Characters of representations for molecular motions

Motion	E	2C5	2(C5)2	5C'2	σh	2S5	2(S5)3	5σv
Cartesian 3N	33	1.618	-0.618	-1	1	-0.382	-2.618	3
Translation (x,y,z)	3	1.618	-0.618	-1	1	-0.382	-2.618	1
Rotation (Rx,Ry,Rz)	3	1.618	-0.618	-1	-1	0.382	2.618	-1
Vibration	27	-1.618	0.618	1	1	-0.382	-2.618	3

Decomposition to irreducible representations

Motion	A'1	A'2	E'1	E'2	A''1	A''2	E''1	E''2	Total
Cartesian 3N	2	1	4	3	1	3	3	3	20
Translation (x,y,z)	0	0	1	0	0	1	0	0	2
Rotation (Rx,Ry,Rz)	0	1	0	0	0	0	1	0	2
Vibration	2	0	3	3	1	2	2	3	16

Molecular parameter

Number of Atoms (N)	11
Number of internal coordinates	27
Number of independant internal coordinates	2
Number of vibrational modes	16

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Force field analysis

Allowed / forbidden vibronational transitions

Operator	A'1	A'2	E'1	E'2	A''1	A''2	E''1	E''2	Total
Linear (IR)	2	0	3	3	1	2	2	3	5 / 11
Quadratic (Raman)	2	0	3	3	1	2	2	3	7 / 9
IR + Raman	- - - -	0	- - - -	- - - -	1	- - - -	- - - -	3	0 / 4

Characters of force fields
(Symmetric powers of vibration representation)

Force field	E	2C5	2(C5)2	5C'2	σh	2S5	2(S5)3	5σv
linear	27	-1.618	0.618	1	1	-0.382	-2.618	3
quadratic	378	1.618	-0.618	14	14	0.382	2.618	18
cubic	3.654	-1.000	-1.000	14	14	-1.000	-1.000	46
quartic	27.405	-0.000	0.000	105	105	0.000	0.000	165
quintic	169.911	6.000	6.000	105	105	0.000	0.000	375
sextic	906.192	-9.708	3.708	560	560	0.000	-0.000	1.040

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

Force field	A'1	A'2	E'1	E'2	A''1	A''2	E''1	E''2
linear	2	0	3	3	1	2	2	3
quadratic	28	12	39	39	17	19	37	36
cubic	198	168	367	367	174	190	364	364
quartic	1.443	1.308	2.751	2.751	1.350	1.380	2.730	2.730
quintic	8.622	8.382	17.001	17.001	8.424	8.559	16.980	16.980
sextic	45.737	44.937	90.674	90.677	45.161	45.401	90.562	90.565

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_5h

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''2)
..3.	A'1A'1.	..6.	E'1E'1.	..6.	E'2E'2.	..1.	A''1A''1.	..3.	A''2A''2.	..3.	E''1E''1.	..6.	E''2E''2.
Subtotal: 28 / 7 / 8
Irrep combinations (i,j) with indices: pos(A'1) ≤ i ≤ j ≤ pos(E''2)
Subtotal: 0 / 0 / 28
Total: 28 / 7 / 36

Contributions to nonvanishing cubic force field constants

Irrep combinations (i,i,i) with indices: pos(A'1) ≤ i ≤ pos(E''2)
..4.	A'1A'1A'1.
Subtotal: 4 / 1 / 8
Irrep combinations (i,i,j) (i,j,j) with indices: pos(A'1) ≤ i ≤ j ≤ pos(E''2)
..18.	E'1E'1E'2.	..12.	A'1E'1E'1.	..12.	A'1E'2E'2.	..2.	A'1A''1A''1.	..6.	A'1A''2A''2.	..6.	A'1E''1E''1.	..12.	A'1E''2E''2.	..18.	E'1E'2E'2.	..18.	E'1E''2E''2.	..9.	E'2E''1E''1.
Subtotal: 113 / 10 / 56
Irrep combinations (i,j,k) with indices: pos(A'1) ≤ i ≤ j ≤ k ≤ pos(E''2)
..6.	E'1A''1E''1.	..12.	E'1A''2E''1.	..18.	E'1E''1E''2.	..9.	E'2A''1E''2.	..18.	E'2A''2E''2.	..18.	E'2E''1E''2.
Subtotal: 81 / 6 / 56
Total: 198 / 17 / 120

Contributions to nonvanishing quartic force field constants

Irrep combinations (i,i,i,i) with indices: pos(A'1) ≤ i ≤ pos(E''2)
..5.	A'1A'1A'1A'1.	..21.	E'1E'1E'1E'1.	..21.	E'2E'2E'2E'2.	..1.	A''1A''1A''1A''1.	..5.	A''2A''2A''2A''2.	..6.	E''1E''1E''1E''1.	..21.	E''2E''2E''2E''2.
Subtotal: 80 / 7 / 8
Irrep combinations (i,i,i,j) (i,j,j,j) with indices: pos(A'1) ≤ i ≤ j ≤ pos(E''2)
..30.	E'1E'1E'1E'2.	..12.	E''1E''1E''1E''2.	..30.	E'1E'2E'2E'2.	..20.	E''1E''2E''2E''2.
Subtotal: 92 / 4 / 56
Irrep combinations (i,i,j,j) with indices: pos(A'1) ≤ i ≤ j ≤ pos(E''2)
..18.	A'1A'1E'1E'1.	..18.	A'1A'1E'2E'2.	..3.	A'1A'1A''1A''1.	..9.	A'1A'1A''2A''2.	..9.	A'1A'1E''1E''1.	..18.	A'1A'1E''2E''2.	..45.	E'1E'1E'2E'2.	..6.	E'1E'1A''1A''1.	..18.	E'1E'1A''2A''2.	..39.	E'1E'1E''1E''1.
..45.	E'1E'1E''2E''2.	..6.	E'2E'2A''1A''1.	..18.	E'2E'2A''2A''2.	..21.	E'2E'2E''1E''1.	..81.	E'2E'2E''2E''2.	..3.	A''1A''1A''2A''2.	..3.	A''1A''1E''1E''1.	..6.	A''1A''1E''2E''2.	..9.	A''2A''2E''1E''1.	..18.	A''2A''2E''2E''2.
..21.	E''1E''1E''2E''2.
Subtotal: 414 / 21 / 28
Irrep combinations (i,i,j,k) (i,j,j,k) (i,j,k,k) with indices: pos(A'1) ≤ i ≤ j ≤ k ≤ pos(E''2)
..6.	E'1E'1A''1A''2.	..18.	E'1E'1A''1E''2.	..36.	E'1E'1A''2E''2.	..36.	E'1E'1E''1E''2.	..6.	E'2E'2A''1A''2.	..12.	E'2E'2A''1E''1.	..24.	E'2E'2A''2E''1.	..36.	E'2E'2E''1E''2.	..36.	A'1E'1E'1E'2.	..9.	A''1E''1E''1E''2.
..18.	A''2E''1E''1E''2.	..36.	A'1E'1E'2E'2.	..36.	A'1E'1E''2E''2.	..18.	A'1E'2E''1E''1.	..27.	E'1E'2E''1E''1.	..54.	E'1E'2E''2E''2.	..2.	A''1A''2E''1E''1.	..6.	A''1A''2E''2E''2.	..12.	A''1E''1E''2E''2.	..24.	A''2E''1E''2E''2.
Subtotal: 452 / 20 / 168
Irrep combinations (i,j,k,l) with indices: pos(A'1) ≤ i ≤ j ≤ k ≤ l ≤ pos(E''2)
..12.	A'1E'1A''1E''1.	..24.	A'1E'1A''2E''1.	..36.	A'1E'1E''1E''2.	..18.	A'1E'2A''1E''2.	..36.	A'1E'2A''2E''2.	..36.	A'1E'2E''1E''2.	..18.	E'1E'2A''1E''1.	..27.	E'1E'2A''1E''2.	..36.	E'1E'2A''2E''1.	..54.	E'1E'2A''2E''2.
..108.	E'1E'2E''1E''2.
Subtotal: 405 / 11 / 70
Total: 1.443 / 63 / 330

Calculate contributions to

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

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