Characters of representations for molecular motions

Motion	E	C2.(z)	σv.(xz)	σd.(yz)
Cartesian 3N	24	0	8	0
Translation (x,y,z)	3	-1	1	1
Rotation (Rx,Ry,Rz)	3	-1	-1	-1
Vibration	18	2	8	0

Decomposition to irreducible representations

Motion	A1	A2	B1	B2	Total
Cartesian 3N	8	4	8	4	24
Translation (x,y,z)	1	0	1	1	3
Rotation (Rx,Ry,Rz)	0	1	1	1	3
Vibration	7	3	6	2	18

Molecular parameter

Number of Atoms (N)	8
Number of internal coordinates	18
Number of independant internal coordinates	7
Number of vibrational modes	18

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Force field analysis

Allowed / forbidden vibronational transitions

Operator	A1	A2	B1	B2	Total
Linear (IR)	7	3	6	2	15 / 3
Quadratic (Raman)	7	3	6	2	18 / 0
IR + Raman	7	- - - -	6	2	15 / 0

Characters of force fields
(Symmetric powers of vibration representation)

Force field	E	C2.(z)	σv.(xz)	σd.(yz)
linear	18	2	8	0
quadratic	171	11	41	9
cubic	1.140	20	160	0
quartic	5.985	65	525	45
quintic	26.334	110	1.512	0
sextic	100.947	275	3.941	165

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

Force field	A1	A2	B1	B2
linear	7	3	6	2
quadratic	58	33	48	32
cubic	330	250	320	240
quartic	1.655	1.370	1.600	1.360
quintic	6.989	6.233	6.934	6.178
sextic	26.332	24.279	26.112	24.224

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 C_2v

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(B2)
..28.	A1A1.	..6.	A2A2.	..21.	B1B1.	..3.	B2B2.
Subtotal: 58 / 4 / 4
Irrep combinations (i,j) with indices: pos(A1) ≤ i ≤ j ≤ pos(B2)
Subtotal: 0 / 0 / 6
Total: 58 / 4 / 10

Contributions to nonvanishing cubic force field constants

Irrep combinations (i,i,i) with indices: pos(A1) ≤ i ≤ pos(B2)
..84.	A1A1A1.
Subtotal: 84 / 1 / 4
Irrep combinations (i,i,j) (i,j,j) with indices: pos(A1) ≤ i ≤ j ≤ pos(B2)
..42.	A1A2A2.	..147.	A1B1B1.	..21.	A1B2B2.
Subtotal: 210 / 3 / 12
Irrep combinations (i,j,k) with indices: pos(A1) ≤ i ≤ j ≤ k ≤ pos(B2)
..36.	A2B1B2.
Subtotal: 36 / 1 / 4
Total: 330 / 5 / 20

Contributions to nonvanishing quartic force field constants

Irrep combinations (i,i,i,i) with indices: pos(A1) ≤ i ≤ pos(B2)
..210.	A1A1A1A1.	..15.	A2A2A2A2.	..126.	B1B1B1B1.	..5.	B2B2B2B2.
Subtotal: 356 / 4 / 4
Irrep combinations (i,i,i,j) (i,j,j,j) with indices: pos(A1) ≤ i ≤ j ≤ pos(B2)
Subtotal: 0 / 0 / 12
Irrep combinations (i,i,j,j) with indices: pos(A1) ≤ i ≤ j ≤ pos(B2)
..168.	A1A1A2A2.	..588.	A1A1B1B1.	..84.	A1A1B2B2.	..126.	A2A2B1B1.	..18.	A2A2B2B2.	..63.	B1B1B2B2.
Subtotal: 1.047 / 6 / 6
Irrep combinations (i,i,j,k) (i,j,j,k) (i,j,k,k) with indices: pos(A1) ≤ i ≤ j ≤ k ≤ pos(B2)
Subtotal: 0 / 0 / 12
Irrep combinations (i,j,k,l) with indices: pos(A1) ≤ i ≤ j ≤ k ≤ l ≤ pos(B2)
..252.	A1A2B1B2.
Subtotal: 252 / 1 / 1
Total: 1.655 / 11 / 35

Calculate contributions to

A1	A2	B1	B2

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