Characters of representations for molecular motions

Motion	E	C4	C2	(C4)3
Cartesian 3N	36	4	-4	4
Translation (x,y,z)	3	1	-1	1
Rotation (Rx,Ry,Rz)	3	1	-1	1
Vibration	30	2	-2	2

Decomposition to irreducible representations

Motion	A	B	E*	Total
Cartesian 3N	10	6	10	26
Translation (x,y,z)	1	0	1	2
Rotation (Rx,Ry,Rz)	1	0	1	2
Vibration	8	6	8	22

Molecular parameter

Number of Atoms (N)	12
Number of internal coordinates	30
Number of independant internal coordinates	8
Number of vibrational modes	22

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Force field analysis

Allowed / forbidden vibronational transitions

Operator	A	B	E*	Total
Linear (IR)	8	6	8	16 / 6
Quadratic (Raman)	8	6	8	22 / 0
IR + Raman	8	- - - -	8	16 / 0

Characters of force fields
(Symmetric powers of vibration representation)

Force field	E	C4	C2	(C4)3
linear	30	2	-2	2
quadratic	465	1	17	1
cubic	4.960	0	-32	0
quartic	40.920	8	152	8
quintic	278.256	16	-272	16
sextic	1.623.160	8	952	8

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

Force field	A	B	E*
linear	8	6	8
quadratic	121	120	112
cubic	1.232	1.232	1.248
quartic	10.272	10.264	10.192
quintic	69.504	69.488	69.632
sextic	406.032	406.024	405.552

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)
..36.	AA.	..21.	BB.	..64.	EE.
Subtotal: 121 / 3 / 3
Irrep combinations (i,j) with indices: pos(A) ≤ i ≤ j ≤ pos(E)
Subtotal: 0 / 0 / 3
Total: 121 / 3 / 6

Contributions to nonvanishing cubic force field constants

Irrep combinations (i,i,i) with indices: pos(A) ≤ i ≤ pos(E)
..120.	AAA.
Subtotal: 120 / 1 / 3
Irrep combinations (i,i,j) (i,j,j) with indices: pos(A) ≤ i ≤ j ≤ pos(E)
..168.	ABB.	..512.	AEE.	..432.	BEE.
Subtotal: 1.112 / 3 / 6
Irrep combinations (i,j,k) with indices: pos(A) ≤ i ≤ j ≤ k ≤ pos(E)
Subtotal: 0 / 0 / 1
Total: 1.232 / 4 / 10

Contributions to nonvanishing quartic force field constants

Irrep combinations (i,i,i,i) with indices: pos(A) ≤ i ≤ pos(E)
..330.	AAAA.	..126.	BBBB.	..1.956.	EEEE.
Subtotal: 2.412 / 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)
..756.	AABB.	..2.304.	AAEE.	..1.344.	BBEE.
Subtotal: 4.404 / 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)
..3.456.	ABEE.
Subtotal: 3.456 / 1 / 3
Irrep combinations (i,j,k,l) with indices: pos(A) ≤ i ≤ j ≤ k ≤ l ≤ pos(E)
Subtotal: 0 / 0 / 0
Total: 10.272 / 7 / 15

Calculate contributions to

A	B	E

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