Results for Point Group D4h



Characters of representations for molecular motions
Motion E 2C4 C2 2C'2 2C''2 i 2S4 σh 2σv 2σd
Cartesian 3N 21 3 -3 -3 -1 -3 -1 5 5 3
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 15 1 -1 -1 1 -3 -1 5 5 3


Decomposition to irreducible representations
Motion A1g A2g B1g B2g Eg A1u A2u B1u B2u Eu Total
Cartesian 3N 2 1 1 1 2 0 3 0 1 4 15
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 2 0 1 1 1 0 2 0 1 3 11



Molecular parameter
Number of Atoms (N) 7
Number of internal coordinates 15
Number of independant internal coordinates 2
Number of vibrational modes 11

Force field analysis


Allowed / forbidden vibronational transitions
Operator A1g A2g B1g B2g Eg A1u A2u B1u B2u Eu Total
Linear (IR) 2 0 1 1 1 0 2 0 1 3 5 / 6
Quadratic (Raman) 2 0 1 1 1 0 2 0 1 3 5 / 6
IR + Raman - - - - 0 - - - - - - - - - - - - 0 - - - - 0 1 - - - - 0* / 1
* 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 2σv 2σd
linear 15 1 -1 -1 1 -3 -1 5 5 3
quadratic 120 0 8 8 8 12 0 20 20 12
cubic 680 0 -8 -8 8 -28 0 60 60 28
quartic 3.060 4 36 36 36 72 4 160 160 72
quintic 11.628 4 -36 -36 36 -144 -4 376 376 144
sextic 38.760 0 120 120 120 300 0 820 820 300


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 2 0 1 1 1 0 2 0 1 3
quadratic 16 4 11 9 13 4 8 5 7 15
cubic 55 33 46 42 75 29 51 34 46 97
quartic 247 171 218 196 367 159 199 168 190 389
quintic 804 674 759 719 1.393 646 776 671 747 1.523
sextic 2.670 2.330 2.565 2.435 4.765 2.250 2.470 2.295 2.425 4.895


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)
..3. A1gA1g...1. B1gB1g...1. B2gB2g...1. EgEg...3. A2uA2u...1. B2uB2u...6. EuEu.
Subtotal: 16 / 7 / 10
Irrep combinations (i,j) with indices: pos(A1g) ≤ i ≤ j ≤ pos(Eu)
Subtotal: 0 / 0 / 45
Total: 16 / 7 / 55


Contributions to nonvanishing cubic force field constants
Irrep combinations (i,i,i) with indices: pos(A1g) ≤ i ≤ pos(Eu)
..4. A1gA1gA1g.
Subtotal: 4 / 1 / 10
Irrep combinations (i,i,j) (i,j,j) with indices: pos(A1g) ≤ i ≤ j ≤ pos(Eu)
..2. A1gB1gB1g...2. A1gB2gB2g...2. A1gEgEg...6. A1gA2uA2u...2. A1gB2uB2u...12. A1gEuEu...1. B1gEgEg...6. B1gEuEu...1. B2gEgEg...6. B2gEuEu.
Subtotal: 40 / 10 / 90
Irrep combinations (i,j,k) with indices: pos(A1g) ≤ i ≤ j ≤ k ≤ pos(Eu)
..2. B1gA2uB2u...6. EgA2uEu...3. EgB2uEu.
Subtotal: 11 / 3 / 120
Total: 55 / 14 / 220


Contributions to nonvanishing quartic force field constants
Irrep combinations (i,i,i,i) with indices: pos(A1g) ≤ i ≤ pos(Eu)
..5. A1gA1gA1gA1g...1. B1gB1gB1gB1g...1. B2gB2gB2gB2g...2. EgEgEgEg...5. A2uA2uA2uA2u...1. B2uB2uB2uB2u...36. EuEuEuEu.
Subtotal: 51 / 7 / 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)
..3. A1gA1gB1gB1g...3. A1gA1gB2gB2g...3. A1gA1gEgEg...9. A1gA1gA2uA2u...3. A1gA1gB2uB2u...18. A1gA1gEuEu...1. B1gB1gB2gB2g...1. B1gB1gEgEg...3. B1gB1gA2uA2u...1. B1gB1gB2uB2u.
..6. B1gB1gEuEu...1. B2gB2gEgEg...3. B2gB2gA2uA2u...1. B2gB2gB2uB2u...6. B2gB2gEuEu...3. EgEgA2uA2u...1. EgEgB2uB2u...18. EgEgEuEu...3. A2uA2uB2uB2u...18. A2uA2uEuEu.
..6. B2uB2uEuEu.
Subtotal: 111 / 21 / 45
Irrep combinations (i,i,j,k) (i,j,j,k) (i,j,k,k) with indices: pos(A1g) ≤ i ≤ j ≤ k ≤ pos(Eu)
..2. EgEgA2uB2u...2. A1gB1gEgEg...12. A1gB1gEuEu...2. A1gB2gEgEg...12. A1gB2gEuEu...3. B1gB2gEuEu...12. A2uB2uEuEu.
Subtotal: 45 / 7 / 360
Irrep combinations (i,j,k,l) with indices: pos(A1g) ≤ i ≤ j ≤ k ≤ l ≤ pos(Eu)
..4. A1gB1gA2uB2u...12. A1gEgA2uEu...6. A1gEgB2uEu...6. B1gEgA2uEu...3. B1gEgB2uEu...6. B2gEgA2uEu...3. B2gEgB2uEu.
Subtotal: 40 / 7 / 210
Total: 247 / 42 / 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