Results for Point Group Ih



Characters of representations for molecular motions
Motion E 12C5 12(C5)2 20C3 15C2 i 12S10 12(S10)3 20S6 15σd
Cartesian 3N 36 3.236 -1.236 0 0 0 0.000 -0.000 0 4
Translation (x,y,z) 3 1.618 -0.618 0 -1 -3 0.618 -1.618 0 1
Rotation (Rx,Ry,Rz) 3 1.618 -0.618 0 -1 3 -0.618 1.618 0 -1
Vibration 30 -0.000 -0.000 0 2 0 0.000 0.000 0 4


Decomposition to irreducible representations
Motion Ag T1g T2g Gg Hg Au T1u T2u Gu Hu Total
Cartesian 3N 1 1 0 1 2 0 2 1 1 1 10
Translation (x,y,z) 0 0 0 0 0 0 1 0 0 0 1
Rotation (Rx,Ry,Rz) 0 1 0 0 0 0 0 0 0 0 1
Vibration 1 0 0 1 2 0 1 1 1 1 8



Molecular parameter
Number of Atoms (N) 12
Number of internal coordinates 30
Number of independant internal coordinates 1
Number of vibrational modes 8

Force field analysis


Allowed / forbidden vibronational transitions
Operator Ag T1g T2g Gg Hg Au T1u T2u Gu Hu Total
Linear (IR) 1 0 0 1 2 0 1 1 1 1 1 / 7
Quadratic (Raman) 1 0 0 1 2 0 1 1 1 1 3 / 5
IR + Raman - - - - 0 0 1 - - - - 0 - - - - 1 1 1 0* / 4
* Parity Mutual Exclusion Principle


Characters of force fields
(Symmetric powers of vibration representation)
Force field E 12C5 12(C5)2 20C3 15C2 i 12S10 12(S10)3 20S6 15σd
linear 30 -0.000 -0.000 0 2 0 0.000 0.000 0 4
quadratic 465 -0.000 -0.000 0 17 15 -0.000 -0.000 0 23
cubic 4.960 -0.000 -0.000 10 32 0 0.000 0.000 0 72
quartic 40.920 -0.000 -0.000 0 152 120 -0.000 -0.000 0 256
quintic 278.256 6.000 6.000 0 272 0 -0.000 -0.000 0 680
sextic 1.623.160 -0.000 -0.000 55 952 680 -0.000 -0.000 5 1.904


Decomposition to irreducible representations
Column with number of nonvanshing force constants highlighted
Force field Ag T1g T2g Gg Hg Au T1u T2u Gu Hu
linear 1 0 0 1 2 0 1 1 1 1
quadratic 9 7 7 16 25 3 12 12 15 18
cubic 56 111 111 167 218 38 129 129 167 200
quartic 393 975 975 1.368 1.761 327 1.033 1.033 1.360 1.687
quintic 2.439 6.838 6.838 9.274 11.713 2.269 7.008 7.008 9.274 11.543
sextic 13.899 40.239 40.239 54.138 68.007 13.410 40.681 40.681 54.091 67.476


Further Reading


Contributions to nonvanishing force field constants


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

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(Hu)
..1. AgAg...1. GgGg...3. HgHg...1. T1uT1u...1. T2uT2u...1. GuGu...1. HuHu.
Subtotal: 9 / 7 / 10
Irrep combinations (i,j) with indices: pos(Ag) ≤ i ≤ j ≤ pos(Hu)
Subtotal: 0 / 0 / 45
Total: 9 / 7 / 55


Contributions to nonvanishing cubic force field constants
Irrep combinations (i,i,i) with indices: pos(Ag) ≤ i ≤ pos(Hu)
..1. AgAgAg...1. GgGgGg...8. HgHgHg.
Subtotal: 10 / 3 / 10
Irrep combinations (i,i,j) (i,j,j) with indices: pos(Ag) ≤ i ≤ j ≤ pos(Hu)
..2. GgGgHg...1. AgGgGg...3. AgHgHg...1. AgT1uT1u...1. AgT2uT2u...1. AgGuGu...1. AgHuHu...4. GgHgHg...1. GgGuGu...1. GgHuHu.
..2. HgT1uT1u...2. HgT2uT2u...2. HgGuGu...4. HgHuHu.
Subtotal: 26 / 14 / 90
Irrep combinations (i,j,k) with indices: pos(Ag) ≤ i ≤ j ≤ k ≤ pos(Hu)
..1. GgT1uT2u...1. GgT1uGu...1. GgT1uHu...1. GgT2uGu...1. GgT2uHu...1. GgGuHu...2. HgT1uT2u...2. HgT1uGu...2. HgT1uHu...2. HgT2uGu.
..2. HgT2uHu...4. HgGuHu.
Subtotal: 20 / 12 / 120
Total: 56 / 29 / 220


Contributions to nonvanishing quartic force field constants
Irrep combinations (i,i,i,i) with indices: pos(Ag) ≤ i ≤ pos(Hu)
..1. AgAgAgAg...2. GgGgGgGg...16. HgHgHgHg...1. T1uT1uT1uT1u...1. T2uT2uT2uT2u...2. GuGuGuGu...2. HuHuHuHu.
Subtotal: 25 / 7 / 10
Irrep combinations (i,i,i,j) (i,j,j,j) with indices: pos(Ag) ≤ i ≤ j ≤ pos(Hu)
..2. GgGgGgHg...1. T1uT1uT1uT2u...1. T1uT1uT1uGu...1. T2uT2uT2uGu...1. GuGuGuHu...1. AgGgGgGg...8. AgHgHgHg...16. GgHgHgHg...1. T1uT2uT2uT2u...1. T1uGuGuGu.
..1. T1uHuHuHu...1. T2uGuGuGu...1. T2uHuHuHu...3. GuHuHuHu.
Subtotal: 39 / 14 / 90
Irrep combinations (i,i,j,j) with indices: pos(Ag) ≤ i ≤ j ≤ pos(Hu)
..1. AgAgGgGg...3. AgAgHgHg...1. AgAgT1uT1u...1. AgAgT2uT2u...1. AgAgGuGu...1. AgAgHuHu...13. GgGgHgHg...2. GgGgT1uT1u...2. GgGgT2uT2u...3. GgGgGuGu.
..4. GgGgHuHu...9. HgHgT1uT1u...9. HgHgT2uT2u...13. HgHgGuGu...19. HgHgHuHu...2. T1uT1uT2uT2u...2. T1uT1uGuGu...3. T1uT1uHuHu...2. T2uT2uGuGu...3. T2uT2uHuHu.
..4. GuGuHuHu.
Subtotal: 98 / 21 / 45
Irrep combinations (i,i,j,k) (i,j,j,k) (i,j,k,k) with indices: pos(Ag) ≤ i ≤ j ≤ k ≤ pos(Hu)
..2. GgGgT1uT2u...2. GgGgT1uGu...2. GgGgT1uHu...2. GgGgT2uGu...2. GgGgT2uHu...3. GgGgGuHu...10. HgHgT1uT2u...11. HgHgT1uGu...12. HgHgT1uHu...11. HgHgT2uGu.
..12. HgHgT2uHu...18. HgHgGuHu...1. T1uT1uT2uGu...1. T1uT1uT2uHu...2. T1uT1uGuHu...2. T2uT2uGuHu...2. AgGgGgHg...1. T1uT2uT2uGu...1. T1uT2uT2uHu...2. T1uGuGuHu.
..2. T2uGuGuHu...4. AgGgHgHg...1. AgGgGuGu...1. AgGgHuHu...2. AgHgT1uT1u...2. AgHgT2uT2u...2. AgHgGuGu...4. AgHgHuHu...4. GgHgT1uT1u...4. GgHgT2uT2u.
..6. GgHgGuGu...10. GgHgHuHu...2. T1uT2uGuGu...3. T1uT2uHuHu...3. T1uGuHuHu...3. T2uGuHuHu.
Subtotal: 152 / 36 / 360
Irrep combinations (i,j,k,l) with indices: pos(Ag) ≤ i ≤ j ≤ k ≤ l ≤ pos(Hu)
..1. AgGgT1uT2u...1. AgGgT1uGu...1. AgGgT1uHu...1. AgGgT2uGu...1. AgGgT2uHu...1. AgGgGuHu...2. AgHgT1uT2u...2. AgHgT1uGu...2. AgHgT1uHu...2. AgHgT2uGu.
..2. AgHgT2uHu...4. AgHgGuHu...6. GgHgT1uT2u...8. GgHgT1uGu...10. GgHgT1uHu...8. GgHgT2uGu...10. GgHgT2uHu...14. GgHgGuHu...3. T1uT2uGuHu.
Subtotal: 79 / 19 / 210
Total: 393 / 97 / 715


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Ag T1g T2g Gg Hg Au T1u T2u Gu Hu
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Last update November, 13th 2023 by A. Gelessus, Impressum, Datenschutzerklärung/DataPrivacyStatement