Characters of representations for molecular motions

Motion	E	4C3	4(C3)2	3C2
Cartesian 3N	21	0	0	-3
Translation (x,y,z)	3	0	0	-1
Rotation (Rx,Ry,Rz)	3	0	0	-1
Vibration	15	0	0	-1

Decomposition to irreducible representations

Motion	A	E*	T	Total
Cartesian 3N	1	1	6	8
Translation (x,y,z)	0	0	1	1
Rotation (Rx,Ry,Rz)	0	0	1	1
Vibration	1	1	4	6

Molecular parameter

Number of Atoms (N)	7
Number of internal coordinates	15
Number of independant internal coordinates	1
Number of vibrational modes	6

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Force field analysis

Allowed / forbidden vibronational transitions

Operator	A	E*	T	Total
Linear (IR)	1	1	4	4 / 2
Quadratic (Raman)	1	1	4	6 / 0
IR + Raman	- - - -	- - - -	4	4 / 0

Characters of force fields
(Symmetric powers of vibration representation)

Force field	E	4C3	4(C3)2	3C2
linear	15	0	0	-1
quadratic	120	0	0	8
cubic	680	5	5	-8
quartic	3.060	0	0	36
quintic	11.628	0	0	-36
sextic	38.760	15	15	120

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

Force field	A	E*	T
linear	1	1	4
quadratic	12	12	28
cubic	58	53	172
quartic	264	264	756
quintic	960	960	2.916
sextic	3.270	3.255	9.660

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)
..1.	AA.	..1.	EE.	..10.	TT.
Subtotal: 12 / 3 / 3
Irrep combinations (i,j) with indices: pos(A) ≤ i ≤ j ≤ pos(T)
Subtotal: 0 / 0 / 3
Total: 12 / 3 / 6

Contributions to nonvanishing cubic force field constants

Irrep combinations (i,i,i) with indices: pos(A) ≤ i ≤ pos(T)
..1.	AAA.	..2.	EEE.	..24.	TTT.
Subtotal: 27 / 3 / 3
Irrep combinations (i,i,j) (i,j,j) with indices: pos(A) ≤ i ≤ j ≤ pos(T)
..1.	AEE.	..10.	ATT.	..20.	ETT.
Subtotal: 31 / 3 / 6
Irrep combinations (i,j,k) with indices: pos(A) ≤ i ≤ j ≤ k ≤ pos(T)
Subtotal: 0 / 0 / 1
Total: 58 / 6 / 10

Contributions to nonvanishing quartic force field constants

Irrep combinations (i,i,i,i) with indices: pos(A) ≤ i ≤ pos(T)
..1.	AAAA.	..1.	EEEE.	..135.	TTTT.
Subtotal: 137 / 3 / 3
Irrep combinations (i,i,i,j) (i,j,j,j) with indices: pos(A) ≤ i ≤ j ≤ pos(T)
..2.	AEEE.	..24.	ATTT.	..40.	ETTT.
Subtotal: 66 / 3 / 6
Irrep combinations (i,i,j,j) with indices: pos(A) ≤ i ≤ j ≤ pos(T)
..1.	AAEE.	..10.	AATT.	..30.	EETT.
Subtotal: 41 / 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)
..20.	AETT.
Subtotal: 20 / 1 / 3
Irrep combinations (i,j,k,l) with indices: pos(A) ≤ i ≤ j ≤ k ≤ l ≤ pos(T)
Subtotal: 0 / 0 / 0
Total: 264 / 10 / 15

Calculate contributions to

A	E	T

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