Characters of representations for molecular motions
Motion |
E |
C2.(z) |
C2.(y) |
C2.(x) |
Cartesian 3N |
48 |
0 |
0 |
0 |
Translation (x,y,z) |
3 |
-1 |
-1 |
-1 |
Rotation (Rx,Ry,Rz) |
3 |
-1 |
-1 |
-1 |
Vibration |
42 |
2 |
2 |
2 |
Decomposition to irreducible representations
Motion |
A |
B1 |
B2 |
B3 |
Total |
Cartesian 3N |
12 |
12 |
12 |
12 |
48 |
Translation (x,y,z) |
0 |
1 |
1 |
1 |
3 |
Rotation (Rx,Ry,Rz) |
0 |
1 |
1 |
1 |
3 |
Vibration |
12 |
10 |
10 |
10 |
42 |
Molecular parameter
Number of Atoms (N) |
16
|
Number of internal coordinates |
42
|
Number of independant internal coordinates |
12
|
Number of vibrational modes |
42
|
Force field analysis
Allowed / forbidden vibronational transitions
Operator |
A |
B1 |
B2 |
B3 |
Total |
Linear (IR) |
12 |
10 |
10 |
10 |
30 / 12 |
Quadratic (Raman) |
12 |
10 |
10 |
10 |
42 / 0 |
IR + Raman |
- - - - |
10 |
10 |
10 |
30 / 0 |
Characters of force fields
(Symmetric powers of vibration representation)
Force field |
E |
C2.(z) |
C2.(y) |
C2.(x) |
linear |
42 |
2 |
2 |
2 |
quadratic |
903 |
23 |
23 |
23 |
cubic |
13.244 |
44 |
44 |
44 |
quartic |
148.995 |
275 |
275 |
275 |
quintic |
1.370.754 |
506 |
506 |
506 |
sextic |
10.737.573 |
2.277 |
2.277 |
2.277 |
Decomposition to irreducible representations
Column with number of nonvanshing force constants highlighted
Force field |
A |
B1 |
B2 |
B3 |
linear |
12 |
10 |
10 |
10 |
quadratic |
243 |
220 |
220 |
220 |
cubic |
3.344 |
3.300 |
3.300 |
3.300 |
quartic |
37.455 |
37.180 |
37.180 |
37.180 |
quintic |
343.068 |
342.562 |
342.562 |
342.562 |
sextic |
2.686.101 |
2.683.824 |
2.683.824 |
2.683.824 |
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 D
2
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(B3) |
..78. |
AA. | ..55. |
B1B1. | ..55. |
B2B2. | ..55. |
B3B3. | | |
| |
| |
| |
| |
| |
Subtotal: 243 / 4 / 4 |
Irrep combinations (i,j) with indices: pos(A) ≤ i ≤ j ≤ pos(B3) |
Subtotal: 0 / 0 / 6 |
Total: 243 / 4 / 10 |
Contributions to nonvanishing cubic force field constants
Irrep combinations (i,i,i) with indices: pos(A) ≤ i ≤ pos(B3) |
..364. |
AAA. | | |
| |
| |
| |
| |
| |
| |
| |
| |
Subtotal: 364 / 1 / 4 |
Irrep combinations (i,i,j) (i,j,j) with indices: pos(A) ≤ i ≤ j ≤ pos(B3) |
..660. |
AB1B1. | ..660. |
AB2B2. | ..660. |
AB3B3. | | |
| |
| |
| |
| |
| |
| |
Subtotal: 1.980 / 3 / 12 |
Irrep combinations (i,j,k) with indices: pos(A) ≤ i ≤ j ≤ k ≤ pos(B3) |
..1.000. |
B1B2B3. | | |
| |
| |
| |
| |
| |
| |
| |
| |
Subtotal: 1.000 / 1 / 4 |
Total: 3.344 / 5 / 20 |
Contributions to nonvanishing quartic force field constants
Irrep combinations (i,i,i,i) with indices: pos(A) ≤ i ≤ pos(B3) |
..1.365. |
AAAA. | ..715. |
B1B1B1B1. | ..715. |
B2B2B2B2. | ..715. |
B3B3B3B3. | | |
| |
| |
| |
| |
| |
Subtotal: 3.510 / 4 / 4 |
Irrep combinations (i,i,i,j) (i,j,j,j) with indices: pos(A) ≤ i ≤ j ≤ pos(B3) |
Subtotal: 0 / 0 / 12 |
Irrep combinations (i,i,j,j) with indices: pos(A) ≤ i ≤ j ≤ pos(B3) |
..4.290. |
AAB1B1. | ..4.290. |
AAB2B2. | ..4.290. |
AAB3B3. | ..3.025. |
B1B1B2B2. | ..3.025. |
B1B1B3B3. | ..3.025. |
B2B2B3B3. | | |
| |
| |
| |
Subtotal: 21.945 / 6 / 6 |
Irrep combinations (i,i,j,k) (i,j,j,k) (i,j,k,k) with indices: pos(A) ≤ i ≤ j ≤ k ≤ pos(B3) |
Subtotal: 0 / 0 / 12 |
Irrep combinations (i,j,k,l) with indices: pos(A) ≤ i ≤ j ≤ k ≤ l ≤ pos(B3) |
..12.000. |
AB1B2B3. | | |
| |
| |
| |
| |
| |
| |
| |
| |
Subtotal: 12.000 / 1 / 1 |
Total: 37.455 / 11 / 35 |
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