Characters of representations for molecular motions
Motion |
E |
C2.(z) |
σv.(xz) |
σd.(yz) |
Cartesian 3N |
156 |
0 |
4 |
4 |
Translation (x,y,z) |
3 |
-1 |
1 |
1 |
Rotation (Rx,Ry,Rz) |
3 |
-1 |
-1 |
-1 |
Vibration |
150 |
2 |
4 |
4 |
Decomposition to irreducible representations
Motion |
A1 |
A2 |
B1 |
B2 |
Total |
Cartesian 3N |
41 |
37 |
39 |
39 |
156 |
Translation (x,y,z) |
1 |
0 |
1 |
1 |
3 |
Rotation (Rx,Ry,Rz) |
0 |
1 |
1 |
1 |
3 |
Vibration |
40 |
36 |
37 |
37 |
150 |
Molecular parameter
Number of Atoms (N) |
52
|
Number of internal coordinates |
150
|
Number of independant internal coordinates |
40
|
Number of vibrational modes |
150
|
Force field analysis
Allowed / forbidden vibronational transitions
Operator |
A1 |
A2 |
B1 |
B2 |
Total |
Linear (IR) |
40 |
36 |
37 |
37 |
114 / 36 |
Quadratic (Raman) |
40 |
36 |
37 |
37 |
150 / 0 |
IR + Raman |
40 |
- - - - |
37 |
37 |
114 / 0 |
Characters of force fields
(Symmetric powers of vibration representation)
Force field |
E |
C2.(z) |
σv.(xz) |
σd.(yz) |
linear |
150 |
2 |
4 |
4 |
quadratic |
11.325 |
77 |
83 |
83 |
cubic |
573.800 |
152 |
312 |
312 |
quartic |
21.947.850 |
3.002 |
3.466 |
3.466 |
quintic |
675.993.780 |
5.852 |
12.320 |
12.320 |
sextic |
17.463.172.650 |
79.002 |
97.174 |
97.174 |
Decomposition to irreducible representations
Column with number of nonvanshing force constants highlighted
Force field |
A1 |
A2 |
B1 |
B2 |
linear |
40 |
36 |
37 |
37 |
quadratic |
2.892 |
2.809 |
2.812 |
2.812 |
cubic |
143.644 |
143.332 |
143.412 |
143.412 |
quartic |
5.489.446 |
5.485.980 |
5.486.212 |
5.486.212 |
quintic |
169.006.068 |
168.993.748 |
168.996.982 |
168.996.982 |
sextic |
4.365.861.500 |
4.365.764.326 |
4.365.773.412 |
4.365.773.412 |
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) |
..820. |
A1A1. | ..666. |
A2A2. | ..703. |
B1B1. | ..703. |
B2B2. | | |
| |
| |
| |
| |
| |
Subtotal: 2.892 / 4 / 4 |
Irrep combinations (i,j) with indices: pos(A1) ≤ i ≤ j ≤ pos(B2) |
Subtotal: 0 / 0 / 6 |
Total: 2.892 / 4 / 10 |
Contributions to nonvanishing cubic force field constants
Irrep combinations (i,i,i) with indices: pos(A1) ≤ i ≤ pos(B2) |
..11.480. |
A1A1A1. | | |
| |
| |
| |
| |
| |
| |
| |
| |
Subtotal: 11.480 / 1 / 4 |
Irrep combinations (i,i,j) (i,j,j) with indices: pos(A1) ≤ i ≤ j ≤ pos(B2) |
..26.640. |
A1A2A2. | ..28.120. |
A1B1B1. | ..28.120. |
A1B2B2. | | |
| |
| |
| |
| |
| |
| |
Subtotal: 82.880 / 3 / 12 |
Irrep combinations (i,j,k) with indices: pos(A1) ≤ i ≤ j ≤ k ≤ pos(B2) |
..49.284. |
A2B1B2. | | |
| |
| |
| |
| |
| |
| |
| |
| |
Subtotal: 49.284 / 1 / 4 |
Total: 143.644 / 5 / 20 |
Contributions to nonvanishing quartic force field constants
Irrep combinations (i,i,i,i) with indices: pos(A1) ≤ i ≤ pos(B2) |
..123.410. |
A1A1A1A1. | ..82.251. |
A2A2A2A2. | ..91.390. |
B1B1B1B1. | ..91.390. |
B2B2B2B2. | | |
| |
| |
| |
| |
| |
Subtotal: 388.441 / 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) |
..546.120. |
A1A1A2A2. | ..576.460. |
A1A1B1B1. | ..576.460. |
A1A1B2B2. | ..468.198. |
A2A2B1B1. | ..468.198. |
A2A2B2B2. | ..494.209. |
B1B1B2B2. | | |
| |
| |
| |
Subtotal: 3.129.645 / 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) |
..1.971.360. |
A1A2B1B2. | | |
| |
| |
| |
| |
| |
| |
| |
| |
Subtotal: 1.971.360 / 1 / 1 |
Total: 5.489.446 / 11 / 35 |
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