Results for Point Group Oh



Characters of representations for molecular motions
Motion E 8C3 6C2 6C4 3C2 i 6S4 8S6 h d
Cartesian 3N 48 0 0 0 0 0 0 0 0 8
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 42 0 2 -2 2 0 0 0 0 8


Decomposition to irreducible representations
Motion A1g A2g Eg T1g T2g A1u A2u Eu T1u T2u Total
Cartesian 3N 2 0 2 2 4 0 2 2 4 2 20
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 4 0 2 2 3 2 18



Molecular parameter
Number of Atoms (N) 16
Number of internal coordinates 42
Number of independant internal coordinates 2
Number of vibrational modes 18


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 4 0 2 2 3 2 3 / 15
Quadratic (Raman) 2 0 2 1 4 0 2 2 3 2 8 / 10
IR + Raman - - - - 0 - - - - 1 - - - - 0 2 2 - - - - 2 0* / 7
* 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 42 0 2 -2 2 0 0 0 0 8
quadratic 903 0 23 3 23 21 1 0 21 53
cubic 13.244 14 44 -4 44 0 0 0 0 256
quartic 148.995 0 275 15 275 231 11 0 231 1.095
quintic 1.370.754 0 506 -26 506 0 0 0 0 4.056
sextic 10.737.573 105 2.277 37 2.277 1.771 11 7 1.771 13.803


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 4 0 2 2 3 2
quadratic 32 12 44 46 64 15 22 37 59 51
cubic 318 244 555 787 863 254 308 555 851 799
quartic 3.315 2.966 6.281 9.127 9.463 3.000 3.204 6.204 9.398 9.192
quintic 29.156 28.022 57.178 85.067 86.214 28.142 29.036 57.178 86.081 85.200
sextic 226.024 221.992 447.960 668.952 672.960 222.273 225.148 447.372 672.400 669.512


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...10. T2gT2g...3. A2uA2u...3. EuEu...6. T1uT1u...3. T2uT2u.
Subtotal: 32 / 8 / 10
Irrep combinations (i,j) with indices: pos(A1g) ≤ i ≤ j ≤ pos(T2u)
Subtotal: 0 / 0 / 45
Total: 32 / 8 / 55


Contributions to nonvanishing cubic force field constants
Irrep combinations (i,i,i) with indices: pos(A1g) ≤ i ≤ pos(T2u)
..4. A1gA1gA1g...4. EgEgEg...20. T2gT2gT2g.
Subtotal: 28 / 3 / 10
Irrep combinations (i,i,j) (i,j,j) with indices: pos(A1g) ≤ i ≤ j ≤ pos(T2u)
..4. T1gT1gT2g...6. A1gEgEg...2. A1gT1gT1g...20. A1gT2gT2g...6. A1gA2uA2u...6. A1gEuEu...12. A1gT1uT1u...6. A1gT2uT2u...2. EgT1gT1g...20. EgT2gT2g.
..6. EgEuEu...12. EgT1uT1u...6. EgT2uT2u...6. T1gT2gT2g...3. T1gT1uT1u...1. T1gT2uT2u...24. T2gT1uT1u...12. T2gT2uT2u.
Subtotal: 154 / 18 / 90
Irrep combinations (i,j,k) with indices: pos(A1g) ≤ i ≤ j ≤ k ≤ pos(T2u)
..8. EgT1gT2g...8. EgA2uEu...12. EgT1uT2u...4. T1gA2uT2u...6. T1gEuT1u...4. T1gEuT2u...6. T1gT1uT2u...24. T2gA2uT1u...24. T2gEuT1u...16. T2gEuT2u.
..24. T2gT1uT2u.
Subtotal: 136 / 11 / 120
Total: 318 / 32 / 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...90. T2gT2gT2gT2g...5. A2uA2uA2uA2u...6. EuEuEuEu...36. T1uT1uT1uT1u...11. T2uT2uT2uT2u.
Subtotal: 161 / 8 / 10
Irrep combinations (i,i,i,j) (i,j,j,j) with indices: pos(A1g) ≤ i ≤ j ≤ pos(T2u)
..4. T1gT1gT1gT2g...36. T1uT1uT1uT2u...8. A1gEgEgEg...40. A1gT2gT2gT2g...40. EgT2gT2gT2g...40. T1gT2gT2gT2g...8. A2uEuEuEu...20. A2uT1uT1uT1u...16. EuT1uT1uT1u...4. EuT2uT2uT2u.
..18. T1uT2uT2uT2u.
Subtotal: 234 / 11 / 90
Irrep combinations (i,i,j,j) with indices: pos(A1g) ≤ i ≤ j ≤ pos(T2u)
..9. A1gA1gEgEg...3. A1gA1gT1gT1g...30. A1gA1gT2gT2g...9. A1gA1gA2uA2u...9. A1gA1gEuEu...18. A1gA1gT1uT1u...9. A1gA1gT2uT2u...6. EgEgT1gT1g...60. EgEgT2gT2g...9. EgEgA2uA2u.
..19. EgEgEuEu...36. EgEgT1uT1u...18. EgEgT2uT2u...30. T1gT1gT2gT2g...3. T1gT1gA2uA2u...6. T1gT1gEuEu...18. T1gT1gT1uT1u...9. T1gT1gT2uT2u...30. T2gT2gA2uA2u...60. T2gT2gEuEu.
..198. T2gT2gT1uT1u...96. T2gT2gT2uT2u...9. A2uA2uEuEu...18. A2uA2uT1uT1u...9. A2uA2uT2uT2u...36. EuEuT1uT1u...18. EuEuT2uT2u...57. T1uT1uT2uT2u.
Subtotal: 832 / 28 / 45
Irrep combinations (i,i,j,k) (i,j,j,k) (i,j,k,k) with indices: pos(A1g) ≤ i ≤ j ≤ k ≤ pos(T2u)
..16. EgEgT1gT2g...12. EgEgA2uEu...24. EgEgT1uT2u...4. T1gT1gA2uEu...6. T1gT1gA2uT1u...6. T1gT1gEuT1u...4. T1gT1gEuT2u...12. T1gT1gT1uT2u...40. T2gT2gA2uEu...60. T2gT2gA2uT1u.
..24. T2gT2gA2uT2u...96. T2gT2gEuT1u...64. T2gT2gEuT2u...156. T2gT2gT1uT2u...24. EuEuT1uT2u...8. A1gT1gT1gT2g...8. EgT1gT1gT2g...12. A2uT1uT1uT2u...36. EuT1uT1uT2u...4. A1gEgT1gT1g.
..40. A1gEgT2gT2g...12. A1gEgEuEu...24. A1gEgT1uT1u...12. A1gEgT2uT2u...12. A1gT1gT2gT2g...6. A1gT1gT1uT1u...2. A1gT1gT2uT2u...48. A1gT2gT1uT1u...24. A1gT2gT2uT2u...32. EgT1gT2gT2g.
..18. EgT1gT1uT1u...8. EgT1gT2uT2u...72. EgT2gT1uT1u...32. EgT2gT2uT2u...16. T1gT2gEuEu...60. T1gT2gT1uT1u...28. T1gT2gT2uT2u...24. A2uEuT1uT1u...12. A2uEuT2uT2u...18. A2uT1uT2uT2u.
..24. EuT1uT2uT2u.
Subtotal: 1.140 / 41 / 360
Irrep combinations (i,j,k,l) with indices: pos(A1g) ≤ i ≤ j ≤ k ≤ l ≤ pos(T2u)
..16. A1gEgT1gT2g...16. A1gEgA2uEu...24. A1gEgT1uT2u...8. A1gT1gA2uT2u...12. A1gT1gEuT1u...8. A1gT1gEuT2u...12. A1gT1gT1uT2u...48. A1gT2gA2uT1u...48. A1gT2gEuT1u...32. A1gT2gEuT2u.
..48. A1gT2gT1uT2u...12. EgT1gA2uT1u...8. EgT1gA2uT2u...24. EgT1gEuT1u...16. EgT1gEuT2u...24. EgT1gT1uT2u...48. EgT2gA2uT1u...32. EgT2gA2uT2u...96. EgT2gEuT1u...64. EgT2gEuT2u.
..96. EgT2gT1uT2u...16. T1gT2gA2uEu...24. T1gT2gA2uT1u...16. T1gT2gA2uT2u...48. T1gT2gEuT1u...32. T1gT2gEuT2u...96. T1gT2gT1uT2u...24. A2uEuT1uT2u.
Subtotal: 948 / 28 / 210
Total: 3.315 / 116 / 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