Characters of representations for molecular motions
| Motion |
E |
8C3 |
6C2 |
6C4 |
3C2 |
| Cartesian 3N |
240 |
0 |
0 |
0 |
0 |
| Translation (x,y,z) |
3 |
0 |
-1 |
1 |
-1 |
| Rotation (Rx,Ry,Rz) |
3 |
0 |
-1 |
1 |
-1 |
| Vibration |
234 |
0 |
2 |
-2 |
2 |
Decomposition to irreducible representations
| Motion |
A1 |
A2 |
E |
T1 |
T2 |
Total |
| Cartesian 3N |
10 |
10 |
20 |
30 |
30 |
100 |
| Translation (x,y,z) |
0 |
0 |
0 |
1 |
0 |
1 |
| Rotation (Rx,Ry,Rz) |
0 |
0 |
0 |
1 |
0 |
1 |
| Vibration |
10 |
10 |
20 |
28 |
30 |
98 |
Molecular parameter
| Number of Atoms (N) |
80
|
| Number of internal coordinates |
234
|
| Number of independant internal coordinates |
10
|
| Number of vibrational modes |
98
|
Force field analysis
Allowed / forbidden vibronational transitions
| Operator |
A1 |
A2 |
E |
T1 |
T2 |
Total |
| Linear (IR) |
10 |
10 |
20 |
28 |
30 |
28 / 70 |
| Quadratic (Raman) |
10 |
10 |
20 |
28 |
30 |
60 / 38 |
| IR + Raman |
- - - - |
10 |
- - - - |
- - - - |
- - - - |
0 / 10 |
Characters of force fields
(Symmetric powers of vibration representation)
| Force field |
E |
8C3 |
6C2 |
6C4 |
3C2 |
| linear |
234 |
0 |
2 |
-2 |
2 |
| quadratic |
27.495 |
0 |
119 |
3 |
119 |
| cubic |
2.162.940 |
78 |
236 |
-4 |
236 |
| quartic |
128.154.195 |
0 |
7.139 |
63 |
7.139 |
| quintic |
6.100.139.682 |
0 |
14.042 |
-122 |
14.042 |
| sextic |
242.988.897.333 |
3.081 |
287.861 |
181 |
287.861 |
Decomposition to irreducible representations
Column with number of nonvanshing force constants highlighted
| Force field |
A1 |
A2 |
E |
T1 |
T2 |
| linear |
10 |
10 |
20 |
28 |
30 |
| quadratic |
1.191 |
1.130 |
2.321 |
3.393 |
3.451 |
| cubic |
90.236 |
90.120 |
180.278 |
270.278 |
270.398 |
| quartic |
5.342.451 |
5.338.850 |
10.681.301 |
16.016.613 |
16.020.151 |
| quintic |
254.177.722 |
254.170.762 |
508.348.484 |
762.512.164 |
762.519.246 |
| sextic |
10.124.646.409 |
10.124.502.388 |
20.249.145.716 |
30.373.504.264 |
30.373.648.104 |
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
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(T2) |
|---|
| ..55. |
A1A1. | ..55. |
A2A2. | ..210. |
EE. | ..406. |
T1T1. | ..465. |
T2T2. | | |
| |
| |
| |
| |
| Subtotal: 1.191 / 5 / 5 |
| Irrep combinations (i,j) with indices: pos(A1) ≤ i ≤ j ≤ pos(T2) |
|---|
| Subtotal: 0 / 0 / 10 |
| Total: 1.191 / 5 / 15 |
Contributions to nonvanishing cubic force field constants
| Irrep combinations (i,i,i) with indices: pos(A1) ≤ i ≤ pos(T2) |
|---|
| ..220. |
A1A1A1. | ..1.540. |
EEE. | ..3.276. |
T1T1T1. | ..4.960. |
T2T2T2. | | |
| |
| |
| |
| |
| |
| Subtotal: 9.996 / 4 / 5 |
| Irrep combinations (i,i,j) (i,j,j) with indices: pos(A1) ≤ i ≤ j ≤ pos(T2) |
|---|
| ..12.180. |
T1T1T2. | ..550. |
A1A2A2. | ..2.100. |
A1EE. | ..4.060. |
A1T1T1. | ..4.650. |
A1T2T2. | ..1.900. |
A2EE. | ..8.120. |
ET1T1. | ..9.300. |
ET2T2. | ..12.180. |
T1T2T2. | | |
| Subtotal: 55.040 / 9 / 20 |
| Irrep combinations (i,j,k) with indices: pos(A1) ≤ i ≤ j ≤ k ≤ pos(T2) |
|---|
| ..8.400. |
A2T1T2. | ..16.800. |
ET1T2. | | |
| |
| |
| |
| |
| |
| |
| |
| Subtotal: 25.200 / 2 / 10 |
| Total: 90.236 / 15 / 35 |
Contributions to nonvanishing quartic force field constants
| Irrep combinations (i,i,i,i) with indices: pos(A1) ≤ i ≤ pos(T2) |
|---|
| ..715. |
A1A1A1A1. | ..715. |
A2A2A2A2. | ..22.155. |
EEEE. | ..114.086. |
T1T1T1T1. | ..149.265. |
T2T2T2T2. | | |
| |
| |
| |
| |
| Subtotal: 286.936 / 5 / 5 |
| Irrep combinations (i,i,i,j) (i,j,j,j) with indices: pos(A1) ≤ i ≤ j ≤ pos(T2) |
|---|
| ..341.040. |
T1T1T1T2. | ..15.400. |
A1EEE. | ..32.760. |
A1T1T1T1. | ..49.600. |
A1T2T2T2. | ..15.400. |
A2EEE. | ..40.600. |
A2T1T1T1. | ..40.600. |
A2T2T2T2. | ..146.160. |
ET1T1T1. | ..179.800. |
ET2T2T2. | ..390.600. |
T1T2T2T2. |
| Subtotal: 1.251.960 / 10 / 20 |
| Irrep combinations (i,i,j,j) with indices: pos(A1) ≤ i ≤ j ≤ pos(T2) |
|---|
| ..3.025. |
A1A1A2A2. | ..11.550. |
A1A1EE. | ..22.330. |
A1A1T1T1. | ..25.575. |
A1A1T2T2. | ..11.550. |
A2A2EE. | ..22.330. |
A2A2T1T1. | ..25.575. |
A2A2T2T2. | ..170.520. |
EET1T1. | ..195.300. |
EET2T2. | ..730.800. |
T1T1T2T2. |
| Subtotal: 1.218.555 / 10 / 10 |
| Irrep combinations (i,i,j,k) (i,j,j,k) (i,j,k,k) with indices: pos(A1) ≤ i ≤ j ≤ k ≤ pos(T2) |
|---|
| ..336.000. |
EET1T2. | ..121.800. |
A1T1T1T2. | ..113.400. |
A2T1T1T2. | ..470.400. |
ET1T1T2. | ..19.000. |
A1A2EE. | ..81.200. |
A1ET1T1. | ..93.000. |
A1ET2T2. | ..121.800. |
A1T1T2T2. | ..81.200. |
A2ET1T1. | ..93.000. |
A2ET2T2. |
| ..130.200. |
A2T1T2T2. | ..504.000. |
ET1T2T2. | | |
| |
| |
| |
| |
| |
| |
| |
| Subtotal: 2.165.000 / 12 / 30 |
| Irrep combinations (i,j,k,l) with indices: pos(A1) ≤ i ≤ j ≤ k ≤ l ≤ pos(T2) |
|---|
| ..84.000. |
A1A2T1T2. | ..168.000. |
A1ET1T2. | ..168.000. |
A2ET1T2. | | |
| |
| |
| |
| |
| |
| |
| Subtotal: 420.000 / 3 / 5 |
| Total: 5.342.451 / 40 / 70 |
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Last update November, 13th 2023 by A. Gelessus, Impressum, Datenschutzerklärung/DataPrivacyStatement