Characters of representations for molecular motions

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

Decomposition to irreducible representations

Motion	A1	A2	B1	B2	Total
Cartesian 3N	10	6	10	10	36
Translation (x,y,z)	1	0	1	1	3
Rotation (Rx,Ry,Rz)	0	1	1	1	3
Vibration	9	5	8	8	30

Molecular parameter

Number of Atoms (N)	12
Number of internal coordinates	30
Number of independant internal coordinates	9
Number of vibrational modes	30

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Force field analysis

Allowed / forbidden vibronational transitions

Operator	A1	A2	B1	B2	Total
Linear (IR)	9	5	8	8	25 / 5
Quadratic (Raman)	9	5	8	8	30 / 0
IR + Raman	9	- - - -	8	8	25 / 0

Characters of force fields
(Symmetric powers of vibration representation)

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

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

Force field	A1	A2	B1	B2
linear	9	5	8	8
quadratic	132	109	112	112
cubic	1.268	1.196	1.248	1.248
quartic	10.396	10.140	10.192	10.192
quintic	69.836	69.156	69.632	69.632
sextic	406.980	405.076	405.552	405.552

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)
..45.	A1A1.	..15.	A2A2.	..36.	B1B1.	..36.	B2B2.
Subtotal: 132 / 4 / 4
Irrep combinations (i,j) with indices: pos(A1) ≤ i ≤ j ≤ pos(B2)
Subtotal: 0 / 0 / 6
Total: 132 / 4 / 10

Contributions to nonvanishing cubic force field constants

Irrep combinations (i,i,i) with indices: pos(A1) ≤ i ≤ pos(B2)
..165.	A1A1A1.
Subtotal: 165 / 1 / 4
Irrep combinations (i,i,j) (i,j,j) with indices: pos(A1) ≤ i ≤ j ≤ pos(B2)
..135.	A1A2A2.	..324.	A1B1B1.	..324.	A1B2B2.
Subtotal: 783 / 3 / 12
Irrep combinations (i,j,k) with indices: pos(A1) ≤ i ≤ j ≤ k ≤ pos(B2)
..320.	A2B1B2.
Subtotal: 320 / 1 / 4
Total: 1.268 / 5 / 20

Contributions to nonvanishing quartic force field constants

Irrep combinations (i,i,i,i) with indices: pos(A1) ≤ i ≤ pos(B2)
..495.	A1A1A1A1.	..70.	A2A2A2A2.	..330.	B1B1B1B1.	..330.	B2B2B2B2.
Subtotal: 1.225 / 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)
..675.	A1A1A2A2.	..1.620.	A1A1B1B1.	..1.620.	A1A1B2B2.	..540.	A2A2B1B1.	..540.	A2A2B2B2.	..1.296.	B1B1B2B2.
Subtotal: 6.291 / 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)
..2.880.	A1A2B1B2.
Subtotal: 2.880 / 1 / 1
Total: 10.396 / 11 / 35

Calculate contributions to

A1	A2	B1	B2

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