Characters of representations for molecular motions
| Motion |
E |
8C3 |
6C2 |
6C4 |
3C2 |
i |
6S4 |
8S6 |
3σh |
6σd |
| Cartesian 3N |
21 |
0 |
-1 |
3 |
-3 |
-3 |
-1 |
0 |
5 |
3 |
| Translation (x,y,z) |
3 |
0 |
-1 |
1 |
-1 |
-3 |
-1 |
0 |
1 |
1 |
| Rotation (Rx,Ry,Rz) |
3 |
0 |
-1 |
1 |
-1 |
3 |
1 |
0 |
-1 |
-1 |
| Vibration |
15 |
0 |
1 |
1 |
-1 |
-3 |
-1 |
0 |
5 |
3 |
Decomposition to irreducible representations
| Motion |
A1g |
A2g |
Eg |
T1g |
T2g |
A1u |
A2u |
Eu |
T1u |
T2u |
Total |
| Cartesian 3N |
1 |
0 |
1 |
1 |
1 |
0 |
0 |
0 |
3 |
1 |
8 |
| Translation (x,y,z) |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
1 |
0 |
1 |
| Rotation (Rx,Ry,Rz) |
0 |
0 |
0 |
1 |
0 |
0 |
0 |
0 |
0 |
0 |
1 |
| Vibration |
1 |
0 |
1 |
0 |
1 |
0 |
0 |
0 |
2 |
1 |
6 |
Molecular parameter
| Number of Atoms (N) |
7
|
| Number of internal coordinates |
15
|
| Number of independant internal coordinates |
1
|
| Number of vibrational modes |
6
|
Force field analysis
Allowed / forbidden vibronational transitions
| Operator |
A1g |
A2g |
Eg |
T1g |
T2g |
A1u |
A2u |
Eu |
T1u |
T2u |
Total |
| Linear (IR) |
1 |
0 |
1 |
0 |
1 |
0 |
0 |
0 |
2 |
1 |
2 / 4 |
| Quadratic (Raman) |
1 |
0 |
1 |
0 |
1 |
0 |
0 |
0 |
2 |
1 |
3 / 3 |
| IR + Raman |
- - - - |
0 |
- - - - |
0 |
- - - - |
0 |
0 |
0 |
- - - - |
1 |
0* / 1 |
* Parity Mutual Exclusion Principle
Characters of force fields
(Symmetric powers of vibration representation)
| Force field |
E |
8C3 |
6C2 |
6C4 |
3C2 |
i |
6S4 |
8S6 |
3σh |
6σd |
| linear |
15 |
0 |
1 |
1 |
-1 |
-3 |
-1 |
0 |
5 |
3 |
| quadratic |
120 |
0 |
8 |
0 |
8 |
12 |
0 |
0 |
20 |
12 |
| cubic |
680 |
5 |
8 |
0 |
-8 |
-28 |
0 |
-1 |
60 |
28 |
| quartic |
3.060 |
0 |
36 |
4 |
36 |
72 |
4 |
0 |
160 |
72 |
| quintic |
11.628 |
0 |
36 |
4 |
-36 |
-144 |
-4 |
0 |
376 |
144 |
| sextic |
38.760 |
15 |
120 |
0 |
120 |
300 |
0 |
3 |
820 |
300 |
Decomposition to irreducible representations
Column with number of nonvanshing force constants highlighted
| Force field |
A1g |
A2g |
Eg |
T1g |
T2g |
A1u |
A2u |
Eu |
T1u |
T2u |
| linear |
1 |
0 |
1 |
0 |
1 |
0 |
0 |
0 |
2 |
1 |
| quadratic |
7 |
2 |
9 |
4 |
9 |
1 |
2 |
3 |
8 |
7 |
| cubic |
22 |
13 |
33 |
33 |
42 |
9 |
14 |
20 |
51 |
46 |
| quartic |
92 |
63 |
155 |
171 |
196 |
50 |
59 |
109 |
199 |
190 |
| quintic |
283 |
238 |
521 |
674 |
719 |
207 |
232 |
439 |
776 |
747 |
| sextic |
928 |
823 |
1.742 |
2.330 |
2.435 |
737 |
782 |
1.513 |
2.470 |
2.425 |
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 O
h
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(T2u) |
|---|
| ..1. |
A1gA1g. | ..1. |
EgEg. | ..1. |
T2gT2g. | ..3. |
T1uT1u. | ..1. |
T2uT2u. | | |
| |
| |
| |
| |
| Subtotal: 7 / 5 / 10 |
| Irrep combinations (i,j) with indices: pos(A1g) ≤ i ≤ j ≤ pos(T2u) |
|---|
| Subtotal: 0 / 0 / 45 |
| Total: 7 / 5 / 55 |
Contributions to nonvanishing cubic force field constants
| Irrep combinations (i,i,i) with indices: pos(A1g) ≤ i ≤ pos(T2u) |
|---|
| ..1. |
A1gA1gA1g. | ..1. |
EgEgEg. | ..1. |
T2gT2gT2g. | | |
| |
| |
| |
| |
| |
| |
| Subtotal: 3 / 3 / 10 |
| Irrep combinations (i,i,j) (i,j,j) with indices: pos(A1g) ≤ i ≤ j ≤ pos(T2u) |
|---|
| ..1. |
A1gEgEg. | ..1. |
A1gT2gT2g. | ..3. |
A1gT1uT1u. | ..1. |
A1gT2uT2u. | ..1. |
EgT2gT2g. | ..3. |
EgT1uT1u. | ..1. |
EgT2uT2u. | ..3. |
T2gT1uT1u. | ..1. |
T2gT2uT2u. | | |
| Subtotal: 15 / 9 / 90 |
| Irrep combinations (i,j,k) with indices: pos(A1g) ≤ i ≤ j ≤ k ≤ pos(T2u) |
|---|
| ..2. |
EgT1uT2u. | ..2. |
T2gT1uT2u. | | |
| |
| |
| |
| |
| |
| |
| |
| Subtotal: 4 / 2 / 120 |
| Total: 22 / 14 / 220 |
Contributions to nonvanishing quartic force field constants
| Irrep combinations (i,i,i,i) with indices: pos(A1g) ≤ i ≤ pos(T2u) |
|---|
| ..1. |
A1gA1gA1gA1g. | ..1. |
EgEgEgEg. | ..2. |
T2gT2gT2gT2g. | ..11. |
T1uT1uT1uT1u. | ..2. |
T2uT2uT2uT2u. | | |
| |
| |
| |
| |
| Subtotal: 17 / 5 / 10 |
| Irrep combinations (i,i,i,j) (i,j,j,j) with indices: pos(A1g) ≤ i ≤ j ≤ pos(T2u) |
|---|
| ..6. |
T1uT1uT1uT2u. | ..1. |
A1gEgEgEg. | ..1. |
A1gT2gT2gT2g. | ..2. |
T1uT2uT2uT2u. | | |
| |
| |
| |
| |
| |
| Subtotal: 10 / 4 / 90 |
| Irrep combinations (i,i,j,j) with indices: pos(A1g) ≤ i ≤ j ≤ pos(T2u) |
|---|
| ..1. |
A1gA1gEgEg. | ..1. |
A1gA1gT2gT2g. | ..3. |
A1gA1gT1uT1u. | ..1. |
A1gA1gT2uT2u. | ..2. |
EgEgT2gT2g. | ..6. |
EgEgT1uT1u. | ..2. |
EgEgT2uT2u. | ..9. |
T2gT2gT1uT1u. | ..3. |
T2gT2gT2uT2u. | ..9. |
T1uT1uT2uT2u. |
| Subtotal: 37 / 10 / 45 |
| Irrep combinations (i,i,j,k) (i,j,j,k) (i,j,k,k) with indices: pos(A1g) ≤ i ≤ j ≤ k ≤ pos(T2u) |
|---|
| ..2. |
EgEgT1uT2u. | ..4. |
T2gT2gT1uT2u. | ..1. |
A1gEgT2gT2g. | ..3. |
A1gEgT1uT1u. | ..1. |
A1gEgT2uT2u. | ..3. |
A1gT2gT1uT1u. | ..1. |
A1gT2gT2uT2u. | ..4. |
EgT2gT1uT1u. | ..1. |
EgT2gT2uT2u. | | |
| Subtotal: 20 / 9 / 360 |
| Irrep combinations (i,j,k,l) with indices: pos(A1g) ≤ i ≤ j ≤ k ≤ l ≤ pos(T2u) |
|---|
| ..2. |
A1gEgT1uT2u. | ..2. |
A1gT2gT1uT2u. | ..4. |
EgT2gT1uT2u. | | |
| |
| |
| |
| |
| |
| |
| Subtotal: 8 / 3 / 210 |
| Total: 92 / 31 / 715 |
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Last update November, 13th 2023 by A. Gelessus, Impressum, Datenschutzerklärung/DataPrivacyStatement