Characters of representations for molecular motions
Motion |
E |
C2.(z) |
σv.(xz) |
σd.(yz) |
Cartesian 3N |
39 |
-5 |
5 |
5 |
Translation (x,y,z) |
3 |
-1 |
1 |
1 |
Rotation (Rx,Ry,Rz) |
3 |
-1 |
-1 |
-1 |
Vibration |
33 |
-3 |
5 |
5 |
Decomposition to irreducible representations
Motion |
A1 |
A2 |
B1 |
B2 |
Total |
Cartesian 3N |
11 |
6 |
11 |
11 |
39 |
Translation (x,y,z) |
1 |
0 |
1 |
1 |
3 |
Rotation (Rx,Ry,Rz) |
0 |
1 |
1 |
1 |
3 |
Vibration |
10 |
5 |
9 |
9 |
33 |
Molecular parameter
Number of Atoms (N) |
13
|
Number of internal coordinates |
33
|
Number of independant internal coordinates |
10
|
Number of vibrational modes |
33
|
Force field analysis
Allowed / forbidden vibronational transitions
Operator |
A1 |
A2 |
B1 |
B2 |
Total |
Linear (IR) |
10 |
5 |
9 |
9 |
28 / 5 |
Quadratic (Raman) |
10 |
5 |
9 |
9 |
33 / 0 |
IR + Raman |
10 |
- - - - |
9 |
9 |
28 / 0 |
Characters of force fields
(Symmetric powers of vibration representation)
Force field |
E |
C2.(z) |
σv.(xz) |
σd.(yz) |
linear |
33 |
-3 |
5 |
5 |
quadratic |
561 |
21 |
29 |
29 |
cubic |
6.545 |
-55 |
105 |
105 |
quartic |
58.905 |
225 |
385 |
385 |
quintic |
435.897 |
-531 |
1.141 |
1.141 |
sextic |
2.760.681 |
1.653 |
3.325 |
3.325 |
Decomposition to irreducible representations
Column with number of nonvanshing force constants highlighted
Force field |
A1 |
A2 |
B1 |
B2 |
linear |
10 |
5 |
9 |
9 |
quadratic |
160 |
131 |
135 |
135 |
cubic |
1.675 |
1.570 |
1.650 |
1.650 |
quartic |
14.975 |
14.590 |
14.670 |
14.670 |
quintic |
109.412 |
108.271 |
109.107 |
109.107 |
sextic |
692.246 |
688.921 |
689.757 |
689.757 |
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) |
..55. |
A1A1. | ..15. |
A2A2. | ..45. |
B1B1. | ..45. |
B2B2. | | |
| |
| |
| |
| |
| |
Subtotal: 160 / 4 / 4 |
Irrep combinations (i,j) with indices: pos(A1) ≤ i ≤ j ≤ pos(B2) |
Subtotal: 0 / 0 / 6 |
Total: 160 / 4 / 10 |
Contributions to nonvanishing cubic force field constants
Irrep combinations (i,i,i) with indices: pos(A1) ≤ i ≤ pos(B2) |
..220. |
A1A1A1. | | |
| |
| |
| |
| |
| |
| |
| |
| |
Subtotal: 220 / 1 / 4 |
Irrep combinations (i,i,j) (i,j,j) with indices: pos(A1) ≤ i ≤ j ≤ pos(B2) |
..150. |
A1A2A2. | ..450. |
A1B1B1. | ..450. |
A1B2B2. | | |
| |
| |
| |
| |
| |
| |
Subtotal: 1.050 / 3 / 12 |
Irrep combinations (i,j,k) with indices: pos(A1) ≤ i ≤ j ≤ k ≤ pos(B2) |
..405. |
A2B1B2. | | |
| |
| |
| |
| |
| |
| |
| |
| |
Subtotal: 405 / 1 / 4 |
Total: 1.675 / 5 / 20 |
Contributions to nonvanishing quartic force field constants
Irrep combinations (i,i,i,i) with indices: pos(A1) ≤ i ≤ pos(B2) |
..715. |
A1A1A1A1. | ..70. |
A2A2A2A2. | ..495. |
B1B1B1B1. | ..495. |
B2B2B2B2. | | |
| |
| |
| |
| |
| |
Subtotal: 1.775 / 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) |
..825. |
A1A1A2A2. | ..2.475. |
A1A1B1B1. | ..2.475. |
A1A1B2B2. | ..675. |
A2A2B1B1. | ..675. |
A2A2B2B2. | ..2.025. |
B1B1B2B2. | | |
| |
| |
| |
Subtotal: 9.150 / 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) |
..4.050. |
A1A2B1B2. | | |
| |
| |
| |
| |
| |
| |
| |
| |
Subtotal: 4.050 / 1 / 1 |
Total: 14.975 / 11 / 35 |
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