Characters of representations for molecular motions

Motion	E	σh
Cartesian 3N	48	0
Translation (x,y,z)	3	1
Rotation (Rx,Ry,Rz)	3	-1
Vibration	42	0

Decomposition to irreducible representations

Motion	A'	A''	Total
Cartesian 3N	24	24	48
Translation (x,y,z)	2	1	3
Rotation (Rx,Ry,Rz)	1	2	3
Vibration	21	21	42

Molecular parameter

Number of Atoms (N)	16
Number of internal coordinates	42
Number of independant internal coordinates	21
Number of vibrational modes	42

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Force field analysis

Allowed / forbidden vibronational transitions

Operator	A'	A''	Total
Linear (IR)	21	21	42 / 0
Quadratic (Raman)	21	21	42 / 0
IR + Raman	21	21	42 / 0

Characters of force fields
(Symmetric powers of vibration representation)

Force field	E	σh
linear	42	0
quadratic	903	21
cubic	13.244	0
quartic	148.995	231
quintic	1.370.754	0
sextic	10.737.573	1.771

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

Force field	A'	A''
linear	21	21
quadratic	462	441
cubic	6.622	6.622
quartic	74.613	74.382
quintic	685.377	685.377
sextic	5.369.672	5.367.901

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'')
..231.	A'A'.	..231.	A''A''.
Subtotal: 462 / 2 / 2
Irrep combinations (i,j) with indices: pos(A') ≤ i ≤ j ≤ pos(A'')
Subtotal: 0 / 0 / 1
Total: 462 / 2 / 3

Contributions to nonvanishing cubic force field constants

Irrep combinations (i,i,i) with indices: pos(A') ≤ i ≤ pos(A'')
..1.771.	A'A'A'.
Subtotal: 1.771 / 1 / 2
Irrep combinations (i,i,j) (i,j,j) with indices: pos(A') ≤ i ≤ j ≤ pos(A'')
..4.851.	A'A''A''.
Subtotal: 4.851 / 1 / 2
Irrep combinations (i,j,k) with indices: pos(A') ≤ i ≤ j ≤ k ≤ pos(A'')
Subtotal: 0 / 0 / 0
Total: 6.622 / 2 / 4

Contributions to nonvanishing quartic force field constants

Irrep combinations (i,i,i,i) with indices: pos(A') ≤ i ≤ pos(A'')
..10.626.	A'A'A'A'.	..10.626.	A''A''A''A''.
Subtotal: 21.252 / 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'')
..53.361.	A'A'A''A''.
Subtotal: 53.361 / 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: 74.613 / 3 / 5

Calculate contributions to

A'	A''

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