14.601 |
13.455 |
11.700 |
9.543 |
7.224 |
4.978 |
3 |
1.418 |
0.282 |
-0.433 |
-0.810 |
-0.961 |
-0.997 |
-1 |
-1 |
-1 |
quintic |
-6 |
-5.900 |
-5.604 |
-5.127 |
-4.494 |
-3.736 |
-2.890 |
-2 |
-1.110 |
-0.264 |
0.494 |
1.127 |
1.604 |
1.900 |
2 |
2 |
2 |
sextic |
1 |
1.000 |
1.000 |
1.000 |
1.000 |
1.000 |
1.000 |
1 |
1.000 |
1.000 |
1.000 |
1.000 |
1.000 |
1.000 |
1 |
1 |
1 |
Decomposition to irreducible representations
Column with number of nonvanshing force constants highlighted
Force field |
A1 |
A2 |
B1 |
B2 |
E1 |
E2 |
E3 |
E4 |
E5 |
E6 |
E7 |
E8 |
E9 |
E10 |
E11 |
E12 |
E13 |
linear |
0 |
-2 |
0 |
0 |
-2 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
quadratic |
2 |
3 |
0 |
0 |
4 |
1 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
cubic |
-6 |
-2 |
0 |
0 |
-4 |
-2 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
quartic |
2 |
3 |
0 |
0 |
4 |
1 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
quintic |
0 |
-2 |
0 |
0 |
-2 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
sextic |
1 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
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
28
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(E13) |
..1. |
A2A2. | ..1. |
E1E1. | | |
| |
| |
| |
| |
| |
| |
| |
Subtotal: 2 / 2 / 17 |
Irrep combinations (i,j) with indices: pos(A1) ≤ i ≤ j ≤ pos(E13) |
Subtotal: 0 / 0 / 136 |
Total: 2 / 2 / 153 |
Contributions to nonvanishing cubic force field constants
Irrep combinations (i,i,i) with indices: pos(A1) ≤ i ≤ pos(E13) |
Subtotal: 0 / 0 / 17 |
Irrep combinations (i,i,j) (i,j,j) with indices: pos(A1) ≤ i ≤ j ≤ pos(E13) |
Subtotal: -6 / 0 / 272 |
Irrep combinations (i,j,k) with indices: pos(A1) ≤ i ≤ j ≤ k ≤ pos(E13) |
Subtotal: 0 / 0 / 680 |
Total: -6 / 0 / 969 |
Contributions to nonvanishing quartic force field constants
Irrep combinations (i,i,i,i) with indices: pos(A1) ≤ i ≤ pos(E13) |
..1. |
E1E1E1E1. | | |
| |
| |
| |
| |
| |
| |
| |
| |
Subtotal: 1 / 1 / 17 |
Irrep combinations (i,i,i,j) (i,j,j,j) with indices: pos(A1) ≤ i ≤ j ≤ pos(E13) |
Subtotal: 0 / 0 / 272 |
Irrep combinations (i,i,j,j) with indices: pos(A1) ≤ i ≤ j ≤ pos(E13) |
..1. |
A2A2E1E1. | | |
| |
| |
| |
| |
| |
| |
| |
| |
Subtotal: 1 / 1 / 136 |
Irrep combinations (i,i,j,k) (i,j,j,k) (i,j,k,k) with indices: pos(A1) ≤ i ≤ j ≤ k ≤ pos(E13) |
Subtotal: 0 / 0 / 2.040 |
Irrep combinations (i,j,k,l) with indices: pos(A1) ≤ i ≤ j ≤ k ≤ l ≤ pos(E13) |
Subtotal: 0 / 0 / 2.380 |
Total: 2 / 2 / 4.845 |
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Last update November, 13th 2023 by A. Gelessus, Impressum, Datenschutzerklärung/DataPrivacyStatement