Characters of representations for molecular motions

Motion	E	2S8	2C4	2(S8)3	C2	4C'2	4σd
Cartesian 3N	24	0.000	0	-0.000	0	0	2
Translation (x,y,z)	3	0.414	1	-2.414	-1	-1	1
Rotation (Rx,Ry,Rz)	3	-0.414	1	2.414	-1	-1	-1
Vibration	18	0.000	-2	0.000	2	2	2

Decomposition to irreducible representations

Motion	A1	A2	B1	B2	E1	E2	E3	Total
Cartesian 3N	2	1	1	2	3	3	3	15
Translation (x,y,z)	0	0	0	1	1	0	0	2
Rotation (Rx,Ry,Rz)	0	1	0	0	0	0	1	2
Vibration	2	0	1	1	2	3	2	11

Molecular parameter

Number of Atoms (N)	8
Number of internal coordinates	18
Number of independant internal coordinates	2
Number of vibrational modes	11

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Force field analysis

Allowed / forbidden vibronational transitions

Operator	A1	A2	B1	B2	E1	E2	E3	Total
Linear (IR)	2	0	1	1	2	3	2	3 / 8
Quadratic (Raman)	2	0	1	1	2	3	2	7 / 4
IR + Raman	- - - -	0	1	- - - -	- - - -	- - - -	- - - -	0 / 1

Characters of force fields
(Symmetric powers of vibration representation)

Force field	E	2S8	2C4	2(S8)3	C2	4C'2	4σd
linear	18	0.000	-2	0.000	2	2	2
quadratic	171	-1.000	3	-1.000	11	11	11
cubic	1.140	-0.000	-4	-0.000	20	20	20
quartic	5.985	1.000	9	1.000	65	65	65
quintic	26.334	0.000	-14	0.000	110	110	110
sextic	100.947	-1.000	19	-1.000	275	275	275

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

Force field	A1	A2	B1	B2	E1	E2	E3
linear	2	0	1	1	2	3	2
quadratic	17	6	12	12	20	22	20
cubic	82	62	72	72	140	146	140
quartic	412	347	379	379	740	754	740
quintic	1.706	1.596	1.651	1.651	3.278	3.309	3.278
sextic	6.466	6.191	6.329	6.329	12.584	12.648	12.584

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_4d

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(A1) ≤ i ≤ pos(E3)
..3.	A1A1.	..1.	B1B1.	..1.	B2B2.	..3.	E1E1.	..6.	E2E2.	..3.	E3E3.
Subtotal: 17 / 6 / 7
Irrep combinations (i,j) with indices: pos(A1) ≤ i ≤ j ≤ pos(E3)
Subtotal: 0 / 0 / 21
Total: 17 / 6 / 28

Contributions to nonvanishing cubic force field constants

Irrep combinations (i,i,i) with indices: pos(A1) ≤ i ≤ pos(E3)
..4.	A1A1A1.
Subtotal: 4 / 1 / 7
Irrep combinations (i,i,j) (i,j,j) with indices: pos(A1) ≤ i ≤ j ≤ pos(E3)
..9.	E1E1E2.	..2.	A1B1B1.	..2.	A1B2B2.	..6.	A1E1E1.	..12.	A1E2E2.	..6.	A1E3E3.	..6.	B1E2E2.	..6.	B2E2E2.	..9.	E2E3E3.
Subtotal: 58 / 9 / 42
Irrep combinations (i,j,k) with indices: pos(A1) ≤ i ≤ j ≤ k ≤ pos(E3)
..4.	B1E1E3.	..4.	B2E1E3.	..12.	E1E2E3.
Subtotal: 20 / 3 / 35
Total: 82 / 13 / 84

Contributions to nonvanishing quartic force field constants

Irrep combinations (i,i,i,i) with indices: pos(A1) ≤ i ≤ pos(E3)
..5.	A1A1A1A1.	..1.	B1B1B1B1.	..1.	B2B2B2B2.	..6.	E1E1E1E1.	..36.	E2E2E2E2.	..6.	E3E3E3E3.
Subtotal: 55 / 6 / 7
Irrep combinations (i,i,i,j) (i,j,j,j) with indices: pos(A1) ≤ i ≤ j ≤ pos(E3)
..8.	E1E1E1E3.	..8.	E1E3E3E3.
Subtotal: 16 / 2 / 42
Irrep combinations (i,i,j,j) with indices: pos(A1) ≤ i ≤ j ≤ pos(E3)
..3.	A1A1B1B1.	..3.	A1A1B2B2.	..9.	A1A1E1E1.	..18.	A1A1E2E2.	..9.	A1A1E3E3.	..1.	B1B1B2B2.	..3.	B1B1E1E1.	..6.	B1B1E2E2.	..3.	B1B1E3E3.	..3.	B2B2E1E1.
..6.	B2B2E2E2.	..3.	B2B2E3E3.	..21.	E1E1E2E2.	..19.	E1E1E3E3.	..21.	E2E2E3E3.
Subtotal: 128 / 15 / 21
Irrep combinations (i,i,j,k) (i,j,j,k) (i,j,k,k) with indices: pos(A1) ≤ i ≤ j ≤ k ≤ pos(E3)
..18.	A1E1E1E2.	..9.	B1E1E1E2.	..9.	B2E1E1E2.	..48.	E1E2E2E3.	..12.	A1B1E2E2.	..12.	A1B2E2E2.	..18.	A1E2E3E3.	..1.	B1B2E1E1.	..3.	B1B2E2E2.	..1.	B1B2E3E3.
..9.	B1E2E3E3.	..9.	B2E2E3E3.
Subtotal: 149 / 12 / 105
Irrep combinations (i,j,k,l) with indices: pos(A1) ≤ i ≤ j ≤ k ≤ l ≤ pos(E3)
..8.	A1B1E1E3.	..8.	A1B2E1E3.	..24.	A1E1E2E3.	..12.	B1E1E2E3.	..12.	B2E1E2E3.
Subtotal: 64 / 5 / 35
Total: 412 / 40 / 210

Calculate contributions to

A1	A2	B1	B2	E1	E2	E3

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