Characters of representations for molecular motions

Motion	E	σh
Cartesian 3N	51	17
Translation (x,y,z)	3	1
Rotation (Rx,Ry,Rz)	3	-1
Vibration	45	17

Decomposition to irreducible representations

Motion	A'	A''	Total
Cartesian 3N	34	17	51
Translation (x,y,z)	2	1	3
Rotation (Rx,Ry,Rz)	1	2	3
Vibration	31	14	45

Molecular parameter

Number of Atoms (N)	17
Number of internal coordinates	45
Number of independant internal coordinates	31
Number of vibrational modes	45

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Force field analysis

Allowed / forbidden vibronational transitions

Operator	A'	A''	Total
Linear (IR)	31	14	45 / 0
Quadratic (Raman)	31	14	45 / 0
IR + Raman	31	14	45 / 0

Characters of force fields
(Symmetric powers of vibration representation)

Force field	E	σh
linear	45	17
quadratic	1.035	167
cubic	16.215	1.207
quartic	194.580	7.092
quintic	1.906.884	35.700
sextic	15.890.700	159.068

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

Force field	A'	A''
linear	31	14
quadratic	601	434
cubic	8.711	7.504
quartic	100.836	93.744
quintic	971.292	935.592
sextic	8.024.884	7.865.816

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'')
..496.	A'A'.	..105.	A''A''.
Subtotal: 601 / 2 / 2
Irrep combinations (i,j) with indices: pos(A') ≤ i ≤ j ≤ pos(A'')
Subtotal: 0 / 0 / 1
Total: 601 / 2 / 3

Contributions to nonvanishing cubic force field constants

Irrep combinations (i,i,i) with indices: pos(A') ≤ i ≤ pos(A'')
..5.456.	A'A'A'.
Subtotal: 5.456 / 1 / 2
Irrep combinations (i,i,j) (i,j,j) with indices: pos(A') ≤ i ≤ j ≤ pos(A'')
..3.255.	A'A''A''.
Subtotal: 3.255 / 1 / 2
Irrep combinations (i,j,k) with indices: pos(A') ≤ i ≤ j ≤ k ≤ pos(A'')
Subtotal: 0 / 0 / 0
Total: 8.711 / 2 / 4

Contributions to nonvanishing quartic force field constants

Irrep combinations (i,i,i,i) with indices: pos(A') ≤ i ≤ pos(A'')
..46.376.	A'A'A'A'.	..2.380.	A''A''A''A''.
Subtotal: 48.756 / 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'')
..52.080.	A'A'A''A''.
Subtotal: 52.080 / 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: 100.836 / 3 / 5

Calculate contributions to

A'	A''

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