Results for Point Group D6h



Characters of representations for molecular motions
Motion E 2C6 2C3 C2 3C'2 3C''2 i 2S3 2S6 σh d v
Cartesian 3N 75 2 0 -1 -1 -1 -3 -2 0 1 1 9
Translation (x,y,z) 3 2 0 -1 -1 -1 -3 -2 0 1 1 1
Rotation (Rx,Ry,Rz) 3 2 0 -1 -1 -1 3 2 0 -1 -1 -1
Vibration 69 -2 0 1 1 1 -3 -2 0 1 1 9


Decomposition to irreducible representations
Motion A1g A2g B1g B2g E1g E2g A1u A2u B1u B2u E1u E2u Total
Cartesian 3N 4 2 2 4 6 6 2 5 4 2 7 6 50
Translation (x,y,z) 0 0 0 0 0 0 0 1 0 0 1 0 2
Rotation (Rx,Ry,Rz) 0 1 0 0 1 0 0 0 0 0 0 0 2
Vibration 4 1 2 4 5 6 2 4 4 2 6 6 46



Molecular parameter
Number of Atoms (N) 25
Number of internal coordinates 69
Number of independant internal coordinates 4
Number of vibrational modes 46


Force field analysis


Allowed / forbidden vibronational transitions
Operator A1g A2g B1g B2g E1g E2g A1u A2u B1u B2u E1u E2u Total
Linear (IR) 4 1 2 4 5 6 2 4 4 2 6 6 10 / 36
Quadratic (Raman) 4 1 2 4 5 6 2 4 4 2 6 6 15 / 31
IR + Raman - - - - 1 2 4 - - - - - - - - 2 - - - - 4 2 - - - - 6 0* / 21
* Parity Mutual Exclusion Principle


Characters of force fields
(Symmetric powers of vibration representation)
Force field E 2C6 2C3 C2 3C'2 3C''2 i 2S3 2S6 σh d v
linear 69 -2 0 1 1 1 -3 -2 0 1 1 9
quadratic 2.415 2 0 35 35 35 39 2 0 35 35 75
cubic 57.155 -1 23 35 35 35 -109 -1 -1 35 35 435
quartic 1.028.790 0 0 630 630 630 774 0 0 630 630 2.310
quintic 15.020.334 0 0 630 630 630 -2.034 0 0 630 630 10.422
sextic 185.250.786 12 276 7.770 7.770 7.770 10.434 12 12 7.770 7.770 43.738


Decomposition to irreducible representations
Column with number of nonvanshing force constants highlighted
Force field A1g A2g B1g B2g E1g E2g A1u A2u B1u B2u E1u E2u
linear 4 1 2 4 5 6 2 4 4 2 6 6
quadratic 128 83 94 104 199 210 94 104 104 94 198 198
cubic 2.449 2.314 2.326 2.426 4.746 4.758 2.338 2.438 2.438 2.338 4.770 4.770
quartic 43.476 42.426 42.636 43.056 85.692 85.902 42.624 43.044 43.044 42.624 85.668 85.668
quintic 627.354 624.276 624.486 626.934 1.251.420 1.251.630 624.708 627.156 627.156 624.708 1.251.864 1.251.864
sextic 7.728.272 7.711.510 7.714.096 7.723.088 15.437.118 15.439.704 7.713.874 7.722.866 7.722.866 7.713.874 15.436.674 15.436.674


Further Reading



Contributions to nonvanishing force field constants


pos(X) : Position of irreducible representation (irrep) X in character table of D6h

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(E2u)
..10. A1gA1g...1. A2gA2g...3. B1gB1g...10. B2gB2g...15. E1gE1g...21. E2gE2g...3. A1uA1u...10. A2uA2u...10. B1uB1u...3. B2uB2u.
..21. E1uE1u...21. E2uE2u.
Subtotal: 128 / 12 / 12
Irrep combinations (i,j) with indices: pos(A1g) ≤ i ≤ j ≤ pos(E2u)
Subtotal: 0 / 0 / 66
Total: 128 / 12 / 78


Contributions to nonvanishing cubic force field constants
Irrep combinations (i,i,i) with indices: pos(A1g) ≤ i ≤ pos(E2u)
..20. A1gA1gA1g...56. E2gE2gE2g.
Subtotal: 76 / 2 / 12
Irrep combinations (i,i,j) (i,j,j) with indices: pos(A1g) ≤ i ≤ j ≤ pos(E2u)
..90. E1gE1gE2g...4. A1gA2gA2g...12. A1gB1gB1g...40. A1gB2gB2g...60. A1gE1gE1g...84. A1gE2gE2g...12. A1gA1uA1u...40. A1gA2uA2u...40. A1gB1uB1u...12. A1gB2uB2u.
..84. A1gE1uE1u...84. A1gE2uE2u...10. A2gE1gE1g...15. A2gE2gE2g...15. A2gE1uE1u...15. A2gE2uE2u...126. E2gE1uE1u...126. E2gE2uE2u.
Subtotal: 869 / 18 / 132
Irrep combinations (i,j,k) with indices: pos(A1g) ≤ i ≤ j ≤ k ≤ pos(E2u)
..8. A2gB1gB2g...8. A2gA1uA2u...8. A2gB1uB2u...60. B1gE1gE2g...16. B1gA1uB1u...16. B1gA2uB2u...72. B1gE1uE2u...120. B2gE1gE2g...16. B2gA1uB2u...64. B2gA2uB1u.
..144. B2gE1uE2u...60. E1gA1uE1u...120. E1gA2uE1u...120. E1gB1uE2u...60. E1gB2uE2u...180. E1gE1uE2u...72. E2gA1uE2u...144. E2gA2uE2u...144. E2gB1uE1u...72. E2gB2uE1u.
Subtotal: 1.504 / 20 / 220
Total: 2.449 / 40 / 364


Contributions to nonvanishing quartic force field constants
Irrep combinations (i,i,i,i) with indices: pos(A1g) ≤ i ≤ pos(E2u)
..35. A1gA1gA1gA1g...1. A2gA2gA2gA2g...5. B1gB1gB1gB1g...35. B2gB2gB2gB2g...120. E1gE1gE1gE1g...231. E2gE2gE2gE2g...5. A1uA1uA1uA1u...35. A2uA2uA2uA2u...35. B1uB1uB1uB1u...5. B2uB2uB2uB2u.
..231. E1uE1uE1uE1u...231. E2uE2uE2uE2u.
Subtotal: 969 / 12 / 12
Irrep combinations (i,i,i,j) (i,j,j,j) with indices: pos(A1g) ≤ i ≤ j ≤ pos(E2u)
..224. A1gE2gE2gE2g...56. A2gE2gE2gE2g...70. B1gE1gE1gE1g...140. B2gE1gE1gE1g...112. A1uE2uE2uE2u...224. A2uE2uE2uE2u...224. B1uE1uE1uE1u...112. B2uE1uE1uE1u.
Subtotal: 1.162 / 8 / 132
Irrep combinations (i,i,j,j) with indices: pos(A1g) ≤ i ≤ j ≤ pos(E2u)
..10. A1gA1gA2gA2g...30. A1gA1gB1gB1g...100. A1gA1gB2gB2g...150. A1gA1gE1gE1g...210. A1gA1gE2gE2g...30. A1gA1gA1uA1u...100. A1gA1gA2uA2u...100. A1gA1gB1uB1u...30. A1gA1gB2uB2u...210. A1gA1gE1uE1u.
..210. A1gA1gE2uE2u...3. A2gA2gB1gB1g...10. A2gA2gB2gB2g...15. A2gA2gE1gE1g...21. A2gA2gE2gE2g...3. A2gA2gA1uA1u...10. A2gA2gA2uA2u...10. A2gA2gB1uB1u...3. A2gA2gB2uB2u...21. A2gA2gE1uE1u.
..21. A2gA2gE2uE2u...30. B1gB1gB2gB2g...45. B1gB1gE1gE1g...63. B1gB1gE2gE2g...9. B1gB1gA1uA1u...30. B1gB1gA2uA2u...30. B1gB1gB1uB1u...9. B1gB1gB2uB2u...63. B1gB1gE1uE1u...63. B1gB1gE2uE2u.
..150. B2gB2gE1gE1g...210. B2gB2gE2gE2g...30. B2gB2gA1uA1u...100. B2gB2gA2uA2u...100. B2gB2gB1uB1u...30. B2gB2gB2uB2u...210. B2gB2gE1uE1u...210. B2gB2gE2uE2u...780. E1gE1gE2gE2g...45. E1gE1gA1uA1u.
..150. E1gE1gA2uA2u...150. E1gE1gB1uB1u...45. E1gE1gB2uB2u...780. E1gE1gE1uE1u...780. E1gE1gE2uE2u...63. E2gE2gA1uA1u...210. E2gE2gA2uA2u...210. E2gE2gB1uB1u...63. E2gE2gB2uB2u...1.107. E2gE2gE1uE1u.
..1.107. E2gE2gE2uE2u...30. A1uA1uA2uA2u...30. A1uA1uB1uB1u...9. A1uA1uB2uB2u...63. A1uA1uE1uE1u...63. A1uA1uE2uE2u...100. A2uA2uB1uB1u...30. A2uA2uB2uB2u...210. A2uA2uE1uE1u...210. A2uA2uE2uE2u.
..30. B1uB1uB2uB2u...210. B1uB1uE1uE1u...210. B1uB1uE2uE2u...63. B2uB2uE1uE1u...63. B2uB2uE2uE2u...1.107. E1uE1uE2uE2u.
Subtotal: 10.597 / 66 / 66
Irrep combinations (i,i,j,k) (i,j,j,k) (i,j,k,k) with indices: pos(A1g) ≤ i ≤ j ≤ k ≤ pos(E2u)
..80. E1gE1gA1uA2u...180. E1gE1gA1uE2u...360. E1gE1gA2uE2u...80. E1gE1gB1uB2u...360. E1gE1gB1uE1u...180. E1gE1gB2uE1u...120. E2gE2gA1uA2u...252. E2gE2gA1uE2u...504. E2gE2gA2uE2u...120. E2gE2gB1uB2u.
..504. E2gE2gB1uE1u...252. E2gE2gB2uE1u...360. A1gE1gE1gE2g...90. A2gE1gE1gE2g...252. A1uE1uE1uE2u...504. A2uE1uE1uE2u...40. A1gA2gE1gE1g...60. A1gA2gE2gE2g...60. A1gA2gE1uE1u...60. A1gA2gE2uE2u.
..504. A1gE2gE1uE1u...504. A1gE2gE2uE2u...126. A2gE2gE1uE1u...126. A2gE2gE2uE2u...80. B1gB2gE1gE1g...120. B1gB2gE2gE2g...120. B1gB2gE1uE1u...120. B1gB2gE2uE2u...210. B1gE1gE2gE2g...210. B1gE1gE1uE1u.
..210. B1gE1gE2uE2u...420. B2gE1gE2gE2g...420. B2gE1gE1uE1u...420. B2gE1gE2uE2u...120. A1uA2uE1uE1u...120. A1uA2uE2uE2u...120. B1uB2uE1uE1u...120. B1uB2uE2uE2u...504. B1uE1uE2uE2u...252. B2uE1uE2uE2u.
Subtotal: 9.244 / 40 / 660
Irrep combinations (i,j,k,l) with indices: pos(A1g) ≤ i ≤ j ≤ k ≤ l ≤ pos(E2u)
..32. A1gA2gB1gB2g...32. A1gA2gA1uA2u...32. A1gA2gB1uB2u...240. A1gB1gE1gE2g...64. A1gB1gA1uB1u...64. A1gB1gA2uB2u...288. A1gB1gE1uE2u...480. A1gB2gE1gE2g...64. A1gB2gA1uB2u...256. A1gB2gA2uB1u.
..576. A1gB2gE1uE2u...240. A1gE1gA1uE1u...480. A1gE1gA2uE1u...480. A1gE1gB1uE2u...240. A1gE1gB2uE2u...720. A1gE1gE1uE2u...288. A1gE2gA1uE2u...576. A1gE2gA2uE2u...576. A1gE2gB1uE1u...288. A1gE2gB2uE1u.
..60. A2gB1gE1gE2g...8. A2gB1gA1uB2u...32. A2gB1gA2uB1u...72. A2gB1gE1uE2u...120. A2gB2gE1gE2g...32. A2gB2gA1uB1u...32. A2gB2gA2uB2u...144. A2gB2gE1uE2u...60. A2gE1gA1uE1u...120. A2gE1gA2uE1u.
..120. A2gE1gB1uE2u...60. A2gE1gB2uE2u...180. A2gE1gE1uE2u...72. A2gE2gA1uE2u...144. A2gE2gA2uE2u...144. A2gE2gB1uE1u...72. A2gE2gB2uE1u...64. B1gB2gA1uA2u...64. B1gB2gB1uB2u...120. B1gE1gA1uE2u.
..240. B1gE1gA2uE2u...240. B1gE1gB1uE1u...120. B1gE1gB2uE1u...144. B1gE2gA1uE1u...288. B1gE2gA2uE1u...288. B1gE2gB1uE2u...144. B1gE2gB2uE2u...432. B1gE2gE1uE2u...240. B2gE1gA1uE2u...480. B2gE1gA2uE2u.
..480. B2gE1gB1uE1u...240. B2gE1gB2uE1u...288. B2gE2gA1uE1u...576. B2gE2gA2uE1u...576. B2gE2gB1uE2u...288. B2gE2gB2uE2u...864. B2gE2gE1uE2u...240. E1gE2gA1uB1u...120. E1gE2gA1uB2u...360. E1gE2gA1uE1u.
..480. E1gE2gA2uB1u...240. E1gE2gA2uB2u...720. E1gE2gA2uE1u...720. E1gE2gB1uE2u...360. E1gE2gB2uE2u...3.240. E1gE2gE1uE2u...64. A1uA2uB1uB2u...288. A1uB1uE1uE2u...144. A1uB2uE1uE2u...576. A2uB1uE1uE2u.
..288. A2uB2uE1uE2u.
Subtotal: 21.504 / 71 / 495
Total: 43.476 / 197 / 1.365


Calculate contributions to

A1g A2g B1g B2g E1g E2g A1u A2u B1u B2u E1u E2u
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Last update January, 3rd 2020 by A. Gelessus, Impressum, Datenschutzerklärung/DataPrivacyStatement