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 252 0 0 0 0 0 0 4
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 246 0 0 2 0 0 0 4


Decomposition to irreducible representations
Motion Ag Eg* Tg Au Eu* Tu Total
Cartesian 3N 11 11 31 10 10 32 105
Translation (x,y,z) 0 0 0 0 0 1 1
Rotation (Rx,Ry,Rz) 0 0 1 0 0 0 1
Vibration 11 11 30 10 10 31 103



Molecular parameter
Number of Atoms (N) 84
Number of internal coordinates 246
Number of independant internal coordinates 11
Number of vibrational modes 103

Force field analysis


Allowed / forbidden vibronational transitions
Operator Ag Eg* Tg Au Eu* Tu Total
Linear (IR) 11 11 30 10 10 31 31 / 72
Quadratic (Raman) 11 11 30 10 10 31 52 / 51
IR + Raman - - - - - - - - - - - - 10 10 - - - - 0* / 20
* 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 246 0 0 2 0 0 0 4
quadratic 30.381 0 0 125 123 0 0 131
cubic 2.511.496 82 82 248 0 0 0 504
quartic 156.340.626 0 0 7.874 7.626 0 0 8.626
quintic 7.817.031.300 0 0 15.500 0 0 0 32.000
sextic 327.012.476.050 3.403 3.403 333.250 317.750 41 41 380.750


Decomposition to irreducible representations
Column with number of nonvanshing force constants highlighted
Force field Ag Eg* Tg Au Eu* Tu
linear 11 11 30 10 10 31
quadratic 1.303 1.303 3.781 1.260 1.260 3.783
cubic 104.767 104.726 313.843 104.641 104.600 313.969
quartic 6.516.573 6.516.573 19.541.469 6.513.781 6.513.781 19.541.719
quintic 325.715.575 325.715.575 977.122.975 325.707.575 325.707.575 977.130.975
sextic 13.625.623.473 13.625.621.751 40.876.509.975 13.625.501.779 13.625.500.098 40.876.525.725


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)
..66. AgAg...121. EgEg...465. TgTg...55. AuAu...100. EuEu...496. TuTu.
Subtotal: 1.303 / 6 / 6
Irrep combinations (i,j) with indices: pos(Ag) ≤ i ≤ j ≤ pos(Tu)
Subtotal: 0 / 0 / 15
Total: 1.303 / 6 / 21


Contributions to nonvanishing cubic force field constants
Irrep combinations (i,i,i) with indices: pos(Ag) ≤ i ≤ pos(Tu)
..286. AgAgAg...572. EgEgEg...9.020. TgTgTg.
Subtotal: 9.878 / 3 / 6
Irrep combinations (i,i,j) (i,j,j) with indices: pos(Ag) ≤ i ≤ j ≤ pos(Tu)
..1.331. AgEgEg...5.115. AgTgTg...605. AgAuAu...1.100. AgEuEu...5.456. AgTuTu...10.230. EgTgTg...1.210. EgEuEu...10.912. EgTuTu...28.830. TgTuTu.
Subtotal: 64.789 / 9 / 30
Irrep combinations (i,j,k) with indices: pos(Ag) ≤ i ≤ j ≤ k ≤ pos(Tu)
..2.200. EgAuEu...9.300. TgAuTu...18.600. TgEuTu.
Subtotal: 30.100 / 3 / 20
Total: 104.767 / 15 / 56


Contributions to nonvanishing quartic force field constants
Irrep combinations (i,i,i,i) with indices: pos(Ag) ≤ i ≤ pos(Tu)
..1.001. AgAgAgAg...4.356. EgEgEgEg...257.145. TgTgTgTg...715. AuAuAuAu...3.025. EuEuEuEu...292.392. TuTuTuTu.
Subtotal: 558.634 / 6 / 6
Irrep combinations (i,i,i,j) (i,j,j,j) with indices: pos(Ag) ≤ i ≤ j ≤ pos(Tu)
..6.292. AgEgEgEg...99.220. AgTgTgTg...197.780. EgTgTgTg...4.400. AuEuEuEu...99.510. AuTuTuTu...198.400. EuTuTuTu.
Subtotal: 605.602 / 6 / 30
Irrep combinations (i,i,j,j) with indices: pos(Ag) ≤ i ≤ j ≤ pos(Tu)
..7.986. AgAgEgEg...30.690. AgAgTgTg...3.630. AgAgAuAu...6.600. AgAgEuEu...32.736. AgAgTuTu...117.645. EgEgTgTg...6.655. EgEgAuAu...19.360. EgEgEuEu...125.488. EgEgTuTu...25.575. TgTgAuAu.
..97.650. TgTgEuEu...1.556.820. TgTgTuTu...5.500. AuAuEuEu...27.280. AuAuTuTu...104.160. EuEuTuTu.
Subtotal: 2.167.775 / 15 / 15
Irrep combinations (i,i,j,k) (i,j,j,k) (i,j,k,k) with indices: pos(Ag) ≤ i ≤ j ≤ k ≤ pos(Tu)
..13.200. EgEgAuEu...93.000. TgTgAuEu...279.000. TgTgAuTu...558.000. TgTgEuTu...112.530. AgEgTgTg...13.310. AgEgEuEu...120.032. AgEgTuTu...317.130. AgTgTuTu...634.260. EgTgTuTu...99.200. AuEuTuTu.
Subtotal: 2.239.662 / 10 / 60
Irrep combinations (i,j,k,l) with indices: pos(Ag) ≤ i ≤ j ≤ k ≤ l ≤ pos(Tu)
..24.200. AgEgAuEu...102.300. AgTgAuTu...204.600. AgTgEuTu...204.600. EgTgAuTu...409.200. EgTgEuTu.
Subtotal: 944.900 / 5 / 15
Total: 6.516.573 / 42 / 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