Results for Point Group D4h



Characters of representations for molecular motions
Motion E 2C4 C2 2C'2 2C''2 i 2S4 σh v d
Cartesian 3N 39 5 -5 -5 -1 -3 -1 9 9 5
Translation (x,y,z) 3 1 -1 -1 -1 -3 -1 1 1 1
Rotation (Rx,Ry,Rz) 3 1 -1 -1 -1 3 1 -1 -1 -1
Vibration 33 3 -3 -3 1 -3 -1 9 9 5


Decomposition to irreducible representations
Motion A1g A2g B1g B2g Eg A1u A2u B1u B2u Eu Total
Cartesian 3N 4 2 2 2 4 0 5 0 2 7 28
Translation (x,y,z) 0 0 0 0 0 0 1 0 0 1 2
Rotation (Rx,Ry,Rz) 0 1 0 0 1 0 0 0 0 0 2
Vibration 4 1 2 2 3 0 4 0 2 6 24



Molecular parameter
Number of Atoms (N) 13
Number of internal coordinates 33
Number of independant internal coordinates 4
Number of vibrational modes 24


Force field analysis


Allowed / forbidden vibronational transitions
Operator A1g A2g B1g B2g Eg A1u A2u B1u B2u Eu Total
Linear (IR) 4 1 2 2 3 0 4 0 2 6 10 / 14
Quadratic (Raman) 4 1 2 2 3 0 4 0 2 6 11 / 13
IR + Raman - - - - 1 - - - - - - - - - - - - 0 - - - - 0 2 - - - - 0* / 3
* Parity Mutual Exclusion Principle


Characters of force fields
(Symmetric powers of vibration representation)
Force field E 2C4 C2 2C'2 2C''2 i 2S4 σh v d
linear 33 3 -3 -3 1 -3 -1 9 9 5
quadratic 561 3 21 21 17 21 -1 57 57 29
cubic 6.545 1 -55 -55 17 -55 1 273 273 105
quartic 58.905 9 225 225 153 225 9 1.113 1.113 385
quintic 435.897 27 -531 -531 153 -531 -9 3.969 3.969 1.141
sextic 2.760.681 27 1.653 1.653 969 1.653 -9 12.817 12.817 3.325


Decomposition to irreducible representations
Column with number of nonvanshing force constants highlighted
Force field A1g A2g B1g B2g Eg A1u A2u B1u B2u Eu
linear 4 1 2 2 3 0 4 0 2 6
quadratic 57 26 45 37 63 26 38 28 34 72
cubic 462 377 431 407 784 340 444 362 422 866
quartic 4.016 3.547 3.877 3.677 7.224 3.472 3.752 3.530 3.694 7.446
quintic 28.019 26.836 27.691 27.155 53.991 26.314 27.686 26.552 27.430 55.116
sextic 175.898 171.207 174.820 172.276 343.483 170.056 173.436 170.636 172.838 346.274


Further Reading



Contributions to nonvanishing force field constants


pos(X) : Position of irreducible representation (irrep) X in character table of D4h

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(Eu)
..10. A1gA1g...1. A2gA2g...3. B1gB1g...3. B2gB2g...6. EgEg...10. A2uA2u...3. B2uB2u...21. EuEu.
Subtotal: 57 / 8 / 10
Irrep combinations (i,j) with indices: pos(A1g) ≤ i ≤ j ≤ pos(Eu)
Subtotal: 0 / 0 / 45
Total: 57 / 8 / 55


Contributions to nonvanishing cubic force field constants
Irrep combinations (i,i,i) with indices: pos(A1g) ≤ i ≤ pos(Eu)
..20. A1gA1gA1g.
Subtotal: 20 / 1 / 10
Irrep combinations (i,i,j) (i,j,j) with indices: pos(A1g) ≤ i ≤ j ≤ pos(Eu)
..4. A1gA2gA2g...12. A1gB1gB1g...12. A1gB2gB2g...24. A1gEgEg...40. A1gA2uA2u...12. A1gB2uB2u...84. A1gEuEu...3. A2gEgEg...15. A2gEuEu...12. B1gEgEg.
..42. B1gEuEu...12. B2gEgEg...42. B2gEuEu.
Subtotal: 314 / 13 / 90
Irrep combinations (i,j,k) with indices: pos(A1g) ≤ i ≤ j ≤ k ≤ pos(Eu)
..4. A2gB1gB2g...16. B1gA2uB2u...72. EgA2uEu...36. EgB2uEu.
Subtotal: 128 / 4 / 120
Total: 462 / 18 / 220


Contributions to nonvanishing quartic force field constants
Irrep combinations (i,i,i,i) with indices: pos(A1g) ≤ i ≤ pos(Eu)
..35. A1gA1gA1gA1g...1. A2gA2gA2gA2g...5. B1gB1gB1gB1g...5. B2gB2gB2gB2g...36. EgEgEgEg...35. A2uA2uA2uA2u...5. B2uB2uB2uB2u...357. EuEuEuEu.
Subtotal: 479 / 8 / 10
Irrep combinations (i,i,i,j) (i,j,j,j) with indices: pos(A1g) ≤ i ≤ j ≤ pos(Eu)
Subtotal: 0 / 0 / 90
Irrep combinations (i,i,j,j) with indices: pos(A1g) ≤ i ≤ j ≤ pos(Eu)
..10. A1gA1gA2gA2g...30. A1gA1gB1gB1g...30. A1gA1gB2gB2g...60. A1gA1gEgEg...100. A1gA1gA2uA2u...30. A1gA1gB2uB2u...210. A1gA1gEuEu...3. A2gA2gB1gB1g...3. A2gA2gB2gB2g...6. A2gA2gEgEg.
..10. A2gA2gA2uA2u...3. A2gA2gB2uB2u...21. A2gA2gEuEu...9. B1gB1gB2gB2g...18. B1gB1gEgEg...30. B1gB1gA2uA2u...9. B1gB1gB2uB2u...63. B1gB1gEuEu...18. B2gB2gEgEg...30. B2gB2gA2uA2u.
..9. B2gB2gB2uB2u...63. B2gB2gEuEu...60. EgEgA2uA2u...18. EgEgB2uB2u...423. EgEgEuEu...30. A2uA2uB2uB2u...210. A2uA2uEuEu...63. B2uB2uEuEu.
Subtotal: 1.569 / 28 / 45
Irrep combinations (i,i,j,k) (i,j,j,k) (i,j,k,k) with indices: pos(A1g) ≤ i ≤ j ≤ k ≤ pos(Eu)
..48. EgEgA2uB2u...12. A1gA2gEgEg...60. A1gA2gEuEu...48. A1gB1gEgEg...168. A1gB1gEuEu...48. A1gB2gEgEg...168. A1gB2gEuEu...12. A2gB1gEgEg...42. A2gB1gEuEu...12. A2gB2gEgEg.
..42. A2gB2gEuEu...12. B1gB2gEgEg...60. B1gB2gEuEu...168. A2uB2uEuEu.
Subtotal: 900 / 14 / 360
Irrep combinations (i,j,k,l) with indices: pos(A1g) ≤ i ≤ j ≤ k ≤ l ≤ pos(Eu)
..16. A1gA2gB1gB2g...64. A1gB1gA2uB2u...288. A1gEgA2uEu...144. A1gEgB2uEu...16. A2gB2gA2uB2u...72. A2gEgA2uEu...36. A2gEgB2uEu...144. B1gEgA2uEu...72. B1gEgB2uEu...144. B2gEgA2uEu.
..72. B2gEgB2uEu.
Subtotal: 1.068 / 11 / 210
Total: 4.016 / 61 / 715


Calculate contributions to

A1g A2g B1g B2g Eg A1u A2u B1u B2u Eu
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Last update November, 13th 2023 by A. Gelessus, Impressum, Datenschutzerklärung/DataPrivacyStatement