Characters of representations for molecular motions

Motion	E	C3	(C3)2
Cartesian 3N	105	0	0
Translation (x,y,z)	3	0	0
Rotation (Rx,Ry,Rz)	3	0	0
Vibration	99	0	0

Decomposition to irreducible representations

Motion	A	E*	Total
Cartesian 3N	35	35	70
Translation (x,y,z)	1	1	2
Rotation (Rx,Ry,Rz)	1	1	2
Vibration	33	33	66

Molecular parameter

Number of Atoms (N)	35
Number of internal coordinates	99
Number of independant internal coordinates	33
Number of vibrational modes	66

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Force field analysis

Allowed / forbidden vibronational transitions

Operator	A	E*	Total
Linear (IR)	33	33	66 / 0
Quadratic (Raman)	33	33	66 / 0
IR + Raman	33	33	66 / 0

Characters of force fields
(Symmetric powers of vibration representation)

Force field	E	C3	(C3)2
linear	99	0	0
quadratic	4.950	0	0
cubic	166.650	33	33
quartic	4.249.575	0	0
quintic	87.541.245	0	0
sextic	1.517.381.580	561	561

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

Force field	A	E*
linear	33	33
quadratic	1.650	1.650
cubic	55.572	55.539
quartic	1.416.525	1.416.525
quintic	29.180.415	29.180.415
sextic	505.794.234	505.793.673

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)
..561.	AA.	..1.089.	EE.
Subtotal: 1.650 / 2 / 2
Irrep combinations (i,j) with indices: pos(A) ≤ i ≤ j ≤ pos(E)
Subtotal: 0 / 0 / 1
Total: 1.650 / 2 / 3

Contributions to nonvanishing cubic force field constants

Irrep combinations (i,i,i) with indices: pos(A) ≤ i ≤ pos(E)
..6.545.	AAA.	..13.090.	EEE.
Subtotal: 19.635 / 2 / 2
Irrep combinations (i,i,j) (i,j,j) with indices: pos(A) ≤ i ≤ j ≤ pos(E)
..35.937.	AEE.
Subtotal: 35.937 / 1 / 2
Irrep combinations (i,j,k) with indices: pos(A) ≤ i ≤ j ≤ k ≤ pos(E)
Subtotal: 0 / 0 / 0
Total: 55.572 / 3 / 4

Contributions to nonvanishing quartic force field constants

Irrep combinations (i,i,i,i) with indices: pos(A) ≤ i ≤ pos(E)
..58.905.	AAAA.	..314.721.	EEEE.
Subtotal: 373.626 / 2 / 2
Irrep combinations (i,i,i,j) (i,j,j,j) with indices: pos(A) ≤ i ≤ j ≤ pos(E)
..431.970.	AEEE.
Subtotal: 431.970 / 1 / 2
Irrep combinations (i,i,j,j) with indices: pos(A) ≤ i ≤ j ≤ pos(E)
..610.929.	AAEE.
Subtotal: 610.929 / 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: 1.416.525 / 4 / 5

Calculate contributions to

A	E

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