Characters of representations for molecular motions

Motion	E	S4	C2	(S4)3
Cartesian 3N	135	-1	-1	-1
Translation (x,y,z)	3	-1	-1	-1
Rotation (Rx,Ry,Rz)	3	1	-1	1
Vibration	129	-1	1	-1

Decomposition to irreducible representations

Motion	A	B	E*	Total
Cartesian 3N	33	34	34	101
Translation (x,y,z)	0	1	1	2
Rotation (Rx,Ry,Rz)	1	0	1	2
Vibration	32	33	32	97

Molecular parameter

Number of Atoms (N)	45
Number of internal coordinates	129
Number of independant internal coordinates	32
Number of vibrational modes	97

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Force field analysis

Allowed / forbidden vibronational transitions

Operator	A	B	E*	Total
Linear (IR)	32	33	32	65 / 32
Quadratic (Raman)	32	33	32	97 / 0
IR + Raman	- - - -	33	32	65 / 0

Characters of force fields
(Symmetric powers of vibration representation)

Force field	E	S4	C2	(S4)3
linear	129	-1	1	-1
quadratic	8.385	1	65	1
cubic	366.145	-1	65	-1
quartic	12.082.785	33	2.145	33
quintic	321.402.081	-33	2.145	-33
sextic	7.177.979.809	33	47.905	33

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

Force field	A	B	E*
linear	32	33	32
quadratic	2.113	2.112	2.080
cubic	91.552	91.553	91.520
quartic	3.021.249	3.021.216	3.020.160
quintic	80.351.040	80.351.073	80.349.984
sextic	1.794.506.945	1.794.506.912	1.794.482.976

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 S₄

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(E)
..528.	AA.	..561.	BB.	..1.024.	EE.
Subtotal: 2.113 / 3 / 3
Irrep combinations (i,j) with indices: pos(A) ≤ i ≤ j ≤ pos(E)
Subtotal: 0 / 0 / 3
Total: 2.113 / 3 / 6

Contributions to nonvanishing cubic force field constants

Irrep combinations (i,i,i) with indices: pos(A) ≤ i ≤ pos(E)
..5.984.	AAA.
Subtotal: 5.984 / 1 / 3
Irrep combinations (i,i,j) (i,j,j) with indices: pos(A) ≤ i ≤ j ≤ pos(E)
..17.952.	ABB.	..32.768.	AEE.	..34.848.	BEE.
Subtotal: 85.568 / 3 / 6
Irrep combinations (i,j,k) with indices: pos(A) ≤ i ≤ j ≤ k ≤ pos(E)
Subtotal: 0 / 0 / 1
Total: 91.552 / 4 / 10

Contributions to nonvanishing quartic force field constants

Irrep combinations (i,i,i,i) with indices: pos(A) ≤ i ≤ pos(E)
..52.360.	AAAA.	..58.905.	BBBB.	..383.504.	EEEE.
Subtotal: 494.769 / 3 / 3
Irrep combinations (i,i,i,j) (i,j,j,j) with indices: pos(A) ≤ i ≤ j ≤ pos(E)
Subtotal: 0 / 0 / 6
Irrep combinations (i,i,j,j) with indices: pos(A) ≤ i ≤ j ≤ pos(E)
..296.208.	AABB.	..540.672.	AAEE.	..574.464.	BBEE.
Subtotal: 1.411.344 / 3 / 3
Irrep combinations (i,i,j,k) (i,j,j,k) (i,j,k,k) with indices: pos(A) ≤ i ≤ j ≤ k ≤ pos(E)
..1.115.136.	ABEE.
Subtotal: 1.115.136 / 1 / 3
Irrep combinations (i,j,k,l) with indices: pos(A) ≤ i ≤ j ≤ k ≤ l ≤ pos(E)
Subtotal: 0 / 0 / 0
Total: 3.021.249 / 7 / 15

Calculate contributions to

A	B	E

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