Characters of representations for molecular motions

Motion	E	2C3	3C'2	σh	2S3	3σv
Cartesian 3N	33	0	-3	7	-2	7
Translation (x,y,z)	3	0	-1	1	-2	1
Rotation (Rx,Ry,Rz)	3	0	-1	-1	2	-1
Vibration	27	0	-1	7	-2	7

Decomposition to irreducible representations

Motion	A'1	A'2	E'	A''1	A''2	E''	Total
Cartesian 3N	4	2	7	0	5	4	22
Translation (x,y,z)	0	0	1	0	1	0	2
Rotation (Rx,Ry,Rz)	0	1	0	0	0	1	2
Vibration	4	1	6	0	4	3	18

Molecular parameter

Number of Atoms (N)	11
Number of internal coordinates	27
Number of independant internal coordinates	4
Number of vibrational modes	18

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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	1	6	0	4	3	10 / 8
Quadratic (Raman)	4	1	6	0	4	3	13 / 5
IR + Raman	- - - -	1	6	0	- - - -	- - - -	6 / 1

Characters of force fields
(Symmetric powers of vibration representation)

Force field	E	2C3	3C'2	σh	2S3	3σv
linear	27	0	-1	7	-2	7
quadratic	378	0	14	38	2	38
cubic	3.654	9	-14	154	1	154
quartic	27.405	0	105	545	-4	545
quintic	169.911	0	-105	1.687	4	1.687
sextic	906.192	45	560	4.784	5	4.784

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	1	6	0	4	3
quadratic	48	22	69	22	34	57
cubic	354	284	633	251	335	582
quartic	2.491	2.166	4.659	2.129	2.349	4.476
quintic	14.696	13.905	28.599	13.570	14.466	28.038
sextic	77.259	74.587	151.821	74.068	76.180	150.228

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.	..1.	A'2A'2.	..21.	E'E'.	..10.	A''2A''2.	..6.	E''E''.
Subtotal: 48 / 5 / 6
Irrep combinations (i,j) with indices: pos(A'1) ≤ i ≤ j ≤ pos(E'')
Subtotal: 0 / 0 / 15
Total: 48 / 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.	..56.	E'E'E'.
Subtotal: 76 / 2 / 6
Irrep combinations (i,i,j) (i,j,j) with indices: pos(A'1) ≤ i ≤ j ≤ pos(E'')
..4.	A'1A'2A'2.	..84.	A'1E'E'.	..40.	A'1A''2A''2.	..24.	A'1E''E''.	..15.	A'2E'E'.	..3.	A'2E''E''.	..36.	E'E''E''.
Subtotal: 206 / 7 / 30
Irrep combinations (i,j,k) with indices: pos(A'1) ≤ i ≤ j ≤ k ≤ pos(E'')
..72.	E'A''2E''.
Subtotal: 72 / 1 / 20
Total: 354 / 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.	..1.	A'2A'2A'2A'2.	..231.	E'E'E'E'.	..35.	A''2A''2A''2A''2.	..21.	E''E''E''E''.
Subtotal: 323 / 5 / 6
Irrep combinations (i,i,i,j) (i,j,j,j) with indices: pos(A'1) ≤ i ≤ j ≤ pos(E'')
..224.	A'1E'E'E'.	..56.	A'2E'E'E'.	..40.	A''2E''E''E''.
Subtotal: 320 / 3 / 30
Irrep combinations (i,i,j,j) with indices: pos(A'1) ≤ i ≤ j ≤ pos(E'')
..10.	A'1A'1A'2A'2.	..210.	A'1A'1E'E'.	..100.	A'1A'1A''2A''2.	..60.	A'1A'1E''E''.	..21.	A'2A'2E'E'.	..10.	A'2A'2A''2A''2.	..6.	A'2A'2E''E''.	..210.	E'E'A''2A''2.	..297.	E'E'E''E''.	..60.	A''2A''2E''E''.
Subtotal: 984 / 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''.	..60.	A'1A'2E'E'.	..12.	A'1A'2E''E''.	..144.	A'1E'E''E''.	..36.	A'2E'E''E''.
Subtotal: 504 / 5 / 60
Irrep combinations (i,j,k,l) with indices: pos(A'1) ≤ i ≤ j ≤ k ≤ l ≤ pos(E'')
..288.	A'1E'A''2E''.	..72.	A'2E'A''2E''.
Subtotal: 360 / 2 / 15
Total: 2.491 / 25 / 126

Calculate contributions to

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

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