Characters of representations for molecular motions

Motion	E	S4	C2	(S4)3
Cartesian 3N	39	-1	-5	-1
Translation (x,y,z)	3	-1	-1	-1
Rotation (Rx,Ry,Rz)	3	1	-1	1
Vibration	33	-1	-3	-1

Decomposition to irreducible representations

Motion	A	B	E*	Total
Cartesian 3N	8	9	11	28
Translation (x,y,z)	0	1	1	2
Rotation (Rx,Ry,Rz)	1	0	1	2
Vibration	7	8	9	24

Molecular parameter

Number of Atoms (N)	13
Number of internal coordinates	33
Number of independant internal coordinates	7
Number of vibrational modes	24

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Force field analysis

Allowed / forbidden vibronational transitions

Operator	A	B	E*	Total
Linear (IR)	7	8	9	17 / 7
Quadratic (Raman)	7	8	9	24 / 0
IR + Raman	- - - -	8	9	17 / 0

Characters of force fields
(Symmetric powers of vibration representation)

Force field	E	S4	C2	(S4)3
linear	33	-1	-3	-1
quadratic	561	-1	21	-1
cubic	6.545	1	-55	1
quartic	58.905	9	225	9
quintic	435.897	-9	-531	-9
sextic	2.760.681	-9	1.653	-9

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

Force field	A	B	E*
linear	7	8	9
quadratic	145	146	135
cubic	1.623	1.622	1.650
quartic	14.787	14.778	14.670
quintic	108.837	108.846	109.107
sextic	690.579	690.588	689.757

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)
..28.	AA.	..36.	BB.	..81.	EE.
Subtotal: 145 / 3 / 3
Irrep combinations (i,j) with indices: pos(A) ≤ i ≤ j ≤ pos(E)
Subtotal: 0 / 0 / 3
Total: 145 / 3 / 6

Contributions to nonvanishing cubic force field constants

Irrep combinations (i,i,i) with indices: pos(A) ≤ i ≤ pos(E)
..84.	AAA.
Subtotal: 84 / 1 / 3
Irrep combinations (i,i,j) (i,j,j) with indices: pos(A) ≤ i ≤ j ≤ pos(E)
..252.	ABB.	..567.	AEE.	..720.	BEE.
Subtotal: 1.539 / 3 / 6
Irrep combinations (i,j,k) with indices: pos(A) ≤ i ≤ j ≤ k ≤ pos(E)
Subtotal: 0 / 0 / 1
Total: 1.623 / 4 / 10

Contributions to nonvanishing quartic force field constants

Irrep combinations (i,i,i,i) with indices: pos(A) ≤ i ≤ pos(E)
..210.	AAAA.	..330.	BBBB.	..3.015.	EEEE.
Subtotal: 3.555 / 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)
..1.008.	AABB.	..2.268.	AAEE.	..2.916.	BBEE.
Subtotal: 6.192 / 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)
..5.040.	ABEE.
Subtotal: 5.040 / 1 / 3
Irrep combinations (i,j,k,l) with indices: pos(A) ≤ i ≤ j ≤ k ≤ l ≤ pos(E)
Subtotal: 0 / 0 / 0
Total: 14.787 / 7 / 15

Calculate contributions to

A	B	E

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