Characters of representations for molecular motions

Motion	E	C3	(C3)2
Cartesian 3N	24	0	0
Translation (x,y,z)	3	0	0
Rotation (Rx,Ry,Rz)	3	0	0
Vibration	18	0	0

Decomposition to irreducible representations

Motion	A	E*	Total
Cartesian 3N	8	8	16
Translation (x,y,z)	1	1	2
Rotation (Rx,Ry,Rz)	1	1	2
Vibration	6	6	12

Molecular parameter

Number of Atoms (N)	8
Number of internal coordinates	18
Number of independant internal coordinates	6
Number of vibrational modes	12

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Force field analysis

Allowed / forbidden vibronational transitions

Operator	A	E*	Total
Linear (IR)	6	6	12 / 0
Quadratic (Raman)	6	6	12 / 0
IR + Raman	6	6	12 / 0

Characters of force fields
(Symmetric powers of vibration representation)

Force field	E	C3	(C3)2
linear	18	0	0
quadratic	171	0	0
cubic	1.140	6	6
quartic	5.985	0	0
quintic	26.334	0	0
sextic	100.947	21	21

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

Force field	A	E*
linear	6	6
quadratic	57	57
cubic	384	378
quartic	1.995	1.995
quintic	8.778	8.778
sextic	33.663	33.642

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 C₃

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)
..21.	AA.	..36.	EE.
Subtotal: 57 / 2 / 2
Irrep combinations (i,j) with indices: pos(A) ≤ i ≤ j ≤ pos(E)
Subtotal: 0 / 0 / 1
Total: 57 / 2 / 3

Contributions to nonvanishing cubic force field constants

Irrep combinations (i,i,i) with indices: pos(A) ≤ i ≤ pos(E)
..56.	AAA.	..112.	EEE.
Subtotal: 168 / 2 / 2
Irrep combinations (i,i,j) (i,j,j) with indices: pos(A) ≤ i ≤ j ≤ pos(E)
..216.	AEE.
Subtotal: 216 / 1 / 2
Irrep combinations (i,j,k) with indices: pos(A) ≤ i ≤ j ≤ k ≤ pos(E)
Subtotal: 0 / 0 / 0
Total: 384 / 3 / 4

Contributions to nonvanishing quartic force field constants

Irrep combinations (i,i,i,i) with indices: pos(A) ≤ i ≤ pos(E)
..126.	AAAA.	..441.	EEEE.
Subtotal: 567 / 2 / 2
Irrep combinations (i,i,i,j) (i,j,j,j) with indices: pos(A) ≤ i ≤ j ≤ pos(E)
..672.	AEEE.
Subtotal: 672 / 1 / 2
Irrep combinations (i,i,j,j) with indices: pos(A) ≤ i ≤ j ≤ pos(E)
..756.	AAEE.
Subtotal: 756 / 1 / 1
Irrep combinations (i,i,j,k) (i,j,j,k) (i,j,k,k) with indices: pos(A) ≤ i ≤ j ≤ k ≤ pos(E)
Subtotal: 0 / 0 / 0
Irrep combinations (i,j,k,l) with indices: pos(A) ≤ i ≤ j ≤ k ≤ l ≤ pos(E)
Subtotal: 0 / 0 / 0
Total: 1.995 / 4 / 5

Calculate contributions to

A	E

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