Characters of representations for molecular motions

Motion	E	S4	C2	(S4)3
Cartesian 3N	48	0	0	0
Translation (x,y,z)	3	-1	-1	-1
Rotation (Rx,Ry,Rz)	3	1	-1	1
Vibration	42	0	2	0

Decomposition to irreducible representations

Motion	A	B	E*	Total
Cartesian 3N	12	12	12	36
Translation (x,y,z)	0	1	1	2
Rotation (Rx,Ry,Rz)	1	0	1	2
Vibration	11	11	10	32

Molecular parameter

Number of Atoms (N)	16
Number of internal coordinates	42
Number of independant internal coordinates	11
Number of vibrational modes	32

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Force field analysis

Allowed / forbidden vibronational transitions

Operator	A	B	E*	Total
Linear (IR)	11	11	10	21 / 11
Quadratic (Raman)	11	11	10	32 / 0
IR + Raman	- - - -	11	10	21 / 0

Characters of force fields
(Symmetric powers of vibration representation)

Force field	E	S4	C2	(S4)3
linear	42	0	2	0
quadratic	903	1	23	1
cubic	13.244	0	44	0
quartic	148.995	11	275	11
quintic	1.370.754	0	506	0
sextic	10.737.573	11	2.277	11

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

Force field	A	B	E*
linear	11	11	10
quadratic	232	231	220
cubic	3.322	3.322	3.300
quartic	37.323	37.312	37.180
quintic	342.815	342.815	342.562
sextic	2.684.968	2.684.957	2.683.824

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)
..66.	AA.	..66.	BB.	..100.	EE.
Subtotal: 232 / 3 / 3
Irrep combinations (i,j) with indices: pos(A) ≤ i ≤ j ≤ pos(E)
Subtotal: 0 / 0 / 3
Total: 232 / 3 / 6

Contributions to nonvanishing cubic force field constants

Irrep combinations (i,i,i) with indices: pos(A) ≤ i ≤ pos(E)
..286.	AAA.
Subtotal: 286 / 1 / 3
Irrep combinations (i,i,j) (i,j,j) with indices: pos(A) ≤ i ≤ j ≤ pos(E)
..726.	ABB.	..1.100.	AEE.	..1.210.	BEE.
Subtotal: 3.036 / 3 / 6
Irrep combinations (i,j,k) with indices: pos(A) ≤ i ≤ j ≤ k ≤ pos(E)
Subtotal: 0 / 0 / 1
Total: 3.322 / 4 / 10

Contributions to nonvanishing quartic force field constants

Irrep combinations (i,i,i,i) with indices: pos(A) ≤ i ≤ pos(E)
..1.001.	AAAA.	..1.001.	BBBB.	..4.455.	EEEE.
Subtotal: 6.457 / 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)
..4.356.	AABB.	..6.600.	AAEE.	..6.600.	BBEE.
Subtotal: 17.556 / 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)
..13.310.	ABEE.
Subtotal: 13.310 / 1 / 3
Irrep combinations (i,j,k,l) with indices: pos(A) ≤ i ≤ j ≤ k ≤ l ≤ pos(E)
Subtotal: 0 / 0 / 0
Total: 37.323 / 7 / 15

Calculate contributions to

A	B	E

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