Characters of representations for molecular motions

Motion	E	C4	C2	(C4)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	-2	2	-2

Decomposition to irreducible representations

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

Molecular parameter

Number of Atoms (N)	16
Number of internal coordinates	42
Number of independant internal coordinates	10
Number of vibrational modes	32

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Force field analysis

Allowed / forbidden vibronational transitions

Operator	A	B	E*	Total
Linear (IR)	10	12	10	20 / 12
Quadratic (Raman)	10	12	10	32 / 0
IR + Raman	10	- - - -	10	20 / 0

Characters of force fields
(Symmetric powers of vibration representation)

Force field	E	C4	C2	(C4)3
linear	42	-2	2	-2
quadratic	903	3	23	3
cubic	13.244	-4	44	-4
quartic	148.995	15	275	15
quintic	1.370.754	-26	506	-26
sextic	10.737.573	37	2.277	37

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

Force field	A	B	E*
linear	10	12	10
quadratic	233	230	220
cubic	3.320	3.324	3.300
quartic	37.325	37.310	37.180
quintic	342.802	342.828	342.562
sextic	2.684.981	2.684.944	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 C₄

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)
..55.	AA.	..78.	BB.	..100.	EE.
Subtotal: 233 / 3 / 3
Irrep combinations (i,j) with indices: pos(A) ≤ i ≤ j ≤ pos(E)
Subtotal: 0 / 0 / 3
Total: 233 / 3 / 6

Contributions to nonvanishing cubic force field constants

Irrep combinations (i,i,i) with indices: pos(A) ≤ i ≤ pos(E)
..220.	AAA.
Subtotal: 220 / 1 / 3
Irrep combinations (i,i,j) (i,j,j) with indices: pos(A) ≤ i ≤ j ≤ pos(E)
..780.	ABB.	..1.000.	AEE.	..1.320.	BEE.
Subtotal: 3.100 / 3 / 6
Irrep combinations (i,j,k) with indices: pos(A) ≤ i ≤ j ≤ k ≤ pos(E)
Subtotal: 0 / 0 / 1
Total: 3.320 / 4 / 10

Contributions to nonvanishing quartic force field constants

Irrep combinations (i,i,i,i) with indices: pos(A) ≤ i ≤ pos(E)
..715.	AAAA.	..1.365.	BBBB.	..4.455.	EEEE.
Subtotal: 6.535 / 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)
..4.290.	AABB.	..5.500.	AAEE.	..7.800.	BBEE.
Subtotal: 17.590 / 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)
..13.200.	ABEE.
Subtotal: 13.200 / 1 / 3
Irrep combinations (i,j,k,l) with indices: pos(A) ≤ i ≤ j ≤ k ≤ l ≤ pos(E)
Subtotal: 0 / 0 / 0
Total: 37.325 / 7 / 15

Calculate contributions to

A	B	E

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