Characters of representations for molecular motions
| Motion |
E |
2S12 |
2C6 |
2S4 |
2C3 |
2(S12)5 |
C2 |
6C'2 |
6σd |
| Cartesian 3N |
216 |
0.000 |
0 |
0 |
0 |
-0.000 |
0 |
0 |
4 |
| Translation (x,y,z) |
3 |
0.732 |
2 |
-1 |
0 |
-2.732 |
-1 |
-1 |
1 |
| Rotation (Rx,Ry,Rz) |
3 |
-0.732 |
2 |
1 |
0 |
2.732 |
-1 |
-1 |
-1 |
| Vibration |
210 |
0.000 |
-4 |
0 |
0 |
-0.000 |
2 |
2 |
4 |
Decomposition to irreducible representations
| Motion |
A1 |
A2 |
B1 |
B2 |
E1 |
E2 |
E3 |
E4 |
E5 |
Total |
| Cartesian 3N |
10 |
8 |
8 |
10 |
18 |
18 |
18 |
18 |
18 |
126 |
| Translation (x,y,z) |
0 |
0 |
0 |
1 |
1 |
0 |
0 |
0 |
0 |
2 |
| Rotation (Rx,Ry,Rz) |
0 |
1 |
0 |
0 |
0 |
0 |
0 |
0 |
1 |
2 |
| Vibration |
10 |
7 |
8 |
9 |
17 |
18 |
18 |
18 |
17 |
122 |
Molecular parameter
| Number of Atoms (N) |
72
|
| Number of internal coordinates |
210
|
| Number of independant internal coordinates |
10
|
| Number of vibrational modes |
122
|
Force field analysis
Allowed / forbidden vibronational transitions
| Operator |
A1 |
A2 |
B1 |
B2 |
E1 |
E2 |
E3 |
E4 |
E5 |
Total |
| Linear (IR) |
10 |
7 |
8 |
9 |
17 |
18 |
18 |
18 |
17 |
26 / 96 |
| Quadratic (Raman) |
10 |
7 |
8 |
9 |
17 |
18 |
18 |
18 |
17 |
45 / 77 |
| IR + Raman |
- - - - |
7 |
8 |
- - - - |
- - - - |
- - - - |
18 |
18 |
- - - - |
0 / 51 |
Characters of force fields
(Symmetric powers of vibration representation)
| Force field |
E |
2S12 |
2C6 |
2S4 |
2C3 |
2(S12)5 |
C2 |
6C'2 |
6σd |
| linear |
210 |
0.000 |
-4 |
0 |
0 |
-0.000 |
2 |
2 |
4 |
| quadratic |
22.155 |
-2.000 |
8 |
1 |
0 |
-2.000 |
107 |
107 |
113 |
| cubic |
1.565.620 |
0.000 |
-10 |
0 |
70 |
0.000 |
212 |
212 |
432 |
| quartic |
83.369.265 |
2.000 |
8 |
53 |
0 |
2.000 |
5.777 |
5.777 |
6.421 |
| quintic |
3.568.204.542 |
-0.000 |
-4 |
0 |
0 |
-0.000 |
11.342 |
11.342 |
23.540 |
| sextic |
127.860.662.755 |
-1.000 |
37 |
53 |
2.485 |
-1.000 |
209.827 |
209.827 |
244.709 |
Decomposition to irreducible representations
Column with number of nonvanshing force constants highlighted
| Force field |
A1 |
A2 |
B1 |
B2 |
E1 |
E2 |
E3 |
E4 |
E5 |
| linear |
10 |
7 |
8 |
9 |
17 |
18 |
18 |
18 |
17 |
| quadratic |
983 |
873 |
927 |
930 |
1.838 |
1.854 |
1.836 |
1.855 |
1.838 |
| cubic |
65.409 |
65.087 |
65.193 |
65.303 |
130.444 |
130.481 |
130.464 |
130.481 |
130.444 |
| quartic |
3.477.015 |
3.470.916 |
3.473.795 |
3.474.117 |
6.946.958 |
6.947.911 |
6.946.956 |
6.947.928 |
6.946.958 |
| quintic |
148.684.382 |
148.666.941 |
148.672.612 |
148.678.711 |
297.349.433 |
297.351.324 |
297.349.434 |
297.351.324 |
297.349.433 |
| sextic |
5.327.650.206 |
5.327.422.938 |
5.327.527.843 |
5.327.545.284 |
10.655.037.540 |
10.655.072.496 |
10.655.038.152 |
10.655.072.514 |
10.655.037.540 |
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
6d
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(E5) |
|---|
| ..55. |
A1A1. | ..28. |
A2A2. | ..36. |
B1B1. | ..45. |
B2B2. | ..153. |
E1E1. | ..171. |
E2E2. | ..171. |
E3E3. | ..171. |
E4E4. | ..153. |
E5E5. | | |
| Subtotal: 983 / 9 / 9 |
| Irrep combinations (i,j) with indices: pos(A1) ≤ i ≤ j ≤ pos(E5) |
|---|
| Subtotal: 0 / 0 / 36 |
| Total: 983 / 9 / 45 |
Contributions to nonvanishing cubic force field constants
| Irrep combinations (i,i,i) with indices: pos(A1) ≤ i ≤ pos(E5) |
|---|
| ..220. |
A1A1A1. | ..1.140. |
E4E4E4. | | |
| |
| |
| |
| |
| |
| |
| |
| Subtotal: 1.360 / 2 / 9 |
| Irrep combinations (i,i,j) (i,j,j) with indices: pos(A1) ≤ i ≤ j ≤ pos(E5) |
|---|
| ..2.754. |
E1E1E2. | ..3.078. |
E2E2E4. | ..280. |
A1A2A2. | ..360. |
A1B1B1. | ..450. |
A1B2B2. | ..1.530. |
A1E1E1. | ..1.710. |
A1E2E2. | ..1.710. |
A1E3E3. | ..1.710. |
A1E4E4. | ..1.530. |
A1E5E5. |
| ..952. |
A2E1E1. | ..1.071. |
A2E2E2. | ..1.071. |
A2E3E3. | ..1.071. |
A2E4E4. | ..952. |
A2E5E5. | ..1.368. |
B1E3E3. | ..1.539. |
B2E3E3. | ..2.754. |
E2E5E5. | | |
| |
| Subtotal: 25.890 / 18 / 72 |
| Irrep combinations (i,j,k) with indices: pos(A1) ≤ i ≤ j ≤ k ≤ pos(E5) |
|---|
| ..504. |
A2B1B2. | ..2.312. |
B1E1E5. | ..2.592. |
B1E2E4. | ..2.601. |
B2E1E5. | ..2.916. |
B2E2E4. | ..5.508. |
E1E2E3. | ..5.508. |
E1E3E4. | ..5.202. |
E1E4E5. | ..5.508. |
E2E3E5. | ..5.508. |
E3E4E5. |
| Subtotal: 38.159 / 10 / 84 |
| Total: 65.409 / 30 / 165 |
Contributions to nonvanishing quartic force field constants
| Irrep combinations (i,i,i,i) with indices: pos(A1) ≤ i ≤ pos(E5) |
|---|
| ..715. |
A1A1A1A1. | ..210. |
A2A2A2A2. | ..330. |
B1B1B1B1. | ..495. |
B2B2B2B2. | ..11.781. |
E1E1E1E1. | ..14.706. |
E2E2E2E2. | ..20.691. |
E3E3E3E3. | ..14.706. |
E4E4E4E4. | ..11.781. |
E5E5E5E5. | | |
| Subtotal: 75.415 / 9 / 9 |
| Irrep combinations (i,i,i,j) (i,j,j,j) with indices: pos(A1) ≤ i ≤ j ≤ pos(E5) |
|---|
| ..17.442. |
E1E1E1E3. | ..11.400. |
A1E4E4E4. | ..7.980. |
A2E4E4E4. | ..9.120. |
B1E2E2E2. | ..10.260. |
B2E2E2E2. | ..17.442. |
E3E5E5E5. | | |
| |
| |
| |
| Subtotal: 73.644 / 6 / 72 |
| Irrep combinations (i,i,j,j) with indices: pos(A1) ≤ i ≤ j ≤ pos(E5) |
|---|
| ..1.540. |
A1A1A2A2. | ..1.980. |
A1A1B1B1. | ..2.475. |
A1A1B2B2. | ..8.415. |
A1A1E1E1. | ..9.405. |
A1A1E2E2. | ..9.405. |
A1A1E3E3. | ..9.405. |
A1A1E4E4. | ..8.415. |
A1A1E5E5. | ..1.008. |
A2A2B1B1. | ..1.260. |
A2A2B2B2. |
| ..4.284. |
A2A2E1E1. | ..4.788. |
A2A2E2E2. | ..4.788. |
A2A2E3E3. | ..4.788. |
A2A2E4E4. | ..4.284. |
A2A2E5E5. | ..1.620. |
B1B1B2B2. | ..5.508. |
B1B1E1E1. | ..6.156. |
B1B1E2E2. | ..6.156. |
B1B1E3E3. | ..6.156. |
B1B1E4E4. |
| ..5.508. |
B1B1E5E5. | ..6.885. |
B2B2E1E1. | ..7.695. |
B2B2E2E2. | ..7.695. |
B2B2E3E3. | ..7.695. |
B2B2E4E4. | ..6.885. |
B2B2E5E5. | ..46.971. |
E1E1E2E2. | ..46.971. |
E1E1E3E3. | ..46.971. |
E1E1E4E4. | ..65.314. |
E1E1E5E5. |
| ..52.650. |
E2E2E3E3. | ..81.891. |
E2E2E4E4. | ..46.971. |
E2E2E5E5. | ..52.650. |
E3E3E4E4. | ..46.971. |
E3E3E5E5. | ..46.971. |
E4E4E5E5. | | |
| |
| |
| |
| Subtotal: 678.530 / 36 / 36 |
| Irrep combinations (i,i,j,k) (i,j,j,k) (i,j,k,k) with indices: pos(A1) ≤ i ≤ j ≤ k ≤ pos(E5) |
|---|
| ..49.572. |
E1E1E2E4. | ..46.818. |
E1E1E3E5. | ..52.326. |
E2E2E3E5. | ..27.540. |
A1E1E1E2. | ..30.780. |
A1E2E2E4. | ..19.278. |
A2E1E1E2. | ..21.546. |
A2E2E2E4. | ..22.032. |
B1E1E1E4. | ..24.786. |
B2E1E1E4. | ..52.326. |
E1E2E2E3. |
| ..49.419. |
E1E2E2E5. | ..98.838. |
E1E3E3E5. | ..49.419. |
E1E4E4E5. | ..110.808. |
E2E3E3E4. | ..52.326. |
E3E4E4E5. | ..9.520. |
A1A2E1E1. | ..10.710. |
A1A2E2E2. | ..10.710. |
A1A2E3E3. | ..10.710. |
A1A2E4E4. | ..9.520. |
A1A2E5E5. |
| ..13.680. |
A1B1E3E3. | ..15.390. |
A1B2E3E3. | ..27.540. |
A1E2E5E5. | ..9.576. |
A2B1E3E3. | ..10.773. |
A2B2E3E3. | ..19.278. |
A2E2E5E5. | ..9.792. |
B1B2E1E1. | ..11.016. |
B1B2E2E2. | ..11.016. |
B1B2E3E3. | ..11.016. |
B1B2E4E4. |
| ..9.792. |
B1B2E5E5. | ..24.624. |
B1E2E4E4. | ..22.032. |
B1E4E5E5. | ..27.702. |
B2E2E4E4. | ..24.786. |
B2E4E5E5. | ..52.326. |
E1E3E4E4. | ..46.818. |
E1E3E5E5. | ..49.572. |
E2E4E5E5. | | |
| |
| Subtotal: 1.155.713 / 38 / 252 |
| Irrep combinations (i,j,k,l) with indices: pos(A1) ≤ i ≤ j ≤ k ≤ l ≤ pos(E5) |
|---|
| ..5.040. |
A1A2B1B2. | ..23.120. |
A1B1E1E5. | ..25.920. |
A1B1E2E4. | ..26.010. |
A1B2E1E5. | ..29.160. |
A1B2E2E4. | ..55.080. |
A1E1E2E3. | ..55.080. |
A1E1E3E4. | ..52.020. |
A1E1E4E5. | ..55.080. |
A1E2E3E5. | ..55.080. |
A1E3E4E5. |
| ..16.184. |
A2B1E1E5. | ..18.144. |
A2B1E2E4. | ..18.207. |
A2B2E1E5. | ..20.412. |
A2B2E2E4. | ..38.556. |
A2E1E2E3. | ..38.556. |
A2E1E3E4. | ..36.414. |
A2E1E4E5. | ..38.556. |
A2E2E3E5. | ..38.556. |
A2E3E4E5. | ..44.064. |
B1E1E2E3. |
| ..41.616. |
B1E1E2E5. | ..44.064. |
B1E1E3E4. | ..44.064. |
B1E2E3E5. | ..44.064. |
B1E3E4E5. | ..49.572. |
B2E1E2E3. | ..46.818. |
B2E1E2E5. | ..49.572. |
B2E1E3E4. | ..49.572. |
B2E2E3E5. | ..49.572. |
B2E3E4E5. | ..99.144. |
E1E2E3E4. |
| ..187.272. |
E1E2E4E5. | ..99.144. |
E2E3E4E5. | | |
| |
| |
| |
| |
| |
| |
| |
| Subtotal: 1.493.713 / 32 / 126 |
| Total: 3.477.015 / 121 / 495 |
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