Characters of representations for molecular motions

Motion	E	2C3	3σv
Cartesian 3N	99	0	3
Translation (x,y,z)	3	0	1
Rotation (Rx,Ry,Rz)	3	0	-1
Vibration	93	0	3

Decomposition to irreducible representations

Motion	A1	A2	E	Total
Cartesian 3N	18	15	33	66
Translation (x,y,z)	1	0	1	2
Rotation (Rx,Ry,Rz)	0	1	1	2
Vibration	17	14	31	62

Molecular parameter

Number of Atoms (N)	33
Number of internal coordinates	93
Number of independant internal coordinates	17
Number of vibrational modes	62

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Force field analysis

Allowed / forbidden vibronational transitions

Operator	A1	A2	E	Total
Linear (IR)	17	14	31	48 / 14
Quadratic (Raman)	17	14	31	48 / 14
IR + Raman	17	14	31	48 / 14

Characters of force fields
(Symmetric powers of vibration representation)

Force field	E	2C3	3σv
linear	93	0	3
quadratic	4.371	0	51
cubic	138.415	31	145
quartic	3.321.960	0	1.320
quintic	64.446.024	0	3.576
sextic	1.052.618.392	496	23.128

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

Force field	A1	A2	E
linear	17	14	31
quadratic	754	703	1.457
cubic	23.152	23.007	46.128
quartic	554.320	553.000	1.107.320
quintic	10.742.792	10.739.216	21.482.008
sextic	175.448.128	175.425.000	350.872.632

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_3v

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(E)
..153.	A1A1.	..105.	A2A2.	..496.	EE.
Subtotal: 754 / 3 / 3
Irrep combinations (i,j) with indices: pos(A1) ≤ i ≤ j ≤ pos(E)
Subtotal: 0 / 0 / 3
Total: 754 / 3 / 6

Contributions to nonvanishing cubic force field constants

Irrep combinations (i,i,i) with indices: pos(A1) ≤ i ≤ pos(E)
..969.	A1A1A1.	..5.456.	EEE.
Subtotal: 6.425 / 2 / 3
Irrep combinations (i,i,j) (i,j,j) with indices: pos(A1) ≤ i ≤ j ≤ pos(E)
..1.785.	A1A2A2.	..8.432.	A1EE.	..6.510.	A2EE.
Subtotal: 16.727 / 3 / 6
Irrep combinations (i,j,k) with indices: pos(A1) ≤ i ≤ j ≤ k ≤ pos(E)
Subtotal: 0 / 0 / 1
Total: 23.152 / 5 / 10

Contributions to nonvanishing quartic force field constants

Irrep combinations (i,i,i,i) with indices: pos(A1) ≤ i ≤ pos(E)
..4.845.	A1A1A1A1.	..2.380.	A2A2A2A2.	..123.256.	EEEE.
Subtotal: 130.481 / 3 / 3
Irrep combinations (i,i,i,j) (i,j,j,j) with indices: pos(A1) ≤ i ≤ j ≤ pos(E)
..92.752.	A1EEE.	..76.384.	A2EEE.
Subtotal: 169.136 / 2 / 6
Irrep combinations (i,i,j,j) with indices: pos(A1) ≤ i ≤ j ≤ pos(E)
..16.065.	A1A1A2A2.	..75.888.	A1A1EE.	..52.080.	A2A2EE.
Subtotal: 144.033 / 3 / 3
Irrep combinations (i,i,j,k) (i,j,j,k) (i,j,k,k) with indices: pos(A1) ≤ i ≤ j ≤ k ≤ pos(E)
..110.670.	A1A2EE.
Subtotal: 110.670 / 1 / 3
Irrep combinations (i,j,k,l) with indices: pos(A1) ≤ i ≤ j ≤ k ≤ l ≤ pos(E)
Subtotal: 0 / 0 / 0
Total: 554.320 / 9 / 15

Calculate contributions to

A1	A2	E

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