Characters of representations for molecular motions

Motion	E	4C3	4(C3)2	3C2
Cartesian 3N	288	0	0	0
Translation (x,y,z)	3	0	0	-1
Rotation (Rx,Ry,Rz)	3	0	0	-1
Vibration	282	0	0	2

Decomposition to irreducible representations

Motion	A	E*	T	Total
Cartesian 3N	24	24	72	120
Translation (x,y,z)	0	0	1	1
Rotation (Rx,Ry,Rz)	0	0	1	1
Vibration	24	24	70	118

Molecular parameter

Number of Atoms (N)	96
Number of internal coordinates	282
Number of independant internal coordinates	24
Number of vibrational modes	118

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Force field analysis

Allowed / forbidden vibronational transitions

Operator	A	E*	T	Total
Linear (IR)	24	24	70	70 / 48
Quadratic (Raman)	24	24	70	118 / 0
IR + Raman	- - - -	- - - -	70	70 / 0

Characters of force fields
(Symmetric powers of vibration representation)

Force field	E	4C3	4(C3)2	3C2
linear	282	0	0	2
quadratic	39.903	0	0	143
cubic	3.777.484	94	94	284
quartic	269.145.735	0	0	10.295
quintic	15.395.136.042	0	0	20.306
sextic	736.400.674.009	4.465	4.465	497.497

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

Force field	A	E*	T
linear	24	24	70
quadratic	3.361	3.361	9.940
cubic	314.924	314.830	944.300
quartic	22.431.385	22.431.385	67.283.860
quintic	1.282.933.080	1.282.933.080	3.848.778.934
sextic	61.366.850.185	61.366.845.720	184.100.044.128

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 T

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(T)
..300.	AA.	..576.	EE.	..2.485.	TT.
Subtotal: 3.361 / 3 / 3
Irrep combinations (i,j) with indices: pos(A) ≤ i ≤ j ≤ pos(T)
Subtotal: 0 / 0 / 3
Total: 3.361 / 3 / 6

Contributions to nonvanishing cubic force field constants

Irrep combinations (i,i,i) with indices: pos(A) ≤ i ≤ pos(T)
..2.600.	AAA.	..5.200.	EEE.	..114.380.	TTT.
Subtotal: 122.180 / 3 / 3
Irrep combinations (i,i,j) (i,j,j) with indices: pos(A) ≤ i ≤ j ≤ pos(T)
..13.824.	AEE.	..59.640.	ATT.	..119.280.	ETT.
Subtotal: 192.744 / 3 / 6
Irrep combinations (i,j,k) with indices: pos(A) ≤ i ≤ j ≤ k ≤ pos(T)
Subtotal: 0 / 0 / 1
Total: 314.924 / 6 / 10

Contributions to nonvanishing quartic force field constants

Irrep combinations (i,i,i,i) with indices: pos(A) ≤ i ≤ pos(T)
..17.550.	AAAA.	..90.000.	EEEE.	..7.263.655.	TTTT.
Subtotal: 7.371.205 / 3 / 3
Irrep combinations (i,i,i,j) (i,j,j,j) with indices: pos(A) ≤ i ≤ j ≤ pos(T)
..124.800.	AEEE.	..2.745.120.	ATTT.	..5.486.880.	ETTT.
Subtotal: 8.356.800 / 3 / 6
Irrep combinations (i,i,j,j) with indices: pos(A) ≤ i ≤ j ≤ pos(T)
..172.800.	AAEE.	..745.500.	AATT.	..2.922.360.	EETT.
Subtotal: 3.840.660 / 3 / 3
Irrep combinations (i,i,j,k) (i,j,j,k) (i,j,k,k) with indices: pos(A) ≤ i ≤ j ≤ k ≤ pos(T)
..2.862.720.	AETT.
Subtotal: 2.862.720 / 1 / 3
Irrep combinations (i,j,k,l) with indices: pos(A) ≤ i ≤ j ≤ k ≤ l ≤ pos(T)
Subtotal: 0 / 0 / 0
Total: 22.431.385 / 10 / 15

Calculate contributions to

A	E	T

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