Characters of representations for molecular motions

Motion	E	C2.(z)	σv.(xz)	σd.(yz)
Cartesian 3N	21	-3	5	5
Translation (x,y,z)	3	-1	1	1
Rotation (Rx,Ry,Rz)	3	-1	-1	-1
Vibration	15	-1	5	5

Decomposition to irreducible representations

Motion	A1	A2	B1	B2	Total
Cartesian 3N	7	2	6	6	21
Translation (x,y,z)	1	0	1	1	3
Rotation (Rx,Ry,Rz)	0	1	1	1	3
Vibration	6	1	4	4	15

Molecular parameter

Number of Atoms (N)	7
Number of internal coordinates	15
Number of independant internal coordinates	6
Number of vibrational modes	15

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Force field analysis

Allowed / forbidden vibronational transitions

Operator	A1	A2	B1	B2	Total
Linear (IR)	6	1	4	4	14 / 1
Quadratic (Raman)	6	1	4	4	15 / 0
IR + Raman	6	- - - -	4	4	14 / 0

Characters of force fields
(Symmetric powers of vibration representation)

Force field	E	C2.(z)	σv.(xz)	σd.(yz)
linear	15	-1	5	5
quadratic	120	8	20	20
cubic	680	-8	60	60
quartic	3.060	36	160	160
quintic	11.628	-36	376	376
sextic	38.760	120	820	820

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

Force field	A1	A2	B1	B2
linear	6	1	4	4
quadratic	42	22	28	28
cubic	198	138	172	172
quartic	854	694	756	756
quintic	3.086	2.710	2.916	2.916
sextic	10.130	9.310	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 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)
..21.	A1A1.	..1.	A2A2.	..10.	B1B1.	..10.	B2B2.
Subtotal: 42 / 4 / 4
Irrep combinations (i,j) with indices: pos(A1) ≤ i ≤ j ≤ pos(B2)
Subtotal: 0 / 0 / 6
Total: 42 / 4 / 10

Contributions to nonvanishing cubic force field constants

Irrep combinations (i,i,i) with indices: pos(A1) ≤ i ≤ pos(B2)
..56.	A1A1A1.
Subtotal: 56 / 1 / 4
Irrep combinations (i,i,j) (i,j,j) with indices: pos(A1) ≤ i ≤ j ≤ pos(B2)
..6.	A1A2A2.	..60.	A1B1B1.	..60.	A1B2B2.
Subtotal: 126 / 3 / 12
Irrep combinations (i,j,k) with indices: pos(A1) ≤ i ≤ j ≤ k ≤ pos(B2)
..16.	A2B1B2.
Subtotal: 16 / 1 / 4
Total: 198 / 5 / 20

Contributions to nonvanishing quartic force field constants

Irrep combinations (i,i,i,i) with indices: pos(A1) ≤ i ≤ pos(B2)
..126.	A1A1A1A1.	..1.	A2A2A2A2.	..35.	B1B1B1B1.	..35.	B2B2B2B2.
Subtotal: 197 / 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)
..21.	A1A1A2A2.	..210.	A1A1B1B1.	..210.	A1A1B2B2.	..10.	A2A2B1B1.	..10.	A2A2B2B2.	..100.	B1B1B2B2.
Subtotal: 561 / 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)
..96.	A1A2B1B2.
Subtotal: 96 / 1 / 1
Total: 854 / 11 / 35

Calculate contributions to

A1	A2	B1	B2

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