Characters of representations for molecular motions
| Motion |
E |
2C29 |
2(C29)2 |
2(C29)3 |
2(C29)4 |
2(C29)5 |
2(C29)6 |
2(C29)7 |
2(C29)8 |
2(C29)9 |
2(C29)10 |
2(C29)11 |
2(C29)12 |
2(C29)13 |
2(C29)14 |
29σv |
| Cartesian 3N |
0 |
0.000 |
0.000 |
0.000 |
0.000 |
0.000 |
0.000 |
0.000 |
0.000 |
0.000 |
0.000 |
0.000 |
0.000 |
0.000 |
0.000 |
0 |
| Translation (x,y,z) |
3 |
2.953 |
2.815 |
2.592 |
2.295 |
1.937 |
1.535 |
1.108 |
0.676 |
0.260 |
-0.122 |
-0.452 |
-0.714 |
-0.895 |
-0.988 |
1 |
| Rotation (Rx,Ry,Rz) |
3 |
2.953 |
2.815 |
2.592 |
2.295 |
1.937 |
1.535 |
1.108 |
0.676 |
0.260 |
-0.122 |
-0.452 |
-0.714 |
-0.895 |
-0.988 |
-1 |
| Vibration |
-6 |
-5.906 |
-5.630 |
-5.184 |
-4.590 |
-3.874 |
-3.070 |
-2.217 |
-1.353 |
-0.519 |
0.245 |
0.904 |
1.427 |
1.791 |
1.977 |
0 |
Decomposition to irreducible representations
| Motion |
A1 |
A2 |
E1 |
E2 |
E3 |
E4 |
E5 |
E6 |
E7 |
E8 |
E9 |
E10 |
E11 |
E12 |
E13 |
E14 |
Total |
| Cartesian 3N |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
| Translation (x,y,z) |
1 |
0 |
1 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
2 |
| Rotation (Rx,Ry,Rz) |
0 |
1 |
1 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
2 |
| Vibration |
-1 |
-1 |
-2 |
0 |
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 |
-1
|
| Number of vibrational modes |
-4
|
Force field analysis
Allowed / forbidden vibronational transitions
| Operator |
A1 |
A2 |
E1 |
E2 |
E3 |
E4 |
E5 |
E6 |
E7 |
E8 |
E9 |
E10 |
E11 |
E12 |
E13 |
E14 |
Total |
| Linear (IR) |
-1 |
-1 |
-2 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
-3 / -1 |
| Quadratic (Raman) |
-1 |
-1 |
-2 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
-3 / -1 |
| IR + Raman |
-1 |
-1 |
-2 |
- - - - |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
-3 / -1 |
Characters of force fields
(Symmetric powers of vibration representation)
| Force field |
E |
2C29 |
2(C29)2 |
2(C29)3 |
2(C29)4 |
2(C29)5 |
2(C29)6 |
2(C29)7 |
2(C29)8 |
2(C29)9 |
2(C29)10 |
2(C29)11 |
2(C29)12 |
2(C29)13 |
2(C29)14 |
29σv |
| linear |
-6 |
-5.906 |
-5.630 |
-5.184 |
-4.590 |
-3.874 |
-3.070 |
-2.217 |
-1.353 |
-0.519 |
0.245 |
0.904 |
1.427 |
1.791 |
1.977 |
0 |
| quadratic |
15 |
14.628 |
13.555 |
11.904 |
9.856 |
7.625 |
5.427 |
3.445 |
1.810 |
0.587 |
-0.230 |
-0.700 |
-0.918 |
-0.989 |
-1.000 |
-3 |
| cubic |
-20 |
-19.443 |
-17.850 |
-15.439 |
-12.532 |
-9.503 |
-6.713 |
-4.457 |
-2.915 |
-2.135 |
-2.030 |
-2.409 |
-3.019 |
-3.603 |
-3.953 |
0 |
| quartic |
15 |
14.628 |
13.555 |
11.904 |
9.856 |
7.625 |
5.427 |
3.445 |
1.810 |
0.587 |
-0.230 |
-0.700 |
-0.918 |
-0.989 |
-1.000 |
3 |
| quintic |
-6 |
-5.906 |
-5.630 |
-5.184 |
-4.590 |
-3.874 |
-3.070 |
-2.217 |
-1.353 |
-0.519 |
0.245 |
0.904 |
1.427 |
1.791 |
1.977 |
0 |
| sextic |
1 |
1.000 |
1.000 |
1.000 |
1.000 |
1.000 |
1.000 |
1.000 |
1.000 |
1.000 |
1.000 |
1.000 |
1.000 |
1.000 |
1.000 |
-1 |
Decomposition to irreducible representations
Column with number of nonvanshing force constants highlighted
| Force field |
A1 |
A2 |
E1 |
E2 |
E3 |
E4 |
E5 |
E6 |
E7 |
E8 |
E9 |
E10 |
E11 |
E12 |
E13 |
E14 |
| linear |
-1 |
-1 |
-2 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
| quadratic |
1 |
4 |
4 |
1 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
| cubic |
-4 |
-4 |
-4 |
-2 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
| quartic |
4 |
1 |
4 |
1 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
| quintic |
-1 |
-1 |
-2 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
| sextic |
0 |
1 |
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 C
29v
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(E14) |
|---|
| ..1. |
E1E1. | | |
| |
| |
| |
| |
| |
| |
| |
| |
| Subtotal: 1 / 1 / 16 |
| Irrep combinations (i,j) with indices: pos(A1) ≤ i ≤ j ≤ pos(E14) |
|---|
| Subtotal: 0 / 0 / 120 |
| Total: 1 / 1 / 136 |
Contributions to nonvanishing cubic force field constants
| Irrep combinations (i,i,i) with indices: pos(A1) ≤ i ≤ pos(E14) |
|---|
| Subtotal: 0 / 0 / 16 |
| Irrep combinations (i,i,j) (i,j,j) with indices: pos(A1) ≤ i ≤ j ≤ pos(E14) |
|---|
| Subtotal: -4 / 0 / 240 |
| Irrep combinations (i,j,k) with indices: pos(A1) ≤ i ≤ j ≤ k ≤ pos(E14) |
|---|
| Subtotal: 0 / 0 / 560 |
| Total: -4 / 0 / 816 |
Contributions to nonvanishing quartic force field constants
| Irrep combinations (i,i,i,i) with indices: pos(A1) ≤ i ≤ pos(E14) |
|---|
| ..1. |
E1E1E1E1. | | |
| |
| |
| |
| |
| |
| |
| |
| |
| Subtotal: 1 / 1 / 16 |
| Irrep combinations (i,i,i,j) (i,j,j,j) with indices: pos(A1) ≤ i ≤ j ≤ pos(E14) |
|---|
| Subtotal: 0 / 0 / 240 |
| Irrep combinations (i,i,j,j) with indices: pos(A1) ≤ i ≤ j ≤ pos(E14) |
|---|
| Subtotal: 0 / 0 / 120 |
| Irrep combinations (i,i,j,k) (i,j,j,k) (i,j,k,k) with indices: pos(A1) ≤ i ≤ j ≤ k ≤ pos(E14) |
|---|
| ..3. |
A1A2E1E1. | | |
| |
| |
| |
| |
| |
| |
| |
| |
| Subtotal: 3 / 1 / 1.680 |
| Irrep combinations (i,j,k,l) with indices: pos(A1) ≤ i ≤ j ≤ k ≤ l ≤ pos(E14) |
|---|
| Subtotal: 0 / 0 / 1.820 |
| Total: 4 / 2 / 3.876 |
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Last update November, 13th 2023 by A. Gelessus, Impressum, Datenschutzerklärung/DataPrivacyStatement