Characters of representations for molecular motions

Motion	E	C2.(z)	C2.(y)	C2.(x)
Cartesian 3N	21	-3	-1	-1
Translation (x,y,z)	3	-1	-1	-1
Rotation (Rx,Ry,Rz)	3	-1	-1	-1
Vibration	15	-1	1	1

Decomposition to irreducible representations

Motion	A	B1	B2	B3	Total
Cartesian 3N	4	5	6	6	21
Translation (x,y,z)	0	1	1	1	3
Rotation (Rx,Ry,Rz)	0	1	1	1	3
Vibration	4	3	4	4	15

Molecular parameter

Number of Atoms (N)	7
Number of internal coordinates	15
Number of independant internal coordinates	4
Number of vibrational modes	15

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Force field analysis

Allowed / forbidden vibronational transitions

Operator	A	B1	B2	B3	Total
Linear (IR)	4	3	4	4	11 / 4
Quadratic (Raman)	4	3	4	4	15 / 0
IR + Raman	- - - -	3	4	4	11 / 0

Characters of force fields
(Symmetric powers of vibration representation)

Force field	E	C2.(z)	C2.(y)	C2.(x)
linear	15	-1	1	1
quadratic	120	8	8	8
cubic	680	-8	8	8
quartic	3.060	36	36	36
quintic	11.628	-36	36	36
sextic	38.760	120	120	120

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

Force field	A	B1	B2	B3
linear	4	3	4	4
quadratic	36	28	28	28
cubic	172	164	172	172
quartic	792	756	756	756
quintic	2.916	2.880	2.916	2.916
sextic	9.780	9.660	9.660	9.660

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

Contributions to nonvanishing cubic force field constants

Irrep combinations (i,i,i) with indices: pos(A) ≤ i ≤ pos(B3)
..20.	AAA.
Subtotal: 20 / 1 / 4
Irrep combinations (i,i,j) (i,j,j) with indices: pos(A) ≤ i ≤ j ≤ pos(B3)
..24.	AB1B1.	..40.	AB2B2.	..40.	AB3B3.
Subtotal: 104 / 3 / 12
Irrep combinations (i,j,k) with indices: pos(A) ≤ i ≤ j ≤ k ≤ pos(B3)
..48.	B1B2B3.
Subtotal: 48 / 1 / 4
Total: 172 / 5 / 20

Contributions to nonvanishing quartic force field constants

Irrep combinations (i,i,i,i) with indices: pos(A) ≤ i ≤ pos(B3)
..35.	AAAA.	..15.	B1B1B1B1.	..35.	B2B2B2B2.	..35.	B3B3B3B3.
Subtotal: 120 / 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)
..60.	AAB1B1.	..100.	AAB2B2.	..100.	AAB3B3.	..60.	B1B1B2B2.	..60.	B1B1B3B3.	..100.	B2B2B3B3.
Subtotal: 480 / 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)
..192.	AB1B2B3.
Subtotal: 192 / 1 / 1
Total: 792 / 11 / 35

Calculate contributions to

A	B1	B2	B3

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