Characters of representations for molecular motions
Motion |
E |
C2.(z) |
σv.(xz) |
σd.(yz) |
Cartesian 3N |
48 |
0 |
0 |
8 |
Translation (x,y,z) |
3 |
-1 |
1 |
1 |
Rotation (Rx,Ry,Rz) |
3 |
-1 |
-1 |
-1 |
Vibration |
42 |
2 |
0 |
8 |
Decomposition to irreducible representations
Motion |
A1 |
A2 |
B1 |
B2 |
Total |
Cartesian 3N |
14 |
10 |
10 |
14 |
48 |
Translation (x,y,z) |
1 |
0 |
1 |
1 |
3 |
Rotation (Rx,Ry,Rz) |
0 |
1 |
1 |
1 |
3 |
Vibration |
13 |
9 |
8 |
12 |
42 |
Molecular parameter
Number of Atoms (N) |
16
|
Number of internal coordinates |
42
|
Number of independant internal coordinates |
13
|
Number of vibrational modes |
42
|
Force field analysis
Allowed / forbidden vibronational transitions
Operator |
A1 |
A2 |
B1 |
B2 |
Total |
Linear (IR) |
13 |
9 |
8 |
12 |
33 / 9 |
Quadratic (Raman) |
13 |
9 |
8 |
12 |
42 / 0 |
IR + Raman |
13 |
- - - - |
8 |
12 |
33 / 0 |
Characters of force fields
(Symmetric powers of vibration representation)
Force field |
E |
C2.(z) |
σv.(xz) |
σd.(yz) |
linear |
42 |
2 |
0 |
8 |
quadratic |
903 |
23 |
21 |
53 |
cubic |
13.244 |
44 |
0 |
256 |
quartic |
148.995 |
275 |
231 |
1.095 |
quintic |
1.370.754 |
506 |
0 |
4.056 |
sextic |
10.737.573 |
2.277 |
1.771 |
13.803 |
Decomposition to irreducible representations
Column with number of nonvanshing force constants highlighted
Force field |
A1 |
A2 |
B1 |
B2 |
linear |
13 |
9 |
8 |
12 |
quadratic |
250 |
213 |
212 |
228 |
cubic |
3.386 |
3.258 |
3.236 |
3.364 |
quartic |
37.649 |
36.986 |
36.964 |
37.396 |
quintic |
343.829 |
341.801 |
341.548 |
343.576 |
sextic |
2.688.856 |
2.681.069 |
2.680.816 |
2.686.832 |
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) |
..91. |
A1A1. | ..45. |
A2A2. | ..36. |
B1B1. | ..78. |
B2B2. | | |
| |
| |
| |
| |
| |
Subtotal: 250 / 4 / 4 |
Irrep combinations (i,j) with indices: pos(A1) ≤ i ≤ j ≤ pos(B2) |
Subtotal: 0 / 0 / 6 |
Total: 250 / 4 / 10 |
Contributions to nonvanishing cubic force field constants
Irrep combinations (i,i,i) with indices: pos(A1) ≤ i ≤ pos(B2) |
..455. |
A1A1A1. | | |
| |
| |
| |
| |
| |
| |
| |
| |
Subtotal: 455 / 1 / 4 |
Irrep combinations (i,i,j) (i,j,j) with indices: pos(A1) ≤ i ≤ j ≤ pos(B2) |
..585. |
A1A2A2. | ..468. |
A1B1B1. | ..1.014. |
A1B2B2. | | |
| |
| |
| |
| |
| |
| |
Subtotal: 2.067 / 3 / 12 |
Irrep combinations (i,j,k) with indices: pos(A1) ≤ i ≤ j ≤ k ≤ pos(B2) |
..864. |
A2B1B2. | | |
| |
| |
| |
| |
| |
| |
| |
| |
Subtotal: 864 / 1 / 4 |
Total: 3.386 / 5 / 20 |
Contributions to nonvanishing quartic force field constants
Irrep combinations (i,i,i,i) with indices: pos(A1) ≤ i ≤ pos(B2) |
..1.820. |
A1A1A1A1. | ..495. |
A2A2A2A2. | ..330. |
B1B1B1B1. | ..1.365. |
B2B2B2B2. | | |
| |
| |
| |
| |
| |
Subtotal: 4.010 / 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) |
..4.095. |
A1A1A2A2. | ..3.276. |
A1A1B1B1. | ..7.098. |
A1A1B2B2. | ..1.620. |
A2A2B1B1. | ..3.510. |
A2A2B2B2. | ..2.808. |
B1B1B2B2. | | |
| |
| |
| |
Subtotal: 22.407 / 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) |
..11.232. |
A1A2B1B2. | | |
| |
| |
| |
| |
| |
| |
| |
| |
Subtotal: 11.232 / 1 / 1 |
Total: 37.649 / 11 / 35 |
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