Characters of representations for molecular motions

Motion	E	σh
Cartesian 3N	156	14
Translation (x,y,z)	3	1
Rotation (Rx,Ry,Rz)	3	-1
Vibration	150	14

Decomposition to irreducible representations

Motion	A'	A''	Total
Cartesian 3N	85	71	156
Translation (x,y,z)	2	1	3
Rotation (Rx,Ry,Rz)	1	2	3
Vibration	82	68	150

Molecular parameter

Number of Atoms (N)	52
Number of internal coordinates	150
Number of independant internal coordinates	82
Number of vibrational modes	150

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Force field analysis

Allowed / forbidden vibronational transitions

Operator	A'	A''	Total
Linear (IR)	82	68	150 / 0
Quadratic (Raman)	82	68	150 / 0
IR + Raman	82	68	150 / 0

Characters of force fields
(Symmetric powers of vibration representation)

Force field	E	σh
linear	150	14
quadratic	11.325	173
cubic	573.800	1.512
quartic	21.947.850	11.866
quintic	675.993.780	79.492
sextic	17.463.172.650	490.042

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

Force field	A'	A''
linear	82	68
quadratic	5.749	5.576
cubic	287.656	286.144
quartic	10.979.858	10.967.992
quintic	338.036.636	337.957.144
sextic	8.731.831.346	8.731.341.304

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'')
..3.403.	A'A'.	..2.346.	A''A''.
Subtotal: 5.749 / 2 / 2
Irrep combinations (i,j) with indices: pos(A') ≤ i ≤ j ≤ pos(A'')
Subtotal: 0 / 0 / 1
Total: 5.749 / 2 / 3

Contributions to nonvanishing cubic force field constants

Irrep combinations (i,i,i) with indices: pos(A') ≤ i ≤ pos(A'')
..95.284.	A'A'A'.
Subtotal: 95.284 / 1 / 2
Irrep combinations (i,i,j) (i,j,j) with indices: pos(A') ≤ i ≤ j ≤ pos(A'')
..192.372.	A'A''A''.
Subtotal: 192.372 / 1 / 2
Irrep combinations (i,j,k) with indices: pos(A') ≤ i ≤ j ≤ k ≤ pos(A'')
Subtotal: 0 / 0 / 0
Total: 287.656 / 2 / 4

Contributions to nonvanishing quartic force field constants

Irrep combinations (i,i,i,i) with indices: pos(A') ≤ i ≤ pos(A'')
..2.024.785.	A'A'A'A'.	..971.635.	A''A''A''A''.
Subtotal: 2.996.420 / 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'')
..7.983.438.	A'A'A''A''.
Subtotal: 7.983.438 / 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: 10.979.858 / 3 / 5

Calculate contributions to

A'	A''

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