Results for Point Group D4



Characters of representations for molecular motions
Motion E 2C4 C2 2C'2 2C''2
Cartesian 3N 36 4 -4 -4 0
Translation (x,y,z) 3 1 -1 -1 -1
Rotation (Rx,Ry,Rz) 3 1 -1 -1 -1
Vibration 30 2 -2 -2 2


Decomposition to irreducible representations
Motion A1 A2 B1 B2 E Total
Cartesian 3N 4 6 2 4 10 26
Translation (x,y,z) 0 1 0 0 1 2
Rotation (Rx,Ry,Rz) 0 1 0 0 1 2
Vibration 4 4 2 4 8 22



Molecular parameter
Number of Atoms (N) 12
Number of internal coordinates 30
Number of independant internal coordinates 4
Number of vibrational modes 22


Force field analysis


Allowed / forbidden vibronational transitions
Operator A1 A2 B1 B2 E Total
Linear (IR) 4 4 2 4 8 12 / 10
Quadratic (Raman) 4 4 2 4 8 18 / 4
IR + Raman - - - - - - - - - - - - - - - - 8 8 / 0


Characters of force fields
(Symmetric powers of vibration representation)
Force field E 2C4 C2 2C'2 2C''2
linear 30 2 -2 -2 2
quadratic 465 1 17 17 17
cubic 4.960 0 -32 -32 32
quartic 40.920 8 152 152 152
quintic 278.256 16 -272 -272 272
sextic 1.623.160 8 952 952 952


Decomposition to irreducible representations
Column with number of nonvanshing force constants highlighted
Force field A1 A2 B1 B2 E
linear 4 4 2 4 8
quadratic 69 52 60 60 112
cubic 616 616 600 632 1.248
quartic 5.212 5.060 5.132 5.132 10.192
quintic 34.752 34.752 34.608 34.880 69.632
sextic 203.492 202.540 203.012 203.012 405.552


Further Reading



Contributions to nonvanishing force field constants


pos(X) : Position of irreducible representation (irrep) X in character table of D4

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(E)
..10. A1A1...10. A2A2...3. B1B1...10. B2B2...36. EE.
Subtotal: 69 / 5 / 5
Irrep combinations (i,j) with indices: pos(A1) ≤ i ≤ j ≤ pos(E)
Subtotal: 0 / 0 / 10
Total: 69 / 5 / 15


Contributions to nonvanishing cubic force field constants
Irrep combinations (i,i,i) with indices: pos(A1) ≤ i ≤ pos(E)
..20. A1A1A1.
Subtotal: 20 / 1 / 5
Irrep combinations (i,i,j) (i,j,j) with indices: pos(A1) ≤ i ≤ j ≤ pos(E)
..40. A1A2A2...12. A1B1B1...40. A1B2B2...144. A1EE...112. A2EE...72. B1EE...144. B2EE.
Subtotal: 564 / 7 / 20
Irrep combinations (i,j,k) with indices: pos(A1) ≤ i ≤ j ≤ k ≤ pos(E)
..32. A2B1B2.
Subtotal: 32 / 1 / 10
Total: 616 / 9 / 35


Contributions to nonvanishing quartic force field constants
Irrep combinations (i,i,i,i) with indices: pos(A1) ≤ i ≤ pos(E)
..35. A1A1A1A1...35. A2A2A2A2...5. B1B1B1B1...35. B2B2B2B2...996. EEEE.
Subtotal: 1.106 / 5 / 5
Irrep combinations (i,i,i,j) (i,j,j,j) with indices: pos(A1) ≤ i ≤ j ≤ pos(E)
Subtotal: 0 / 0 / 20
Irrep combinations (i,i,j,j) with indices: pos(A1) ≤ i ≤ j ≤ pos(E)
..100. A1A1A2A2...30. A1A1B1B1...100. A1A1B2B2...360. A1A1EE...30. A2A2B1B1...100. A2A2B2B2...360. A2A2EE...30. B1B1B2B2...108. B1B1EE...360. B2B2EE.
Subtotal: 1.578 / 10 / 10
Irrep combinations (i,i,j,k) (i,j,j,k) (i,j,k,k) with indices: pos(A1) ≤ i ≤ j ≤ k ≤ pos(E)
..448. A1A2EE...288. A1B1EE...576. A1B2EE...288. A2B1EE...576. A2B2EE...224. B1B2EE.
Subtotal: 2.400 / 6 / 30
Irrep combinations (i,j,k,l) with indices: pos(A1) ≤ i ≤ j ≤ k ≤ l ≤ pos(E)
..128. A1A2B1B2.
Subtotal: 128 / 1 / 5
Total: 5.212 / 22 / 70


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Last update November, 13th 2023 by A. Gelessus, Impressum, Datenschutzerklärung/DataPrivacyStatement