Results for Point Group S6



Characters of representations for molecular motions
Motion E C3 (C3)2 i (S6)5 S6
Cartesian 3N 36 0 0 0 0 0
Translation (x,y,z) 3 0 0 -3 0 0
Rotation (Rx,Ry,Rz) 3 0 0 3 0 0
Vibration 30 0 0 0 0 0


Decomposition to irreducible representations
Motion Ag Eg* Au Eu* Total
Cartesian 3N 6 6 6 6 24
Translation (x,y,z) 0 0 1 1 2
Rotation (Rx,Ry,Rz) 1 1 0 0 2
Vibration 5 5 5 5 20



Molecular parameter
Number of Atoms (N) 12
Number of internal coordinates 30
Number of independant internal coordinates 5
Number of vibrational modes 20

Force field analysis


Allowed / forbidden vibronational transitions
Operator Ag Eg* Au Eu* Total
Linear (IR) 5 5 5 5 10 / 10
Quadratic (Raman) 5 5 5 5 10 / 10
IR + Raman - - - - - - - - - - - - - - - - 0* / 0
* Parity Mutual Exclusion Principle


Characters of force fields
(Symmetric powers of vibration representation)
Force field E C3 (C3)2 i (S6)5 S6
linear 30 0 0 0 0 0
quadratic 465 0 0 15 0 0
cubic 4.960 10 10 0 0 0
quartic 40.920 0 0 120 0 0
quintic 278.256 0 0 0 0 0
sextic 1.623.160 55 55 680 5 5


Decomposition to irreducible representations
Column with number of nonvanshing force constants highlighted
Force field Ag Eg* Au Eu*
linear 5 5 5 5
quadratic 80 80 75 75
cubic 830 825 830 825
quartic 6.840 6.840 6.800 6.800
quintic 46.376 46.376 46.376 46.376
sextic 270.660 270.630 270.430 270.405


Further Reading


Contributions to nonvanishing force field constants


pos(X) : Position of irreducible representation (irrep) X in character table of S6

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(Ag) ≤ i ≤ pos(Eu)
..15. AgAg...25. EgEg...15. AuAu...25. EuEu.
Subtotal: 80 / 4 / 4
Irrep combinations (i,j) with indices: pos(Ag) ≤ i ≤ j ≤ pos(Eu)
Subtotal: 0 / 0 / 6
Total: 80 / 4 / 10


Contributions to nonvanishing cubic force field constants
Irrep combinations (i,i,i) with indices: pos(Ag) ≤ i ≤ pos(Eu)
..35. AgAgAg...70. EgEgEg.
Subtotal: 105 / 2 / 4
Irrep combinations (i,i,j) (i,j,j) with indices: pos(Ag) ≤ i ≤ j ≤ pos(Eu)
..125. AgEgEg...75. AgAuAu...125. AgEuEu...150. EgEuEu.
Subtotal: 475 / 4 / 12
Irrep combinations (i,j,k) with indices: pos(Ag) ≤ i ≤ j ≤ k ≤ pos(Eu)
..250. EgAuEu.
Subtotal: 250 / 1 / 4
Total: 830 / 7 / 20


Contributions to nonvanishing quartic force field constants
Irrep combinations (i,i,i,i) with indices: pos(Ag) ≤ i ≤ pos(Eu)
..70. AgAgAgAg...225. EgEgEgEg...70. AuAuAuAu...225. EuEuEuEu.
Subtotal: 590 / 4 / 4
Irrep combinations (i,i,i,j) (i,j,j,j) with indices: pos(Ag) ≤ i ≤ j ≤ pos(Eu)
..350. AgEgEgEg...350. AuEuEuEu.
Subtotal: 700 / 2 / 12
Irrep combinations (i,i,j,j) with indices: pos(Ag) ≤ i ≤ j ≤ pos(Eu)
..375. AgAgEgEg...225. AgAgAuAu...375. AgAgEuEu...375. EgEgAuAu...1.075. EgEgEuEu...375. AuAuEuEu.
Subtotal: 2.800 / 6 / 6
Irrep combinations (i,i,j,k) (i,j,j,k) (i,j,k,k) with indices: pos(Ag) ≤ i ≤ j ≤ k ≤ pos(Eu)
..750. EgEgAuEu...750. AgEgEuEu.
Subtotal: 1.500 / 2 / 12
Irrep combinations (i,j,k,l) with indices: pos(Ag) ≤ i ≤ j ≤ k ≤ l ≤ pos(Eu)
..1.250. AgEgAuEu.
Subtotal: 1.250 / 1 / 1
Total: 6.840 / 15 / 35


Calculate contributions to

Ag Eg Au Eu
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Last update November, 13th 2023 by A. Gelessus, Impressum, Datenschutzerklärung/DataPrivacyStatement