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 39 0 0 -5 -3 0 0 9
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 33 0 0 -3 -3 0 0 9


Decomposition to irreducible representations
Motion Ag Eg* Tg Au Eu* Tu Total
Cartesian 3N 2 2 4 0 0 7 15
Translation (x,y,z) 0 0 0 0 0 1 1
Rotation (Rx,Ry,Rz) 0 0 1 0 0 0 1
Vibration 2 2 3 0 0 6 13



Molecular parameter
Number of Atoms (N) 13
Number of internal coordinates 33
Number of independant internal coordinates 2
Number of vibrational modes 13

Force field analysis


Allowed / forbidden vibronational transitions
Operator Ag Eg* Tg Au Eu* Tu Total
Linear (IR) 2 2 3 0 0 6 6 / 7
Quadratic (Raman) 2 2 3 0 0 6 7 / 6
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 33 0 0 -3 -3 0 0 9
quadratic 561 0 0 21 21 0 0 57
cubic 6.545 11 11 -55 -55 -1 -1 273
quartic 58.905 0 0 225 225 0 0 1.113
quintic 435.897 0 0 -531 -531 0 0 3.969
sextic 2.760.681 66 66 1.653 1.653 6 6 12.817


Decomposition to irreducible representations
Column with number of nonvanshing force constants highlighted
Force field Ag Eg* Tg Au Eu* Tu
linear 2 2 3 0 0 6
quadratic 34 34 63 18 18 72
cubic 301 296 784 238 232 866
quartic 2.631 2.631 7.224 2.334 2.334 7.446
quintic 18.570 18.570 53.991 17.622 17.622 55.116
sextic 116.930 116.894 343.483 113.584 113.554 346.274


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)
..3. AgAg...4. EgEg...6. TgTg...21. TuTu.
Subtotal: 34 / 4 / 6
Irrep combinations (i,j) with indices: pos(Ag) ≤ i ≤ j ≤ pos(Tu)
Subtotal: 0 / 0 / 15
Total: 34 / 4 / 21


Contributions to nonvanishing cubic force field constants
Irrep combinations (i,i,i) with indices: pos(Ag) ≤ i ≤ pos(Tu)
..4. AgAgAg...8. EgEgEg...11. TgTgTg.
Subtotal: 23 / 3 / 6
Irrep combinations (i,i,j) (i,j,j) with indices: pos(Ag) ≤ i ≤ j ≤ pos(Tu)
..8. AgEgEg...12. AgTgTg...42. AgTuTu...24. EgTgTg...84. EgTuTu...108. TgTuTu.
Subtotal: 278 / 6 / 30
Irrep combinations (i,j,k) with indices: pos(Ag) ≤ i ≤ j ≤ k ≤ pos(Tu)
Subtotal: 0 / 0 / 20
Total: 301 / 9 / 56


Contributions to nonvanishing quartic force field constants
Irrep combinations (i,i,i,i) with indices: pos(Ag) ≤ i ≤ pos(Tu)
..5. AgAgAgAg...9. EgEgEgEg...51. TgTgTgTg...567. TuTuTuTu.
Subtotal: 632 / 4 / 6
Irrep combinations (i,i,i,j) (i,j,j,j) with indices: pos(Ag) ≤ i ≤ j ≤ pos(Tu)
..16. AgEgEgEg...22. AgTgTgTg...32. EgTgTgTg.
Subtotal: 70 / 3 / 30
Irrep combinations (i,i,j,j) with indices: pos(Ag) ≤ i ≤ j ≤ pos(Tu)
..12. AgAgEgEg...18. AgAgTgTg...63. AgAgTuTu...60. EgEgTgTg...210. EgEgTuTu...702. TgTgTuTu.
Subtotal: 1.065 / 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)
..48. AgEgTgTg...168. AgEgTuTu...216. AgTgTuTu...432. EgTgTuTu.
Subtotal: 864 / 4 / 60
Irrep combinations (i,j,k,l) with indices: pos(Ag) ≤ i ≤ j ≤ k ≤ l ≤ pos(Tu)
Subtotal: 0 / 0 / 15
Total: 2.631 / 17 / 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