Characters of representations for molecular motions

Motion	E	C2.(z)	σv.(xz)	σd.(yz)
Cartesian 3N	39	-5	5	5
Translation (x,y,z)	3	-1	1	1
Rotation (Rx,Ry,Rz)	3	-1	-1	-1
Vibration	33	-3	5	5

Decomposition to irreducible representations

Motion	A1	A2	B1	B2	Total
Cartesian 3N	11	6	11	11	39
Translation (x,y,z)	1	0	1	1	3
Rotation (Rx,Ry,Rz)	0	1	1	1	3
Vibration	10	5	9	9	33

Molecular parameter

Number of Atoms (N)	13
Number of internal coordinates	33
Number of independant internal coordinates	10
Number of vibrational modes	33

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Force field analysis

Allowed / forbidden vibronational transitions

Operator	A1	A2	B1	B2	Total
Linear (IR)	10	5	9	9	28 / 5
Quadratic (Raman)	10	5	9	9	33 / 0
IR + Raman	10	- - - -	9	9	28 / 0

Characters of force fields
(Symmetric powers of vibration representation)

Force field	E	C2.(z)	σv.(xz)	σd.(yz)
linear	33	-3	5	5
quadratic	561	21	29	29
cubic	6.545	-55	105	105
quartic	58.905	225	385	385
quintic	435.897	-531	1.141	1.141
sextic	2.760.681	1.653	3.325	3.325

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

Force field	A1	A2	B1	B2
linear	10	5	9	9
quadratic	160	131	135	135
cubic	1.675	1.570	1.650	1.650
quartic	14.975	14.590	14.670	14.670
quintic	109.412	108.271	109.107	109.107
sextic	692.246	688.921	689.757	689.757

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)
..55.	A1A1.	..15.	A2A2.	..45.	B1B1.	..45.	B2B2.
Subtotal: 160 / 4 / 4
Irrep combinations (i,j) with indices: pos(A1) ≤ i ≤ j ≤ pos(B2)
Subtotal: 0 / 0 / 6
Total: 160 / 4 / 10

Contributions to nonvanishing cubic force field constants

Irrep combinations (i,i,i) with indices: pos(A1) ≤ i ≤ pos(B2)
..220.	A1A1A1.
Subtotal: 220 / 1 / 4
Irrep combinations (i,i,j) (i,j,j) with indices: pos(A1) ≤ i ≤ j ≤ pos(B2)
..150.	A1A2A2.	..450.	A1B1B1.	..450.	A1B2B2.
Subtotal: 1.050 / 3 / 12
Irrep combinations (i,j,k) with indices: pos(A1) ≤ i ≤ j ≤ k ≤ pos(B2)
..405.	A2B1B2.
Subtotal: 405 / 1 / 4
Total: 1.675 / 5 / 20

Contributions to nonvanishing quartic force field constants

Irrep combinations (i,i,i,i) with indices: pos(A1) ≤ i ≤ pos(B2)
..715.	A1A1A1A1.	..70.	A2A2A2A2.	..495.	B1B1B1B1.	..495.	B2B2B2B2.
Subtotal: 1.775 / 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)
..825.	A1A1A2A2.	..2.475.	A1A1B1B1.	..2.475.	A1A1B2B2.	..675.	A2A2B1B1.	..675.	A2A2B2B2.	..2.025.	B1B1B2B2.
Subtotal: 9.150 / 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)
..4.050.	A1A2B1B2.
Subtotal: 4.050 / 1 / 1
Total: 14.975 / 11 / 35

Calculate contributions to

A1	A2	B1	B2

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