Characters of representations for molecular motions

Motion	E	4C3	4(C3)2	3C2
Cartesian 3N	39	0	0	-5
Translation (x,y,z)	3	0	0	-1
Rotation (Rx,Ry,Rz)	3	0	0	-1
Vibration	33	0	0	-3

Decomposition to irreducible representations

Motion	A	E*	T	Total
Cartesian 3N	2	2	11	15
Translation (x,y,z)	0	0	1	1
Rotation (Rx,Ry,Rz)	0	0	1	1
Vibration	2	2	9	13

Molecular parameter

Number of Atoms (N)	13
Number of internal coordinates	33
Number of independant internal coordinates	2
Number of vibrational modes	13

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Force field analysis

Allowed / forbidden vibronational transitions

Operator	A	E*	T	Total
Linear (IR)	2	2	9	9 / 4
Quadratic (Raman)	2	2	9	13 / 0
IR + Raman	- - - -	- - - -	9	9 / 0

Characters of force fields
(Symmetric powers of vibration representation)

Force field	E	4C3	4(C3)2	3C2
linear	33	0	0	-3
quadratic	561	0	0	21
cubic	6.545	11	11	-55
quartic	58.905	0	0	225
quintic	435.897	0	0	-531
sextic	2.760.681	66	66	1.653

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

Force field	A	E*	T
linear	2	2	9
quadratic	52	52	135
cubic	539	528	1.650
quartic	4.965	4.965	14.670
quintic	36.192	36.192	109.107
sextic	230.514	230.448	689.757

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)
..3.	AA.	..4.	EE.	..45.	TT.
Subtotal: 52 / 3 / 3
Irrep combinations (i,j) with indices: pos(A) ≤ i ≤ j ≤ pos(T)
Subtotal: 0 / 0 / 3
Total: 52 / 3 / 6

Contributions to nonvanishing cubic force field constants

Irrep combinations (i,i,i) with indices: pos(A) ≤ i ≤ pos(T)
..4.	AAA.	..8.	EEE.	..249.	TTT.
Subtotal: 261 / 3 / 3
Irrep combinations (i,i,j) (i,j,j) with indices: pos(A) ≤ i ≤ j ≤ pos(T)
..8.	AEE.	..90.	ATT.	..180.	ETT.
Subtotal: 278 / 3 / 6
Irrep combinations (i,j,k) with indices: pos(A) ≤ i ≤ j ≤ k ≤ pos(T)
Subtotal: 0 / 0 / 1
Total: 539 / 6 / 10

Contributions to nonvanishing quartic force field constants

Irrep combinations (i,i,i,i) with indices: pos(A) ≤ i ≤ pos(T)
..5.	AAAA.	..9.	EEEE.	..2.520.	TTTT.
Subtotal: 2.534 / 3 / 3
Irrep combinations (i,i,i,j) (i,j,j,j) with indices: pos(A) ≤ i ≤ j ≤ pos(T)
..16.	AEEE.	..498.	ATTT.	..960.	ETTT.
Subtotal: 1.474 / 3 / 6
Irrep combinations (i,i,j,j) with indices: pos(A) ≤ i ≤ j ≤ pos(T)
..12.	AAEE.	..135.	AATT.	..450.	EETT.
Subtotal: 597 / 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)
..360.	AETT.
Subtotal: 360 / 1 / 3
Irrep combinations (i,j,k,l) with indices: pos(A) ≤ i ≤ j ≤ k ≤ l ≤ pos(T)
Subtotal: 0 / 0 / 0
Total: 4.965 / 10 / 15

Calculate contributions to

A	E	T

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