Characters of representations for molecular motions

Motion	E	C3	(C3)2	σh	S3	(S3)5
Cartesian 3N	0	0	0	0	0	0
Translation (x,y,z)	3	0	0	1	-2	-2
Rotation (Rx,Ry,Rz)	3	0	0	-1	2	2
Vibration	-6	0	0	0	0	0

Decomposition to irreducible representations

Motion	A'	E'*	A''	E''*	Total
Cartesian 3N	0	0	0	0	0
Translation (x,y,z)	0	1	1	0	2
Rotation (Rx,Ry,Rz)	1	0	0	1	2
Vibration	-1	-1	-1	-1	-4

Molecular parameter

Number of Atoms (N)	0
Number of internal coordinates	-6
Number of independant internal coordinates	-1
Number of vibrational modes	-4

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Force field analysis

Allowed / forbidden vibronational transitions

Operator	A'	E'*	A''	E''*	Total
Linear (IR)	-1	-1	-1	-1	-2 / -2
Quadratic (Raman)	-1	-1	-1	-1	-3 / -1
IR + Raman	- - - -	-1	- - - -	- - - -	-1 / 0

Characters of force fields
(Symmetric powers of vibration representation)

Force field	E	C3	(C3)2	σh	S3	(S3)5
linear	-6	0	0	0	0	0
quadratic	15	0	0	-3	0	0
cubic	-20	-2	-2	0	0	0
quartic	15	0	0	3	0	0
quintic	-6	0	0	0	0	0
sextic	1	1	1	-1	-1	-1

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

Force field	A'	E'*	A''	E''*
linear	-1	-1	-1	-1
quadratic	2	2	3	3
cubic	-4	-3	-4	-3
quartic	3	3	2	2
quintic	-1	-1	-1	-1
sextic	0	0	1	0

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 C_3h

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'')
..1.	E'E'.	..1.	E''E''.
Subtotal: 2 / 2 / 4
Irrep combinations (i,j) with indices: pos(A') ≤ i ≤ j ≤ pos(E'')
Subtotal: 0 / 0 / 6
Total: 2 / 2 / 10

Contributions to nonvanishing cubic force field constants

Irrep combinations (i,i,i) with indices: pos(A') ≤ i ≤ pos(E'')
Subtotal: 0 / 0 / 4
Irrep combinations (i,i,j) (i,j,j) with indices: pos(A') ≤ i ≤ j ≤ pos(E'')
Subtotal: -2 / 0 / 12
Irrep combinations (i,j,k) with indices: pos(A') ≤ i ≤ j ≤ k ≤ pos(E'')
Subtotal: -2 / 0 / 4
Total: -4 / 0 / 20

Contributions to nonvanishing quartic force field constants

Irrep combinations (i,i,i,i) with indices: pos(A') ≤ i ≤ pos(E'')
Subtotal: 0 / 0 / 4
Irrep combinations (i,i,i,j) (i,j,j,j) with indices: pos(A') ≤ i ≤ j ≤ pos(E'')
Subtotal: 0 / 0 / 12
Irrep combinations (i,i,j,j) with indices: pos(A') ≤ i ≤ j ≤ pos(E'')
..1.	E'E'E''E''.
Subtotal: 1 / 1 / 6
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 / 12
Irrep combinations (i,j,k,l) with indices: pos(A') ≤ i ≤ j ≤ k ≤ l ≤ pos(E'')
..2.	A'E'A''E''.
Subtotal: 2 / 1 / 1
Total: 3 / 2 / 35

Calculate contributions to

A'	E'	A''	E''

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