Characters of representations for molecular motions
| Motion |
E |
2C11 |
2(C11)2 |
2(C11)3 |
2(C11)4 |
2(C11)5 |
11C'2 |
σh |
2S11 |
2(S11)9 |
2(S11)3 |
2(S11)7 |
2(S11)5 |
11σv |
| Cartesian 3N |
0 |
0.000 |
0.000 |
0.000 |
0.000 |
0.000 |
0 |
0 |
0.000 |
0.000 |
0.000 |
0.000 |
0.000 |
0 |
| Translation (x,y,z) |
3 |
2.683 |
1.831 |
0.715 |
-0.310 |
-0.919 |
-1 |
1 |
0.683 |
-0.169 |
-1.285 |
-2.310 |
-2.919 |
1 |
| Rotation (Rx,Ry,Rz) |
3 |
2.683 |
1.831 |
0.715 |
-0.310 |
-0.919 |
-1 |
-1 |
-0.683 |
0.169 |
1.285 |
2.310 |
2.919 |
-1 |
| Vibration |
-6 |
-5.365 |
-3.662 |
-1.431 |
0.619 |
1.838 |
2 |
0 |
0.000 |
0.000 |
0.000 |
0.000 |
0.000 |
0 |
Decomposition to irreducible representations
| Motion |
A'1 |
A'2 |
E'1 |
E'2 |
E'3 |
E'4 |
E'5 |
A''1 |
A''2 |
E''1 |
E''2 |
E''3 |
E''4 |
E''5 |
Total |
| Cartesian 3N |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
| Translation (x,y,z) |
0 |
0 |
1 |
0 |
0 |
0 |
0 |
0 |
1 |
0 |
0 |
0 |
0 |
0 |
2 |
| Rotation (Rx,Ry,Rz) |
0 |
1 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
1 |
0 |
0 |
0 |
0 |
2 |
| Vibration |
0 |
-1 |
-1 |
0 |
0 |
0 |
0 |
0 |
-1 |
-1 |
0 |
0 |
0 |
0 |
-4 |
Molecular parameter
| Number of Atoms (N) |
0
|
| Number of internal coordinates |
-6
|
| Number of independant internal coordinates |
0
|
| Number of vibrational modes |
-4
|
Force field analysis
Allowed / forbidden vibronational transitions
| Operator |
A'1 |
A'2 |
E'1 |
E'2 |
E'3 |
E'4 |
E'5 |
A''1 |
A''2 |
E''1 |
E''2 |
E''3 |
E''4 |
E''5 |
Total |
| Linear (IR) |
0 |
-1 |
-1 |
0 |
0 |
0 |
0 |
0 |
-1 |
-1 |
0 |
0 |
0 |
0 |
-2 / -2 |
| Quadratic (Raman) |
0 |
-1 |
-1 |
0 |
0 |
0 |
0 |
0 |
-1 |
-1 |
0 |
0 |
0 |
0 |
-1 / -3 |
| IR + Raman |
- - - - |
-1 |
- - - - |
- - - - |
0 |
0 |
0 |
0 |
- - - - |
- - - - |
0 |
0 |
0 |
0 |
0 / -1 |
Characters of force fields
(Symmetric powers of vibration representation)
| Force field |
E |
2C11 |
2(C11)2 |
2(C11)3 |
2(C11)4 |
2(C11)5 |
11C'2 |
σh |
2S11 |
2(S11)9 |
2(S11)3 |
2(S11)7 |
2(S11)5 |
11σv |
| linear |
-6 |
-5.365 |
-3.662 |
-1.431 |
0.619 |
1.838 |
2 |
0 |
0.000 |
0.000 |
0.000 |
0.000 |
0.000 |
0 |
| quadratic |
15 |
12.561 |
7.014 |
1.942 |
-0.524 |
-0.993 |
-1 |
-3 |
-1.831 |
0.310 |
0.919 |
-0.715 |
-2.683 |
-3 |
| cubic |
-20 |
-16.392 |
-8.704 |
-3.024 |
-2.192 |
-3.689 |
-4 |
0 |
0.000 |
0.000 |
0.000 |
0.000 |
0.000 |
0 |
| quartic |
15 |
12.561 |
7.014 |
1.942 |
-0.524 |
-0.993 |
-1 |
3 |
1.831 |
-0.310 |
-0.919 |
0.715 |
2.683 |
3 |
| quintic |
-6 |
-5.365 |
-3.662 |
-1.431 |
0.619 |
1.838 |
2 |
0 |
0.000 |
0.000 |
0.000 |
0.000 |
0.000 |
0 |
| sextic |
1 |
1.000 |
1.000 |
1.000 |
1.000 |
1.000 |
1 |
-1 |
-1.000 |
-1.000 |
-1.000 |
-1.000 |
-1.000 |
-1 |
Decomposition to irreducible representations
Column with number of nonvanshing force constants highlighted
| Force field |
A'1 |
A'2 |
E'1 |
E'2 |
E'3 |
E'4 |
E'5 |
A''1 |
A''2 |
E''1 |
E''2 |
E''3 |
E''4 |
E''5 |
| linear |
0 |
-1 |
-1 |
0 |
0 |
0 |
0 |
0 |
-1 |
-1 |
0 |
0 |
0 |
0 |
| quadratic |
0 |
2 |
2 |
0 |
0 |
0 |
0 |
2 |
1 |
2 |
1 |
0 |
0 |
0 |
| cubic |
-3 |
-1 |
-2 |
-1 |
0 |
0 |
0 |
-3 |
-1 |
-2 |
-1 |
0 |
0 |
0 |
| quartic |
2 |
1 |
2 |
1 |
0 |
0 |
0 |
0 |
2 |
2 |
0 |
0 |
0 |
0 |
| quintic |
0 |
-1 |
-1 |
0 |
0 |
0 |
0 |
0 |
-1 |
-1 |
0 |
0 |
0 |
0 |
| sextic |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
1 |
0 |
0 |
0 |
0 |
0 |
0 |
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
11h
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(A'1) ≤ i ≤ pos(E''5) |
|---|
| Subtotal: 0 / 0 / 14 |
| Irrep combinations (i,j) with indices: pos(A'1) ≤ i ≤ j ≤ pos(E''5) |
|---|
| Subtotal: 0 / 0 / 91 |
| Total: 0 / 0 / 105 |
Contributions to nonvanishing cubic force field constants
| Irrep combinations (i,i,i) with indices: pos(A'1) ≤ i ≤ pos(E''5) |
|---|
| Subtotal: 0 / 0 / 14 |
| Irrep combinations (i,i,j) (i,j,j) with indices: pos(A'1) ≤ i ≤ j ≤ pos(E''5) |
|---|
| Subtotal: -2 / 0 / 182 |
| Irrep combinations (i,j,k) with indices: pos(A'1) ≤ i ≤ j ≤ k ≤ pos(E''5) |
|---|
| Subtotal: -1 / 0 / 364 |
| Total: -3 / 0 / 560 |
Contributions to nonvanishing quartic force field constants
| Irrep combinations (i,i,i,i) with indices: pos(A'1) ≤ i ≤ pos(E''5) |
|---|
| Subtotal: 0 / 0 / 14 |
| Irrep combinations (i,i,i,j) (i,j,j,j) with indices: pos(A'1) ≤ i ≤ j ≤ pos(E''5) |
|---|
| Subtotal: 0 / 0 / 182 |
| Irrep combinations (i,i,j,j) with indices: pos(A'1) ≤ i ≤ j ≤ pos(E''5) |
|---|
| ..1. |
E'1E'1E''1E''1. | | |
| |
| |
| |
| |
| |
| |
| |
| |
| Subtotal: 1 / 1 / 91 |
| Irrep combinations (i,i,j,k) (i,j,j,k) (i,j,k,k) with indices: pos(A'1) ≤ i ≤ j ≤ k ≤ pos(E''5) |
|---|
| Subtotal: 0 / 0 / 1.092 |
| Irrep combinations (i,j,k,l) with indices: pos(A'1) ≤ i ≤ j ≤ k ≤ l ≤ pos(E''5) |
|---|
| ..1. |
A'2E'1A''2E''1. | | |
| |
| |
| |
| |
| |
| |
| |
| |
| Subtotal: 1 / 1 / 1.001 |
| Total: 2 / 2 / 2.380 |
Calculate contributions to
Last update November, 13th 2023 by A. Gelessus, Impressum, Datenschutzerklärung/DataPrivacyStatement