Characters of representations for molecular motions

Motion	E	C2.(z)	σv.(xz)	σd.(yz)
Cartesian 3N	39	-1	9	5
Translation (x,y,z)	3	-1	1	1
Rotation (Rx,Ry,Rz)	3	-1	-1	-1
Vibration	33	1	9	5

Decomposition to irreducible representations

Motion	A1	A2	B1	B2	Total
Cartesian 3N	13	6	11	9	39
Translation (x,y,z)	1	0	1	1	3
Rotation (Rx,Ry,Rz)	0	1	1	1	3
Vibration	12	5	9	7	33

Molecular parameter

Number of Atoms (N)	13
Number of internal coordinates	33
Number of independant internal coordinates	12
Number of vibrational modes	33

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Force field analysis

Allowed / forbidden vibronational transitions

Operator	A1	A2	B1	B2	Total
Linear (IR)	12	5	9	7	28 / 5
Quadratic (Raman)	12	5	9	7	33 / 0
IR + Raman	12	- - - -	9	7	28 / 0

Characters of force fields
(Symmetric powers of vibration representation)

Force field	E	C2.(z)	σv.(xz)	σd.(yz)
linear	33	1	9	5
quadratic	561	17	57	29
cubic	6.545	17	273	105
quartic	58.905	153	1.113	385
quintic	435.897	153	3.969	1.141
sextic	2.760.681	969	12.817	3.325

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

Force field	A1	A2	B1	B2
linear	12	5	9	7
quadratic	166	123	143	129
cubic	1.735	1.546	1.674	1.590
quartic	15.139	14.390	14.870	14.506
quintic	110.290	107.735	109.643	108.229
sextic	694.448	686.377	692.301	687.555

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)
..78.	A1A1.	..15.	A2A2.	..45.	B1B1.	..28.	B2B2.
Subtotal: 166 / 4 / 4
Irrep combinations (i,j) with indices: pos(A1) ≤ i ≤ j ≤ pos(B2)
Subtotal: 0 / 0 / 6
Total: 166 / 4 / 10

Contributions to nonvanishing cubic force field constants

Irrep combinations (i,i,i) with indices: pos(A1) ≤ i ≤ pos(B2)
..364.	A1A1A1.
Subtotal: 364 / 1 / 4
Irrep combinations (i,i,j) (i,j,j) with indices: pos(A1) ≤ i ≤ j ≤ pos(B2)
..180.	A1A2A2.	..540.	A1B1B1.	..336.	A1B2B2.
Subtotal: 1.056 / 3 / 12
Irrep combinations (i,j,k) with indices: pos(A1) ≤ i ≤ j ≤ k ≤ pos(B2)
..315.	A2B1B2.
Subtotal: 315 / 1 / 4
Total: 1.735 / 5 / 20

Contributions to nonvanishing quartic force field constants

Irrep combinations (i,i,i,i) with indices: pos(A1) ≤ i ≤ pos(B2)
..1.365.	A1A1A1A1.	..70.	A2A2A2A2.	..495.	B1B1B1B1.	..210.	B2B2B2B2.
Subtotal: 2.140 / 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)
..1.170.	A1A1A2A2.	..3.510.	A1A1B1B1.	..2.184.	A1A1B2B2.	..675.	A2A2B1B1.	..420.	A2A2B2B2.	..1.260.	B1B1B2B2.
Subtotal: 9.219 / 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)
..3.780.	A1A2B1B2.
Subtotal: 3.780 / 1 / 1
Total: 15.139 / 11 / 35

Calculate contributions to

A1	A2	B1	B2

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