Characters of representations for molecular motions

Motion	E	2C7	2(C7)2	2(C7)3	7C'2	σh	2S7	2(S7)5	2(S7)3	7σv
Cartesian 3N	42	0.000	0.000	-0.000	-2	14	0.000	-0.000	-0.000	2
Translation (x,y,z)	3	2.247	0.555	-0.802	-1	1	0.247	-1.445	-2.802	1
Rotation (Rx,Ry,Rz)	3	2.247	0.555	-0.802	-1	-1	-0.247	1.445	2.802	-1
Vibration	36	-4.494	-1.110	1.604	0	14	0.000	-0.000	0.000	2

Decomposition to irreducible representations

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

Molecular parameter

Number of Atoms (N)	14
Number of internal coordinates	36
Number of independant internal coordinates	2
Number of vibrational modes	20

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Force field analysis

Allowed / forbidden vibronational transitions

Operator	A'1	A'2	E'1	E'2	E'3	A''1	A''2	E''1	E''2	E''3	Total
Linear (IR)	2	1	3	4	4	0	1	1	2	2	4 / 16
Quadratic (Raman)	2	1	3	4	4	0	1	1	2	2	7 / 13
IR + Raman	- - - -	1	- - - -	- - - -	4	0	- - - -	- - - -	2	2	0 / 9

Characters of force fields
(Symmetric powers of vibration representation)

Force field	E	2C7	2(C7)2	2(C7)3	7C'2	σh	2S7	2(S7)5	2(S7)3	7σv
linear	36	-4.494	-1.110	1.604	0	14	0.000	-0.000	0.000	2
quadratic	666	9.543	1.418	-0.961	18	116	-0.555	0.802	-2.247	20
cubic	8.436	-12.098	-2.616	-3.286	0	714	-0.000	0.000	-0.000	38
quartic	82.251	9.543	1.418	-0.961	171	3.601	0.555	-0.802	2.247	209
quintic	658.008	-4.494	-1.110	1.604	0	15.652	0.000	0.000	0.000	380
sextic	4.496.388	1.000	1.000	1.000	1.140	60.528	-1.000	-1.000	-1.000	1.520

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

Force field	A'1	A'2	E'1	E'2	E'3	A''1	A''2	E''1	E''2	E''3
linear	2	1	3	4	4	0	1	1	2	2
quadratic	38	19	57	55	55	20	21	40	39	38
cubic	335	316	653	654	655	265	284	551	552	553
quartic	3.162	2.972	6.133	6.132	6.131	2.800	2.819	5.619	5.617	5.617
quintic	24.154	23.964	48.118	48.119	48.119	22.846	23.036	45.882	45.883	45.883
sextic	163.412	162.082	325.494	325.494	325.494	158.329	158.519	316.847	316.847	316.847

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_7h

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''3)
..3.	A'1A'1.	..1.	A'2A'2.	..6.	E'1E'1.	..10.	E'2E'2.	..10.	E'3E'3.	..1.	A''2A''2.	..1.	E''1E''1.	..3.	E''2E''2.	..3.	E''3E''3.
Subtotal: 38 / 9 / 10
Irrep combinations (i,j) with indices: pos(A'1) ≤ i ≤ j ≤ pos(E''3)
Subtotal: 0 / 0 / 45
Total: 38 / 9 / 55

Contributions to nonvanishing cubic force field constants

Irrep combinations (i,i,i) with indices: pos(A'1) ≤ i ≤ pos(E''3)
..4.	A'1A'1A'1.
Subtotal: 4 / 1 / 10
Irrep combinations (i,i,j) (i,j,j) with indices: pos(A'1) ≤ i ≤ j ≤ pos(E''3)
..24.	E'1E'1E'2.	..40.	E'2E'2E'3.	..2.	A'1A'2A'2.	..12.	A'1E'1E'1.	..20.	A'1E'2E'2.	..20.	A'1E'3E'3.	..2.	A'1A''2A''2.	..2.	A'1E''1E''1.	..6.	A'1E''2E''2.	..6.	A'1E''3E''3.
..3.	A'2E'1E'1.	..6.	A'2E'2E'2.	..6.	A'2E'3E'3.	..1.	A'2E''2E''2.	..1.	A'2E''3E''3.	..30.	E'1E'3E'3.	..9.	E'1E''3E''3.	..4.	E'2E''1E''1.	..12.	E'3E''2E''2.
Subtotal: 206 / 19 / 90
Irrep combinations (i,j,k) with indices: pos(A'1) ≤ i ≤ j ≤ k ≤ pos(E''3)
..48.	E'1E'2E'3.	..3.	E'1A''2E''1.	..6.	E'1E''1E''2.	..12.	E'1E''2E''3.	..8.	E'2A''2E''2.	..8.	E'2E''1E''3.	..16.	E'2E''2E''3.	..8.	E'3A''2E''3.	..8.	E'3E''1E''2.	..8.	E'3E''1E''3.
Subtotal: 125 / 10 / 120
Total: 335 / 30 / 220

Contributions to nonvanishing quartic force field constants

Irrep combinations (i,i,i,i) with indices: pos(A'1) ≤ i ≤ pos(E''3)
..5.	A'1A'1A'1A'1.	..1.	A'2A'2A'2A'2.	..21.	E'1E'1E'1E'1.	..55.	E'2E'2E'2E'2.	..55.	E'3E'3E'3E'3.	..1.	A''2A''2A''2A''2.	..1.	E''1E''1E''1E''1.	..6.	E''2E''2E''2E''2.	..6.	E''3E''3E''3E''3.
Subtotal: 151 / 9 / 10
Irrep combinations (i,i,i,j) (i,j,j,j) with indices: pos(A'1) ≤ i ≤ j ≤ pos(E''3)
..40.	E'1E'1E'1E'3.	..2.	E''1E''1E''1E''3.	..60.	E'1E'2E'2E'2.	..80.	E'2E'3E'3E'3.	..4.	E''1E''2E''2E''2.	..8.	E''2E''3E''3E''3.
Subtotal: 194 / 6 / 90
Irrep combinations (i,i,j,j) with indices: pos(A'1) ≤ i ≤ j ≤ pos(E''3)
..3.	A'1A'1A'2A'2.	..18.	A'1A'1E'1E'1.	..30.	A'1A'1E'2E'2.	..30.	A'1A'1E'3E'3.	..3.	A'1A'1A''2A''2.	..3.	A'1A'1E''1E''1.	..9.	A'1A'1E''2E''2.	..9.	A'1A'1E''3E''3.	..6.	A'2A'2E'1E'1.	..10.	A'2A'2E'2E'2.
..10.	A'2A'2E'3E'3.	..1.	A'2A'2A''2A''2.	..1.	A'2A'2E''1E''1.	..3.	A'2A'2E''2E''2.	..3.	A'2A'2E''3E''3.	..78.	E'1E'1E'2E'2.	..78.	E'1E'1E'3E'3.	..6.	E'1E'1A''2A''2.	..12.	E'1E'1E''1E''1.	..21.	E'1E'1E''2E''2.
..21.	E'1E'1E''3E''3.	..136.	E'2E'2E'3E'3.	..10.	E'2E'2A''2A''2.	..10.	E'2E'2E''1E''1.	..66.	E'2E'2E''2E''2.	..36.	E'2E'2E''3E''3.	..10.	E'3E'3A''2A''2.	..10.	E'3E'3E''1E''1.	..36.	E'3E'3E''2E''2.	..66.	E'3E'3E''3E''3.
..1.	A''2A''2E''1E''1.	..3.	A''2A''2E''2E''2.	..3.	A''2A''2E''3E''3.	..3.	E''1E''1E''2E''2.	..3.	E''1E''1E''3E''3.	..10.	E''2E''2E''3E''3.
Subtotal: 758 / 36 / 45
Irrep combinations (i,i,j,k) (i,j,j,k) (i,j,k,k) with indices: pos(A'1) ≤ i ≤ j ≤ k ≤ pos(E''3)
..96.	E'1E'1E'2E'3.	..12.	E'1E'1A''2E''2.	..12.	E'1E'1E''1E''3.	..24.	E'1E'1E''2E''3.	..20.	E'2E'2A''2E''3.	..20.	E'2E'2E''1E''2.	..20.	E'2E'2E''1E''3.	..10.	E'3E'3A''2E''1.	..20.	E'3E'3E''1E''2.	..40.	E'3E'3E''2E''3.
..4.	E''1E''1E''2E''3.	..48.	A'1E'1E'1E'2.	..80.	A'1E'2E'2E'3.	..24.	A'2E'1E'1E'2.	..40.	A'2E'2E'2E'3.	..120.	E'1E'2E'2E'3.	..2.	A''2E''1E''1E''2.	..6.	A''2E''2E''2E''3.	..6.	E''1E''2E''2E''3.	..6.	A'1A'2E'1E'1.
..12.	A'1A'2E'2E'2.	..12.	A'1A'2E'3E'3.	..2.	A'1A'2E''2E''2.	..2.	A'1A'2E''3E''3.	..60.	A'1E'1E'3E'3.	..18.	A'1E'1E''3E''3.	..8.	A'1E'2E''1E''1.	..24.	A'1E'3E''2E''2.	..30.	A'2E'1E'3E'3.	..9.	A'2E'1E''3E''3.
..4.	A'2E'2E''1E''1.	..12.	A'2E'3E''2E''2.	..120.	E'1E'2E'3E'3.	..36.	E'1E'2E''2E''2.	..36.	E'1E'2E''3E''3.	..12.	E'1E'3E''1E''1.	..36.	E'1E'3E''2E''2.	..16.	E'2E'3E''1E''1.	..48.	E'2E'3E''3E''3.	..3.	A''2E''1E''3E''3.
..6.	E''1E''2E''3E''3.
Subtotal: 1.116 / 41 / 360
Irrep combinations (i,j,k,l) with indices: pos(A'1) ≤ i ≤ j ≤ k ≤ l ≤ pos(E''3)
..96.	A'1E'1E'2E'3.	..6.	A'1E'1A''2E''1.	..12.	A'1E'1E''1E''2.	..24.	A'1E'1E''2E''3.	..16.	A'1E'2A''2E''2.	..16.	A'1E'2E''1E''3.	..32.	A'1E'2E''2E''3.	..16.	A'1E'3A''2E''3.	..16.	A'1E'3E''1E''2.	..16.	A'1E'3E''1E''3.
..48.	A'2E'1E'2E'3.	..3.	A'2E'1A''2E''1.	..6.	A'2E'1E''1E''2.	..12.	A'2E'1E''2E''3.	..8.	A'2E'2A''2E''2.	..8.	A'2E'2E''1E''3.	..16.	A'2E'2E''2E''3.	..8.	A'2E'3A''2E''3.	..8.	A'2E'3E''1E''2.	..8.	A'2E'3E''1E''3.
..12.	E'1E'2A''2E''1.	..24.	E'1E'2A''2E''3.	..48.	E'1E'2E''1E''2.	..24.	E'1E'2E''1E''3.	..48.	E'1E'2E''2E''3.	..24.	E'1E'3A''2E''2.	..24.	E'1E'3A''2E''3.	..24.	E'1E'3E''1E''2.	..48.	E'1E'3E''1E''3.	..48.	E'1E'3E''2E''3.
..16.	E'2E'3A''2E''1.	..32.	E'2E'3A''2E''2.	..32.	E'2E'3E''1E''2.	..32.	E'2E'3E''1E''3.	..128.	E'2E'3E''2E''3.	..4.	A''2E''1E''2E''3.
Subtotal: 943 / 36 / 210
Total: 3.162 / 128 / 715

Calculate contributions to

A'1	A'2	E'1	E'2	E'3	A''1	A''2	E''1	E''2	E''3

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