Results for Point Group Th



Characters of representations for molecular motions
Motion E 4C3 4(C3)2 3C2 i 4(S6)5 4S6 3σh
Cartesian 3N 21 0 0 -3 -3 0 0 5
Translation (x,y,z) 3 0 0 -1 -3 0 0 1
Rotation (Rx,Ry,Rz) 3 0 0 -1 3 0 0 -1
Vibration 15 0 0 -1 -3 0 0 5


Decomposition to irreducible representations
Motion Ag Eg* Tg Au Eu* Tu Total
Cartesian 3N 1 1 2 0 0 4 8
Translation (x,y,z) 0 0 0 0 0 1 1
Rotation (Rx,Ry,Rz) 0 0 1 0 0 0 1
Vibration 1 1 1 0 0 3 6



Molecular parameter
Number of Atoms (N) 7
Number of internal coordinates 15
Number of independant internal coordinates 1
Number of vibrational modes 6

Force field analysis


Allowed / forbidden vibronational transitions
Operator Ag Eg* Tg Au Eu* Tu Total
Linear (IR) 1 1 1 0 0 3 3 / 3
Quadratic (Raman) 1 1 1 0 0 3 3 / 3
IR + Raman - - - - - - - - - - - - 0 0 - - - - 0* / 0
* Parity Mutual Exclusion Principle


Characters of force fields
(Symmetric powers of vibration representation)
Force field E 4C3 4(C3)2 3C2 i 4(S6)5 4S6 3σh
linear 15 0 0 -1 -3 0 0 5
quadratic 120 0 0 8 12 0 0 20
cubic 680 5 5 -8 -28 -1 -1 60
quartic 3.060 0 0 36 72 0 0 160
quintic 11.628 0 0 -36 -144 0 0 376
sextic 38.760 15 15 120 300 3 3 820


Decomposition to irreducible representations
Column with number of nonvanshing force constants highlighted
Force field Ag Eg* Tg Au Eu* Tu
linear 1 1 1 0 0 3
quadratic 9 9 13 3 3 15
cubic 35 33 75 23 20 97
quartic 155 155 367 109 109 389
quintic 521 521 1.393 439 439 1.523
sextic 1.751 1.742 4.765 1.519 1.513 4.895


Further Reading


Contributions to nonvanishing force field constants


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

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(Tu)
..1. AgAg...1. EgEg...1. TgTg...6. TuTu.
Subtotal: 9 / 4 / 6
Irrep combinations (i,j) with indices: pos(Ag) ≤ i ≤ j ≤ pos(Tu)
Subtotal: 0 / 0 / 15
Total: 9 / 4 / 21


Contributions to nonvanishing cubic force field constants
Irrep combinations (i,i,i) with indices: pos(Ag) ≤ i ≤ pos(Tu)
..1. AgAgAg...2. EgEgEg...1. TgTgTg.
Subtotal: 4 / 3 / 6
Irrep combinations (i,i,j) (i,j,j) with indices: pos(Ag) ≤ i ≤ j ≤ pos(Tu)
..1. AgEgEg...1. AgTgTg...6. AgTuTu...2. EgTgTg...12. EgTuTu...9. TgTuTu.
Subtotal: 31 / 6 / 30
Irrep combinations (i,j,k) with indices: pos(Ag) ≤ i ≤ j ≤ k ≤ pos(Tu)
Subtotal: 0 / 0 / 20
Total: 35 / 9 / 56


Contributions to nonvanishing quartic force field constants
Irrep combinations (i,i,i,i) with indices: pos(Ag) ≤ i ≤ pos(Tu)
..1. AgAgAgAg...1. EgEgEgEg...2. TgTgTgTg...51. TuTuTuTu.
Subtotal: 55 / 4 / 6
Irrep combinations (i,i,i,j) (i,j,j,j) with indices: pos(Ag) ≤ i ≤ j ≤ pos(Tu)
..2. AgEgEgEg...1. AgTgTgTg.
Subtotal: 3 / 2 / 30
Irrep combinations (i,i,j,j) with indices: pos(Ag) ≤ i ≤ j ≤ pos(Tu)
..1. AgAgEgEg...1. AgAgTgTg...6. AgAgTuTu...3. EgEgTgTg...18. EgEgTuTu...27. TgTgTuTu.
Subtotal: 56 / 6 / 15
Irrep combinations (i,i,j,k) (i,j,j,k) (i,j,k,k) with indices: pos(Ag) ≤ i ≤ j ≤ k ≤ pos(Tu)
..2. AgEgTgTg...12. AgEgTuTu...9. AgTgTuTu...18. EgTgTuTu.
Subtotal: 41 / 4 / 60
Irrep combinations (i,j,k,l) with indices: pos(Ag) ≤ i ≤ j ≤ k ≤ l ≤ pos(Tu)
Subtotal: 0 / 0 / 15
Total: 155 / 16 / 126


Calculate contributions to

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