Results for Point Group Oh



Characters of representations for molecular motions
Motion E 8C3 6C2 6C4 3C2 i 6S4 8S6 h d
Cartesian 3N 21 0 -1 3 -3 -3 -1 0 5 3
Translation (x,y,z) 3 0 -1 1 -1 -3 -1 0 1 1
Rotation (Rx,Ry,Rz) 3 0 -1 1 -1 3 1 0 -1 -1
Vibration 15 0 1 1 -1 -3 -1 0 5 3


Decomposition to irreducible representations
Motion A1g A2g Eg T1g T2g A1u A2u Eu T1u T2u Total
Cartesian 3N 1 0 1 1 1 0 0 0 3 1 8
Translation (x,y,z) 0 0 0 0 0 0 0 0 1 0 1
Rotation (Rx,Ry,Rz) 0 0 0 1 0 0 0 0 0 0 1
Vibration 1 0 1 0 1 0 0 0 2 1 6



Molecular parameter
Number of Atoms (N) 7
Number of internal coordinates 15
Number of independant internal coordinates 1
Number of vibrational modes 6


Force field analysis


Allowed / forbidden vibronational transitions
Operator A1g A2g Eg T1g T2g A1u A2u Eu T1u T2u Total
Linear (IR) 1 0 1 0 1 0 0 0 2 1 2 / 4
Quadratic (Raman) 1 0 1 0 1 0 0 0 2 1 3 / 3
IR + Raman - - - - 0 - - - - 0 - - - - 0 0 0 - - - - 1 0* / 1
* Parity Mutual Exclusion Principle


Characters of force fields
(Symmetric powers of vibration representation)
Force field E 8C3 6C2 6C4 3C2 i 6S4 8S6 h d
linear 15 0 1 1 -1 -3 -1 0 5 3
quadratic 120 0 8 0 8 12 0 0 20 12
cubic 680 5 8 0 -8 -28 0 -1 60 28
quartic 3.060 0 36 4 36 72 4 0 160 72
quintic 11.628 0 36 4 -36 -144 -4 0 376 144
sextic 38.760 15 120 0 120 300 0 3 820 300


Decomposition to irreducible representations
Column with number of nonvanshing force constants highlighted
Force field A1g A2g Eg T1g T2g A1u A2u Eu T1u T2u
linear 1 0 1 0 1 0 0 0 2 1
quadratic 7 2 9 4 9 1 2 3 8 7
cubic 22 13 33 33 42 9 14 20 51 46
quartic 92 63 155 171 196 50 59 109 199 190
quintic 283 238 521 674 719 207 232 439 776 747
sextic 928 823 1.742 2.330 2.435 737 782 1.513 2.470 2.425


Further Reading



Contributions to nonvanishing force field constants


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

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(T2u)
..1. A1gA1g...1. EgEg...1. T2gT2g...3. T1uT1u...1. T2uT2u.
Subtotal: 7 / 5 / 10
Irrep combinations (i,j) with indices: pos(A1g) ≤ i ≤ j ≤ pos(T2u)
Subtotal: 0 / 0 / 45
Total: 7 / 5 / 55


Contributions to nonvanishing cubic force field constants
Irrep combinations (i,i,i) with indices: pos(A1g) ≤ i ≤ pos(T2u)
..1. A1gA1gA1g...1. EgEgEg...1. T2gT2gT2g.
Subtotal: 3 / 3 / 10
Irrep combinations (i,i,j) (i,j,j) with indices: pos(A1g) ≤ i ≤ j ≤ pos(T2u)
..1. A1gEgEg...1. A1gT2gT2g...3. A1gT1uT1u...1. A1gT2uT2u...1. EgT2gT2g...3. EgT1uT1u...1. EgT2uT2u...3. T2gT1uT1u...1. T2gT2uT2u.
Subtotal: 15 / 9 / 90
Irrep combinations (i,j,k) with indices: pos(A1g) ≤ i ≤ j ≤ k ≤ pos(T2u)
..2. EgT1uT2u...2. T2gT1uT2u.
Subtotal: 4 / 2 / 120
Total: 22 / 14 / 220


Contributions to nonvanishing quartic force field constants
Irrep combinations (i,i,i,i) with indices: pos(A1g) ≤ i ≤ pos(T2u)
..1. A1gA1gA1gA1g...1. EgEgEgEg...2. T2gT2gT2gT2g...11. T1uT1uT1uT1u...2. T2uT2uT2uT2u.
Subtotal: 17 / 5 / 10
Irrep combinations (i,i,i,j) (i,j,j,j) with indices: pos(A1g) ≤ i ≤ j ≤ pos(T2u)
..6. T1uT1uT1uT2u...1. A1gEgEgEg...1. A1gT2gT2gT2g...2. T1uT2uT2uT2u.
Subtotal: 10 / 4 / 90
Irrep combinations (i,i,j,j) with indices: pos(A1g) ≤ i ≤ j ≤ pos(T2u)
..1. A1gA1gEgEg...1. A1gA1gT2gT2g...3. A1gA1gT1uT1u...1. A1gA1gT2uT2u...2. EgEgT2gT2g...6. EgEgT1uT1u...2. EgEgT2uT2u...9. T2gT2gT1uT1u...3. T2gT2gT2uT2u...9. T1uT1uT2uT2u.
Subtotal: 37 / 10 / 45
Irrep combinations (i,i,j,k) (i,j,j,k) (i,j,k,k) with indices: pos(A1g) ≤ i ≤ j ≤ k ≤ pos(T2u)
..2. EgEgT1uT2u...4. T2gT2gT1uT2u...1. A1gEgT2gT2g...3. A1gEgT1uT1u...1. A1gEgT2uT2u...3. A1gT2gT1uT1u...1. A1gT2gT2uT2u...4. EgT2gT1uT1u...1. EgT2gT2uT2u.
Subtotal: 20 / 9 / 360
Irrep combinations (i,j,k,l) with indices: pos(A1g) ≤ i ≤ j ≤ k ≤ l ≤ pos(T2u)
..2. A1gEgT1uT2u...2. A1gT2gT1uT2u...4. EgT2gT1uT2u.
Subtotal: 8 / 3 / 210
Total: 92 / 31 / 715


Calculate contributions to

A1g A2g Eg T1g T2g A1u A2u Eu T1u T2u
Show only nonzero contributions Show all contributions
Up to quartic force fieldUp to quintic force fieldUp to sextic force field






Last update January, 3rd 2020 by A. Gelessus, Impressum, Datenschutzerklärung/DataPrivacyStatement