Results for Point Group D3d



Characters of representations for molecular motions
Motion E 2C3 3C'2 i 2S6 d
Cartesian 3N 48 0 0 0 0 8
Translation (x,y,z) 3 0 -1 -3 0 1
Rotation (Rx,Ry,Rz) 3 0 -1 3 0 -1
Vibration 42 0 2 0 0 8


Decomposition to irreducible representations
Motion A1g A2g Eg A1u A2u Eu Total
Cartesian 3N 6 2 8 2 6 8 32
Translation (x,y,z) 0 0 0 0 1 1 2
Rotation (Rx,Ry,Rz) 0 1 1 0 0 0 2
Vibration 6 1 7 2 5 7 28



Molecular parameter
Number of Atoms (N) 16
Number of internal coordinates 42
Number of independant internal coordinates 6
Number of vibrational modes 28


Force field analysis


Allowed / forbidden vibronational transitions
Operator A1g A2g Eg A1u A2u Eu Total
Linear (IR) 6 1 7 2 5 7 12 / 16
Quadratic (Raman) 6 1 7 2 5 7 13 / 15
IR + Raman - - - - 1 - - - - 2 - - - - - - - - 0* / 3
* Parity Mutual Exclusion Principle


Characters of force fields
(Symmetric powers of vibration representation)
Force field E 2C3 3C'2 i 2S6 d
linear 42 0 2 0 0 8
quadratic 903 0 23 21 0 53
cubic 13.244 14 44 0 0 256
quartic 148.995 0 275 231 0 1.095
quintic 1.370.754 0 506 0 0 4.056
sextic 10.737.573 105 2.277 1.771 7 13.803


Decomposition to irreducible representations
Column with number of nonvanshing force constants highlighted
Force field A1g A2g Eg A1u A2u Eu
linear 6 1 7 2 5 7
quadratic 96 58 154 66 81 147
cubic 1.181 1.031 2.205 1.053 1.159 2.205
quartic 12.778 12.093 24.871 12.192 12.602 24.794
quintic 115.370 113.089 228.459 113.342 115.117 228.459
sextic 898.984 890.944 1.789.872 891.785 897.548 1.789.284


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)
..21. A1gA1g...1. A2gA2g...28. EgEg...3. A1uA1u...15. A2uA2u...28. EuEu.
Subtotal: 96 / 6 / 6
Irrep combinations (i,j) with indices: pos(A1g) ≤ i ≤ j ≤ pos(Eu)
Subtotal: 0 / 0 / 15
Total: 96 / 6 / 21


Contributions to nonvanishing cubic force field constants
Irrep combinations (i,i,i) with indices: pos(A1g) ≤ i ≤ pos(Eu)
..56. A1gA1gA1g...84. EgEgEg.
Subtotal: 140 / 2 / 6
Irrep combinations (i,i,j) (i,j,j) with indices: pos(A1g) ≤ i ≤ j ≤ pos(Eu)
..6. A1gA2gA2g...168. A1gEgEg...18. A1gA1uA1u...90. A1gA2uA2u...168. A1gEuEu...21. A2gEgEg...21. A2gEuEu...196. EgEuEu.
Subtotal: 688 / 8 / 30
Irrep combinations (i,j,k) with indices: pos(A1g) ≤ i ≤ j ≤ k ≤ pos(Eu)
..10. A2gA1uA2u...98. EgA1uEu...245. EgA2uEu.
Subtotal: 353 / 3 / 20
Total: 1.181 / 13 / 56


Contributions to nonvanishing quartic force field constants
Irrep combinations (i,i,i,i) with indices: pos(A1g) ≤ i ≤ pos(Eu)
..126. A1gA1gA1gA1g...1. A2gA2gA2gA2g...406. EgEgEgEg...5. A1uA1uA1uA1u...70. A2uA2uA2uA2u...406. EuEuEuEu.
Subtotal: 1.014 / 6 / 6
Irrep combinations (i,i,i,j) (i,j,j,j) with indices: pos(A1g) ≤ i ≤ j ≤ pos(Eu)
..504. A1gEgEgEg...84. A2gEgEgEg...168. A1uEuEuEu...420. A2uEuEuEu.
Subtotal: 1.176 / 4 / 30
Irrep combinations (i,i,j,j) with indices: pos(A1g) ≤ i ≤ j ≤ pos(Eu)
..21. A1gA1gA2gA2g...588. A1gA1gEgEg...63. A1gA1gA1uA1u...315. A1gA1gA2uA2u...588. A1gA1gEuEu...28. A2gA2gEgEg...3. A2gA2gA1uA1u...15. A2gA2gA2uA2u...28. A2gA2gEuEu...84. EgEgA1uA1u.
..420. EgEgA2uA2u...2.009. EgEgEuEu...45. A1uA1uA2uA2u...84. A1uA1uEuEu...420. A2uA2uEuEu.
Subtotal: 4.711 / 15 / 15
Irrep combinations (i,i,j,k) (i,j,j,k) (i,j,k,k) with indices: pos(A1g) ≤ i ≤ j ≤ k ≤ pos(Eu)
..210. EgEgA1uA2u...392. EgEgA1uEu...980. EgEgA2uEu...126. A1gA2gEgEg...126. A1gA2gEuEu...1.176. A1gEgEuEu...196. A2gEgEuEu...210. A1uA2uEuEu.
Subtotal: 3.416 / 8 / 60
Irrep combinations (i,j,k,l) with indices: pos(A1g) ≤ i ≤ j ≤ k ≤ l ≤ pos(Eu)
..60. A1gA2gA1uA2u...588. A1gEgA1uEu...1.470. A1gEgA2uEu...98. A2gEgA1uEu...245. A2gEgA2uEu.
Subtotal: 2.461 / 5 / 15
Total: 12.778 / 38 / 126


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Last update November, 13th 2023 by A. Gelessus, Impressum, Datenschutzerklärung/DataPrivacyStatement