Characters of representations for molecular motions

Motion	E	C2.(z)	C2.(y)	C2.(x)
Cartesian 3N	36	-4	-4	-4
Translation (x,y,z)	3	-1	-1	-1
Rotation (Rx,Ry,Rz)	3	-1	-1	-1
Vibration	30	-2	-2	-2

Decomposition to irreducible representations

Motion	A	B1	B2	B3	Total
Cartesian 3N	6	10	10	10	36
Translation (x,y,z)	0	1	1	1	3
Rotation (Rx,Ry,Rz)	0	1	1	1	3
Vibration	6	8	8	8	30

Molecular parameter

Number of Atoms (N)	12
Number of internal coordinates	30
Number of independant internal coordinates	6
Number of vibrational modes	30

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Force field analysis

Allowed / forbidden vibronational transitions

Operator	A	B1	B2	B3	Total
Linear (IR)	6	8	8	8	24 / 6
Quadratic (Raman)	6	8	8	8	30 / 0
IR + Raman	- - - -	8	8	8	24 / 0

Characters of force fields
(Symmetric powers of vibration representation)

Force field	E	C2.(z)	C2.(y)	C2.(x)
linear	30	-2	-2	-2
quadratic	465	17	17	17
cubic	4.960	-32	-32	-32
quartic	40.920	152	152	152
quintic	278.256	-272	-272	-272
sextic	1.623.160	952	952	952

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

Force field	A	B1	B2	B3
linear	6	8	8	8
quadratic	129	112	112	112
cubic	1.216	1.248	1.248	1.248
quartic	10.344	10.192	10.192	10.192
quintic	69.360	69.632	69.632	69.632
sextic	406.504	405.552	405.552	405.552

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)
..21.	AA.	..36.	B1B1.	..36.	B2B2.	..36.	B3B3.
Subtotal: 129 / 4 / 4
Irrep combinations (i,j) with indices: pos(A) ≤ i ≤ j ≤ pos(B3)
Subtotal: 0 / 0 / 6
Total: 129 / 4 / 10

Contributions to nonvanishing cubic force field constants

Irrep combinations (i,i,i) with indices: pos(A) ≤ i ≤ pos(B3)
..56.	AAA.
Subtotal: 56 / 1 / 4
Irrep combinations (i,i,j) (i,j,j) with indices: pos(A) ≤ i ≤ j ≤ pos(B3)
..216.	AB1B1.	..216.	AB2B2.	..216.	AB3B3.
Subtotal: 648 / 3 / 12
Irrep combinations (i,j,k) with indices: pos(A) ≤ i ≤ j ≤ k ≤ pos(B3)
..512.	B1B2B3.
Subtotal: 512 / 1 / 4
Total: 1.216 / 5 / 20

Contributions to nonvanishing quartic force field constants

Irrep combinations (i,i,i,i) with indices: pos(A) ≤ i ≤ pos(B3)
..126.	AAAA.	..330.	B1B1B1B1.	..330.	B2B2B2B2.	..330.	B3B3B3B3.
Subtotal: 1.116 / 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)
..756.	AAB1B1.	..756.	AAB2B2.	..756.	AAB3B3.	..1.296.	B1B1B2B2.	..1.296.	B1B1B3B3.	..1.296.	B2B2B3B3.
Subtotal: 6.156 / 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)
..3.072.	AB1B2B3.
Subtotal: 3.072 / 1 / 1
Total: 10.344 / 11 / 35

Calculate contributions to

A	B1	B2	B3

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