Characters of representations for molecular motions
Motion |
E |
C2.(z) |
σv.(xz) |
σd.(yz) |
Cartesian 3N |
36 |
-4 |
8 |
8 |
Translation (x,y,z) |
3 |
-1 |
1 |
1 |
Rotation (Rx,Ry,Rz) |
3 |
-1 |
-1 |
-1 |
Vibration |
30 |
-2 |
8 |
8 |
Decomposition to irreducible representations
Motion |
A1 |
A2 |
B1 |
B2 |
Total |
Cartesian 3N |
12 |
4 |
10 |
10 |
36 |
Translation (x,y,z) |
1 |
0 |
1 |
1 |
3 |
Rotation (Rx,Ry,Rz) |
0 |
1 |
1 |
1 |
3 |
Vibration |
11 |
3 |
8 |
8 |
30 |
Molecular parameter
Number of Atoms (N) |
12
|
Number of internal coordinates |
30
|
Number of independant internal coordinates |
11
|
Number of vibrational modes |
30
|
Force field analysis
Allowed / forbidden vibronational transitions
Operator |
A1 |
A2 |
B1 |
B2 |
Total |
Linear (IR) |
11 |
3 |
8 |
8 |
27 / 3 |
Quadratic (Raman) |
11 |
3 |
8 |
8 |
30 / 0 |
IR + Raman |
11 |
- - - - |
8 |
8 |
27 / 0 |
Characters of force fields
(Symmetric powers of vibration representation)
Force field |
E |
C2.(z) |
σv.(xz) |
σd.(yz) |
linear |
30 |
-2 |
8 |
8 |
quadratic |
465 |
17 |
47 |
47 |
cubic |
4.960 |
-32 |
208 |
208 |
quartic |
40.920 |
152 |
792 |
792 |
quintic |
278.256 |
-272 |
2.640 |
2.640 |
sextic |
1.623.160 |
952 |
8.008 |
8.008 |
Decomposition to irreducible representations
Column with number of nonvanshing force constants highlighted
Force field |
A1 |
A2 |
B1 |
B2 |
linear |
11 |
3 |
8 |
8 |
quadratic |
144 |
97 |
112 |
112 |
cubic |
1.336 |
1.128 |
1.248 |
1.248 |
quartic |
10.664 |
9.872 |
10.192 |
10.192 |
quintic |
70.816 |
68.176 |
69.632 |
69.632 |
sextic |
410.032 |
402.024 |
405.552 |
405.552 |
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) |
..66. |
A1A1. | ..6. |
A2A2. | ..36. |
B1B1. | ..36. |
B2B2. | | |
| |
| |
| |
| |
| |
Subtotal: 144 / 4 / 4 |
Irrep combinations (i,j) with indices: pos(A1) ≤ i ≤ j ≤ pos(B2) |
Subtotal: 0 / 0 / 6 |
Total: 144 / 4 / 10 |
Contributions to nonvanishing cubic force field constants
Irrep combinations (i,i,i) with indices: pos(A1) ≤ i ≤ pos(B2) |
..286. |
A1A1A1. | | |
| |
| |
| |
| |
| |
| |
| |
| |
Subtotal: 286 / 1 / 4 |
Irrep combinations (i,i,j) (i,j,j) with indices: pos(A1) ≤ i ≤ j ≤ pos(B2) |
..66. |
A1A2A2. | ..396. |
A1B1B1. | ..396. |
A1B2B2. | | |
| |
| |
| |
| |
| |
| |
Subtotal: 858 / 3 / 12 |
Irrep combinations (i,j,k) with indices: pos(A1) ≤ i ≤ j ≤ k ≤ pos(B2) |
..192. |
A2B1B2. | | |
| |
| |
| |
| |
| |
| |
| |
| |
Subtotal: 192 / 1 / 4 |
Total: 1.336 / 5 / 20 |
Contributions to nonvanishing quartic force field constants
Irrep combinations (i,i,i,i) with indices: pos(A1) ≤ i ≤ pos(B2) |
..1.001. |
A1A1A1A1. | ..15. |
A2A2A2A2. | ..330. |
B1B1B1B1. | ..330. |
B2B2B2B2. | | |
| |
| |
| |
| |
| |
Subtotal: 1.676 / 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) |
..396. |
A1A1A2A2. | ..2.376. |
A1A1B1B1. | ..2.376. |
A1A1B2B2. | ..216. |
A2A2B1B1. | ..216. |
A2A2B2B2. | ..1.296. |
B1B1B2B2. | | |
| |
| |
| |
Subtotal: 6.876 / 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) |
..2.112. |
A1A2B1B2. | | |
| |
| |
| |
| |
| |
| |
| |
| |
Subtotal: 2.112 / 1 / 1 |
Total: 10.664 / 11 / 35 |
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