Characters of representations for molecular motions
Motion |
E |
C2 |
Cartesian 3N |
39 |
-5 |
Translation (x,y,z) |
3 |
-1 |
Rotation (Rx,Ry,Rz) |
3 |
-1 |
Vibration |
33 |
-3 |
Decomposition to irreducible representations
Motion |
A |
B |
Total |
Cartesian 3N |
17 |
22 |
39 |
Translation (x,y,z) |
1 |
2 |
3 |
Rotation (Rx,Ry,Rz) |
1 |
2 |
3 |
Vibration |
15 |
18 |
33 |
Molecular parameter
Number of Atoms (N) |
13
|
Number of internal coordinates |
33
|
Number of independant internal coordinates |
15
|
Number of vibrational modes |
33
|
Force field analysis
Allowed / forbidden vibronational transitions
Operator |
A |
B |
Total |
Linear (IR) |
15 |
18 |
33 / 0 |
Quadratic (Raman) |
15 |
18 |
33 / 0 |
IR + Raman |
15 |
18 |
33 / 0 |
Characters of force fields
(Symmetric powers of vibration representation)
Force field |
E |
C2 |
linear |
33 |
-3 |
quadratic |
561 |
21 |
cubic |
6.545 |
-55 |
quartic |
58.905 |
225 |
quintic |
435.897 |
-531 |
sextic |
2.760.681 |
1.653 |
Decomposition to irreducible representations
Column with number of nonvanshing force constants highlighted
Force field |
A |
B |
linear |
15 |
18 |
quadratic |
291 |
270 |
cubic |
3.245 |
3.300 |
quartic |
29.565 |
29.340 |
quintic |
217.683 |
218.214 |
sextic |
1.381.167 |
1.379.514 |
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
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(B) |
..120. |
AA. | ..171. |
BB. | | |
| |
| |
| |
| |
| |
| |
| |
Subtotal: 291 / 2 / 2 |
Irrep combinations (i,j) with indices: pos(A) ≤ i ≤ j ≤ pos(B) |
Subtotal: 0 / 0 / 1 |
Total: 291 / 2 / 3 |
Contributions to nonvanishing cubic force field constants
Irrep combinations (i,i,i) with indices: pos(A) ≤ i ≤ pos(B) |
..680. |
AAA. | | |
| |
| |
| |
| |
| |
| |
| |
| |
Subtotal: 680 / 1 / 2 |
Irrep combinations (i,i,j) (i,j,j) with indices: pos(A) ≤ i ≤ j ≤ pos(B) |
..2.565. |
ABB. | | |
| |
| |
| |
| |
| |
| |
| |
| |
Subtotal: 2.565 / 1 / 2 |
Irrep combinations (i,j,k) with indices: pos(A) ≤ i ≤ j ≤ k ≤ pos(B) |
Subtotal: 0 / 0 / 0 |
Total: 3.245 / 2 / 4 |
Contributions to nonvanishing quartic force field constants
Irrep combinations (i,i,i,i) with indices: pos(A) ≤ i ≤ pos(B) |
..3.060. |
AAAA. | ..5.985. |
BBBB. | | |
| |
| |
| |
| |
| |
| |
| |
Subtotal: 9.045 / 2 / 2 |
Irrep combinations (i,i,i,j) (i,j,j,j) with indices: pos(A) ≤ i ≤ j ≤ pos(B) |
Subtotal: 0 / 0 / 2 |
Irrep combinations (i,i,j,j) with indices: pos(A) ≤ i ≤ j ≤ pos(B) |
..20.520. |
AABB. | | |
| |
| |
| |
| |
| |
| |
| |
| |
Subtotal: 20.520 / 1 / 1 |
Irrep combinations (i,i,j,k) (i,j,j,k) (i,j,k,k) with indices: pos(A) ≤ i ≤ j ≤ k ≤ pos(B) |
Subtotal: 0 / 0 / 0 |
Irrep combinations (i,j,k,l) with indices: pos(A) ≤ i ≤ j ≤ k ≤ l ≤ pos(B) |
Subtotal: 0 / 0 / 0 |
Total: 29.565 / 3 / 5 |
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