Results for Point Group D3d



Characters of representations for molecular motions
Motion E 2C3 3C'2 i 2S6 3σd
Cartesian 3N 21 0 -1 -3 0 3
Translation (x,y,z) 3 0 -1 -3 0 1
Rotation (Rx,Ry,Rz) 3 0 -1 3 0 -1
Vibration 15 0 1 -3 0 3


Decomposition to irreducible representations
Motion A1g A2g Eg A1u A2u Eu Total
Cartesian 3N 2 1 3 1 3 4 14
Translation (x,y,z) 0 0 0 0 1 1 2
Rotation (Rx,Ry,Rz) 0 1 1 0 0 0 2
Vibration 2 0 2 1 2 3 10



Molecular parameter
Number of Atoms (N) 7
Number of internal coordinates 15
Number of independant internal coordinates 2
Number of vibrational modes 10

Force field analysis


Allowed / forbidden vibronational transitions
Operator A1g A2g Eg A1u A2u Eu Total
Linear (IR) 2 0 2 1 2 3 5 / 5
Quadratic (Raman) 2 0 2 1 2 3 4 / 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 15 0 1 -3 0 3
quadratic 120 0 8 12 0 12
cubic 680 5 8 -28 -1 28
quartic 3.060 0 36 72 0 72
quintic 11.628 0 36 -144 0 144
sextic 38.760 15 120 300 3 300


Decomposition to irreducible representations
Column with number of nonvanshing force constants highlighted
Force field A1g A2g Eg A1u A2u Eu
linear 2 0 2 1 2 3
quadratic 16 6 22 8 10 18
cubic 64 46 108 55 65 117
quartic 288 234 522 240 258 498
quintic 1.002 912 1.914 954 1.008 1.962
sextic 3.363 3.153 6.507 3.162 3.252 6.408


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)
..3. A1gA1g...3. EgEg...1. A1uA1u...3. A2uA2u...6. EuEu.
Subtotal: 16 / 5 / 6
Irrep combinations (i,j) with indices: pos(A1g) ≤ i ≤ j ≤ pos(Eu)
Subtotal: 0 / 0 / 15
Total: 16 / 5 / 21


Contributions to nonvanishing cubic force field constants
Irrep combinations (i,i,i) with indices: pos(A1g) ≤ i ≤ pos(Eu)
..4. A1gA1gA1g...4. EgEgEg.
Subtotal: 8 / 2 / 6
Irrep combinations (i,i,j) (i,j,j) with indices: pos(A1g) ≤ i ≤ j ≤ pos(Eu)
..6. A1gEgEg...2. A1gA1uA1u...6. A1gA2uA2u...12. A1gEuEu...12. EgEuEu.
Subtotal: 38 / 5 / 30
Irrep combinations (i,j,k) with indices: pos(A1g) ≤ i ≤ j ≤ k ≤ pos(Eu)
..6. EgA1uEu...12. EgA2uEu.
Subtotal: 18 / 2 / 20
Total: 64 / 9 / 56


Contributions to nonvanishing quartic force field constants
Irrep combinations (i,i,i,i) with indices: pos(A1g) ≤ i ≤ pos(Eu)
..5. A1gA1gA1gA1g...6. EgEgEgEg...1. A1uA1uA1uA1u...5. A2uA2uA2uA2u...21. EuEuEuEu.
Subtotal: 38 / 5 / 6
Irrep combinations (i,i,i,j) (i,j,j,j) with indices: pos(A1g) ≤ i ≤ j ≤ pos(Eu)
..8. A1gEgEgEg...10. A1uEuEuEu...20. A2uEuEuEu.
Subtotal: 38 / 3 / 30
Irrep combinations (i,i,j,j) with indices: pos(A1g) ≤ i ≤ j ≤ pos(Eu)
..9. A1gA1gEgEg...3. A1gA1gA1uA1u...9. A1gA1gA2uA2u...18. A1gA1gEuEu...3. EgEgA1uA1u...9. EgEgA2uA2u...39. EgEgEuEu...3. A1uA1uA2uA2u...6. A1uA1uEuEu...18. A2uA2uEuEu.
Subtotal: 117 / 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)
..2. EgEgA1uA2u...9. EgEgA1uEu...18. EgEgA2uEu...24. A1gEgEuEu...6. A1uA2uEuEu.
Subtotal: 59 / 5 / 60
Irrep combinations (i,j,k,l) with indices: pos(A1g) ≤ i ≤ j ≤ k ≤ l ≤ pos(Eu)
..12. A1gEgA1uEu...24. A1gEgA2uEu.
Subtotal: 36 / 2 / 15
Total: 288 / 25 / 126


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