Results for Point Group D3d



Characters of representations for molecular motions
Motion E 2C3 3C'2 i 2S6 3σd
Cartesian 3N 18 0 0 0 0 2
Translation (x,y,z) 3 0 -1 -3 0 1
Rotation (Rx,Ry,Rz) 3 0 -1 3 0 -1
Vibration 12 0 2 0 0 2


Decomposition to irreducible representations
Motion A1g A2g Eg A1u A2u Eu Total
Cartesian 3N 2 1 3 1 2 3 12
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 1 2 8



Molecular parameter
Number of Atoms (N) 6
Number of internal coordinates 12
Number of independant internal coordinates 2
Number of vibrational modes 8

Force field analysis


Allowed / forbidden vibronational transitions
Operator A1g A2g Eg A1u A2u Eu Total
Linear (IR) 2 0 2 1 1 2 3 / 5
Quadratic (Raman) 2 0 2 1 1 2 4 / 4
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 12 0 2 0 0 2
quadratic 78 0 8 6 0 8
cubic 364 4 14 0 0 14
quartic 1.365 0 35 21 0 35
quintic 4.368 0 56 0 0 56
sextic 12.376 10 112 56 2 112


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 1 2
quadratic 11 3 14 6 6 12
cubic 38 24 60 31 31 60
quartic 133 98 231 112 112 224
quintic 392 336 728 364 364 728
sextic 1.094 982 2.070 1.028 1.028 2.052


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...1. A2uA2u...3. EuEu.
Subtotal: 11 / 5 / 6
Irrep combinations (i,j) with indices: pos(A1g) ≤ i ≤ j ≤ pos(Eu)
Subtotal: 0 / 0 / 15
Total: 11 / 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...2. A1gA2uA2u...6. A1gEuEu...6. EgEuEu.
Subtotal: 22 / 5 / 30
Irrep combinations (i,j,k) with indices: pos(A1g) ≤ i ≤ j ≤ k ≤ pos(Eu)
..4. EgA1uEu...4. EgA2uEu.
Subtotal: 8 / 2 / 20
Total: 38 / 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...1. A2uA2uA2uA2u...6. EuEuEuEu.
Subtotal: 19 / 5 / 6
Irrep combinations (i,i,i,j) (i,j,j,j) with indices: pos(A1g) ≤ i ≤ j ≤ pos(Eu)
..8. A1gEgEgEg...4. A1uEuEuEu...4. A2uEuEuEu.
Subtotal: 16 / 3 / 30
Irrep combinations (i,i,j,j) with indices: pos(A1g) ≤ i ≤ j ≤ pos(Eu)
..9. A1gA1gEgEg...3. A1gA1gA1uA1u...3. A1gA1gA2uA2u...9. A1gA1gEuEu...3. EgEgA1uA1u...3. EgEgA2uA2u...19. EgEgEuEu...1. A1uA1uA2uA2u...3. A1uA1uEuEu...3. A2uA2uEuEu.
Subtotal: 56 / 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)
..1. EgEgA1uA2u...6. EgEgA1uEu...6. EgEgA2uEu...12. A1gEgEuEu...1. A1uA2uEuEu.
Subtotal: 26 / 5 / 60
Irrep combinations (i,j,k,l) with indices: pos(A1g) ≤ i ≤ j ≤ k ≤ l ≤ pos(Eu)
..8. A1gEgA1uEu...8. A1gEgA2uEu.
Subtotal: 16 / 2 / 15
Total: 133 / 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