Characters of representations for molecular motions

Motion	E	C2.(z)	C2.(y)	C2.(x)
Cartesian 3N	78	0	0	-2
Translation (x,y,z)	3	-1	-1	-1
Rotation (Rx,Ry,Rz)	3	-1	-1	-1
Vibration	72	2	2	0

Decomposition to irreducible representations

Motion	A	B1	B2	B3	Total
Cartesian 3N	19	20	20	19	78
Translation (x,y,z)	0	1	1	1	3
Rotation (Rx,Ry,Rz)	0	1	1	1	3
Vibration	19	18	18	17	72

Molecular parameter

Number of Atoms (N)	26
Number of internal coordinates	72
Number of independant internal coordinates	19
Number of vibrational modes	72

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Force field analysis

Allowed / forbidden vibronational transitions

Operator	A	B1	B2	B3	Total
Linear (IR)	19	18	18	17	53 / 19
Quadratic (Raman)	19	18	18	17	72 / 0
IR + Raman	- - - -	18	18	17	53 / 0

Characters of force fields
(Symmetric powers of vibration representation)

Force field	E	C2.(z)	C2.(y)	C2.(x)
linear	72	2	2	0
quadratic	2.628	38	38	36
cubic	64.824	74	74	0
quartic	1.215.450	740	740	666
quintic	18.474.840	1.406	1.406	0
sextic	237.093.780	9.842	9.842	8.436

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

Force field	A	B1	B2	B3
linear	19	18	18	17
quadratic	685	648	648	647
cubic	16.243	16.206	16.206	16.169
quartic	304.399	303.696	303.696	303.659
quintic	4.619.413	4.618.710	4.618.710	4.618.007
sextic	59.280.475	59.271.336	59.271.336	59.270.633

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 D₂

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(A) ≤ i ≤ pos(B3)
..190.	AA.	..171.	B1B1.	..171.	B2B2.	..153.	B3B3.
Subtotal: 685 / 4 / 4
Irrep combinations (i,j) with indices: pos(A) ≤ i ≤ j ≤ pos(B3)
Subtotal: 0 / 0 / 6
Total: 685 / 4 / 10

Contributions to nonvanishing cubic force field constants

Irrep combinations (i,i,i) with indices: pos(A) ≤ i ≤ pos(B3)
..1.330.	AAA.
Subtotal: 1.330 / 1 / 4
Irrep combinations (i,i,j) (i,j,j) with indices: pos(A) ≤ i ≤ j ≤ pos(B3)
..3.249.	AB1B1.	..3.249.	AB2B2.	..2.907.	AB3B3.
Subtotal: 9.405 / 3 / 12
Irrep combinations (i,j,k) with indices: pos(A) ≤ i ≤ j ≤ k ≤ pos(B3)
..5.508.	B1B2B3.
Subtotal: 5.508 / 1 / 4
Total: 16.243 / 5 / 20

Contributions to nonvanishing quartic force field constants

Irrep combinations (i,i,i,i) with indices: pos(A) ≤ i ≤ pos(B3)
..7.315.	AAAA.	..5.985.	B1B1B1B1.	..5.985.	B2B2B2B2.	..4.845.	B3B3B3B3.
Subtotal: 24.130 / 4 / 4
Irrep combinations (i,i,i,j) (i,j,j,j) with indices: pos(A) ≤ i ≤ j ≤ pos(B3)
Subtotal: 0 / 0 / 12
Irrep combinations (i,i,j,j) with indices: pos(A) ≤ i ≤ j ≤ pos(B3)
..32.490.	AAB1B1.	..32.490.	AAB2B2.	..29.070.	AAB3B3.	..29.241.	B1B1B2B2.	..26.163.	B1B1B3B3.	..26.163.	B2B2B3B3.
Subtotal: 175.617 / 6 / 6
Irrep combinations (i,i,j,k) (i,j,j,k) (i,j,k,k) with indices: pos(A) ≤ i ≤ j ≤ k ≤ pos(B3)
Subtotal: 0 / 0 / 12
Irrep combinations (i,j,k,l) with indices: pos(A) ≤ i ≤ j ≤ k ≤ l ≤ pos(B3)
..104.652.	AB1B2B3.
Subtotal: 104.652 / 1 / 1
Total: 304.399 / 11 / 35

Calculate contributions to

A	B1	B2	B3

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