Characters of representations for molecular motions

Motion	E	S4	C2	(S4)3
Cartesian 3N	51	-1	-1	-1
Translation (x,y,z)	3	-1	-1	-1
Rotation (Rx,Ry,Rz)	3	1	-1	1
Vibration	45	-1	1	-1

Decomposition to irreducible representations

Motion	A	B	E*	Total
Cartesian 3N	12	13	13	38
Translation (x,y,z)	0	1	1	2
Rotation (Rx,Ry,Rz)	1	0	1	2
Vibration	11	12	11	34

Molecular parameter

Number of Atoms (N)	17
Number of internal coordinates	45
Number of independant internal coordinates	11
Number of vibrational modes	34

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Force field analysis

Allowed / forbidden vibronational transitions

Operator	A	B	E*	Total
Linear (IR)	11	12	11	23 / 11
Quadratic (Raman)	11	12	11	34 / 0
IR + Raman	- - - -	12	11	23 / 0

Characters of force fields
(Symmetric powers of vibration representation)

Force field	E	S4	C2	(S4)3
linear	45	-1	1	-1
quadratic	1.035	1	23	1
cubic	16.215	-1	23	-1
quartic	194.580	12	276	12
quintic	1.906.884	-12	276	-12
sextic	15.890.700	12	2.300	12

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

Force field	A	B	E*
linear	11	12	11
quadratic	265	264	253
cubic	4.059	4.060	4.048
quartic	48.720	48.708	48.576
quintic	476.784	476.796	476.652
sextic	3.973.256	3.973.244	3.972.100

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.	..78.	BB.	..121.	EE.
Subtotal: 265 / 3 / 3
Irrep combinations (i,j) with indices: pos(A) ≤ i ≤ j ≤ pos(E)
Subtotal: 0 / 0 / 3
Total: 265 / 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)
..858.	ABB.	..1.331.	AEE.	..1.584.	BEE.
Subtotal: 3.773 / 3 / 6
Irrep combinations (i,j,k) with indices: pos(A) ≤ i ≤ j ≤ k ≤ pos(E)
Subtotal: 0 / 0 / 1
Total: 4.059 / 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.	..1.365.	BBBB.	..6.358.	EEEE.
Subtotal: 8.724 / 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)
..5.148.	AABB.	..7.986.	AAEE.	..9.438.	BBEE.
Subtotal: 22.572 / 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)
..17.424.	ABEE.
Subtotal: 17.424 / 1 / 3
Irrep combinations (i,j,k,l) with indices: pos(A) ≤ i ≤ j ≤ k ≤ l ≤ pos(E)
Subtotal: 0 / 0 / 0
Total: 48.720 / 7 / 15

Calculate contributions to

A	B	E

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