Results for Point Group D3h



Characters of representations for molecular motions
Motion E 2C3 3C'2 σh 2S3 3σv
Cartesian 3N 18 0 -2 4 -2 4
Translation (x,y,z) 3 0 -1 1 -2 1
Rotation (Rx,Ry,Rz) 3 0 -1 -1 2 -1
Vibration 12 0 0 4 -2 4


Decomposition to irreducible representations
Motion A'1 A'2 E' A''1 A''2 E'' Total
Cartesian 3N 2 1 4 0 3 2 12
Translation (x,y,z) 0 0 1 0 1 0 2
Rotation (Rx,Ry,Rz) 0 1 0 0 0 1 2
Vibration 2 0 3 0 2 1 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 A'1 A'2 E' A''1 A''2 E'' Total
Linear (IR) 2 0 3 0 2 1 5 / 3
Quadratic (Raman) 2 0 3 0 2 1 6 / 2
IR + Raman - - - - 0 3 0 - - - - - - - - 3 / 0


Characters of force fields
(Symmetric powers of vibration representation)
Force field E 2C3 3C'2 σh 2S3 3σv
linear 12 0 0 4 -2 4
quadratic 78 0 6 14 2 14
cubic 364 4 0 36 0 36
quartic 1.365 0 21 85 -2 85
quintic 4.368 0 0 176 2 176
sextic 12.376 10 56 344 2 344


Decomposition to irreducible representations
Column with number of nonvanshing force constants highlighted
Force field A'1 A'2 E' A''1 A''2 E''
linear 2 0 3 0 2 1
quadratic 13 3 15 3 7 11
cubic 43 25 66 19 37 54
quartic 147 94 242 91 123 213
quintic 423 335 757 305 393 699
sextic 1.162 962 2.118 932 1.076 2.004


Further Reading


Contributions to nonvanishing force field constants


pos(X) : Position of irreducible representation (irrep) X in character table of D3h

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(A'1) ≤ i ≤ pos(E'')
..3. A'1A'1...6. E'E'...3. A''2A''2...1. E''E''.
Subtotal: 13 / 4 / 6
Irrep combinations (i,j) with indices: pos(A'1) ≤ i ≤ j ≤ pos(E'')
Subtotal: 0 / 0 / 15
Total: 13 / 4 / 21


Contributions to nonvanishing cubic force field constants
Irrep combinations (i,i,i) with indices: pos(A'1) ≤ i ≤ pos(E'')
..4. A'1A'1A'1...10. E'E'E'.
Subtotal: 14 / 2 / 6
Irrep combinations (i,i,j) (i,j,j) with indices: pos(A'1) ≤ i ≤ j ≤ pos(E'')
..12. A'1E'E'...6. A'1A''2A''2...2. A'1E''E''...3. E'E''E''.
Subtotal: 23 / 4 / 30
Irrep combinations (i,j,k) with indices: pos(A'1) ≤ i ≤ j ≤ k ≤ pos(E'')
..6. E'A''2E''.
Subtotal: 6 / 1 / 20
Total: 43 / 7 / 56


Contributions to nonvanishing quartic force field constants
Irrep combinations (i,i,i,i) with indices: pos(A'1) ≤ i ≤ pos(E'')
..5. A'1A'1A'1A'1...21. E'E'E'E'...5. A''2A''2A''2A''2...1. E''E''E''E''.
Subtotal: 32 / 4 / 6
Irrep combinations (i,i,i,j) (i,j,j,j) with indices: pos(A'1) ≤ i ≤ j ≤ pos(E'')
..20. A'1E'E'E'...2. A''2E''E''E''.
Subtotal: 22 / 2 / 30
Irrep combinations (i,i,j,j) with indices: pos(A'1) ≤ i ≤ j ≤ pos(E'')
..18. A'1A'1E'E'...9. A'1A'1A''2A''2...3. A'1A'1E''E''...18. E'E'A''2A''2...12. E'E'E''E''...3. A''2A''2E''E''.
Subtotal: 63 / 6 / 15
Irrep combinations (i,i,j,k) (i,j,j,k) (i,j,k,k) with indices: pos(A'1) ≤ i ≤ j ≤ k ≤ pos(E'')
..12. E'E'A''2E''...6. A'1E'E''E''.
Subtotal: 18 / 2 / 60
Irrep combinations (i,j,k,l) with indices: pos(A'1) ≤ i ≤ j ≤ k ≤ l ≤ pos(E'')
..12. A'1E'A''2E''.
Subtotal: 12 / 1 / 15
Total: 147 / 15 / 126


Calculate contributions to

A'1 A'2 E' A''1 A''2 E''
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Last update November, 13th 2023 by A. Gelessus, Impressum, Datenschutzerklärung/DataPrivacyStatement