Characters of representations for molecular motions

Motion	E	C2.(z)	σv.(xz)	σd.(yz)
Cartesian 3N	48	0	8	8
Translation (x,y,z)	3	-1	1	1
Rotation (Rx,Ry,Rz)	3	-1	-1	-1
Vibration	42	2	8	8

Decomposition to irreducible representations

Motion	A1	A2	B1	B2	Total
Cartesian 3N	16	8	12	12	48
Translation (x,y,z)	1	0	1	1	3
Rotation (Rx,Ry,Rz)	0	1	1	1	3
Vibration	15	7	10	10	42

Molecular parameter

Number of Atoms (N)	16
Number of internal coordinates	42
Number of independant internal coordinates	15
Number of vibrational modes	42

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Force field analysis

Allowed / forbidden vibronational transitions

Operator	A1	A2	B1	B2	Total
Linear (IR)	15	7	10	10	35 / 7
Quadratic (Raman)	15	7	10	10	42 / 0
IR + Raman	15	- - - -	10	10	35 / 0

Characters of force fields
(Symmetric powers of vibration representation)

Force field	E	C2.(z)	σv.(xz)	σd.(yz)
linear	42	2	8	8
quadratic	903	23	53	53
cubic	13.244	44	256	256
quartic	148.995	275	1.095	1.095
quintic	1.370.754	506	4.056	4.056
sextic	10.737.573	2.277	13.803	13.803

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

Force field	A1	A2	B1	B2
linear	15	7	10	10
quadratic	258	205	220	220
cubic	3.450	3.194	3.300	3.300
quartic	37.865	36.770	37.180	37.180
quintic	344.843	340.787	342.562	342.562
sextic	2.691.864	2.678.061	2.683.824	2.683.824

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)
..120.	A1A1.	..28.	A2A2.	..55.	B1B1.	..55.	B2B2.
Subtotal: 258 / 4 / 4
Irrep combinations (i,j) with indices: pos(A1) ≤ i ≤ j ≤ pos(B2)
Subtotal: 0 / 0 / 6
Total: 258 / 4 / 10

Contributions to nonvanishing cubic force field constants

Irrep combinations (i,i,i) with indices: pos(A1) ≤ i ≤ pos(B2)
..680.	A1A1A1.
Subtotal: 680 / 1 / 4
Irrep combinations (i,i,j) (i,j,j) with indices: pos(A1) ≤ i ≤ j ≤ pos(B2)
..420.	A1A2A2.	..825.	A1B1B1.	..825.	A1B2B2.
Subtotal: 2.070 / 3 / 12
Irrep combinations (i,j,k) with indices: pos(A1) ≤ i ≤ j ≤ k ≤ pos(B2)
..700.	A2B1B2.
Subtotal: 700 / 1 / 4
Total: 3.450 / 5 / 20

Contributions to nonvanishing quartic force field constants

Irrep combinations (i,i,i,i) with indices: pos(A1) ≤ i ≤ pos(B2)
..3.060.	A1A1A1A1.	..210.	A2A2A2A2.	..715.	B1B1B1B1.	..715.	B2B2B2B2.
Subtotal: 4.700 / 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)
..3.360.	A1A1A2A2.	..6.600.	A1A1B1B1.	..6.600.	A1A1B2B2.	..1.540.	A2A2B1B1.	..1.540.	A2A2B2B2.	..3.025.	B1B1B2B2.
Subtotal: 22.665 / 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)
..10.500.	A1A2B1B2.
Subtotal: 10.500 / 1 / 1
Total: 37.865 / 11 / 35

Calculate contributions to

A1	A2	B1	B2

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