Characters of representations for molecular motions

Motion	E	C2.(z)	σv.(xz)	σd.(yz)
Cartesian 3N	36	0	8	4
Translation (x,y,z)	3	-1	1	1
Rotation (Rx,Ry,Rz)	3	-1	-1	-1
Vibration	30	2	8	4

Decomposition to irreducible representations

Motion	A1	A2	B1	B2	Total
Cartesian 3N	12	6	10	8	36
Translation (x,y,z)	1	0	1	1	3
Rotation (Rx,Ry,Rz)	0	1	1	1	3
Vibration	11	5	8	6	30

Molecular parameter

Number of Atoms (N)	12
Number of internal coordinates	30
Number of independant internal coordinates	11
Number of vibrational modes	30

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Force field analysis

Allowed / forbidden vibronational transitions

Operator	A1	A2	B1	B2	Total
Linear (IR)	11	5	8	6	25 / 5
Quadratic (Raman)	11	5	8	6	30 / 0
IR + Raman	11	- - - -	8	6	25 / 0

Characters of force fields
(Symmetric powers of vibration representation)

Force field	E	C2.(z)	σv.(xz)	σd.(yz)
linear	30	2	8	4
quadratic	465	17	47	23
cubic	4.960	32	208	72
quartic	40.920	152	792	256
quintic	278.256	272	2.640	680
sextic	1.623.160	952	8.008	1.904

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

Force field	A1	A2	B1	B2
linear	11	5	8	6
quadratic	138	103	118	106
cubic	1.318	1.178	1.266	1.198
quartic	10.530	10.006	10.326	10.058
quintic	70.462	68.802	69.986	69.006
sextic	408.506	403.550	407.078	404.026

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)
..66.	A1A1.	..15.	A2A2.	..36.	B1B1.	..21.	B2B2.
Subtotal: 138 / 4 / 4
Irrep combinations (i,j) with indices: pos(A1) ≤ i ≤ j ≤ pos(B2)
Subtotal: 0 / 0 / 6
Total: 138 / 4 / 10

Contributions to nonvanishing cubic force field constants

Irrep combinations (i,i,i) with indices: pos(A1) ≤ i ≤ pos(B2)
..286.	A1A1A1.
Subtotal: 286 / 1 / 4
Irrep combinations (i,i,j) (i,j,j) with indices: pos(A1) ≤ i ≤ j ≤ pos(B2)
..165.	A1A2A2.	..396.	A1B1B1.	..231.	A1B2B2.
Subtotal: 792 / 3 / 12
Irrep combinations (i,j,k) with indices: pos(A1) ≤ i ≤ j ≤ k ≤ pos(B2)
..240.	A2B1B2.
Subtotal: 240 / 1 / 4
Total: 1.318 / 5 / 20

Contributions to nonvanishing quartic force field constants

Irrep combinations (i,i,i,i) with indices: pos(A1) ≤ i ≤ pos(B2)
..1.001.	A1A1A1A1.	..70.	A2A2A2A2.	..330.	B1B1B1B1.	..126.	B2B2B2B2.
Subtotal: 1.527 / 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)
..990.	A1A1A2A2.	..2.376.	A1A1B1B1.	..1.386.	A1A1B2B2.	..540.	A2A2B1B1.	..315.	A2A2B2B2.	..756.	B1B1B2B2.
Subtotal: 6.363 / 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)
..2.640.	A1A2B1B2.
Subtotal: 2.640 / 1 / 1
Total: 10.530 / 11 / 35

Calculate contributions to

A1	A2	B1	B2

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