Characters of representations for molecular motions

Motion	E	σh
Cartesian 3N	39	5
Translation (x,y,z)	3	1
Rotation (Rx,Ry,Rz)	3	-1
Vibration	33	5

Decomposition to irreducible representations

Motion	A'	A''	Total
Cartesian 3N	22	17	39
Translation (x,y,z)	2	1	3
Rotation (Rx,Ry,Rz)	1	2	3
Vibration	19	14	33

Molecular parameter

Number of Atoms (N)	13
Number of internal coordinates	33
Number of independant internal coordinates	19
Number of vibrational modes	33

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Force field analysis

Allowed / forbidden vibronational transitions

Operator	A'	A''	Total
Linear (IR)	19	14	33 / 0
Quadratic (Raman)	19	14	33 / 0
IR + Raman	19	14	33 / 0

Characters of force fields
(Symmetric powers of vibration representation)

Force field	E	σh
linear	33	5
quadratic	561	29
cubic	6.545	105
quartic	58.905	385
quintic	435.897	1.141
sextic	2.760.681	3.325

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

Force field	A'	A''
linear	19	14
quadratic	295	266
cubic	3.325	3.220
quartic	29.645	29.260
quintic	218.519	217.378
sextic	1.382.003	1.378.678

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'')
..190.	A'A'.	..105.	A''A''.
Subtotal: 295 / 2 / 2
Irrep combinations (i,j) with indices: pos(A') ≤ i ≤ j ≤ pos(A'')
Subtotal: 0 / 0 / 1
Total: 295 / 2 / 3

Contributions to nonvanishing cubic force field constants

Irrep combinations (i,i,i) with indices: pos(A') ≤ i ≤ pos(A'')
..1.330.	A'A'A'.
Subtotal: 1.330 / 1 / 2
Irrep combinations (i,i,j) (i,j,j) with indices: pos(A') ≤ i ≤ j ≤ pos(A'')
..1.995.	A'A''A''.
Subtotal: 1.995 / 1 / 2
Irrep combinations (i,j,k) with indices: pos(A') ≤ i ≤ j ≤ k ≤ pos(A'')
Subtotal: 0 / 0 / 0
Total: 3.325 / 2 / 4

Contributions to nonvanishing quartic force field constants

Irrep combinations (i,i,i,i) with indices: pos(A') ≤ i ≤ pos(A'')
..7.315.	A'A'A'A'.	..2.380.	A''A''A''A''.
Subtotal: 9.695 / 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'')
..19.950.	A'A'A''A''.
Subtotal: 19.950 / 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: 29.645 / 3 / 5

Calculate contributions to

A'	A''

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