Characters of representations for molecular motions

Motion	E	2C3	3C'2	σh	2S3	3σv
Cartesian 3N	36	0	-4	12	0	4
Translation (x,y,z)	3	0	-1	1	-2	1
Rotation (Rx,Ry,Rz)	3	0	-1	-1	2	-1
Vibration	30	0	-2	12	0	4

Decomposition to irreducible representations

Motion	A'1	A'2	E'	A''1	A''2	E''	Total
Cartesian 3N	4	4	8	0	4	4	24
Translation (x,y,z)	0	0	1	0	1	0	2
Rotation (Rx,Ry,Rz)	0	1	0	0	0	1	2
Vibration	4	3	7	0	3	3	20

Molecular parameter

Number of Atoms (N)	12
Number of internal coordinates	30
Number of independant internal coordinates	4
Number of vibrational modes	20

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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)	4	3	7	0	3	3	10 / 10
Quadratic (Raman)	4	3	7	0	3	3	14 / 6
IR + Raman	- - - -	3	7	0	- - - -	- - - -	7 / 3

Characters of force fields
(Symmetric powers of vibration representation)

Force field	E	2C3	3C'2	σh	2S3	3σv
linear	30	0	-2	12	0	4
quadratic	465	0	17	87	0	23
cubic	4.960	10	-32	472	4	72
quartic	40.920	0	152	2.112	0	256
quintic	278.256	0	-272	8.184	0	680
sextic	1.623.160	55	952	28.336	13	1.904

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	4	3	7	0	3	3
quadratic	56	36	92	30	33	63
cubic	465	445	903	349	401	747
quartic	3.688	3.484	7.172	3.208	3.260	6.468
quintic	23.972	23.768	47.740	22.268	22.744	45.012
sextic	138.350	136.922	275.238	132.671	133.147	265.797

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'')
..10.	A'1A'1.	..6.	A'2A'2.	..28.	E'E'.	..6.	A''2A''2.	..6.	E''E''.
Subtotal: 56 / 5 / 6
Irrep combinations (i,j) with indices: pos(A'1) ≤ i ≤ j ≤ pos(E'')
Subtotal: 0 / 0 / 15
Total: 56 / 5 / 21

Contributions to nonvanishing cubic force field constants

Irrep combinations (i,i,i) with indices: pos(A'1) ≤ i ≤ pos(E'')
..20.	A'1A'1A'1.	..84.	E'E'E'.
Subtotal: 104 / 2 / 6
Irrep combinations (i,i,j) (i,j,j) with indices: pos(A'1) ≤ i ≤ j ≤ pos(E'')
..24.	A'1A'2A'2.	..112.	A'1E'E'.	..24.	A'1A''2A''2.	..24.	A'1E''E''.	..63.	A'2E'E'.	..9.	A'2E''E''.	..42.	E'E''E''.
Subtotal: 298 / 7 / 30
Irrep combinations (i,j,k) with indices: pos(A'1) ≤ i ≤ j ≤ k ≤ pos(E'')
..63.	E'A''2E''.
Subtotal: 63 / 1 / 20
Total: 465 / 10 / 56

Contributions to nonvanishing quartic force field constants

Irrep combinations (i,i,i,i) with indices: pos(A'1) ≤ i ≤ pos(E'')
..35.	A'1A'1A'1A'1.	..15.	A'2A'2A'2A'2.	..406.	E'E'E'E'.	..15.	A''2A''2A''2A''2.	..21.	E''E''E''E''.
Subtotal: 492 / 5 / 6
Irrep combinations (i,i,i,j) (i,j,j,j) with indices: pos(A'1) ≤ i ≤ j ≤ pos(E'')
..336.	A'1E'E'E'.	..252.	A'2E'E'E'.	..30.	A''2E''E''E''.
Subtotal: 618 / 3 / 30
Irrep combinations (i,i,j,j) with indices: pos(A'1) ≤ i ≤ j ≤ pos(E'')
..60.	A'1A'1A'2A'2.	..280.	A'1A'1E'E'.	..60.	A'1A'1A''2A''2.	..60.	A'1A'1E''E''.	..168.	A'2A'2E'E'.	..36.	A'2A'2A''2A''2.	..36.	A'2A'2E''E''.	..168.	E'E'A''2A''2.	..399.	E'E'E''E''.	..36.	A''2A''2E''E''.
Subtotal: 1.303 / 10 / 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'')
..252.	E'E'A''2E''.	..252.	A'1A'2E'E'.	..36.	A'1A'2E''E''.	..168.	A'1E'E''E''.	..126.	A'2E'E''E''.
Subtotal: 834 / 5 / 60
Irrep combinations (i,j,k,l) with indices: pos(A'1) ≤ i ≤ j ≤ k ≤ l ≤ pos(E'')
..252.	A'1E'A''2E''.	..189.	A'2E'A''2E''.
Subtotal: 441 / 2 / 15
Total: 3.688 / 25 / 126

Calculate contributions to

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

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