Results for Point Group Oh



Characters of representations for molecular motions
Motion E 8C3 6C2 6C4 3C2 i 6S4 8S6 h d
Cartesian 3N 39 0 -1 5 -5 -3 -1 0 9 5
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 33 0 1 3 -3 -3 -1 0 9 5


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 5 2 15
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 4 2 13



Molecular parameter
Number of Atoms (N) 13
Number of internal coordinates 33
Number of independant internal coordinates 2
Number of vibrational modes 13


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 4 2 4 / 9
Quadratic (Raman) 2 0 2 1 2 0 0 0 4 2 6 / 7
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 33 0 1 3 -3 -3 -1 0 9 5
quadratic 561 0 17 3 21 21 -1 0 57 29
cubic 6.545 11 17 1 -55 -55 1 -1 273 105
quartic 58.905 0 153 9 225 225 9 0 1.113 385
quintic 435.897 0 153 27 -531 -531 -9 0 3.969 1.141
sextic 2.760.681 66 969 27 1.653 1.653 -9 6 12.817 3.325


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 4 2
quadratic 23 11 34 26 37 8 10 18 38 34
cubic 166 135 296 377 407 108 130 232 444 422
quartic 1.385 1.246 2.631 3.547 3.677 1.138 1.196 2.334 3.752 3.694
quintic 9.449 9.121 18.570 26.836 27.155 8.692 8.930 17.622 27.686 27.430
sextic 59.004 57.926 116.894 171.207 172.276 56.502 57.082 113.554 173.436 172.838


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...10. T1uT1u...3. T2uT2u.
Subtotal: 23 / 6 / 10
Irrep combinations (i,j) with indices: pos(A1g) ≤ i ≤ j ≤ pos(T2u)
Subtotal: 0 / 0 / 45
Total: 23 / 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...20. A1gT1uT1u...6. A1gT2uT2u...2. EgT1gT1g...6. EgT2gT2g...20. EgT1uT1u...6. EgT2uT2u.
..1. T1gT2gT2g...6. T1gT1uT1u...1. T1gT2uT2u...20. T2gT1uT1u...6. T2gT2uT2u.
Subtotal: 110 / 15 / 90
Irrep combinations (i,j,k) with indices: pos(A1g) ≤ i ≤ j ≤ k ≤ pos(T2u)
..4. EgT1gT2g...16. EgT1uT2u...8. T1gT1uT2u...16. T2gT1uT2u.
Subtotal: 44 / 4 / 120
Total: 166 / 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...90. T1uT1uT1uT1u...11. T2uT2uT2uT2u.
Subtotal: 125 / 6 / 10
Irrep combinations (i,i,i,j) (i,j,j,j) with indices: pos(A1g) ≤ i ≤ j ≤ pos(T2u)
..2. T1gT1gT1gT2g...80. T1uT1uT1uT2u...8. A1gEgEgEg...8. A1gT2gT2gT2g...4. EgT2gT2gT2g...6. T1gT2gT2gT2g...24. T1uT2uT2uT2u.
Subtotal: 132 / 7 / 90
Irrep combinations (i,i,j,j) with indices: pos(A1g) ≤ i ≤ j ≤ pos(T2u)
..9. A1gA1gEgEg...3. A1gA1gT1gT1g...9. A1gA1gT2gT2g...30. A1gA1gT1uT1u...9. A1gA1gT2uT2u...6. EgEgT1gT1g...18. EgEgT2gT2g...60. EgEgT1uT1u...18. EgEgT2uT2u...9. T1gT1gT2gT2g.
..30. T1gT1gT1uT1u...9. T1gT1gT2uT2u...96. T2gT2gT1uT1u...28. T2gT2gT2uT2u...96. T1uT1uT2uT2u.
Subtotal: 430 / 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...32. EgEgT1uT2u...16. T1gT1gT1uT2u...56. T2gT2gT1uT2u...4. A1gT1gT1gT2g...4. EgT1gT1gT2g...4. A1gEgT1gT1g...12. A1gEgT2gT2g...40. A1gEgT1uT1u...12. A1gEgT2uT2u.
..2. A1gT1gT2gT2g...12. A1gT1gT1uT1u...2. A1gT1gT2uT2u...40. A1gT2gT1uT1u...12. A1gT2gT2uT2u...8. EgT1gT2gT2g...32. EgT1gT1uT1u...8. EgT1gT2uT2u...64. EgT2gT1uT1u...16. EgT2gT2uT2u.
..52. T1gT2gT1uT1u...14. T1gT2gT2uT2u.
Subtotal: 450 / 22 / 360
Irrep combinations (i,j,k,l) with indices: pos(A1g) ≤ i ≤ j ≤ k ≤ l ≤ pos(T2u)
..8. A1gEgT1gT2g...32. A1gEgT1uT2u...16. A1gT1gT1uT2u...32. A1gT2gT1uT2u...32. EgT1gT1uT2u...64. EgT2gT1uT2u...64. T1gT2gT1uT2u.
Subtotal: 248 / 7 / 210
Total: 1.385 / 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