Characters of representations for molecular motions

Motion	E	C4	C2	(C4)3
Cartesian 3N	21	3	-3	3
Translation (x,y,z)	3	1	-1	1
Rotation (Rx,Ry,Rz)	3	1	-1	1
Vibration	15	1	-1	1

Decomposition to irreducible representations

Motion	A	B	E*	Total
Cartesian 3N	6	3	6	15
Translation (x,y,z)	1	0	1	2
Rotation (Rx,Ry,Rz)	1	0	1	2
Vibration	4	3	4	11

Molecular parameter

Number of Atoms (N)	7
Number of internal coordinates	15
Number of independant internal coordinates	4
Number of vibrational modes	11

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Force field analysis

Allowed / forbidden vibronational transitions

Operator	A	B	E*	Total
Linear (IR)	4	3	4	8 / 3
Quadratic (Raman)	4	3	4	11 / 0
IR + Raman	4	- - - -	4	8 / 0

Characters of force fields
(Symmetric powers of vibration representation)

Force field	E	C4	C2	(C4)3
linear	15	1	-1	1
quadratic	120	0	8	0
cubic	680	0	-8	0
quartic	3.060	4	36	4
quintic	11.628	4	-36	4
sextic	38.760	0	120	0

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

Force field	A	B	E*
linear	4	3	4
quadratic	32	32	28
cubic	168	168	172
quartic	776	772	756
quintic	2.900	2.896	2.916
sextic	9.720	9.720	9.660

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)
..10.	AA.	..6.	BB.	..16.	EE.
Subtotal: 32 / 3 / 3
Irrep combinations (i,j) with indices: pos(A) ≤ i ≤ j ≤ pos(E)
Subtotal: 0 / 0 / 3
Total: 32 / 3 / 6

Contributions to nonvanishing cubic force field constants

Irrep combinations (i,i,i) with indices: pos(A) ≤ i ≤ pos(E)
..20.	AAA.
Subtotal: 20 / 1 / 3
Irrep combinations (i,i,j) (i,j,j) with indices: pos(A) ≤ i ≤ j ≤ pos(E)
..24.	ABB.	..64.	AEE.	..60.	BEE.
Subtotal: 148 / 3 / 6
Irrep combinations (i,j,k) with indices: pos(A) ≤ i ≤ j ≤ k ≤ pos(E)
Subtotal: 0 / 0 / 1
Total: 168 / 4 / 10

Contributions to nonvanishing quartic force field constants

Irrep combinations (i,i,i,i) with indices: pos(A) ≤ i ≤ pos(E)
..35.	AAAA.	..15.	BBBB.	..170.	EEEE.
Subtotal: 220 / 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)
..60.	AABB.	..160.	AAEE.	..96.	BBEE.
Subtotal: 316 / 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)
..240.	ABEE.
Subtotal: 240 / 1 / 3
Irrep combinations (i,j,k,l) with indices: pos(A) ≤ i ≤ j ≤ k ≤ l ≤ pos(E)
Subtotal: 0 / 0 / 0
Total: 776 / 7 / 15

Calculate contributions to

A	B	E

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