Results for Point Group Oh



Characters of representations for molecular motions
Motion E 8C3 6C2 6C4 3C2 i 6S4 8S6 h d
Cartesian 3N 156 0 -2 0 0 0 0 0 4 14
Translation (x,y,z) 3 0 -1 1 -1 -3 -1 0 1 1
Rotation (Rx,Ry,Rz) 3 0 -1 1 -1 3 1 0 -1 -1
Vibration 150 0 0 -2 2 0 0 0 4 14


Decomposition to irreducible representations
Motion A1g A2g Eg T1g T2g A1u A2u Eu T1u T2u Total
Cartesian 3N 5 2 7 8 11 1 5 6 12 8 65
Translation (x,y,z) 0 0 0 0 0 0 0 0 1 0 1
Rotation (Rx,Ry,Rz) 0 0 0 1 0 0 0 0 0 0 1
Vibration 5 2 7 7 11 1 5 6 11 8 63



Molecular parameter
Number of Atoms (N) 52
Number of internal coordinates 150
Number of independant internal coordinates 5
Number of vibrational modes 63


Force field analysis


Allowed / forbidden vibronational transitions
Operator A1g A2g Eg T1g T2g A1u A2u Eu T1u T2u Total
Linear (IR) 5 2 7 7 11 1 5 6 11 8 11 / 52
Quadratic (Raman) 5 2 7 7 11 1 5 6 11 8 23 / 40
IR + Raman - - - - 2 - - - - 7 - - - - 1 5 6 - - - - 8 0* / 29
* Parity Mutual Exclusion Principle


Characters of force fields
(Symmetric powers of vibration representation)
Force field E 8C3 6C2 6C4 3C2 i 6S4 8S6 h d
linear 150 0 0 -2 2 0 0 0 4 14
quadratic 11.325 0 75 3 77 75 1 0 83 173
cubic 573.800 50 0 -4 152 0 0 0 312 1.512
quartic 21.947.850 0 2.850 42 3.002 2.850 38 0 3.466 11.866
quintic 675.993.780 0 0 -80 5.852 0 0 0 12.320 79.492
sextic 17.463.172.650 1.275 73.150 118 79.002 73.150 38 25 97.174 490.042


Decomposition to irreducible representations
Column with number of nonvanshing force constants highlighted
Force field A1g A2g Eg T1g T2g A1u A2u Eu T1u T2u
linear 5 2 7 7 11 1 5 6 11 8
quadratic 279 216 495 672 733 222 246 468 716 691
cubic 12.180 11.803 23.958 35.644 36.023 11.763 12.142 23.880 36.061 35.684
quartic 459.560 455.861 915.421 1.369.685 1.373.344 456.032 458.285 914.317 1.372.719 1.370.464
quintic 14.094.266 14.074.413 28.168.679 42.238.529 42.258.422 14.072.853 14.092.746 28.165.599 42.259.942 42.240.089
sextic 363.899.267 363.758.430 727.657.047 1.091.371.472 1.091.512.231 363.761.544 363.865.747 727.626.666 1.091.496.976 1.091.392.733


Further Reading



Contributions to nonvanishing force field constants


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

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(A1g) ≤ i ≤ pos(T2u)
..15. A1gA1g...3. A2gA2g...28. EgEg...28. T1gT1g...66. T2gT2g...1. A1uA1u...15. A2uA2u...21. EuEu...66. T1uT1u...36. T2uT2u.
Subtotal: 279 / 10 / 10
Irrep combinations (i,j) with indices: pos(A1g) ≤ i ≤ j ≤ pos(T2u)
Subtotal: 0 / 0 / 45
Total: 279 / 10 / 55


Contributions to nonvanishing cubic force field constants
Irrep combinations (i,i,i) with indices: pos(A1g) ≤ i ≤ pos(T2u)
..35. A1gA1gA1g...84. EgEgEg...35. T1gT1gT1g...286. T2gT2gT2g.
Subtotal: 440 / 4 / 10
Irrep combinations (i,i,j) (i,j,j) with indices: pos(A1g) ≤ i ≤ j ≤ pos(T2u)
..308. T1gT1gT2g...15. A1gA2gA2g...140. A1gEgEg...140. A1gT1gT1g...330. A1gT2gT2g...5. A1gA1uA1u...75. A1gA2uA2u...105. A1gEuEu...330. A1gT1uT1u...180. A1gT2uT2u.
..42. A2gEgEg...30. A2gEuEu...196. EgT1gT1g...462. EgT2gT2g...147. EgEuEu...462. EgT1uT1u...252. EgT2uT2u...385. T1gT2gT2g...385. T1gT1uT1u...196. T1gT2uT2u.
..726. T2gT1uT1u...396. T2gT2uT2u.
Subtotal: 5.307 / 22 / 90
Irrep combinations (i,j,k) with indices: pos(A1g) ≤ i ≤ j ≤ k ≤ pos(T2u)
..154. A2gT1gT2g...10. A2gA1uA2u...176. A2gT1uT2u...539. EgT1gT2g...42. EgA1uEu...210. EgA2uEu...616. EgT1uT2u...77. T1gA1uT1u...280. T1gA2uT2u...462. T1gEuT1u.
..336. T1gEuT2u...616. T1gT1uT2u...88. T2gA1uT2u...605. T2gA2uT1u...726. T2gEuT1u...528. T2gEuT2u...968. T2gT1uT2u.
Subtotal: 6.433 / 17 / 120
Total: 12.180 / 43 / 220


Contributions to nonvanishing quartic force field constants
Irrep combinations (i,i,i,i) with indices: pos(A1g) ≤ i ≤ pos(T2u)
..70. A1gA1gA1gA1g...5. A2gA2gA2gA2g...406. EgEgEgEg...616. T1gT1gT1gT1g...3.212. T2gT2gT2gT2g...1. A1uA1uA1uA1u...70. A2uA2uA2uA2u...231. EuEuEuEu...3.212. T1uT1uT1uT1u...996. T2uT2uT2uT2u.
Subtotal: 8.819 / 10 / 10
Irrep combinations (i,i,i,j) (i,j,j,j) with indices: pos(A1g) ≤ i ≤ j ≤ pos(T2u)
..2.156. T1gT1gT1gT2g...5.808. T1uT1uT1uT2u...420. A1gEgEgEg...175. A1gT1gT1gT1g...1.430. A1gT2gT2gT2g...168. A2gEgEgEg...168. A2gT1gT1gT1g...330. A2gT2gT2gT2g...784. EgT1gT1gT1g...3.080. EgT2gT2gT2g.
..5.082. T1gT2gT2gT2g...56. A1uEuEuEu...165. A1uT1uT1uT1u...120. A1uT2uT2uT2u...280. A2uEuEuEu...1.430. A2uT1uT1uT1u...280. A2uT2uT2uT2u...2.640. EuT1uT1uT1u...1.008. EuT2uT2uT2u...3.168. T1uT2uT2uT2u.
Subtotal: 28.748 / 20 / 90
Irrep combinations (i,i,j,j) with indices: pos(A1g) ≤ i ≤ j ≤ pos(T2u)
..45. A1gA1gA2gA2g...420. A1gA1gEgEg...420. A1gA1gT1gT1g...990. A1gA1gT2gT2g...15. A1gA1gA1uA1u...225. A1gA1gA2uA2u...315. A1gA1gEuEu...990. A1gA1gT1uT1u...540. A1gA1gT2uT2u...84. A2gA2gEgEg.
..84. A2gA2gT1gT1g...198. A2gA2gT2gT2g...3. A2gA2gA1uA1u...45. A2gA2gA2uA2u...63. A2gA2gEuEu...198. A2gA2gT1uT1u...108. A2gA2gT2uT2u...1.568. EgEgT1gT1g...3.696. EgEgT2gT2g...28. EgEgA1uA1u.
..420. EgEgA2uA2u...1.491. EgEgEuEu...3.696. EgEgT1uT1u...2.016. EgEgT2uT2u...6.699. T1gT1gT2gT2g...28. T1gT1gA1uA1u...420. T1gT1gA2uA2u...1.176. T1gT1gEuEu...6.699. T1gT1gT1uT1u...3.612. T1gT1gT2uT2u.
..66. T2gT2gA1uA1u...990. T2gT2gA2uA2u...2.772. T2gT2gEuEu...16.093. T2gT2gT1uT1u...8.668. T2gT2gT2uT2u...15. A1uA1uA2uA2u...21. A1uA1uEuEu...66. A1uA1uT1uT1u...36. A1uA1uT2uT2u...315. A2uA2uEuEu.
..990. A2uA2uT1uT1u...540. A2uA2uT2uT2u...2.772. EuEuT1uT1u...1.512. EuEuT2uT2u...8.668. T1uT1uT2uT2u.
Subtotal: 79.816 / 45 / 45
Irrep combinations (i,i,j,k) (i,j,j,k) (i,j,k,k) with indices: pos(A1g) ≤ i ≤ j ≤ k ≤ pos(T2u)
..3.773. EgEgT1gT2g...105. EgEgA1uA2u...168. EgEgA1uEu...840. EgEgA2uEu...4.312. EgEgT1uT2u...168. T1gT1gA1uEu...231. T1gT1gA1uT1u...224. T1gT1gA1uT2u...840. T1gT1gA2uEu...1.540. T1gT1gA2uT1u.
..840. T1gT1gA2uT2u...3.234. T1gT1gEuT1u...2.352. T1gT1gEuT2u...6.776. T1gT1gT1uT2u...396. T2gT2gA1uEu...605. T2gT2gA1uT1u...528. T2gT2gA1uT2u...1.980. T2gT2gA2uEu...3.630. T2gT2gA2uT1u...2.200. T2gT2gA2uT2u.
..7.986. T2gT2gEuT1u...5.808. T2gT2gEuT2u...16.456. T2gT2gT1uT2u...3.168. EuEuT1uT2u...1.540. A1gT1gT1gT2g...462. A2gT1gT1gT2g...3.773. EgT1gT1gT2g...528. A1uT1uT1uT2u...2.200. A2uT1uT1uT2u...5.808. EuT1uT1uT2u.
..210. A1gA2gEgEg...150. A1gA2gEuEu...980. A1gEgT1gT1g...2.310. A1gEgT2gT2g...735. A1gEgEuEu...2.310. A1gEgT1uT1u...1.260. A1gEgT2uT2u...1.925. A1gT1gT2gT2g...1.925. A1gT1gT1uT1u...980. A1gT1gT2uT2u.
..3.630. A1gT2gT1uT1u...1.980. A1gT2gT2uT2u...392. A2gEgT1gT1g...924. A2gEgT2gT2g...294. A2gEgEuEu...924. A2gEgT1uT1u...504. A2gEgT2uT2u...924. A2gT1gT2gT2g...924. A2gT1gT1uT1u...504. A2gT1gT2uT2u.
..1.210. A2gT2gT1uT1u...616. A2gT2gT2uT2u...5.929. EgT1gT2gT2g...5.929. EgT1gT1uT1u...3.136. EgT1gT2uT2u...9.317. EgT2gT1uT1u...4.928. EgT2gT2uT2u...2.772. T1gT2gEuEu...14.399. T1gT2gT1uT1u...7.700. T1gT2gT2uT2u.
..75. A1uA2uEuEu...396. A1uEuT1uT1u...216. A1uEuT2uT2u...308. A1uT1uT2uT2u...1.980. A2uEuT1uT1u...1.080. A2uEuT2uT2u...1.980. A2uT1uT2uT2u...4.224. EuT1uT2uT2u.
Subtotal: 172.451 / 68 / 360
Irrep combinations (i,j,k,l) with indices: pos(A1g) ≤ i ≤ j ≤ k ≤ l ≤ pos(T2u)
..770. A1gA2gT1gT2g...50. A1gA2gA1uA2u...880. A1gA2gT1uT2u...2.695. A1gEgT1gT2g...210. A1gEgA1uEu...1.050. A1gEgA2uEu...3.080. A1gEgT1uT2u...385. A1gT1gA1uT1u...1.400. A1gT1gA2uT2u...2.310. A1gT1gEuT1u.
..1.680. A1gT1gEuT2u...3.080. A1gT1gT1uT2u...440. A1gT2gA1uT2u...3.025. A1gT2gA2uT1u...3.630. A1gT2gEuT1u...2.640. A1gT2gEuT2u...4.840. A1gT2gT1uT2u...1.078. A2gEgT1gT2g...84. A2gEgA1uEu...420. A2gEgA2uEu.
..1.232. A2gEgT1uT2u...112. A2gT1gA1uT2u...770. A2gT1gA2uT1u...924. A2gT1gEuT1u...672. A2gT1gEuT2u...1.232. A2gT1gT1uT2u...242. A2gT2gA1uT1u...880. A2gT2gA2uT2u...1.452. A2gT2gEuT1u...1.056. A2gT2gEuT2u.
..1.936. A2gT2gT1uT2u...539. EgT1gA1uT1u...392. EgT1gA1uT2u...2.695. EgT1gA2uT1u...1.960. EgT1gA2uT2u...6.468. EgT1gEuT1u...4.704. EgT1gEuT2u...8.624. EgT1gT1uT2u...847. EgT2gA1uT1u...616. EgT2gA1uT2u.
..4.235. EgT2gA2uT1u...3.080. EgT2gA2uT2u...10.164. EgT2gEuT1u...7.392. EgT2gEuT2u...13.552. EgT2gT1uT2u...385. T1gT2gA1uA2u...462. T1gT2gA1uEu...847. T1gT2gA1uT1u...616. T1gT2gA1uT2u...2.310. T1gT2gA2uEu.
..4.235. T1gT2gA2uT1u...3.080. T1gT2gA2uT2u...10.164. T1gT2gEuT1u...7.392. T1gT2gEuT2u...27.104. T1gT2gT1uT2u...440. A1uA2uT1uT2u...528. A1uEuT1uT2u...2.640. A2uEuT1uT2u.
Subtotal: 169.726 / 58 / 210
Total: 459.560 / 201 / 715


Calculate contributions to

A1g A2g Eg T1g T2g A1u A2u Eu T1u T2u
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Last update January, 3rd 2020 by A. Gelessus, Impressum, Datenschutzerklärung/DataPrivacyStatement