Characters of representations for molecular motions

Motion	E	2C3	3C'2	σh	2S3	3σv
Cartesian 3N	27	0	-1	3	0	3
Translation (x,y,z)	3	0	-1	1	-2	1
Rotation (Rx,Ry,Rz)	3	0	-1	-1	2	-1
Vibration	21	0	1	3	0	3

Decomposition to irreducible representations

Motion	A'1	A'2	E'	A''1	A''2	E''	Total
Cartesian 3N	3	2	5	1	3	4	18
Translation (x,y,z)	0	0	1	0	1	0	2
Rotation (Rx,Ry,Rz)	0	1	0	0	0	1	2
Vibration	3	1	4	1	2	3	14

Molecular parameter

Number of Atoms (N)	9
Number of internal coordinates	21
Number of independant internal coordinates	3
Number of vibrational modes	14

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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)	3	1	4	1	2	3	6 / 8
Quadratic (Raman)	3	1	4	1	2	3	10 / 4
IR + Raman	- - - -	1	4	1	- - - -	- - - -	4 / 2

Characters of force fields
(Symmetric powers of vibration representation)

Force field	E	2C3	3C'2	σh	2S3	3σv
linear	21	0	1	3	0	3
quadratic	231	0	11	15	0	15
cubic	1.771	7	11	37	1	37
quartic	10.626	0	66	114	0	114
quintic	53.130	0	66	246	0	246
sextic	230.230	28	286	598	4	598

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	3	1	4	1	2	3
quadratic	27	14	41	17	19	36
cubic	164	140	300	139	152	288
quartic	940	850	1.790	864	888	1.752
quintic	4.526	4.370	8.896	4.362	4.452	8.814
sextic	19.462	19.020	38.466	19.062	19.218	38.268

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'')
..6.	A'1A'1.	..1.	A'2A'2.	..10.	E'E'.	..1.	A''1A''1.	..3.	A''2A''2.	..6.	E''E''.
Subtotal: 27 / 6 / 6
Irrep combinations (i,j) with indices: pos(A'1) ≤ i ≤ j ≤ pos(E'')
Subtotal: 0 / 0 / 15
Total: 27 / 6 / 21

Contributions to nonvanishing cubic force field constants

Irrep combinations (i,i,i) with indices: pos(A'1) ≤ i ≤ pos(E'')
..10.	A'1A'1A'1.	..20.	E'E'E'.
Subtotal: 30 / 2 / 6
Irrep combinations (i,i,j) (i,j,j) with indices: pos(A'1) ≤ i ≤ j ≤ pos(E'')
..3.	A'1A'2A'2.	..30.	A'1E'E'.	..3.	A'1A''1A''1.	..9.	A'1A''2A''2.	..18.	A'1E''E''.	..6.	A'2E'E'.	..3.	A'2E''E''.	..24.	E'E''E''.
Subtotal: 96 / 8 / 30
Irrep combinations (i,j,k) with indices: pos(A'1) ≤ i ≤ j ≤ k ≤ pos(E'')
..2.	A'2A''1A''2.	..12.	E'A''1E''.	..24.	E'A''2E''.
Subtotal: 38 / 3 / 20
Total: 164 / 13 / 56

Contributions to nonvanishing quartic force field constants

Irrep combinations (i,i,i,i) with indices: pos(A'1) ≤ i ≤ pos(E'')
..15.	A'1A'1A'1A'1.	..1.	A'2A'2A'2A'2.	..55.	E'E'E'E'.	..1.	A''1A''1A''1A''1.	..5.	A''2A''2A''2A''2.	..21.	E''E''E''E''.
Subtotal: 98 / 6 / 6
Irrep combinations (i,i,i,j) (i,j,j,j) with indices: pos(A'1) ≤ i ≤ j ≤ pos(E'')
..60.	A'1E'E'E'.	..20.	A'2E'E'E'.	..10.	A''1E''E''E''.	..20.	A''2E''E''E''.
Subtotal: 110 / 4 / 30
Irrep combinations (i,i,j,j) with indices: pos(A'1) ≤ i ≤ j ≤ pos(E'')
..6.	A'1A'1A'2A'2.	..60.	A'1A'1E'E'.	..6.	A'1A'1A''1A''1.	..18.	A'1A'1A''2A''2.	..36.	A'1A'1E''E''.	..10.	A'2A'2E'E'.	..1.	A'2A'2A''1A''1.	..3.	A'2A'2A''2A''2.	..6.	A'2A'2E''E''.	..10.	E'E'A''1A''1.
..30.	E'E'A''2A''2.	..138.	E'E'E''E''.	..3.	A''1A''1A''2A''2.	..6.	A''1A''1E''E''.	..18.	A''2A''2E''E''.
Subtotal: 351 / 15 / 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'')
..12.	E'E'A''1A''2.	..30.	E'E'A''1E''.	..60.	E'E'A''2E''.	..18.	A'1A'2E'E'.	..9.	A'1A'2E''E''.	..72.	A'1E'E''E''.	..24.	A'2E'E''E''.	..6.	A''1A''2E''E''.
Subtotal: 231 / 8 / 60
Irrep combinations (i,j,k,l) with indices: pos(A'1) ≤ i ≤ j ≤ k ≤ l ≤ pos(E'')
..6.	A'1A'2A''1A''2.	..36.	A'1E'A''1E''.	..72.	A'1E'A''2E''.	..12.	A'2E'A''1E''.	..24.	A'2E'A''2E''.
Subtotal: 150 / 5 / 15
Total: 940 / 38 / 126

Calculate contributions to

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

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