Characters of representations for molecular motions

Motion	E	σh
Cartesian 3N	24	8
Translation (x,y,z)	3	1
Rotation (Rx,Ry,Rz)	3	-1
Vibration	18	8

Decomposition to irreducible representations

Motion	A'	A''	Total
Cartesian 3N	16	8	24
Translation (x,y,z)	2	1	3
Rotation (Rx,Ry,Rz)	1	2	3
Vibration	13	5	18

Molecular parameter

Number of Atoms (N)	8
Number of internal coordinates	18
Number of independant internal coordinates	13
Number of vibrational modes	18

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Force field analysis

Allowed / forbidden vibronational transitions

Operator	A'	A''	Total
Linear (IR)	13	5	18 / 0
Quadratic (Raman)	13	5	18 / 0
IR + Raman	13	5	18 / 0

Characters of force fields
(Symmetric powers of vibration representation)

Force field	E	σh
linear	18	8
quadratic	171	41
cubic	1.140	160
quartic	5.985	525
quintic	26.334	1.512
sextic	100.947	3.941

Decomposition to irreducible representations
Column with number of nonvanshing force constants highlighted

Force field	A'	A''
linear	13	5
quadratic	106	65
cubic	650	490
quartic	3.255	2.730
quintic	13.923	12.411
sextic	52.444	48.503

Further Reading

• J.K.G. Watson, J. Mol. Spec. 41 229 (1972)
  The Numbers of Structural Parameters and Potential Constants of Molecules
  
  
• X.F. Zhou, P. Pulay. J. Comp. Chem. 10 No. 7, 935-938 (1989)
  Characters for Symmetric and Antisymmetric Higher Powers of Representations:
  Application to the Number of Anharmonic Force Constants in Symmetrical Molecules
  
  
• F. Varga, L. Nemes, J.K.G. Watson. J. Phys. B: At. Mol. Opt. Phys. 10 No. 7, 5043-5048 (1996)
  The number of anharmonic potential constants of the fullerenes C₆₀ and C₇₀

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Contributions to nonvanishing force field constants

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

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(A') ≤ i ≤ pos(A'')
..91.	A'A'.	..15.	A''A''.
Subtotal: 106 / 2 / 2
Irrep combinations (i,j) with indices: pos(A') ≤ i ≤ j ≤ pos(A'')
Subtotal: 0 / 0 / 1
Total: 106 / 2 / 3

Contributions to nonvanishing cubic force field constants

Irrep combinations (i,i,i) with indices: pos(A') ≤ i ≤ pos(A'')
..455.	A'A'A'.
Subtotal: 455 / 1 / 2
Irrep combinations (i,i,j) (i,j,j) with indices: pos(A') ≤ i ≤ j ≤ pos(A'')
..195.	A'A''A''.
Subtotal: 195 / 1 / 2
Irrep combinations (i,j,k) with indices: pos(A') ≤ i ≤ j ≤ k ≤ pos(A'')
Subtotal: 0 / 0 / 0
Total: 650 / 2 / 4

Contributions to nonvanishing quartic force field constants

Irrep combinations (i,i,i,i) with indices: pos(A') ≤ i ≤ pos(A'')
..1.820.	A'A'A'A'.	..70.	A''A''A''A''.
Subtotal: 1.890 / 2 / 2
Irrep combinations (i,i,i,j) (i,j,j,j) with indices: pos(A') ≤ i ≤ j ≤ pos(A'')
Subtotal: 0 / 0 / 2
Irrep combinations (i,i,j,j) with indices: pos(A') ≤ i ≤ j ≤ pos(A'')
..1.365.	A'A'A''A''.
Subtotal: 1.365 / 1 / 1
Irrep combinations (i,i,j,k) (i,j,j,k) (i,j,k,k) with indices: pos(A') ≤ i ≤ j ≤ k ≤ pos(A'')
Subtotal: 0 / 0 / 0
Irrep combinations (i,j,k,l) with indices: pos(A') ≤ i ≤ j ≤ k ≤ l ≤ pos(A'')
Subtotal: 0 / 0 / 0
Total: 3.255 / 3 / 5

Calculate contributions to

A'	A''

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