Characters of representations for molecular motions
| Motion |
E |
C2.(z) |
C2.(y) |
C2.(x) |
i |
σ.(xy) |
σ.(xy) |
σ.(xy) |
| Cartesian 3N |
120 |
0 |
0 |
0 |
0 |
12 |
8 |
4 |
| 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 |
114 |
2 |
2 |
2 |
0 |
12 |
8 |
4 |
Decomposition to irreducible representations
| Motion |
A1g |
B1g |
B2g |
B3g |
A1u |
B1u |
B2u |
B3u |
Total |
| Cartesian 3N |
18 |
15 |
14 |
13 |
12 |
15 |
16 |
17 |
120 |
| 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 |
18 |
14 |
13 |
12 |
12 |
14 |
15 |
16 |
114 |
Molecular parameter
| Number of Atoms (N) |
40
|
| Number of internal coordinates |
114
|
| Number of independant internal coordinates |
18
|
| Number of vibrational modes |
114
|
Force field analysis
Allowed / forbidden vibronational transitions
| Operator |
A1g |
B1g |
B2g |
B3g |
A1u |
B1u |
B2u |
B3u |
Total |
| Linear (IR) |
18 |
14 |
13 |
12 |
12 |
14 |
15 |
16 |
45 / 69 |
| Quadratic (Raman) |
18 |
14 |
13 |
12 |
12 |
14 |
15 |
16 |
57 / 57 |
| IR + Raman |
- - - - |
- - - - |
- - - - |
- - - - |
12 |
- - - - |
- - - - |
- - - - |
0* / 12 |
* 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 |
114 |
2 |
2 |
2 |
0 |
12 |
8 |
4 |
| quadratic |
6.555 |
59 |
59 |
59 |
57 |
129 |
89 |
65 |
| cubic |
253.460 |
116 |
116 |
116 |
0 |
976 |
544 |
240 |
| quartic |
7.413.705 |
1.769 |
1.769 |
1.769 |
1.653 |
6.669 |
3.669 |
2.125 |
| quintic |
174.963.438 |
3.422 |
3.422 |
3.422 |
0 |
38.844 |
18.600 |
7.316 |
| sextic |
3.470.108.187 |
35.931 |
35.931 |
35.931 |
32.509 |
208.845 |
96.957 |
46.669 |
Decomposition to irreducible representations
Column with number of nonvanshing force constants highlighted
| Force field |
A1g |
B1g |
B2g |
B3g |
A1u |
B1u |
B2u |
B3u |
| linear |
18 |
14 |
13 |
12 |
12 |
14 |
15 |
16 |
| quadratic |
884 |
816 |
806 |
800 |
799 |
808 |
818 |
824 |
| cubic |
31.946 |
31.692 |
31.584 |
31.508 |
31.506 |
31.644 |
31.752 |
31.828 |
| quartic |
929.141 |
926.808 |
926.058 |
925.672 |
925.612 |
926.176 |
926.926 |
927.312 |
| quintic |
21.879.808 |
21.871.618 |
21.866.557 |
21.863.736 |
21.863.618 |
21.868.386 |
21.873.447 |
21.876.268 |
| sextic |
433.825.120 |
433.771.248 |
433.743.276 |
433.730.704 |
433.728.875 |
433.746.816 |
433.774.788 |
433.787.360 |
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) |
|---|
| ..171. |
A1gA1g. | ..105. |
B1gB1g. | ..91. |
B2gB2g. | ..78. |
B3gB3g. | ..78. |
A1uA1u. | ..105. |
B1uB1u. | ..120. |
B2uB2u. | ..136. |
B3uB3u. | | |
| |
| Subtotal: 884 / 8 / 8 |
| Irrep combinations (i,j) with indices: pos(A1g) ≤ i ≤ j ≤ pos(B3u) |
|---|
| Subtotal: 0 / 0 / 28 |
| Total: 884 / 8 / 36 |
Contributions to nonvanishing cubic force field constants
| Irrep combinations (i,i,i) with indices: pos(A1g) ≤ i ≤ pos(B3u) |
|---|
| ..1.140. |
A1gA1gA1g. | | |
| |
| |
| |
| |
| |
| |
| |
| |
| Subtotal: 1.140 / 1 / 8 |
| Irrep combinations (i,i,j) (i,j,j) with indices: pos(A1g) ≤ i ≤ j ≤ pos(B3u) |
|---|
| ..1.890. |
A1gB1gB1g. | ..1.638. |
A1gB2gB2g. | ..1.404. |
A1gB3gB3g. | ..1.404. |
A1gA1uA1u. | ..1.890. |
A1gB1uB1u. | ..2.160. |
A1gB2uB2u. | ..2.448. |
A1gB3uB3u. | | |
| |
| |
| Subtotal: 12.834 / 7 / 56 |
| Irrep combinations (i,j,k) with indices: pos(A1g) ≤ i ≤ j ≤ k ≤ pos(B3u) |
|---|
| ..2.184. |
B1gB2gB3g. | ..2.352. |
B1gA1uB1u. | ..3.360. |
B1gB2uB3u. | ..2.340. |
B2gA1uB2u. | ..2.912. |
B2gB1uB3u. | ..2.304. |
B3gA1uB3u. | ..2.520. |
B3gB1uB2u. | | |
| |
| |
| Subtotal: 17.972 / 7 / 56 |
| Total: 31.946 / 15 / 120 |
Contributions to nonvanishing quartic force field constants
| Irrep combinations (i,i,i,i) with indices: pos(A1g) ≤ i ≤ pos(B3u) |
|---|
| ..5.985. |
A1gA1gA1gA1g. | ..2.380. |
B1gB1gB1gB1g. | ..1.820. |
B2gB2gB2gB2g. | ..1.365. |
B3gB3gB3gB3g. | ..1.365. |
A1uA1uA1uA1u. | ..2.380. |
B1uB1uB1uB1u. | ..3.060. |
B2uB2uB2uB2u. | ..3.876. |
B3uB3uB3uB3u. | | |
| |
| Subtotal: 22.231 / 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) |
|---|
| ..17.955. |
A1gA1gB1gB1g. | ..15.561. |
A1gA1gB2gB2g. | ..13.338. |
A1gA1gB3gB3g. | ..13.338. |
A1gA1gA1uA1u. | ..17.955. |
A1gA1gB1uB1u. | ..20.520. |
A1gA1gB2uB2u. | ..23.256. |
A1gA1gB3uB3u. | ..9.555. |
B1gB1gB2gB2g. | ..8.190. |
B1gB1gB3gB3g. | ..8.190. |
B1gB1gA1uA1u. |
| ..11.025. |
B1gB1gB1uB1u. | ..12.600. |
B1gB1gB2uB2u. | ..14.280. |
B1gB1gB3uB3u. | ..7.098. |
B2gB2gB3gB3g. | ..7.098. |
B2gB2gA1uA1u. | ..9.555. |
B2gB2gB1uB1u. | ..10.920. |
B2gB2gB2uB2u. | ..12.376. |
B2gB2gB3uB3u. | ..6.084. |
B3gB3gA1uA1u. | ..8.190. |
B3gB3gB1uB1u. |
| ..9.360. |
B3gB3gB2uB2u. | ..10.608. |
B3gB3gB3uB3u. | ..8.190. |
A1uA1uB1uB1u. | ..9.360. |
A1uA1uB2uB2u. | ..10.608. |
A1uA1uB3uB3u. | ..12.600. |
B1uB1uB2uB2u. | ..14.280. |
B1uB1uB3uB3u. | ..16.320. |
B2uB2uB3uB3u. | | |
| |
| Subtotal: 338.410 / 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) |
|---|
| ..39.312. |
A1gB1gB2gB3g. | ..42.336. |
A1gB1gA1uB1u. | ..60.480. |
A1gB1gB2uB3u. | ..42.120. |
A1gB2gA1uB2u. | ..52.416. |
A1gB2gB1uB3u. | ..41.472. |
A1gB3gA1uB3u. | ..45.360. |
A1gB3gB1uB2u. | ..34.944. |
B1gB2gA1uB3u. | ..38.220. |
B1gB2gB1uB2u. | ..30.240. |
B1gB3gA1uB2u. |
| ..37.632. |
B1gB3gB1uB3u. | ..26.208. |
B2gB3gA1uB1u. | ..37.440. |
B2gB3gB2uB3u. | ..40.320. |
A1uB1uB2uB3u. | | |
| |
| |
| |
| |
| |
| Subtotal: 568.500 / 14 / 70 |
| Total: 929.141 / 50 / 330 |
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