Results for Point Group S6



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


Decomposition to irreducible representations
Motion Ag Eg* Au Eu* Total
Cartesian 3N 6 6 7 7 26
Translation (x,y,z) 0 0 1 1 2
Rotation (Rx,Ry,Rz) 1 1 0 0 2
Vibration 5 5 6 6 22



Molecular parameter
Number of Atoms (N) 13
Number of internal coordinates 33
Number of independant internal coordinates 5
Number of vibrational modes 22

Force field analysis


Allowed / forbidden vibronational transitions
Operator Ag Eg* Au Eu* Total
Linear (IR) 5 5 6 6 12 / 10
Quadratic (Raman) 5 5 6 6 10 / 12
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 33 0 0 -3 0 0
quadratic 561 0 0 21 0 0
cubic 6.545 11 11 -55 -1 -1
quartic 58.905 0 0 225 0 0
quintic 435.897 0 0 -531 0 0
sextic 2.760.681 66 66 1.653 6 6


Decomposition to irreducible representations
Column with number of nonvanshing force constants highlighted
Force field Ag Eg* Au Eu*
linear 5 5 6 6
quadratic 97 97 90 90
cubic 1.085 1.080 1.104 1.098
quartic 9.855 9.855 9.780 9.780
quintic 72.561 72.561 72.738 72.738
sextic 460.413 460.377 459.858 459.828


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...21. AuAu...36. EuEu.
Subtotal: 97 / 4 / 4
Irrep combinations (i,j) with indices: pos(Ag) ≤ i ≤ j ≤ pos(Eu)
Subtotal: 0 / 0 / 6
Total: 97 / 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...105. AgAuAu...180. AgEuEu...210. EgEuEu.
Subtotal: 620 / 4 / 12
Irrep combinations (i,j,k) with indices: pos(Ag) ≤ i ≤ j ≤ k ≤ pos(Eu)
..360. EgAuEu.
Subtotal: 360 / 1 / 4
Total: 1.085 / 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...126. AuAuAuAu...441. EuEuEuEu.
Subtotal: 862 / 4 / 4
Irrep combinations (i,i,i,j) (i,j,j,j) with indices: pos(Ag) ≤ i ≤ j ≤ pos(Eu)
..350. AgEgEgEg...672. AuEuEuEu.
Subtotal: 1.022 / 2 / 12
Irrep combinations (i,i,j,j) with indices: pos(Ag) ≤ i ≤ j ≤ pos(Eu)
..375. AgAgEgEg...315. AgAgAuAu...540. AgAgEuEu...525. EgEgAuAu...1.530. EgEgEuEu...756. AuAuEuEu.
Subtotal: 4.041 / 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)
..1.080. EgEgAuEu...1.050. AgEgEuEu.
Subtotal: 2.130 / 2 / 12
Irrep combinations (i,j,k,l) with indices: pos(Ag) ≤ i ≤ j ≤ k ≤ l ≤ pos(Eu)
..1.800. AgEgAuEu.
Subtotal: 1.800 / 1 / 1
Total: 9.855 / 15 / 35


Calculate contributions to

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