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 156 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 150 0 0 2 0 0 0 4


Decomposition to irreducible representations
Motion Ag Eg* Tg Au Eu* Tu Total
Cartesian 3N 7 7 19 6 6 20 65
Translation (x,y,z) 0 0 0 0 0 1 1
Rotation (Rx,Ry,Rz) 0 0 1 0 0 0 1
Vibration 7 7 18 6 6 19 63



Molecular parameter
Number of Atoms (N) 52
Number of internal coordinates 150
Number of independant internal coordinates 7
Number of vibrational modes 63

Force field analysis


Allowed / forbidden vibronational transitions
Operator Ag Eg* Tg Au Eu* Tu Total
Linear (IR) 7 7 18 6 6 19 19 / 44
Quadratic (Raman) 7 7 18 6 6 19 32 / 31
IR + Raman - - - - - - - - - - - - 6 6 - - - - 0* / 12
* 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 150 0 0 2 0 0 0 4
quadratic 11.325 0 0 77 75 0 0 83
cubic 573.800 50 50 152 0 0 0 312
quartic 21.947.850 0 0 3.002 2.850 0 0 3.466
quintic 675.993.780 0 0 5.852 0 0 0 12.320
sextic 17.463.172.650 1.275 1.275 79.002 73.150 25 25 97.174


Decomposition to irreducible representations
Column with number of nonvanshing force constants highlighted
Force field Ag Eg* Tg Au Eu* Tu
linear 7 7 18 6 6 19
quadratic 495 495 1.405 468 468 1.407
cubic 23.983 23.958 71.667 23.905 23.880 71.745
quartic 915.421 915.421 2.743.029 914.317 914.317 2.743.183
quintic 28.168.679 28.168.679 84.496.951 28.165.599 28.165.599 84.500.031
sextic 727.657.697 727.657.047 2.182.883.703 727.627.291 727.626.666 2.182.889.709


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)
..28. AgAg...49. EgEg...171. TgTg...21. AuAu...36. EuEu...190. TuTu.
Subtotal: 495 / 6 / 6
Irrep combinations (i,j) with indices: pos(Ag) ≤ i ≤ j ≤ pos(Tu)
Subtotal: 0 / 0 / 15
Total: 495 / 6 / 21


Contributions to nonvanishing cubic force field constants
Irrep combinations (i,i,i) with indices: pos(Ag) ≤ i ≤ pos(Tu)
..84. AgAgAg...168. EgEgEg...1.956. TgTgTg.
Subtotal: 2.208 / 3 / 6
Irrep combinations (i,i,j) (i,j,j) with indices: pos(Ag) ≤ i ≤ j ≤ pos(Tu)
..343. AgEgEg...1.197. AgTgTg...147. AgAuAu...252. AgEuEu...1.330. AgTuTu...2.394. EgTgTg...294. EgEuEu...2.660. EgTuTu...6.498. TgTuTu.
Subtotal: 15.115 / 9 / 30
Irrep combinations (i,j,k) with indices: pos(Ag) ≤ i ≤ j ≤ k ≤ pos(Tu)
..504. EgAuEu...2.052. TgAuTu...4.104. TgEuTu.
Subtotal: 6.660 / 3 / 20
Total: 23.983 / 15 / 56


Contributions to nonvanishing quartic force field constants
Irrep combinations (i,i,i,i) with indices: pos(Ag) ≤ i ≤ pos(Tu)
..210. AgAgAgAg...784. EgEgEgEg...35.226. TgTgTgTg...126. AuAuAuAu...441. EuEuEuEu...43.415. TuTuTuTu.
Subtotal: 80.202 / 6 / 6
Irrep combinations (i,i,i,j) (i,j,j,j) with indices: pos(Ag) ≤ i ≤ j ≤ pos(Tu)
..1.176. AgEgEgEg...13.692. AgTgTgTg...27.132. EgTgTgTg...672. AuEuEuEu...13.794. AuTuTuTu...27.360. EuTuTuTu.
Subtotal: 83.826 / 6 / 30
Irrep combinations (i,i,j,j) with indices: pos(Ag) ≤ i ≤ j ≤ pos(Tu)
..1.372. AgAgEgEg...4.788. AgAgTgTg...588. AgAgAuAu...1.008. AgAgEuEu...5.320. AgAgTuTu...17.955. EgEgTgTg...1.029. EgEgAuAu...2.940. EgEgEuEu...19.950. EgEgTuTu...3.591. TgTgAuAu.
..13.338. TgTgEuEu...214.434. TgTgTuTu...756. AuAuEuEu...3.990. AuAuTuTu...14.820. EuEuTuTu.
Subtotal: 305.879 / 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)
..2.016. EgEgAuEu...12.312. TgTgAuEu...36.936. TgTgAuTu...73.872. TgTgEuTu...16.758. AgEgTgTg...2.058. AgEgEuEu...18.620. AgEgTuTu...45.486. AgTgTuTu...90.972. EgTgTuTu...13.680. AuEuTuTu.
Subtotal: 312.710 / 10 / 60
Irrep combinations (i,j,k,l) with indices: pos(Ag) ≤ i ≤ j ≤ k ≤ l ≤ pos(Tu)
..3.528. AgEgAuEu...14.364. AgTgAuTu...28.728. AgTgEuTu...28.728. EgTgAuTu...57.456. EgTgEuTu.
Subtotal: 132.804 / 5 / 15
Total: 915.421 / 42 / 126


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