Characters of representations for molecular motions
| Motion |
E |
2C6 |
2C3 |
C2 |
3C'2 |
3C''2 |
i |
2S3 |
2S6 |
σh |
3σd |
3σv |
| Cartesian 3N |
324 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
24 |
12 |
8 |
| Translation (x,y,z) |
3 |
2 |
0 |
-1 |
-1 |
-1 |
-3 |
-2 |
0 |
1 |
1 |
1 |
| Rotation (Rx,Ry,Rz) |
3 |
2 |
0 |
-1 |
-1 |
-1 |
3 |
2 |
0 |
-1 |
-1 |
-1 |
| Vibration |
318 |
-4 |
0 |
2 |
2 |
2 |
0 |
0 |
0 |
24 |
12 |
8 |
Decomposition to irreducible representations
| Motion |
A1g |
A2g |
B1g |
B2g |
E1g |
E2g |
A1u |
A2u |
B1u |
B2u |
E1u |
E2u |
Total |
| Cartesian 3N |
17 |
12 |
13 |
12 |
25 |
29 |
10 |
15 |
14 |
15 |
29 |
25 |
216 |
| Translation (x,y,z) |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
1 |
0 |
0 |
1 |
0 |
2 |
| Rotation (Rx,Ry,Rz) |
0 |
1 |
0 |
0 |
1 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
2 |
| Vibration |
17 |
11 |
13 |
12 |
24 |
29 |
10 |
14 |
14 |
15 |
28 |
25 |
212 |
Molecular parameter
| Number of Atoms (N) |
108
|
| Number of internal coordinates |
318
|
| Number of independant internal coordinates |
17
|
| Number of vibrational modes |
212
|
Force field analysis
Allowed / forbidden vibronational transitions
| Operator |
A1g |
A2g |
B1g |
B2g |
E1g |
E2g |
A1u |
A2u |
B1u |
B2u |
E1u |
E2u |
Total |
| Linear (IR) |
17 |
11 |
13 |
12 |
24 |
29 |
10 |
14 |
14 |
15 |
28 |
25 |
42 / 170 |
| Quadratic (Raman) |
17 |
11 |
13 |
12 |
24 |
29 |
10 |
14 |
14 |
15 |
28 |
25 |
70 / 142 |
| IR + Raman |
- - - - |
11 |
13 |
12 |
- - - - |
- - - - |
10 |
- - - - |
14 |
15 |
- - - - |
25 |
0* / 100 |
* Parity Mutual Exclusion Principle
Characters of force fields
(Symmetric powers of vibration representation)
| Force field |
E |
2C6 |
2C3 |
C2 |
3C'2 |
3C''2 |
i |
2S3 |
2S6 |
σh |
3σd |
3σv |
| linear |
318 |
-4 |
0 |
2 |
2 |
2 |
0 |
0 |
0 |
24 |
12 |
8 |
| quadratic |
50.721 |
8 |
0 |
161 |
161 |
161 |
159 |
0 |
0 |
447 |
231 |
191 |
| cubic |
5.410.240 |
-10 |
106 |
320 |
320 |
320 |
0 |
8 |
0 |
6.128 |
2.200 |
1.360 |
| quartic |
434.171.760 |
8 |
0 |
13.040 |
13.040 |
13.040 |
12.720 |
0 |
0 |
72.528 |
25.080 |
18.000 |
| quintic |
27.960.661.344 |
-4 |
0 |
25.760 |
25.760 |
25.760 |
0 |
0 |
0 |
741.552 |
201.432 |
116.112 |
| sextic |
1.505.215.602.352 |
55 |
5.671 |
708.400 |
708.400 |
708.400 |
682.640 |
85 |
53 |
6.858.544 |
1.748.824 |
1.120.816 |
Decomposition to irreducible representations
Column with number of nonvanshing force constants highlighted
| Force field |
A1g |
A2g |
B1g |
B2g |
E1g |
E2g |
A1u |
A2u |
B1u |
B2u |
E1u |
E2u |
| linear |
17 |
11 |
13 |
12 |
24 |
29 |
10 |
14 |
14 |
15 |
28 |
25 |
| quadratic |
2.239 |
2.053 |
2.099 |
2.089 |
4.190 |
4.290 |
2.083 |
2.108 |
2.113 |
2.123 |
4.238 |
4.189 |
| cubic |
226.229 |
225.179 |
225.272 |
225.062 |
450.307 |
451.382 |
224.827 |
225.557 |
225.574 |
225.784 |
451.327 |
450.362 |
| quartic |
18.103.231 |
18.085.941 |
18.088.339 |
18.086.569 |
36.174.910 |
36.189.170 |
18.085.357 |
18.089.607 |
18.091.553 |
18.093.323 |
36.184.878 |
36.174.962 |
| quintic |
1.165.105.660 |
1.165.013.394 |
1.165.006.250 |
1.164.984.920 |
2.329.991.169 |
2.330.119.055 |
1.164.964.478 |
1.165.030.984 |
1.165.046.716 |
1.165.068.046 |
2.330.114.761 |
2.329.995.463 |
| sextic |
62.718.196.791 |
62.717.125.181 |
62.717.108.885 |
62.716.951.883 |
125.434.059.372 |
125.435.320.506 |
62.716.850.926 |
62.717.214.136 |
62.717.466.547 |
62.717.623.549 |
125.435.088.684 |
125.434.063.665 |
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
6h
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(E2u) |
|---|
| ..153. |
A1gA1g. | ..66. |
A2gA2g. | ..91. |
B1gB1g. | ..78. |
B2gB2g. | ..300. |
E1gE1g. | ..435. |
E2gE2g. | ..55. |
A1uA1u. | ..105. |
A2uA2u. | ..105. |
B1uB1u. | ..120. |
B2uB2u. |
| ..406. |
E1uE1u. | ..325. |
E2uE2u. | | |
| |
| |
| |
| |
| |
| |
| |
| Subtotal: 2.239 / 12 / 12 |
| Irrep combinations (i,j) with indices: pos(A1g) ≤ i ≤ j ≤ pos(E2u) |
|---|
| Subtotal: 0 / 0 / 66 |
| Total: 2.239 / 12 / 78 |
Contributions to nonvanishing cubic force field constants
| Irrep combinations (i,i,i) with indices: pos(A1g) ≤ i ≤ pos(E2u) |
|---|
| ..969. |
A1gA1gA1g. | ..4.495. |
E2gE2gE2g. | | |
| |
| |
| |
| |
| |
| |
| |
| Subtotal: 5.464 / 2 / 12 |
| Irrep combinations (i,i,j) (i,j,j) with indices: pos(A1g) ≤ i ≤ j ≤ pos(E2u) |
|---|
| ..8.700. |
E1gE1gE2g. | ..1.122. |
A1gA2gA2g. | ..1.547. |
A1gB1gB1g. | ..1.326. |
A1gB2gB2g. | ..5.100. |
A1gE1gE1g. | ..7.395. |
A1gE2gE2g. | ..935. |
A1gA1uA1u. | ..1.785. |
A1gA2uA2u. | ..1.785. |
A1gB1uB1u. | ..2.040. |
A1gB2uB2u. |
| ..6.902. |
A1gE1uE1u. | ..5.525. |
A1gE2uE2u. | ..3.036. |
A2gE1gE1g. | ..4.466. |
A2gE2gE2g. | ..4.158. |
A2gE1uE1u. | ..3.300. |
A2gE2uE2u. | ..11.774. |
E2gE1uE1u. | ..9.425. |
E2gE2uE2u. | | |
| |
| Subtotal: 80.321 / 18 / 132 |
| Irrep combinations (i,j,k) with indices: pos(A1g) ≤ i ≤ j ≤ k ≤ pos(E2u) |
|---|
| ..1.716. |
A2gB1gB2g. | ..1.540. |
A2gA1uA2u. | ..2.310. |
A2gB1uB2u. | ..9.048. |
B1gE1gE2g. | ..1.820. |
B1gA1uB1u. | ..2.730. |
B1gA2uB2u. | ..9.100. |
B1gE1uE2u. | ..8.352. |
B2gE1gE2g. | ..1.800. |
B2gA1uB2u. | ..2.352. |
B2gA2uB1u. |
| ..8.400. |
B2gE1uE2u. | ..6.720. |
E1gA1uE1u. | ..9.408. |
E1gA2uE1u. | ..8.400. |
E1gB1uE2u. | ..9.000. |
E1gB2uE2u. | ..16.800. |
E1gE1uE2u. | ..7.250. |
E2gA1uE2u. | ..10.150. |
E2gA2uE2u. | ..11.368. |
E2gB1uE1u. | ..12.180. |
E2gB2uE1u. |
| Subtotal: 140.444 / 20 / 220 |
| Total: 226.229 / 40 / 364 |
Contributions to nonvanishing quartic force field constants
| Irrep combinations (i,i,i,i) with indices: pos(A1g) ≤ i ≤ pos(E2u) |
|---|
| ..4.845. |
A1gA1gA1gA1g. | ..1.001. |
A2gA2gA2gA2g. | ..1.820. |
B1gB1gB1gB1g. | ..1.365. |
B2gB2gB2gB2g. | ..45.150. |
E1gE1gE1gE1g. | ..94.830. |
E2gE2gE2gE2g. | ..715. |
A1uA1uA1uA1u. | ..2.380. |
A2uA2uA2uA2u. | ..2.380. |
B1uB1uB1uB1u. | ..3.060. |
B2uB2uB2uB2u. |
| ..82.621. |
E1uE1uE1uE1u. | ..52.975. |
E2uE2uE2uE2u. | | |
| |
| |
| |
| |
| |
| |
| |
| Subtotal: 293.142 / 12 / 12 |
| Irrep combinations (i,i,i,j) (i,j,j,j) with indices: pos(A1g) ≤ i ≤ j ≤ pos(E2u) |
|---|
| ..76.415. |
A1gE2gE2gE2g. | ..49.445. |
A2gE2gE2gE2g. | ..33.800. |
B1gE1gE1gE1g. | ..31.200. |
B2gE1gE1gE1g. | ..29.250. |
A1uE2uE2uE2u. | ..40.950. |
A2uE2uE2uE2u. | ..56.840. |
B1uE1uE1uE1u. | ..60.900. |
B2uE1uE1uE1u. | | |
| |
| Subtotal: 378.800 / 8 / 132 |
| Irrep combinations (i,i,j,j) with indices: pos(A1g) ≤ i ≤ j ≤ pos(E2u) |
|---|
| ..10.098. |
A1gA1gA2gA2g. | ..13.923. |
A1gA1gB1gB1g. | ..11.934. |
A1gA1gB2gB2g. | ..45.900. |
A1gA1gE1gE1g. | ..66.555. |
A1gA1gE2gE2g. | ..8.415. |
A1gA1gA1uA1u. | ..16.065. |
A1gA1gA2uA2u. | ..16.065. |
A1gA1gB1uB1u. | ..18.360. |
A1gA1gB2uB2u. | ..62.118. |
A1gA1gE1uE1u. |
| ..49.725. |
A1gA1gE2uE2u. | ..6.006. |
A2gA2gB1gB1g. | ..5.148. |
A2gA2gB2gB2g. | ..19.800. |
A2gA2gE1gE1g. | ..28.710. |
A2gA2gE2gE2g. | ..3.630. |
A2gA2gA1uA1u. | ..6.930. |
A2gA2gA2uA2u. | ..6.930. |
A2gA2gB1uB1u. | ..7.920. |
A2gA2gB2uB2u. | ..26.796. |
A2gA2gE1uE1u. |
| ..21.450. |
A2gA2gE2uE2u. | ..7.098. |
B1gB1gB2gB2g. | ..27.300. |
B1gB1gE1gE1g. | ..39.585. |
B1gB1gE2gE2g. | ..5.005. |
B1gB1gA1uA1u. | ..9.555. |
B1gB1gA2uA2u. | ..9.555. |
B1gB1gB1uB1u. | ..10.920. |
B1gB1gB2uB2u. | ..36.946. |
B1gB1gE1uE1u. | ..29.575. |
B1gB1gE2uE2u. |
| ..23.400. |
B2gB2gE1gE1g. | ..33.930. |
B2gB2gE2gE2g. | ..4.290. |
B2gB2gA1uA1u. | ..8.190. |
B2gB2gA2uA2u. | ..8.190. |
B2gB2gB1uB1u. | ..9.360. |
B2gB2gB2uB2u. | ..31.668. |
B2gB2gE1uE1u. | ..25.350. |
B2gB2gE2uE2u. | ..373.056. |
E1gE1gE2gE2g. | ..16.500. |
E1gE1gA1uA1u. |
| ..31.500. |
E1gE1gA2uA2u. | ..31.500. |
E1gE1gB1uB1u. | ..36.000. |
E1gE1gB2uB2u. | ..347.928. |
E1gE1gE1uE1u. | ..277.800. |
E1gE1gE2uE2u. | ..23.925. |
E2gE2gA1uA1u. | ..45.675. |
E2gE2gA2uA2u. | ..45.675. |
E2gE2gB1uB1u. | ..52.200. |
E2gE2gB2uB2u. | ..506.688. |
E2gE2gE1uE1u. |
| ..404.550. |
E2gE2gE2uE2u. | ..5.775. |
A1uA1uA2uA2u. | ..5.775. |
A1uA1uB1uB1u. | ..6.600. |
A1uA1uB2uB2u. | ..22.330. |
A1uA1uE1uE1u. | ..17.875. |
A1uA1uE2uE2u. | ..11.025. |
A2uA2uB1uB1u. | ..12.600. |
A2uA2uB2uB2u. | ..42.630. |
A2uA2uE1uE1u. | ..34.125. |
A2uA2uE2uE2u. |
| ..12.600. |
B1uB1uB2uB2u. | ..42.630. |
B1uB1uE1uE1u. | ..34.125. |
B1uB1uE2uE2u. | ..48.720. |
B2uB2uE1uE1u. | ..39.000. |
B2uB2uE2uE2u. | ..377.300. |
E1uE1uE2uE2u. | | |
| |
| |
| |
| Subtotal: 3.678.502 / 66 / 66 |
| Irrep combinations (i,i,j,k) (i,j,j,k) (i,j,k,k) with indices: pos(A1g) ≤ i ≤ j ≤ k ≤ pos(E2u) |
|---|
| ..38.640. |
E1gE1gA1uA2u. | ..75.000. |
E1gE1gA1uE2u. | ..105.000. |
E1gE1gA2uE2u. | ..57.960. |
E1gE1gB1uB2u. | ..117.600. |
E1gE1gB1uE1u. | ..126.000. |
E1gE1gB2uE1u. | ..56.840. |
E2gE2gA1uA2u. | ..108.750. |
E2gE2gA1uE2u. | ..152.250. |
E2gE2gA2uE2u. | ..85.260. |
E2gE2gB1uB2u. |
| ..170.520. |
E2gE2gB1uE1u. | ..182.700. |
E2gE2gB2uE1u. | ..147.900. |
A1gE1gE1gE2g. | ..95.700. |
A2gE1gE1gE2g. | ..101.500. |
A1uE1uE1uE2u. | ..142.100. |
A2uE1uE1uE2u. | ..51.612. |
A1gA2gE1gE1g. | ..75.922. |
A1gA2gE2gE2g. | ..70.686. |
A1gA2gE1uE1u. | ..56.100. |
A1gA2gE2uE2u. |
| ..200.158. |
A1gE2gE1uE1u. | ..160.225. |
A1gE2gE2uE2u. | ..129.514. |
A2gE2gE1uE1u. | ..103.675. |
A2gE2gE2uE2u. | ..43.056. |
B1gB2gE1gE1g. | ..63.336. |
B1gB2gE2gE2g. | ..58.968. |
B1gB2gE1uE1u. | ..46.800. |
B1gB2gE2uE2u. | ..135.720. |
B1gE1gE2gE2g. | ..126.672. |
B1gE1gE1uE1u. |
| ..101.400. |
B1gE1gE2uE2u. | ..125.280. |
B2gE1gE2gE2g. | ..116.928. |
B2gE1gE1uE1u. | ..93.600. |
B2gE1gE2uE2u. | ..52.920. |
A1uA2uE1uE1u. | ..42.000. |
A1uA2uE2uE2u. | ..79.380. |
B1uB2uE1uE1u. | ..63.000. |
B1uB2uE2uE2u. | ..127.400. |
B1uE1uE2uE2u. | ..136.500. |
B2uE1uE2uE2u. |
| Subtotal: 4.024.572 / 40 / 660 |
| Irrep combinations (i,j,k,l) with indices: pos(A1g) ≤ i ≤ j ≤ k ≤ l ≤ pos(E2u) |
|---|
| ..29.172. |
A1gA2gB1gB2g. | ..26.180. |
A1gA2gA1uA2u. | ..39.270. |
A1gA2gB1uB2u. | ..153.816. |
A1gB1gE1gE2g. | ..30.940. |
A1gB1gA1uB1u. | ..46.410. |
A1gB1gA2uB2u. | ..154.700. |
A1gB1gE1uE2u. | ..141.984. |
A1gB2gE1gE2g. | ..30.600. |
A1gB2gA1uB2u. | ..39.984. |
A1gB2gA2uB1u. |
| ..142.800. |
A1gB2gE1uE2u. | ..114.240. |
A1gE1gA1uE1u. | ..159.936. |
A1gE1gA2uE1u. | ..142.800. |
A1gE1gB1uE2u. | ..153.000. |
A1gE1gB2uE2u. | ..285.600. |
A1gE1gE1uE2u. | ..123.250. |
A1gE2gA1uE2u. | ..172.550. |
A1gE2gA2uE2u. | ..193.256. |
A1gE2gB1uE1u. | ..207.060. |
A1gE2gB2uE1u. |
| ..99.528. |
A2gB1gE1gE2g. | ..21.450. |
A2gB1gA1uB2u. | ..28.028. |
A2gB1gA2uB1u. | ..100.100. |
A2gB1gE1uE2u. | ..91.872. |
A2gB2gE1gE2g. | ..18.480. |
A2gB2gA1uB1u. | ..27.720. |
A2gB2gA2uB2u. | ..92.400. |
A2gB2gE1uE2u. | ..73.920. |
A2gE1gA1uE1u. | ..103.488. |
A2gE1gA2uE1u. |
| ..92.400. |
A2gE1gB1uE2u. | ..99.000. |
A2gE1gB2uE2u. | ..184.800. |
A2gE1gE1uE2u. | ..79.750. |
A2gE2gA1uE2u. | ..111.650. |
A2gE2gA2uE2u. | ..125.048. |
A2gE2gB1uE1u. | ..133.980. |
A2gE2gB2uE1u. | ..21.840. |
B1gB2gA1uA2u. | ..32.760. |
B1gB2gB1uB2u. | ..78.000. |
B1gE1gA1uE2u. |
| ..109.200. |
B1gE1gA2uE2u. | ..122.304. |
B1gE1gB1uE1u. | ..131.040. |
B1gE1gB2uE1u. | ..105.560. |
B1gE2gA1uE1u. | ..147.784. |
B1gE2gA2uE1u. | ..131.950. |
B1gE2gB1uE2u. | ..141.375. |
B1gE2gB2uE2u. | ..263.900. |
B1gE2gE1uE2u. | ..72.000. |
B2gE1gA1uE2u. | ..100.800. |
B2gE1gA2uE2u. |
| ..112.896. |
B2gE1gB1uE1u. | ..120.960. |
B2gE1gB2uE1u. | ..97.440. |
B2gE2gA1uE1u. | ..136.416. |
B2gE2gA2uE1u. | ..121.800. |
B2gE2gB1uE2u. | ..130.500. |
B2gE2gB2uE2u. | ..243.600. |
B2gE2gE1uE2u. | ..97.440. |
E1gE2gA1uB1u. | ..104.400. |
E1gE2gA1uB2u. | ..194.880. |
E1gE2gA1uE1u. |
| ..136.416. |
E1gE2gA2uB1u. | ..146.160. |
E1gE2gA2uB2u. | ..272.832. |
E1gE2gA2uE1u. | ..243.600. |
E1gE2gB1uE2u. | ..261.000. |
E1gE2gB2uE2u. | ..1.461.600. |
E1gE2gE1uE2u. | ..29.400. |
A1uA2uB1uB2u. | ..98.000. |
A1uB1uE1uE2u. | ..105.000. |
A1uB2uE1uE2u. | ..137.200. |
A2uB1uE1uE2u. |
| ..147.000. |
A2uB2uE1uE2u. | | |
| |
| |
| |
| |
| |
| |
| |
| |
| Subtotal: 9.728.215 / 71 / 495 |
| Total: 18.103.231 / 197 / 1.365 |
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Last update November, 13th 2023 by A. Gelessus, Impressum, Datenschutzerklärung/DataPrivacyStatement