Characters of representations for molecular motions

Motion	E	4C3	4(C3)2	3C2
Cartesian 3N	51	0	0	-1
Translation (x,y,z)	3	0	0	-1
Rotation (Rx,Ry,Rz)	3	0	0	-1
Vibration	45	0	0	1

Decomposition to irreducible representations

Motion	A	E*	T	Total
Cartesian 3N	4	4	13	21
Translation (x,y,z)	0	0	1	1
Rotation (Rx,Ry,Rz)	0	0	1	1
Vibration	4	4	11	19

Molecular parameter

Number of Atoms (N)	17
Number of internal coordinates	45
Number of independant internal coordinates	4
Number of vibrational modes	19

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Force field analysis

Allowed / forbidden vibronational transitions

Operator	A	E*	T	Total
Linear (IR)	4	4	11	11 / 8
Quadratic (Raman)	4	4	11	19 / 0
IR + Raman	- - - -	- - - -	11	11 / 0

Characters of force fields
(Symmetric powers of vibration representation)

Force field	E	4C3	4(C3)2	3C2
linear	45	0	0	1
quadratic	1.035	0	0	23
cubic	16.215	15	15	23
quartic	194.580	0	0	276
quintic	1.906.884	0	0	276
sextic	15.890.700	120	120	2.300

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

Force field	A	E*	T
linear	4	4	11
quadratic	92	92	253
cubic	1.367	1.352	4.048
quartic	16.284	16.284	48.576
quintic	158.976	158.976	476.652
sextic	1.324.880	1.324.760	3.972.100

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)
..10.	AA.	..16.	EE.	..66.	TT.
Subtotal: 92 / 3 / 3
Irrep combinations (i,j) with indices: pos(A) ≤ i ≤ j ≤ pos(T)
Subtotal: 0 / 0 / 3
Total: 92 / 3 / 6

Contributions to nonvanishing cubic force field constants

Irrep combinations (i,i,i) with indices: pos(A) ≤ i ≤ pos(T)
..20.	AAA.	..40.	EEE.	..451.	TTT.
Subtotal: 511 / 3 / 3
Irrep combinations (i,i,j) (i,j,j) with indices: pos(A) ≤ i ≤ j ≤ pos(T)
..64.	AEE.	..264.	ATT.	..528.	ETT.
Subtotal: 856 / 3 / 6
Irrep combinations (i,j,k) with indices: pos(A) ≤ i ≤ j ≤ k ≤ pos(T)
Subtotal: 0 / 0 / 1
Total: 1.367 / 6 / 10

Contributions to nonvanishing quartic force field constants

Irrep combinations (i,i,i,i) with indices: pos(A) ≤ i ≤ pos(T)
..35.	AAAA.	..100.	EEEE.	..5.357.	TTTT.
Subtotal: 5.492 / 3 / 3
Irrep combinations (i,i,i,j) (i,j,j,j) with indices: pos(A) ≤ i ≤ j ≤ pos(T)
..160.	AEEE.	..1.804.	ATTT.	..3.520.	ETTT.
Subtotal: 5.484 / 3 / 6
Irrep combinations (i,i,j,j) with indices: pos(A) ≤ i ≤ j ≤ pos(T)
..160.	AAEE.	..660.	AATT.	..2.376.	EETT.
Subtotal: 3.196 / 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)
..2.112.	AETT.
Subtotal: 2.112 / 1 / 3
Irrep combinations (i,j,k,l) with indices: pos(A) ≤ i ≤ j ≤ k ≤ l ≤ pos(T)
Subtotal: 0 / 0 / 0
Total: 16.284 / 10 / 15

Calculate contributions to

A	E	T

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