Results for Point Group D3d



Characters of representations for molecular motions
Motion E 2C3 3C'2 i 2S6 3σd
Cartesian 3N 24 0 0 0 0 4
Translation (x,y,z) 3 0 -1 -3 0 1
Rotation (Rx,Ry,Rz) 3 0 -1 3 0 -1
Vibration 18 0 2 0 0 4


Decomposition to irreducible representations
Motion A1g A2g Eg A1u A2u Eu Total
Cartesian 3N 3 1 4 1 3 4 16
Translation (x,y,z) 0 0 0 0 1 1 2
Rotation (Rx,Ry,Rz) 0 1 1 0 0 0 2
Vibration 3 0 3 1 2 3 12



Molecular parameter
Number of Atoms (N) 8
Number of internal coordinates 18
Number of independant internal coordinates 3
Number of vibrational modes 12

Force field analysis


Allowed / forbidden vibronational transitions
Operator A1g A2g Eg A1u A2u Eu Total
Linear (IR) 3 0 3 1 2 3 5 / 7
Quadratic (Raman) 3 0 3 1 2 3 6 / 6
IR + Raman - - - - 0 - - - - 1 - - - - - - - - 0* / 1
* Parity Mutual Exclusion Principle


Characters of force fields
(Symmetric powers of vibration representation)
Force field E 2C3 3C'2 i 2S6 3σd
linear 18 0 2 0 0 4
quadratic 171 0 11 9 0 17
cubic 1.140 6 20 0 0 48
quartic 5.985 0 65 45 0 133
quintic 26.334 0 110 0 0 308
sextic 100.947 21 275 165 3 693


Decomposition to irreducible representations
Column with number of nonvanshing force constants highlighted
Force field A1g A2g Eg A1u A2u Eu
linear 3 0 3 1 2 3
quadratic 22 8 30 12 15 27
cubic 113 79 189 89 103 189
quartic 552 453 1.005 478 512 990
quintic 2.299 2.090 4.389 2.145 2.244 4.389
sextic 8.672 8.188 16.848 8.297 8.506 16.794


Further Reading


Contributions to nonvanishing force field constants


pos(X) : Position of irreducible representation (irrep) X in character table of D3d

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(Eu)
..6. A1gA1g...6. EgEg...1. A1uA1u...3. A2uA2u...6. EuEu.
Subtotal: 22 / 5 / 6
Irrep combinations (i,j) with indices: pos(A1g) ≤ i ≤ j ≤ pos(Eu)
Subtotal: 0 / 0 / 15
Total: 22 / 5 / 21


Contributions to nonvanishing cubic force field constants
Irrep combinations (i,i,i) with indices: pos(A1g) ≤ i ≤ pos(Eu)
..10. A1gA1gA1g...10. EgEgEg.
Subtotal: 20 / 2 / 6
Irrep combinations (i,i,j) (i,j,j) with indices: pos(A1g) ≤ i ≤ j ≤ pos(Eu)
..18. A1gEgEg...3. A1gA1uA1u...9. A1gA2uA2u...18. A1gEuEu...18. EgEuEu.
Subtotal: 66 / 5 / 30
Irrep combinations (i,j,k) with indices: pos(A1g) ≤ i ≤ j ≤ k ≤ pos(Eu)
..9. EgA1uEu...18. EgA2uEu.
Subtotal: 27 / 2 / 20
Total: 113 / 9 / 56


Contributions to nonvanishing quartic force field constants
Irrep combinations (i,i,i,i) with indices: pos(A1g) ≤ i ≤ pos(Eu)
..15. A1gA1gA1gA1g...21. EgEgEgEg...1. A1uA1uA1uA1u...5. A2uA2uA2uA2u...21. EuEuEuEu.
Subtotal: 63 / 5 / 6
Irrep combinations (i,i,i,j) (i,j,j,j) with indices: pos(A1g) ≤ i ≤ j ≤ pos(Eu)
..30. A1gEgEgEg...10. A1uEuEuEu...20. A2uEuEuEu.
Subtotal: 60 / 3 / 30
Irrep combinations (i,i,j,j) with indices: pos(A1g) ≤ i ≤ j ≤ pos(Eu)
..36. A1gA1gEgEg...6. A1gA1gA1uA1u...18. A1gA1gA2uA2u...36. A1gA1gEuEu...6. EgEgA1uA1u...18. EgEgA2uA2u...81. EgEgEuEu...3. A1uA1uA2uA2u...6. A1uA1uEuEu...18. A2uA2uEuEu.
Subtotal: 228 / 10 / 15
Irrep combinations (i,i,j,k) (i,j,j,k) (i,j,k,k) with indices: pos(A1g) ≤ i ≤ j ≤ k ≤ pos(Eu)
..6. EgEgA1uA2u...18. EgEgA1uEu...36. EgEgA2uEu...54. A1gEgEuEu...6. A1uA2uEuEu.
Subtotal: 120 / 5 / 60
Irrep combinations (i,j,k,l) with indices: pos(A1g) ≤ i ≤ j ≤ k ≤ l ≤ pos(Eu)
..27. A1gEgA1uEu...54. A1gEgA2uEu.
Subtotal: 81 / 2 / 15
Total: 552 / 25 / 126


Calculate contributions to

A1g A2g Eg A1u A2u Eu
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Last update November, 13th 2023 by A. Gelessus, Impressum, Datenschutzerklärung/DataPrivacyStatement