Characters of representations for molecular motions

Motion	E	C3	(C3)2
Cartesian 3N	39	0	0
Translation (x,y,z)	3	0	0
Rotation (Rx,Ry,Rz)	3	0	0
Vibration	33	0	0

Decomposition to irreducible representations

Motion	A	E*	Total
Cartesian 3N	13	13	26
Translation (x,y,z)	1	1	2
Rotation (Rx,Ry,Rz)	1	1	2
Vibration	11	11	22

Molecular parameter

Number of Atoms (N)	13
Number of internal coordinates	33
Number of independant internal coordinates	11
Number of vibrational modes	22

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Force field analysis

Allowed / forbidden vibronational transitions

Operator	A	E*	Total
Linear (IR)	11	11	22 / 0
Quadratic (Raman)	11	11	22 / 0
IR + Raman	11	11	22 / 0

Characters of force fields
(Symmetric powers of vibration representation)

Force field	E	C3	(C3)2
linear	33	0	0
quadratic	561	0	0
cubic	6.545	11	11
quartic	58.905	0	0
quintic	435.897	0	0
sextic	2.760.681	66	66

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

Force field	A	E*
linear	11	11
quadratic	187	187
cubic	2.189	2.178
quartic	19.635	19.635
quintic	145.299	145.299
sextic	920.271	920.205

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)
..66.	AA.	..121.	EE.
Subtotal: 187 / 2 / 2
Irrep combinations (i,j) with indices: pos(A) ≤ i ≤ j ≤ pos(E)
Subtotal: 0 / 0 / 1
Total: 187 / 2 / 3

Contributions to nonvanishing cubic force field constants

Irrep combinations (i,i,i) with indices: pos(A) ≤ i ≤ pos(E)
..286.	AAA.	..572.	EEE.
Subtotal: 858 / 2 / 2
Irrep combinations (i,i,j) (i,j,j) with indices: pos(A) ≤ i ≤ j ≤ pos(E)
..1.331.	AEE.
Subtotal: 1.331 / 1 / 2
Irrep combinations (i,j,k) with indices: pos(A) ≤ i ≤ j ≤ k ≤ pos(E)
Subtotal: 0 / 0 / 0
Total: 2.189 / 3 / 4

Contributions to nonvanishing quartic force field constants

Irrep combinations (i,i,i,i) with indices: pos(A) ≤ i ≤ pos(E)
..1.001.	AAAA.	..4.356.	EEEE.
Subtotal: 5.357 / 2 / 2
Irrep combinations (i,i,i,j) (i,j,j,j) with indices: pos(A) ≤ i ≤ j ≤ pos(E)
..6.292.	AEEE.
Subtotal: 6.292 / 1 / 2
Irrep combinations (i,i,j,j) with indices: pos(A) ≤ i ≤ j ≤ pos(E)
..7.986.	AAEE.
Subtotal: 7.986 / 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: 19.635 / 4 / 5

Calculate contributions to

A	E

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