Characters of representations for molecular motions

Motion	E	C2.(z)	σv.(xz)	σd.(yz)
Cartesian 3N	33	-1	5	3
Translation (x,y,z)	3	-1	1	1
Rotation (Rx,Ry,Rz)	3	-1	-1	-1
Vibration	27	1	5	3

Decomposition to irreducible representations

Motion	A1	A2	B1	B2	Total
Cartesian 3N	10	6	9	8	33
Translation (x,y,z)	1	0	1	1	3
Rotation (Rx,Ry,Rz)	0	1	1	1	3
Vibration	9	5	7	6	27

Molecular parameter

Number of Atoms (N)	11
Number of internal coordinates	27
Number of independant internal coordinates	9
Number of vibrational modes	27

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Force field analysis

Allowed / forbidden vibronational transitions

Operator	A1	A2	B1	B2	Total
Linear (IR)	9	5	7	6	22 / 5
Quadratic (Raman)	9	5	7	6	27 / 0
IR + Raman	9	- - - -	7	6	22 / 0

Characters of force fields
(Symmetric powers of vibration representation)

Force field	E	C2.(z)	σv.(xz)	σd.(yz)
linear	27	1	5	3
quadratic	378	14	26	18
cubic	3.654	14	90	46
quartic	27.405	105	301	165
quintic	169.911	105	841	375
sextic	906.192	560	2.256	1.040

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

Force field	A1	A2	B1	B2
linear	9	5	7	6
quadratic	109	87	93	89
cubic	951	883	921	899
quartic	6.994	6.761	6.859	6.791
quintic	42.808	42.200	42.568	42.335
sextic	227.512	225.864	226.712	226.104

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)
..45.	A1A1.	..15.	A2A2.	..28.	B1B1.	..21.	B2B2.
Subtotal: 109 / 4 / 4
Irrep combinations (i,j) with indices: pos(A1) ≤ i ≤ j ≤ pos(B2)
Subtotal: 0 / 0 / 6
Total: 109 / 4 / 10

Contributions to nonvanishing cubic force field constants

Irrep combinations (i,i,i) with indices: pos(A1) ≤ i ≤ pos(B2)
..165.	A1A1A1.
Subtotal: 165 / 1 / 4
Irrep combinations (i,i,j) (i,j,j) with indices: pos(A1) ≤ i ≤ j ≤ pos(B2)
..135.	A1A2A2.	..252.	A1B1B1.	..189.	A1B2B2.
Subtotal: 576 / 3 / 12
Irrep combinations (i,j,k) with indices: pos(A1) ≤ i ≤ j ≤ k ≤ pos(B2)
..210.	A2B1B2.
Subtotal: 210 / 1 / 4
Total: 951 / 5 / 20

Contributions to nonvanishing quartic force field constants

Irrep combinations (i,i,i,i) with indices: pos(A1) ≤ i ≤ pos(B2)
..495.	A1A1A1A1.	..70.	A2A2A2A2.	..210.	B1B1B1B1.	..126.	B2B2B2B2.
Subtotal: 901 / 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)
..675.	A1A1A2A2.	..1.260.	A1A1B1B1.	..945.	A1A1B2B2.	..420.	A2A2B1B1.	..315.	A2A2B2B2.	..588.	B1B1B2B2.
Subtotal: 4.203 / 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)
..1.890.	A1A2B1B2.
Subtotal: 1.890 / 1 / 1
Total: 6.994 / 11 / 35

Calculate contributions to

A1	A2	B1	B2

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