Results for Point Group Td



Characters of representations for molecular motions
Motion E 8C3 3C2 6S4 6σd
Cartesian 3N 27 0 -1 -1 5
Translation (x,y,z) 3 0 -1 -1 1
Rotation (Rx,Ry,Rz) 3 0 -1 1 -1
Vibration 21 0 1 -1 5


Decomposition to irreducible representations
Motion A1 A2 E T1 T2 Total
Cartesian 3N 2 0 2 2 5 11
Translation (x,y,z) 0 0 0 0 1 1
Rotation (Rx,Ry,Rz) 0 0 0 1 0 1
Vibration 2 0 2 1 4 9



Molecular parameter
Number of Atoms (N) 9
Number of internal coordinates 21
Number of independant internal coordinates 2
Number of vibrational modes 9

Force field analysis


Allowed / forbidden vibronational transitions
Operator A1 A2 E T1 T2 Total
Linear (IR) 2 0 2 1 4 4 / 5
Quadratic (Raman) 2 0 2 1 4 8 / 1
IR + Raman - - - - 0 - - - - 1 4 4 / 1


Characters of force fields
(Symmetric powers of vibration representation)
Force field E 8C3 3C2 6S4 6σd
linear 21 0 1 -1 5
quadratic 231 0 11 1 23
cubic 1.771 7 11 -1 75
quartic 10.626 0 66 6 226
quintic 53.130 0 66 -6 586
sextic 230.230 28 286 6 1.430


Decomposition to irreducible representations
Column with number of nonvanshing force constants highlighted
Force field A1 A2 E T1 T2
linear 2 0 2 1 4
quadratic 17 5 22 22 33
cubic 96 59 148 201 239
quartic 509 393 902 1.265 1.375
quintic 2.367 2.077 4.444 6.485 6.781
sextic 9.997 9.279 19.248 28.387 29.099


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...1. T1T1...10. T2T2.
Subtotal: 17 / 4 / 5
Irrep combinations (i,j) with indices: pos(A1) ≤ i ≤ j ≤ pos(T2)
Subtotal: 0 / 0 / 10
Total: 17 / 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...20. T2T2T2.
Subtotal: 28 / 3 / 5
Irrep combinations (i,i,j) (i,j,j) with indices: pos(A1) ≤ i ≤ j ≤ pos(T2)
..4. T1T1T2...6. A1EE...2. A1T1T1...20. A1T2T2...2. ET1T1...20. ET2T2...6. T1T2T2.
Subtotal: 60 / 7 / 20
Irrep combinations (i,j,k) with indices: pos(A1) ≤ i ≤ j ≤ k ≤ pos(T2)
..8. ET1T2.
Subtotal: 8 / 1 / 10
Total: 96 / 11 / 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...2. T1T1T1T1...90. T2T2T2T2.
Subtotal: 103 / 4 / 5
Irrep combinations (i,i,i,j) (i,j,j,j) with indices: pos(A1) ≤ i ≤ j ≤ pos(T2)
..4. T1T1T1T2...8. A1EEE...40. A1T2T2T2...40. ET2T2T2...40. T1T2T2T2.
Subtotal: 132 / 5 / 20
Irrep combinations (i,i,j,j) with indices: pos(A1) ≤ i ≤ j ≤ pos(T2)
..9. A1A1EE...3. A1A1T1T1...30. A1A1T2T2...6. EET1T1...60. EET2T2...30. T1T1T2T2.
Subtotal: 138 / 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)
..16. EET1T2...8. A1T1T1T2...8. ET1T1T2...4. A1ET1T1...40. A1ET2T2...12. A1T1T2T2...32. ET1T2T2.
Subtotal: 120 / 7 / 30
Irrep combinations (i,j,k,l) with indices: pos(A1) ≤ i ≤ j ≤ k ≤ l ≤ pos(T2)
..16. A1ET1T2.
Subtotal: 16 / 1 / 5
Total: 509 / 23 / 70


Calculate contributions to

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