Characters of representations for molecular motions

Motion	E	C2.(z)	σv.(xz)	σd.(yz)
Cartesian 3N	21	-1	7	1
Translation (x,y,z)	3	-1	1	1
Rotation (Rx,Ry,Rz)	3	-1	-1	-1
Vibration	15	1	7	1

Decomposition to irreducible representations

Motion	A1	A2	B1	B2	Total
Cartesian 3N	7	3	7	4	21
Translation (x,y,z)	1	0	1	1	3
Rotation (Rx,Ry,Rz)	0	1	1	1	3
Vibration	6	2	5	2	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	2	5	2	13 / 2
Quadratic (Raman)	6	2	5	2	15 / 0
IR + Raman	6	- - - -	5	2	13 / 0

Characters of force fields
(Symmetric powers of vibration representation)

Force field	E	C2.(z)	σv.(xz)	σd.(yz)
linear	15	1	7	1
quadratic	120	8	32	8
cubic	680	8	112	8
quartic	3.060	36	332	36
quintic	11.628	36	868	36
sextic	38.760	120	2.064	120

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

Force field	A1	A2	B1	B2
linear	6	2	5	2
quadratic	42	22	34	22
cubic	202	142	194	142
quartic	866	682	830	682
quintic	3.142	2.690	3.106	2.690
sextic	10.266	9.174	10.146	9.174

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.	..3.	A2A2.	..15.	B1B1.	..3.	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)
..18.	A1A2A2.	..90.	A1B1B1.	..18.	A1B2B2.
Subtotal: 126 / 3 / 12
Irrep combinations (i,j,k) with indices: pos(A1) ≤ i ≤ j ≤ k ≤ pos(B2)
..20.	A2B1B2.
Subtotal: 20 / 1 / 4
Total: 202 / 5 / 20

Contributions to nonvanishing quartic force field constants

Irrep combinations (i,i,i,i) with indices: pos(A1) ≤ i ≤ pos(B2)
..126.	A1A1A1A1.	..5.	A2A2A2A2.	..70.	B1B1B1B1.	..5.	B2B2B2B2.
Subtotal: 206 / 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)
..63.	A1A1A2A2.	..315.	A1A1B1B1.	..63.	A1A1B2B2.	..45.	A2A2B1B1.	..9.	A2A2B2B2.	..45.	B1B1B2B2.
Subtotal: 540 / 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)
..120.	A1A2B1B2.
Subtotal: 120 / 1 / 1
Total: 866 / 11 / 35

Calculate contributions to

A1	A2	B1	B2

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