Characters of representations for molecular motions

Motion	E	2C3	3C'2	σh	2S3	3σv
Cartesian 3N	12	0	-2	4	-2	2
Translation (x,y,z)	3	0	-1	1	-2	1
Rotation (Rx,Ry,Rz)	3	0	-1	-1	2	-1
Vibration	6	0	0	4	-2	2

Decomposition to irreducible representations

Motion	A'1	A'2	E'	A''1	A''2	E''	Total
Cartesian 3N	1	1	3	0	2	1	8
Translation (x,y,z)	0	0	1	0	1	0	2
Rotation (Rx,Ry,Rz)	0	1	0	0	0	1	2
Vibration	1	0	2	0	1	0	4

Molecular parameter

Number of Atoms (N)	4
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'1	A'2	E'	A''1	A''2	E''	Total
Linear (IR)	1	0	2	0	1	0	3 / 1
Quadratic (Raman)	1	0	2	0	1	0	3 / 1
IR + Raman	- - - -	0	2	0	- - - -	- - - -	2 / 0

Characters of force fields
(Symmetric powers of vibration representation)

Force field	E	2C3	3C'2	σh	2S3	3σv
linear	6	0	0	4	-2	2
quadratic	21	0	3	11	2	5
cubic	56	2	0	24	0	8
quartic	126	0	6	46	-2	14
quintic	252	0	0	80	2	20
sextic	462	3	10	130	1	30

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

Force field	A'1	A'2	E'	A''1	A''2	E''
linear	1	0	2	0	1	0
quadratic	5	1	5	0	1	2
cubic	9	5	13	1	5	5
quartic	19	9	29	5	9	13
quintic	33	23	55	9	19	29
sextic	60	40	98	23	33	55

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 D_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'1) ≤ i ≤ pos(E'')
..1.	A'1A'1.	..3.	E'E'.	..1.	A''2A''2.
Subtotal: 5 / 3 / 6
Irrep combinations (i,j) with indices: pos(A'1) ≤ i ≤ j ≤ pos(E'')
Subtotal: 0 / 0 / 15
Total: 5 / 3 / 21

Contributions to nonvanishing cubic force field constants

Irrep combinations (i,i,i) with indices: pos(A'1) ≤ i ≤ pos(E'')
..1.	A'1A'1A'1.	..4.	E'E'E'.
Subtotal: 5 / 2 / 6
Irrep combinations (i,i,j) (i,j,j) with indices: pos(A'1) ≤ i ≤ j ≤ pos(E'')
..3.	A'1E'E'.	..1.	A'1A''2A''2.
Subtotal: 4 / 2 / 30
Irrep combinations (i,j,k) with indices: pos(A'1) ≤ i ≤ j ≤ k ≤ pos(E'')
Subtotal: 0 / 0 / 20
Total: 9 / 4 / 56

Contributions to nonvanishing quartic force field constants

Irrep combinations (i,i,i,i) with indices: pos(A'1) ≤ i ≤ pos(E'')
..1.	A'1A'1A'1A'1.	..6.	E'E'E'E'.	..1.	A''2A''2A''2A''2.
Subtotal: 8 / 3 / 6
Irrep combinations (i,i,i,j) (i,j,j,j) with indices: pos(A'1) ≤ i ≤ j ≤ pos(E'')
..4.	A'1E'E'E'.
Subtotal: 4 / 1 / 30
Irrep combinations (i,i,j,j) with indices: pos(A'1) ≤ i ≤ j ≤ pos(E'')
..3.	A'1A'1E'E'.	..1.	A'1A'1A''2A''2.	..3.	E'E'A''2A''2.
Subtotal: 7 / 3 / 15
Irrep combinations (i,i,j,k) (i,j,j,k) (i,j,k,k) with indices: pos(A'1) ≤ i ≤ j ≤ k ≤ pos(E'')
Subtotal: 0 / 0 / 60
Irrep combinations (i,j,k,l) with indices: pos(A'1) ≤ i ≤ j ≤ k ≤ l ≤ pos(E'')
Subtotal: 0 / 0 / 15
Total: 19 / 7 / 126

Calculate contributions to

A'1	A'2	E'	A''1	A''2	E''

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