Characters of representations for molecular motions

Motion	E	i
Cartesian 3N	156	0
Translation (x,y,z)	3	-3
Rotation (Rx,Ry,Rz)	3	3
Vibration	150	0

Decomposition to irreducible representations

Motion	Ag	Au	Total
Cartesian 3N	78	78	156
Translation (x,y,z)	0	3	3
Rotation (Rx,Ry,Rz)	3	0	3
Vibration	75	75	150

Molecular parameter

Number of Atoms (N)	52
Number of internal coordinates	150
Number of independant internal coordinates	75
Number of vibrational modes	150

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Force field analysis

Allowed / forbidden vibronational transitions

Operator	Ag	Au	Total
Linear (IR)	75	75	75 / 75
Quadratic (Raman)	75	75	75 / 75
IR + Raman	- - - -	- - - -	0* / 0

* Parity Mutual Exclusion Principle

Characters of force fields
(Symmetric powers of vibration representation)

Force field	E	i
linear	150	0
quadratic	11.325	75
cubic	573.800	0
quartic	21.947.850	2.850
quintic	675.993.780	0
sextic	17.463.172.650	73.150

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

Force field	Ag	Au
linear	75	75
quadratic	5.700	5.625
cubic	286.900	286.900
quartic	10.975.350	10.972.500
quintic	337.996.890	337.996.890
sextic	8.731.622.900	8.731.549.750

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_i

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(Ag) ≤ i ≤ pos(Au)
..2.850.	AgAg.	..2.850.	AuAu.
Subtotal: 5.700 / 2 / 2
Irrep combinations (i,j) with indices: pos(Ag) ≤ i ≤ j ≤ pos(Au)
Subtotal: 0 / 0 / 1
Total: 5.700 / 2 / 3

Contributions to nonvanishing cubic force field constants

Irrep combinations (i,i,i) with indices: pos(Ag) ≤ i ≤ pos(Au)
..73.150.	AgAgAg.
Subtotal: 73.150 / 1 / 2
Irrep combinations (i,i,j) (i,j,j) with indices: pos(Ag) ≤ i ≤ j ≤ pos(Au)
..213.750.	AgAuAu.
Subtotal: 213.750 / 1 / 2
Irrep combinations (i,j,k) with indices: pos(Ag) ≤ i ≤ j ≤ k ≤ pos(Au)
Subtotal: 0 / 0 / 0
Total: 286.900 / 2 / 4

Contributions to nonvanishing quartic force field constants

Irrep combinations (i,i,i,i) with indices: pos(Ag) ≤ i ≤ pos(Au)
..1.426.425.	AgAgAgAg.	..1.426.425.	AuAuAuAu.
Subtotal: 2.852.850 / 2 / 2
Irrep combinations (i,i,i,j) (i,j,j,j) with indices: pos(Ag) ≤ i ≤ j ≤ pos(Au)
Subtotal: 0 / 0 / 2
Irrep combinations (i,i,j,j) with indices: pos(Ag) ≤ i ≤ j ≤ pos(Au)
..8.122.500.	AgAgAuAu.
Subtotal: 8.122.500 / 1 / 1
Irrep combinations (i,i,j,k) (i,j,j,k) (i,j,k,k) with indices: pos(Ag) ≤ i ≤ j ≤ k ≤ pos(Au)
Subtotal: 0 / 0 / 0
Irrep combinations (i,j,k,l) with indices: pos(Ag) ≤ i ≤ j ≤ k ≤ l ≤ pos(Au)
Subtotal: 0 / 0 / 0
Total: 10.975.350 / 3 / 5

Calculate contributions to

Ag	Au

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