Results for Point Group Oh



Characters of representations for molecular motions
Motion E 8C3 6C2 6C4 3C2 i 6S4 8S6 h d
Cartesian 3N 36 0 0 4 -4 0 0 0 8 4
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 30 0 2 2 -2 0 0 0 8 4


Decomposition to irreducible representations
Motion A1g A2g Eg T1g T2g A1u A2u Eu T1u T2u Total
Cartesian 3N 2 0 2 2 2 0 0 0 4 2 14
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 2 0 2 1 2 0 0 0 3 2 12



Molecular parameter
Number of Atoms (N) 12
Number of internal coordinates 30
Number of independant internal coordinates 2
Number of vibrational modes 12


Force field analysis


Allowed / forbidden vibronational transitions
Operator A1g A2g Eg T1g T2g A1u A2u Eu T1u T2u Total
Linear (IR) 2 0 2 1 2 0 0 0 3 2 3 / 9
Quadratic (Raman) 2 0 2 1 2 0 0 0 3 2 6 / 6
IR + Raman - - - - 0 - - - - 1 - - - - 0 0 0 - - - - 2 0* / 3
* Parity Mutual Exclusion Principle


Characters of force fields
(Symmetric powers of vibration representation)
Force field E 8C3 6C2 6C4 3C2 i 6S4 8S6 h d
linear 30 0 2 2 -2 0 0 0 8 4
quadratic 465 0 17 1 17 15 -1 0 47 23
cubic 4.960 10 32 0 -32 0 0 0 208 72
quartic 40.920 0 152 8 152 120 8 0 792 256
quintic 278.256 0 272 16 -272 0 0 0 2.640 680
sextic 1.623.160 55 952 8 952 680 -8 5 8.008 1.904


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 2 0 2 1 2 0 0 0 3 2
quadratic 19 9 28 21 31 7 8 15 31 29
cubic 129 103 227 286 312 85 95 175 330 320
quartic 967 861 1.828 2.457 2.555 797 823 1.620 2.603 2.577
quintic 6.066 5.824 11.890 17.126 17.360 5.566 5.664 11.230 17.626 17.520
sextic 34.757 34.043 68.770 100.573 101.287 33.252 33.486 66.713 101.967 101.725


Further Reading



Contributions to nonvanishing force field constants


pos(X) : Position of irreducible representation (irrep) X in character table of Oh

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)
..3. A1gA1g...3. EgEg...1. T1gT1g...3. T2gT2g...6. T1uT1u...3. T2uT2u.
Subtotal: 19 / 6 / 10
Irrep combinations (i,j) with indices: pos(A1g) ≤ i ≤ j ≤ pos(T2u)
Subtotal: 0 / 0 / 45
Total: 19 / 6 / 55


Contributions to nonvanishing cubic force field constants
Irrep combinations (i,i,i) with indices: pos(A1g) ≤ i ≤ pos(T2u)
..4. A1gA1gA1g...4. EgEgEg...4. T2gT2gT2g.
Subtotal: 12 / 3 / 10
Irrep combinations (i,i,j) (i,j,j) with indices: pos(A1g) ≤ i ≤ j ≤ pos(T2u)
..2. T1gT1gT2g...6. A1gEgEg...2. A1gT1gT1g...6. A1gT2gT2g...12. A1gT1uT1u...6. A1gT2uT2u...2. EgT1gT1g...6. EgT2gT2g...12. EgT1uT1u...6. EgT2uT2u.
..1. T1gT2gT2g...3. T1gT1uT1u...1. T1gT2uT2u...12. T2gT1uT1u...6. T2gT2uT2u.
Subtotal: 83 / 15 / 90
Irrep combinations (i,j,k) with indices: pos(A1g) ≤ i ≤ j ≤ k ≤ pos(T2u)
..4. EgT1gT2g...12. EgT1uT2u...6. T1gT1uT2u...12. T2gT1uT2u.
Subtotal: 34 / 4 / 120
Total: 129 / 22 / 220


Contributions to nonvanishing quartic force field constants
Irrep combinations (i,i,i,i) with indices: pos(A1g) ≤ i ≤ pos(T2u)
..5. A1gA1gA1gA1g...6. EgEgEgEg...2. T1gT1gT1gT1g...11. T2gT2gT2gT2g...36. T1uT1uT1uT1u...11. T2uT2uT2uT2u.
Subtotal: 71 / 6 / 10
Irrep combinations (i,i,i,j) (i,j,j,j) with indices: pos(A1g) ≤ i ≤ j ≤ pos(T2u)
..2. T1gT1gT1gT2g...36. T1uT1uT1uT2u...8. A1gEgEgEg...8. A1gT2gT2gT2g...4. EgT2gT2gT2g...6. T1gT2gT2gT2g...18. T1uT2uT2uT2u.
Subtotal: 82 / 7 / 90
Irrep combinations (i,i,j,j) with indices: pos(A1g) ≤ i ≤ j ≤ pos(T2u)
..9. A1gA1gEgEg...3. A1gA1gT1gT1g...9. A1gA1gT2gT2g...18. A1gA1gT1uT1u...9. A1gA1gT2uT2u...6. EgEgT1gT1g...18. EgEgT2gT2g...36. EgEgT1uT1u...18. EgEgT2uT2u...9. T1gT1gT2gT2g.
..18. T1gT1gT1uT1u...9. T1gT1gT2uT2u...57. T2gT2gT1uT1u...28. T2gT2gT2uT2u...57. T1uT1uT2uT2u.
Subtotal: 304 / 15 / 45
Irrep combinations (i,i,j,k) (i,j,j,k) (i,j,k,k) with indices: pos(A1g) ≤ i ≤ j ≤ k ≤ pos(T2u)
..8. EgEgT1gT2g...24. EgEgT1uT2u...12. T1gT1gT1uT2u...42. T2gT2gT1uT2u...4. A1gT1gT1gT2g...4. EgT1gT1gT2g...4. A1gEgT1gT1g...12. A1gEgT2gT2g...24. A1gEgT1uT1u...12. A1gEgT2uT2u.
..2. A1gT1gT2gT2g...6. A1gT1gT1uT1u...2. A1gT1gT2uT2u...24. A1gT2gT1uT1u...12. A1gT2gT2uT2u...8. EgT1gT2gT2g...18. EgT1gT1uT1u...8. EgT1gT2uT2u...36. EgT2gT1uT1u...16. EgT2gT2uT2u.
..30. T1gT2gT1uT1u...14. T1gT2gT2uT2u.
Subtotal: 322 / 22 / 360
Irrep combinations (i,j,k,l) with indices: pos(A1g) ≤ i ≤ j ≤ k ≤ l ≤ pos(T2u)
..8. A1gEgT1gT2g...24. A1gEgT1uT2u...12. A1gT1gT1uT2u...24. A1gT2gT1uT2u...24. EgT1gT1uT2u...48. EgT2gT1uT2u...48. T1gT2gT1uT2u.
Subtotal: 188 / 7 / 210
Total: 967 / 57 / 715


Calculate contributions to

A1g A2g Eg T1g T2g A1u A2u Eu T1u T2u
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Last update January, 3rd 2020 by A. Gelessus, Impressum, Datenschutzerklärung/DataPrivacyStatement