Results for Point Group D3d



Characters of representations for molecular motions
Motion E 2C3 3C'2 i 2S6 d
Cartesian 3N 39 0 -1 -3 0 5
Translation (x,y,z) 3 0 -1 -3 0 1
Rotation (Rx,Ry,Rz) 3 0 -1 3 0 -1
Vibration 33 0 1 -3 0 5


Decomposition to irreducible representations
Motion A1g A2g Eg A1u A2u Eu Total
Cartesian 3N 4 2 6 2 5 7 26
Translation (x,y,z) 0 0 0 0 1 1 2
Rotation (Rx,Ry,Rz) 0 1 1 0 0 0 2
Vibration 4 1 5 2 4 6 22



Molecular parameter
Number of Atoms (N) 13
Number of internal coordinates 33
Number of independant internal coordinates 4
Number of vibrational modes 22


Force field analysis


Allowed / forbidden vibronational transitions
Operator A1g A2g Eg A1u A2u Eu Total
Linear (IR) 4 1 5 2 4 6 10 / 12
Quadratic (Raman) 4 1 5 2 4 6 9 / 13
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 33 0 1 -3 0 5
quadratic 561 0 17 21 0 29
cubic 6.545 11 17 -55 -1 105
quartic 58.905 0 153 225 0 385
quintic 435.897 0 153 -531 0 1.141
sextic 2.760.681 66 969 1.653 6 3.325


Decomposition to irreducible representations
Column with number of nonvanshing force constants highlighted
Force field A1g A2g Eg A1u A2u Eu
linear 4 1 5 2 4 6
quadratic 60 37 97 42 48 90
cubic 573 512 1.080 530 574 1.098
quartic 5.062 4.793 9.855 4.832 4.948 9.780
quintic 36.604 35.957 72.561 36.122 36.616 72.738
sextic 231.280 229.133 460.377 229.340 230.518 459.828


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)
..10. A1gA1g...1. A2gA2g...15. EgEg...3. A1uA1u...10. A2uA2u...21. EuEu.
Subtotal: 60 / 6 / 6
Irrep combinations (i,j) with indices: pos(A1g) ≤ i ≤ j ≤ pos(Eu)
Subtotal: 0 / 0 / 15
Total: 60 / 6 / 21


Contributions to nonvanishing cubic force field constants
Irrep combinations (i,i,i) with indices: pos(A1g) ≤ i ≤ pos(Eu)
..20. A1gA1gA1g...35. EgEgEg.
Subtotal: 55 / 2 / 6
Irrep combinations (i,i,j) (i,j,j) with indices: pos(A1g) ≤ i ≤ j ≤ pos(Eu)
..4. A1gA2gA2g...60. A1gEgEg...12. A1gA1uA1u...40. A1gA2uA2u...84. A1gEuEu...10. A2gEgEg...15. A2gEuEu...105. EgEuEu.
Subtotal: 330 / 8 / 30
Irrep combinations (i,j,k) with indices: pos(A1g) ≤ i ≤ j ≤ k ≤ pos(Eu)
..8. A2gA1uA2u...60. EgA1uEu...120. EgA2uEu.
Subtotal: 188 / 3 / 20
Total: 573 / 13 / 56


Contributions to nonvanishing quartic force field constants
Irrep combinations (i,i,i,i) with indices: pos(A1g) ≤ i ≤ pos(Eu)
..35. A1gA1gA1gA1g...1. A2gA2gA2gA2g...120. EgEgEgEg...5. A1uA1uA1uA1u...35. A2uA2uA2uA2u...231. EuEuEuEu.
Subtotal: 427 / 6 / 6
Irrep combinations (i,i,i,j) (i,j,j,j) with indices: pos(A1g) ≤ i ≤ j ≤ pos(Eu)
..140. A1gEgEgEg...35. A2gEgEgEg...112. A1uEuEuEu...224. A2uEuEuEu.
Subtotal: 511 / 4 / 30
Irrep combinations (i,i,j,j) with indices: pos(A1g) ≤ i ≤ j ≤ pos(Eu)
..10. A1gA1gA2gA2g...150. A1gA1gEgEg...30. A1gA1gA1uA1u...100. A1gA1gA2uA2u...210. A1gA1gEuEu...15. A2gA2gEgEg...3. A2gA2gA1uA1u...10. A2gA2gA2uA2u...21. A2gA2gEuEu...45. EgEgA1uA1u.
..150. EgEgA2uA2u...780. EgEgEuEu...30. A1uA1uA2uA2u...63. A1uA1uEuEu...210. A2uA2uEuEu.
Subtotal: 1.827 / 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)
..80. EgEgA1uA2u...180. EgEgA1uEu...360. EgEgA2uEu...40. A1gA2gEgEg...60. A1gA2gEuEu...420. A1gEgEuEu...105. A2gEgEuEu...120. A1uA2uEuEu.
Subtotal: 1.365 / 8 / 60
Irrep combinations (i,j,k,l) with indices: pos(A1g) ≤ i ≤ j ≤ k ≤ l ≤ pos(Eu)
..32. A1gA2gA1uA2u...240. A1gEgA1uEu...480. A1gEgA2uEu...60. A2gEgA1uEu...120. A2gEgA2uEu.
Subtotal: 932 / 5 / 15
Total: 5.062 / 38 / 126


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