Characters of representations for molecular motions
| Motion |
E |
C2.(z) |
C2.(y) |
C2.(x) |
i |
σ.(xy) |
σ.(xy) |
σ.(xy) |
| Cartesian 3N |
156 |
0 |
-2 |
-2 |
0 |
4 |
14 |
14 |
| Translation (x,y,z) |
3 |
-1 |
-1 |
-1 |
-3 |
1 |
1 |
1 |
| Rotation (Rx,Ry,Rz) |
3 |
-1 |
-1 |
-1 |
3 |
-1 |
-1 |
-1 |
| Vibration |
150 |
2 |
0 |
0 |
0 |
4 |
14 |
14 |
Decomposition to irreducible representations
| Motion |
A1g |
B1g |
B2g |
B3g |
A1u |
B1u |
B2u |
B3u |
Total |
| Cartesian 3N |
23 |
17 |
19 |
19 |
15 |
23 |
20 |
20 |
156 |
| Translation (x,y,z) |
0 |
0 |
0 |
0 |
0 |
1 |
1 |
1 |
3 |
| Rotation (Rx,Ry,Rz) |
0 |
1 |
1 |
1 |
0 |
0 |
0 |
0 |
3 |
| Vibration |
23 |
16 |
18 |
18 |
15 |
22 |
19 |
19 |
150 |
Molecular parameter
| Number of Atoms (N) |
52
|
| Number of internal coordinates |
150
|
| Number of independant internal coordinates |
23
|
| Number of vibrational modes |
150
|
Force field analysis
Allowed / forbidden vibronational transitions
| Operator |
A1g |
B1g |
B2g |
B3g |
A1u |
B1u |
B2u |
B3u |
Total |
| Linear (IR) |
23 |
16 |
18 |
18 |
15 |
22 |
19 |
19 |
60 / 90 |
| Quadratic (Raman) |
23 |
16 |
18 |
18 |
15 |
22 |
19 |
19 |
75 / 75 |
| IR + Raman |
- - - - |
- - - - |
- - - - |
- - - - |
15 |
- - - - |
- - - - |
- - - - |
0* / 15 |
* Parity Mutual Exclusion Principle
Characters of force fields
(Symmetric powers of vibration representation)
| Force field |
E |
C2.(z) |
C2.(y) |
C2.(x) |
i |
σ.(xy) |
σ.(xy) |
σ.(xy) |
| linear |
150 |
2 |
0 |
0 |
0 |
4 |
14 |
14 |
| quadratic |
11.325 |
77 |
75 |
75 |
75 |
83 |
173 |
173 |
| cubic |
573.800 |
152 |
0 |
0 |
0 |
312 |
1.512 |
1.512 |
| quartic |
21.947.850 |
3.002 |
2.850 |
2.850 |
2.850 |
3.466 |
11.866 |
11.866 |
| quintic |
675.993.780 |
5.852 |
0 |
0 |
0 |
12.320 |
79.492 |
79.492 |
| sextic |
17.463.172.650 |
79.002 |
73.150 |
73.150 |
73.150 |
97.174 |
490.042 |
490.042 |
Decomposition to irreducible representations
Column with number of nonvanshing force constants highlighted
| Force field |
A1g |
B1g |
B2g |
B3g |
A1u |
B1u |
B2u |
B3u |
| linear |
23 |
16 |
18 |
18 |
15 |
22 |
19 |
19 |
| quadratic |
1.507 |
1.383 |
1.405 |
1.405 |
1.381 |
1.430 |
1.407 |
1.407 |
| cubic |
72.161 |
71.405 |
71.667 |
71.667 |
71.327 |
72.083 |
71.745 |
71.745 |
| quartic |
2.748.325 |
2.740.967 |
2.743.029 |
2.743.029 |
2.740.813 |
2.745.321 |
2.743.183 |
2.743.183 |
| quintic |
84.521.367 |
84.481.621 |
84.496.951 |
84.496.951 |
84.478.541 |
84.518.287 |
84.500.031 |
84.500.031 |
| sextic |
2.183.068.545 |
2.182.786.949 |
2.182.883.703 |
2.182.883.703 |
2.182.780.943 |
2.182.989.389 |
2.182.889.709 |
2.182.889.709 |
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
2h
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(A1g) ≤ i ≤ pos(B3u) |
|---|
| ..276. |
A1gA1g. | ..136. |
B1gB1g. | ..171. |
B2gB2g. | ..171. |
B3gB3g. | ..120. |
A1uA1u. | ..253. |
B1uB1u. | ..190. |
B2uB2u. | ..190. |
B3uB3u. | | |
| |
| Subtotal: 1.507 / 8 / 8 |
| Irrep combinations (i,j) with indices: pos(A1g) ≤ i ≤ j ≤ pos(B3u) |
|---|
| Subtotal: 0 / 0 / 28 |
| Total: 1.507 / 8 / 36 |
Contributions to nonvanishing cubic force field constants
| Irrep combinations (i,i,i) with indices: pos(A1g) ≤ i ≤ pos(B3u) |
|---|
| ..2.300. |
A1gA1gA1g. | | |
| |
| |
| |
| |
| |
| |
| |
| |
| Subtotal: 2.300 / 1 / 8 |
| Irrep combinations (i,i,j) (i,j,j) with indices: pos(A1g) ≤ i ≤ j ≤ pos(B3u) |
|---|
| ..3.128. |
A1gB1gB1g. | ..3.933. |
A1gB2gB2g. | ..3.933. |
A1gB3gB3g. | ..2.760. |
A1gA1uA1u. | ..5.819. |
A1gB1uB1u. | ..4.370. |
A1gB2uB2u. | ..4.370. |
A1gB3uB3u. | | |
| |
| |
| Subtotal: 28.313 / 7 / 56 |
| Irrep combinations (i,j,k) with indices: pos(A1g) ≤ i ≤ j ≤ k ≤ pos(B3u) |
|---|
| ..5.184. |
B1gB2gB3g. | ..5.280. |
B1gA1uB1u. | ..5.776. |
B1gB2uB3u. | ..5.130. |
B2gA1uB2u. | ..7.524. |
B2gB1uB3u. | ..5.130. |
B3gA1uB3u. | ..7.524. |
B3gB1uB2u. | | |
| |
| |
| Subtotal: 41.548 / 7 / 56 |
| Total: 72.161 / 15 / 120 |
Contributions to nonvanishing quartic force field constants
| Irrep combinations (i,i,i,i) with indices: pos(A1g) ≤ i ≤ pos(B3u) |
|---|
| ..14.950. |
A1gA1gA1gA1g. | ..3.876. |
B1gB1gB1gB1g. | ..5.985. |
B2gB2gB2gB2g. | ..5.985. |
B3gB3gB3gB3g. | ..3.060. |
A1uA1uA1uA1u. | ..12.650. |
B1uB1uB1uB1u. | ..7.315. |
B2uB2uB2uB2u. | ..7.315. |
B3uB3uB3uB3u. | | |
| |
| Subtotal: 61.136 / 8 / 8 |
| Irrep combinations (i,i,i,j) (i,j,j,j) with indices: pos(A1g) ≤ i ≤ j ≤ pos(B3u) |
|---|
| Subtotal: 0 / 0 / 56 |
| Irrep combinations (i,i,j,j) with indices: pos(A1g) ≤ i ≤ j ≤ pos(B3u) |
|---|
| ..37.536. |
A1gA1gB1gB1g. | ..47.196. |
A1gA1gB2gB2g. | ..47.196. |
A1gA1gB3gB3g. | ..33.120. |
A1gA1gA1uA1u. | ..69.828. |
A1gA1gB1uB1u. | ..52.440. |
A1gA1gB2uB2u. | ..52.440. |
A1gA1gB3uB3u. | ..23.256. |
B1gB1gB2gB2g. | ..23.256. |
B1gB1gB3gB3g. | ..16.320. |
B1gB1gA1uA1u. |
| ..34.408. |
B1gB1gB1uB1u. | ..25.840. |
B1gB1gB2uB2u. | ..25.840. |
B1gB1gB3uB3u. | ..29.241. |
B2gB2gB3gB3g. | ..20.520. |
B2gB2gA1uA1u. | ..43.263. |
B2gB2gB1uB1u. | ..32.490. |
B2gB2gB2uB2u. | ..32.490. |
B2gB2gB3uB3u. | ..20.520. |
B3gB3gA1uA1u. | ..43.263. |
B3gB3gB1uB1u. |
| ..32.490. |
B3gB3gB2uB2u. | ..32.490. |
B3gB3gB3uB3u. | ..30.360. |
A1uA1uB1uB1u. | ..22.800. |
A1uA1uB2uB2u. | ..22.800. |
A1uA1uB3uB3u. | ..48.070. |
B1uB1uB2uB2u. | ..48.070. |
B1uB1uB3uB3u. | ..36.100. |
B2uB2uB3uB3u. | | |
| |
| Subtotal: 983.643 / 28 / 28 |
| Irrep combinations (i,i,j,k) (i,j,j,k) (i,j,k,k) with indices: pos(A1g) ≤ i ≤ j ≤ k ≤ pos(B3u) |
|---|
| Subtotal: 0 / 0 / 168 |
| Irrep combinations (i,j,k,l) with indices: pos(A1g) ≤ i ≤ j ≤ k ≤ l ≤ pos(B3u) |
|---|
| ..119.232. |
A1gB1gB2gB3g. | ..121.440. |
A1gB1gA1uB1u. | ..132.848. |
A1gB1gB2uB3u. | ..117.990. |
A1gB2gA1uB2u. | ..173.052. |
A1gB2gB1uB3u. | ..117.990. |
A1gB3gA1uB3u. | ..173.052. |
A1gB3gB1uB2u. | ..82.080. |
B1gB2gA1uB3u. | ..120.384. |
B1gB2gB1uB2u. | ..82.080. |
B1gB3gA1uB2u. |
| ..120.384. |
B1gB3gB1uB3u. | ..106.920. |
B2gB3gA1uB1u. | ..116.964. |
B2gB3gB2uB3u. | ..119.130. |
A1uB1uB2uB3u. | | |
| |
| |
| |
| |
| |
| Subtotal: 1.703.546 / 14 / 70 |
| Total: 2.748.325 / 50 / 330 |
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