Characters of representations for molecular motions

Motion	E	2C3	3σv
Cartesian 3N	156	0	14
Translation (x,y,z)	3	0	1
Rotation (Rx,Ry,Rz)	3	0	-1
Vibration	150	0	14

Decomposition to irreducible representations

Motion	A1	A2	E	Total
Cartesian 3N	33	19	52	104
Translation (x,y,z)	1	0	1	2
Rotation (Rx,Ry,Rz)	0	1	1	2
Vibration	32	18	50	100

Molecular parameter

Number of Atoms (N)	52
Number of internal coordinates	150
Number of independant internal coordinates	32
Number of vibrational modes	100

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Force field analysis

Allowed / forbidden vibronational transitions

Operator	A1	A2	E	Total
Linear (IR)	32	18	50	82 / 18
Quadratic (Raman)	32	18	50	82 / 18
IR + Raman	32	18	50	82 / 18

Characters of force fields
(Symmetric powers of vibration representation)

Force field	E	2C3	3σv
linear	150	0	14
quadratic	11.325	0	173
cubic	573.800	50	1.512
quartic	21.947.850	0	11.866
quintic	675.993.780	0	79.492
sextic	17.463.172.650	1.275	490.042

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

Force field	A1	A2	E
linear	32	18	50
quadratic	1.974	1.801	3.775
cubic	96.406	94.894	191.250
quartic	3.663.908	3.652.042	7.315.950
quintic	112.705.376	112.625.884	225.331.260
sextic	2.910.774.221	2.910.284.179	5.821.057.125

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)
..528.	A1A1.	..171.	A2A2.	..1.275.	EE.
Subtotal: 1.974 / 3 / 3
Irrep combinations (i,j) with indices: pos(A1) ≤ i ≤ j ≤ pos(E)
Subtotal: 0 / 0 / 3
Total: 1.974 / 3 / 6

Contributions to nonvanishing cubic force field constants

Irrep combinations (i,i,i) with indices: pos(A1) ≤ i ≤ pos(E)
..5.984.	A1A1A1.	..22.100.	EEE.
Subtotal: 28.084 / 2 / 3
Irrep combinations (i,i,j) (i,j,j) with indices: pos(A1) ≤ i ≤ j ≤ pos(E)
..5.472.	A1A2A2.	..40.800.	A1EE.	..22.050.	A2EE.
Subtotal: 68.322 / 3 / 6
Irrep combinations (i,j,k) with indices: pos(A1) ≤ i ≤ j ≤ k ≤ pos(E)
Subtotal: 0 / 0 / 1
Total: 96.406 / 5 / 10

Contributions to nonvanishing quartic force field constants

Irrep combinations (i,i,i,i) with indices: pos(A1) ≤ i ≤ pos(E)
..52.360.	A1A1A1A1.	..5.985.	A2A2A2A2.	..813.450.	EEEE.
Subtotal: 871.795 / 3 / 3
Irrep combinations (i,i,i,j) (i,j,j,j) with indices: pos(A1) ≤ i ≤ j ≤ pos(E)
..707.200.	A1EEE.	..397.800.	A2EEE.
Subtotal: 1.105.000 / 2 / 6
Irrep combinations (i,i,j,j) with indices: pos(A1) ≤ i ≤ j ≤ pos(E)
..90.288.	A1A1A2A2.	..673.200.	A1A1EE.	..218.025.	A2A2EE.
Subtotal: 981.513 / 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)
..705.600.	A1A2EE.
Subtotal: 705.600 / 1 / 3
Irrep combinations (i,j,k,l) with indices: pos(A1) ≤ i ≤ j ≤ k ≤ l ≤ pos(E)
Subtotal: 0 / 0 / 0
Total: 3.663.908 / 9 / 15

Calculate contributions to

A1	A2	E

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