Characters of representations for molecular motions
| Motion |
E |
2C4 |
C2 |
2C'2 |
2C''2 |
| Cartesian 3N |
156 |
0 |
0 |
0 |
-2 |
| Translation (x,y,z) |
3 |
1 |
-1 |
-1 |
-1 |
| Rotation (Rx,Ry,Rz) |
3 |
1 |
-1 |
-1 |
-1 |
| Vibration |
150 |
-2 |
2 |
2 |
0 |
Decomposition to irreducible representations
| Motion |
A1 |
A2 |
B1 |
B2 |
E |
Total |
| Cartesian 3N |
19 |
20 |
20 |
19 |
39 |
117 |
| Translation (x,y,z) |
0 |
1 |
0 |
0 |
1 |
2 |
| Rotation (Rx,Ry,Rz) |
0 |
1 |
0 |
0 |
1 |
2 |
| Vibration |
19 |
18 |
20 |
19 |
37 |
113 |
Molecular parameter
| Number of Atoms (N) |
52
|
| Number of internal coordinates |
150
|
| Number of independant internal coordinates |
19
|
| Number of vibrational modes |
113
|
Force field analysis
Allowed / forbidden vibronational transitions
| Operator |
A1 |
A2 |
B1 |
B2 |
E |
Total |
| Linear (IR) |
19 |
18 |
20 |
19 |
37 |
55 / 58 |
| Quadratic (Raman) |
19 |
18 |
20 |
19 |
37 |
95 / 18 |
| IR + Raman |
- - - - |
- - - - |
- - - - |
- - - - |
37 |
37 / 0 |
Characters of force fields
(Symmetric powers of vibration representation)
| Force field |
E |
2C4 |
C2 |
2C'2 |
2C''2 |
| linear |
150 |
-2 |
2 |
2 |
0 |
| quadratic |
11.325 |
3 |
77 |
77 |
75 |
| cubic |
573.800 |
-4 |
152 |
152 |
0 |
| quartic |
21.947.850 |
42 |
3.002 |
3.002 |
2.850 |
| quintic |
675.993.780 |
-80 |
5.852 |
5.852 |
0 |
| sextic |
17.463.172.650 |
118 |
79.002 |
79.002 |
73.150 |
Decomposition to irreducible representations
Column with number of nonvanshing force constants highlighted
| Force field |
A1 |
A2 |
B1 |
B2 |
E |
| linear |
19 |
18 |
20 |
19 |
37 |
| quadratic |
1.464 |
1.388 |
1.425 |
1.424 |
2.812 |
| cubic |
71.781 |
71.705 |
71.783 |
71.707 |
143.412 |
| quartic |
2.745.330 |
2.742.404 |
2.743.884 |
2.743.808 |
5.486.212 |
| quintic |
84.501.397 |
84.498.471 |
84.501.437 |
84.498.511 |
168.996.982 |
| sextic |
2.182.944.524 |
2.182.868.448 |
2.182.907.890 |
2.182.904.964 |
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 D
4
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(E) |
|---|
| ..190. |
A1A1. | ..171. |
A2A2. | ..210. |
B1B1. | ..190. |
B2B2. | ..703. |
EE. | | |
| |
| |
| |
| |
| Subtotal: 1.464 / 5 / 5 |
| Irrep combinations (i,j) with indices: pos(A1) ≤ i ≤ j ≤ pos(E) |
|---|
| Subtotal: 0 / 0 / 10 |
| Total: 1.464 / 5 / 15 |
Contributions to nonvanishing cubic force field constants
| Irrep combinations (i,i,i) with indices: pos(A1) ≤ i ≤ pos(E) |
|---|
| ..1.330. |
A1A1A1. | | |
| |
| |
| |
| |
| |
| |
| |
| |
| Subtotal: 1.330 / 1 / 5 |
| Irrep combinations (i,i,j) (i,j,j) with indices: pos(A1) ≤ i ≤ j ≤ pos(E) |
|---|
| ..3.249. |
A1A2A2. | ..3.990. |
A1B1B1. | ..3.610. |
A1B2B2. | ..13.357. |
A1EE. | ..11.988. |
A2EE. | ..14.060. |
B1EE. | ..13.357. |
B2EE. | | |
| |
| |
| Subtotal: 63.611 / 7 / 20 |
| Irrep combinations (i,j,k) with indices: pos(A1) ≤ i ≤ j ≤ k ≤ pos(E) |
|---|
| ..6.840. |
A2B1B2. | | |
| |
| |
| |
| |
| |
| |
| |
| |
| Subtotal: 6.840 / 1 / 10 |
| Total: 71.781 / 9 / 35 |
Contributions to nonvanishing quartic force field constants
| Irrep combinations (i,i,i,i) with indices: pos(A1) ≤ i ≤ pos(E) |
|---|
| ..7.315. |
A1A1A1A1. | ..5.985. |
A2A2A2A2. | ..8.855. |
B1B1B1B1. | ..7.315. |
B2B2B2B2. | ..338.846. |
EEEE. | | |
| |
| |
| |
| |
| Subtotal: 368.316 / 5 / 5 |
| Irrep combinations (i,i,i,j) (i,j,j,j) with indices: pos(A1) ≤ i ≤ j ≤ pos(E) |
|---|
| Subtotal: 0 / 0 / 20 |
| Irrep combinations (i,i,j,j) with indices: pos(A1) ≤ i ≤ j ≤ pos(E) |
|---|
| ..32.490. |
A1A1A2A2. | ..39.900. |
A1A1B1B1. | ..36.100. |
A1A1B2B2. | ..133.570. |
A1A1EE. | ..35.910. |
A2A2B1B1. | ..32.490. |
A2A2B2B2. | ..120.213. |
A2A2EE. | ..39.900. |
B1B1B2B2. | ..147.630. |
B1B1EE. | ..133.570. |
B2B2EE. |
| Subtotal: 751.773 / 10 / 10 |
| Irrep combinations (i,i,j,k) (i,j,j,k) (i,j,k,k) with indices: pos(A1) ≤ i ≤ j ≤ k ≤ pos(E) |
|---|
| ..227.772. |
A1A2EE. | ..267.140. |
A1B1EE. | ..253.783. |
A1B2EE. | ..253.080. |
A2B1EE. | ..240.426. |
A2B2EE. | ..253.080. |
B1B2EE. | | |
| |
| |
| |
| Subtotal: 1.495.281 / 6 / 30 |
| Irrep combinations (i,j,k,l) with indices: pos(A1) ≤ i ≤ j ≤ k ≤ l ≤ pos(E) |
|---|
| ..129.960. |
A1A2B1B2. | | |
| |
| |
| |
| |
| |
| |
| |
| |
| Subtotal: 129.960 / 1 / 5 |
| Total: 2.745.330 / 22 / 70 |
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