Characters of representations for molecular motions
Motion |
E |
C30 |
C15 |
C10 |
(C15)2 |
C6 |
C5 |
(C30)7 |
(C15)4 |
(C10)3 |
C3 |
(C30)11 |
(C5)2 |
(C30)13 |
(C15)7 |
C2 |
(C15)8 |
(C30)17 |
(C5)3 |
(C30)19 |
(C3)2 |
(C10)7 |
(C15)11 |
(C30)23 |
(C5)4 |
(C6)5 |
(C15)13 |
(C10)9 |
(C15)14 |
(C30)29 |
Cartesian 3N |
0 |
0.000 |
0.000 |
0.000 |
0.000 |
0 |
0.000 |
0.000 |
0.000 |
0.000 |
0 |
0.000 |
0.000 |
0.000 |
0.000 |
0 |
0.000 |
0.000 |
0.000 |
0.000 |
0 |
0.000 |
0.000 |
0.000 |
0.000 |
0 |
0.000 |
0.000 |
0.000 |
0.000 |
Translation (x,y,z) |
3 |
2.956 |
2.827 |
2.618 |
2.338 |
2 |
1.618 |
1.209 |
0.791 |
0.382 |
0 |
-0.338 |
-0.618 |
-0.827 |
-0.956 |
-1 |
-0.956 |
-0.827 |
-0.618 |
-0.338 |
0 |
0.382 |
0.791 |
1.209 |
1.618 |
2 |
2.338 |
2.618 |
2.827 |
2.956 |
Rotation (Rx,Ry,Rz) |
3 |
2.956 |
2.827 |
2.618 |
2.338 |
2 |
1.618 |
1.209 |
0.791 |
0.382 |
0 |
-0.338 |
-0.618 |
-0.827 |
-0.956 |
-1 |
-0.956 |
-0.827 |
-0.618 |
-0.338 |
0 |
0.382 |
0.791 |
1.209 |
1.618 |
2 |
2.338 |
2.618 |
2.827 |
2.956 |
Vibration |
-6 |
-5.913 |
-5.654 |
-5.236 |
-4.677 |
-4 |
-3.236 |
-2.418 |
-1.582 |
-0.764 |
0 |
0.677 |
1.236 |
1.654 |
1.913 |
2 |
1.913 |
1.654 |
1.236 |
0.677 |
0 |
-0.764 |
-1.582 |
-2.418 |
-3.236 |
-4 |
-4.677 |
-5.236 |
-5.654 |
-5.913 |
Decomposition to irreducible representations
Motion |
A |
B |
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) |
1 |
0 |
1 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
2 |
Vibration |
-2 |
0 |
-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 |
-2
|
Number of vibrational modes |
-4
|
Force field analysis
Allowed / forbidden vibronational transitions
Operator |
A |
B |
E1*
|
E2*
|
E3*
|
E4*
|
E5*
|
E6*
|
E7*
|
E8*
|
E9*
|
E10*
|
E11*
|
E12*
|
E13*
|
E14*
|
Total |
Linear (IR) |
-2 |
0 |
-2 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
-4 / 0 |
Quadratic (Raman) |
-2 |
0 |
-2 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
-4 / 0 |
IR + Raman |
-2 |
0 |
-2 |
- - - - |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
-4 / 0 |
Characters of force fields
(Symmetric powers of vibration representation)
Force field |
E |
C30 |
C15 |
C10 |
(C15)2 |
C6 |
C5 |
(C30)7 |
(C15)4 |
(C10)3 |
C3 |
(C30)11 |
(C5)2 |
(C30)13 |
(C15)7 |
C2 |
(C15)8 |
(C30)17 |
(C5)3 |
(C30)19 |
(C3)2 |
(C10)7 |
(C15)11 |
(C30)23 |
(C5)4 |
(C6)5 |
(C15)13 |
(C10)9 |
(C15)14 |
(C30)29 |
linear |
-6 |
-5.913 |
-5.654 |
-5.236 |
-4.677 |
-4 |
-3.236 |
-2.418 |
-1.582 |
-0.764 |
0 |
0.677 |
1.236 |
1.654 |
1.913 |
2 |
1.913 |
1.654 |
1.236 |
0.677 |
0 |
-0.764 |
-1.582 |
-2.418 |
-3.236 |
-4 |
-4.677 |
-5.236 |
-5.654 |
-5.913 |
quadratic |
15 |
14.652 |
13.647 |
12.090 |
10.144 |
8 |
5.854 |
3.880 |
2.207 |
0.910 |
0 |
-0.562 |
-0.854 |
-0.970 |
-0.998 |
-1 |
-0.998 |
-0.970 |
-0.854 |
-0.562 |
0 |
0.910 |
2.207 |
3.880 |
5.854 |
8 |
10.144 |
12.090 |
13.647 |
14.652 |
cubic |
-20 |
-19.479 |
-17.985 |
-15.708 |
-12.935 |
-10 |
-7.236 |
-4.924 |
-3.251 |
-2.292 |
-2 |
-2.229 |
-2.764 |
-3.368 |
-3.829 |
-4 |
-3.829 |
-3.368 |
-2.764 |
-2.229 |
-2 |
-2.292 |
-3.251 |
-4.924 |
-7.236 |
-10 |
-12.935 |
-15.708 |
-17.985 |
-19.479 |
quartic |
15 |
14.652 |
13.647 |
12.090 |
10.144 |
8 |
5.854 |
3.880 |
2.207 |
0.910 |
0 |
-0.562 |
-0.854 |
-0.970 |
-0.998 |
-1 |
-0.998 |
-0.970 |
-0.854 |
-0.562 |
0 |
0.910 |
2.207 |
3.880 |
5.854 |
8 |
10.144 |
12.090 |
13.647 |
14.652 |
quintic |
-6 |
-5.913 |
-5.654 |
-5.236 |
-4.677 |
-4 |
-3.236 |
-2.418 |
-1.582 |
-0.764 |
0 |
0.677 |
1.236 |
1.654 |
1.913 |
2 |
1.913 |
1.654 |
1.236 |
0.677 |
0 |
-0.764 |
-1.582 |
-2.418 |
-3.236 |
-4 |
-4.677 |
-5.236 |
-5.654 |
-5.913 |
sextic |
1 |
1.000 |
1.000 |
1.000 |
1.000 |
1 |
1.000 |
1.000 |
1.000 |
1.000 |
1 |
1.000 |
1.000 |
1.000 |
1.000 |
1 |
1.000 |
1.000 |
1.000 |
1.000 |
1 |
1.000 |
1.000 |
1.000 |
1.000 |
1 |
1.000 |
1.000 |
1.000 |
1.000 |
Decomposition to irreducible representations
Column with number of nonvanshing force constants highlighted
Force field |
A |
B |
E1*
|
E2*
|
E3*
|
E4*
|
E5*
|
E6*
|
E7*
|
E8*
|
E9*
|
E10*
|
E11*
|
E12*
|
E13*
|
E14*
|
linear |
-2 |
0 |
-2 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
quadratic |
5 |
0 |
4 |
1 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
cubic |
-8 |
0 |
-4 |
-2 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
quartic |
5 |
0 |
4 |
1 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
quintic |
-2 |
0 |
-2 |
0 |
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 |
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
30
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(A) ≤ i ≤ pos(E14) |
..1. |
AA. | ..4. |
E1E1. | | |
| |
| |
| |
| |
| |
| |
| |
Subtotal: 5 / 2 / 16 |
Irrep combinations (i,j) with indices: pos(A) ≤ i ≤ j ≤ pos(E14) |
Subtotal: 0 / 0 / 120 |
Total: 5 / 2 / 136 |
Contributions to nonvanishing cubic force field constants
Irrep combinations (i,i,i) with indices: pos(A) ≤ i ≤ pos(E14) |
Subtotal: 0 / 0 / 16 |
Irrep combinations (i,i,j) (i,j,j) with indices: pos(A) ≤ i ≤ j ≤ pos(E14) |
Subtotal: -8 / 0 / 240 |
Irrep combinations (i,j,k) with indices: pos(A) ≤ i ≤ j ≤ k ≤ pos(E14) |
Subtotal: 0 / 0 / 560 |
Total: -8 / 0 / 816 |
Contributions to nonvanishing quartic force field constants
Irrep combinations (i,i,i,i) with indices: pos(A) ≤ i ≤ pos(E14) |
..1. |
E1E1E1E1. | | |
| |
| |
| |
| |
| |
| |
| |
| |
Subtotal: 1 / 1 / 16 |
Irrep combinations (i,i,i,j) (i,j,j,j) with indices: pos(A) ≤ i ≤ j ≤ pos(E14) |
Subtotal: 0 / 0 / 240 |
Irrep combinations (i,i,j,j) with indices: pos(A) ≤ i ≤ j ≤ pos(E14) |
..4. |
AAE1E1. | | |
| |
| |
| |
| |
| |
| |
| |
| |
Subtotal: 4 / 1 / 120 |
Irrep combinations (i,i,j,k) (i,j,j,k) (i,j,k,k) with indices: pos(A) ≤ i ≤ j ≤ k ≤ pos(E14) |
Subtotal: 0 / 0 / 1.680 |
Irrep combinations (i,j,k,l) with indices: pos(A) ≤ i ≤ j ≤ k ≤ l ≤ pos(E14) |
Subtotal: 0 / 0 / 1.820 |
Total: 5 / 2 / 3.876 |
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Last update November, 13th 2023 by A. Gelessus, Impressum, Datenschutzerklärung/DataPrivacyStatement