Results for Point Group Td



Characters of representations for molecular motions
Motion E 8C3 3C2 6S4 6σd
Cartesian 3N 36 0 -4 0 4
Translation (x,y,z) 3 0 -1 -1 1
Rotation (Rx,Ry,Rz) 3 0 -1 1 -1
Vibration 30 0 -2 0 4


Decomposition to irreducible representations
Motion A1 A2 E T1 T2 Total
Cartesian 3N 2 0 2 4 6 14
Translation (x,y,z) 0 0 0 0 1 1
Rotation (Rx,Ry,Rz) 0 0 0 1 0 1
Vibration 2 0 2 3 5 12



Molecular parameter
Number of Atoms (N) 12
Number of internal coordinates 30
Number of independant internal coordinates 2
Number of vibrational modes 12

Force field analysis


Allowed / forbidden vibronational transitions
Operator A1 A2 E T1 T2 Total
Linear (IR) 2 0 2 3 5 5 / 7
Quadratic (Raman) 2 0 2 3 5 9 / 3
IR + Raman - - - - 0 - - - - 3 5 5 / 3


Characters of force fields
(Symmetric powers of vibration representation)
Force field E 8C3 3C2 6S4 6σd
linear 30 0 -2 0 4
quadratic 465 0 17 -1 23
cubic 4.960 10 -32 0 72
quartic 40.920 0 152 8 256
quintic 278.256 0 -272 0 680
sextic 1.623.160 55 952 -8 1.904


Decomposition to irreducible representations
Column with number of nonvanshing force constants highlighted
Force field A1 A2 E T1 T2
linear 2 0 2 3 5
quadratic 27 16 43 50 62
cubic 224 188 402 606 642
quartic 1.790 1.658 3.448 5.034 5.158
quintic 11.730 11.390 23.120 34.646 34.986
sextic 68.243 67.295 135.483 202.298 203.254


Further Reading


Contributions to nonvanishing force field constants


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

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(A1) ≤ i ≤ pos(T2)
..3. A1A1...3. EE...6. T1T1...15. T2T2.
Subtotal: 27 / 4 / 5
Irrep combinations (i,j) with indices: pos(A1) ≤ i ≤ j ≤ pos(T2)
Subtotal: 0 / 0 / 10
Total: 27 / 4 / 15


Contributions to nonvanishing cubic force field constants
Irrep combinations (i,i,i) with indices: pos(A1) ≤ i ≤ pos(T2)
..4. A1A1A1...4. EEE...1. T1T1T1...35. T2T2T2.
Subtotal: 44 / 4 / 5
Irrep combinations (i,i,j) (i,j,j) with indices: pos(A1) ≤ i ≤ j ≤ pos(T2)
..30. T1T1T2...6. A1EE...12. A1T1T1...30. A1T2T2...12. ET1T1...30. ET2T2...30. T1T2T2.
Subtotal: 150 / 7 / 20
Irrep combinations (i,j,k) with indices: pos(A1) ≤ i ≤ j ≤ k ≤ pos(T2)
..30. ET1T2.
Subtotal: 30 / 1 / 10
Total: 224 / 12 / 35


Contributions to nonvanishing quartic force field constants
Irrep combinations (i,i,i,i) with indices: pos(A1) ≤ i ≤ pos(T2)
..5. A1A1A1A1...6. EEEE...36. T1T1T1T1...190. T2T2T2T2.
Subtotal: 237 / 4 / 5
Irrep combinations (i,i,i,j) (i,j,j,j) with indices: pos(A1) ≤ i ≤ j ≤ pos(T2)
..90. T1T1T1T2...8. A1EEE...2. A1T1T1T1...70. A1T2T2T2...16. ET1T1T1...80. ET2T2T2...225. T1T2T2T2.
Subtotal: 491 / 7 / 20
Irrep combinations (i,i,j,j) with indices: pos(A1) ≤ i ≤ j ≤ pos(T2)
..9. A1A1EE...18. A1A1T1T1...45. A1A1T2T2...36. EET1T1...90. EET2T2...300. T1T1T2T2.
Subtotal: 498 / 6 / 10
Irrep combinations (i,i,j,k) (i,j,j,k) (i,j,k,k) with indices: pos(A1) ≤ i ≤ j ≤ k ≤ pos(T2)
..60. EET1T2...60. A1T1T1T2...90. ET1T1T2...24. A1ET1T1...60. A1ET2T2...60. A1T1T2T2...150. ET1T2T2.
Subtotal: 504 / 7 / 30
Irrep combinations (i,j,k,l) with indices: pos(A1) ≤ i ≤ j ≤ k ≤ l ≤ pos(T2)
..60. A1ET1T2.
Subtotal: 60 / 1 / 5
Total: 1.790 / 25 / 70


Calculate contributions to

A1 A2 E T1 T2
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Last update November, 13th 2023 by A. Gelessus, Impressum, Datenschutzerklärung/DataPrivacyStatement