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 36 0 0 -4 0 0 0 8
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 30 0 0 -2 0 0 0 8


Decomposition to irreducible representations
Motion Ag Eg* Tg Au Eu* Tu Total
Cartesian 3N 2 2 4 0 0 6 14
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 5 12



Molecular parameter
Number of Atoms (N) 12
Number of internal coordinates 30
Number of independant internal coordinates 2
Number of vibrational modes 12

Force field analysis


Allowed / forbidden vibronational transitions
Operator Ag Eg* Tg Au Eu* Tu Total
Linear (IR) 2 2 3 0 0 5 5 / 7
Quadratic (Raman) 2 2 3 0 0 5 7 / 5
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 30 0 0 -2 0 0 0 8
quadratic 465 0 0 17 15 0 0 47
cubic 4.960 10 10 -32 0 0 0 208
quartic 40.920 0 0 152 120 0 0 792
quintic 278.256 0 0 -272 0 0 0 2.640
sextic 1.623.160 55 55 952 680 5 5 8.008


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 5
quadratic 28 28 52 15 15 60
cubic 232 227 598 180 175 650
quartic 1.828 1.828 5.012 1.620 1.620 5.180
quintic 11.890 11.890 34.486 11.230 11.230 35.146
sextic 68.800 68.770 201.860 66.738 66.713 203.692


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...15. TuTu.
Subtotal: 28 / 4 / 6
Irrep combinations (i,j) with indices: pos(Ag) ≤ i ≤ j ≤ pos(Tu)
Subtotal: 0 / 0 / 15
Total: 28 / 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...30. AgTuTu...24. EgTgTg...60. EgTuTu...75. TgTuTu.
Subtotal: 209 / 6 / 30
Irrep combinations (i,j,k) with indices: pos(Ag) ≤ i ≤ j ≤ k ≤ pos(Tu)
Subtotal: 0 / 0 / 20
Total: 232 / 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...295. TuTuTuTu.
Subtotal: 360 / 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...45. AgAgTuTu...60. EgEgTgTg...150. EgEgTuTu...495. TgTgTuTu.
Subtotal: 780 / 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...120. AgEgTuTu...150. AgTgTuTu...300. EgTgTuTu.
Subtotal: 618 / 4 / 60
Irrep combinations (i,j,k,l) with indices: pos(Ag) ≤ i ≤ j ≤ k ≤ l ≤ pos(Tu)
Subtotal: 0 / 0 / 15
Total: 1.828 / 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