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 12 0 0 -2 0 0 0 4 2 0
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 6 -2 2 0 2 0 0 4 2 0


Decomposition to irreducible representations
Motion A1g A2g B1g B2g Eg A1u A2u B1u B2u Eu Total
Cartesian 3N 1 1 1 1 1 0 1 0 1 2 9
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 1 0 1 1 0 0 0 0 1 1 5



Molecular parameter
Number of Atoms (N) 4
Number of internal coordinates 6
Number of independant internal coordinates 1
Number of vibrational modes 5


Force field analysis


Allowed / forbidden vibronational transitions
Operator A1g A2g B1g B2g Eg A1u A2u B1u B2u Eu Total
Linear (IR) 1 0 1 1 0 0 0 0 1 1 1 / 4
Quadratic (Raman) 1 0 1 1 0 0 0 0 1 1 3 / 2
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 v d
linear 6 -2 2 0 2 0 0 4 2 0
quadratic 21 3 5 3 5 3 1 11 5 3
cubic 56 -4 8 0 8 0 0 24 8 0
quartic 126 6 14 6 14 6 2 46 14 6
quintic 252 -8 20 0 20 0 0 80 20 0
sextic 462 10 30 10 30 10 2 130 30 10


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 1 0 1 1 0 0 0 0 1 1
quadratic 5 1 2 2 1 1 1 0 1 3
cubic 7 3 6 6 3 2 2 1 5 9
quartic 18 8 11 11 9 6 6 3 7 19
quintic 26 16 23 23 19 11 11 8 18 39
sextic 51 31 38 38 39 23 23 16 26 69


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)
..1. A1gA1g...1. B1gB1g...1. B2gB2g...1. B2uB2u...1. EuEu.
Subtotal: 5 / 5 / 10
Irrep combinations (i,j) with indices: pos(A1g) ≤ i ≤ j ≤ pos(Eu)
Subtotal: 0 / 0 / 45
Total: 5 / 5 / 55


Contributions to nonvanishing cubic force field constants
Irrep combinations (i,i,i) with indices: pos(A1g) ≤ i ≤ pos(Eu)
..1. A1gA1gA1g.
Subtotal: 1 / 1 / 10
Irrep combinations (i,i,j) (i,j,j) with indices: pos(A1g) ≤ i ≤ j ≤ pos(Eu)
..1. A1gB1gB1g...1. A1gB2gB2g...1. A1gB2uB2u...1. A1gEuEu...1. B1gEuEu...1. B2gEuEu.
Subtotal: 6 / 6 / 90
Irrep combinations (i,j,k) with indices: pos(A1g) ≤ i ≤ j ≤ k ≤ pos(Eu)
Subtotal: 0 / 0 / 120
Total: 7 / 7 / 220


Contributions to nonvanishing quartic force field constants
Irrep combinations (i,i,i,i) with indices: pos(A1g) ≤ i ≤ pos(Eu)
..1. A1gA1gA1gA1g...1. B1gB1gB1gB1g...1. B2gB2gB2gB2g...1. B2uB2uB2uB2u...2. EuEuEuEu.
Subtotal: 6 / 5 / 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)
..1. A1gA1gB1gB1g...1. A1gA1gB2gB2g...1. A1gA1gB2uB2u...1. A1gA1gEuEu...1. B1gB1gB2gB2g...1. B1gB1gB2uB2u...1. B1gB1gEuEu...1. B2gB2gB2uB2u...1. B2gB2gEuEu...1. B2uB2uEuEu.
Subtotal: 10 / 10 / 45
Irrep combinations (i,i,j,k) (i,j,j,k) (i,j,k,k) with indices: pos(A1g) ≤ i ≤ j ≤ k ≤ pos(Eu)
..1. A1gB1gEuEu...1. A1gB2gEuEu.
Subtotal: 2 / 2 / 360
Irrep combinations (i,j,k,l) with indices: pos(A1g) ≤ i ≤ j ≤ k ≤ l ≤ pos(Eu)
Subtotal: 0 / 0 / 210
Total: 18 / 17 / 715


Calculate contributions to

A1g A2g B1g B2g Eg A1u A2u B1u B2u Eu
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