Characters of representations for molecular motions

Motion	E	i
Cartesian 3N	24	0
Translation (x,y,z)	3	-3
Rotation (Rx,Ry,Rz)	3	3
Vibration	18	0

Decomposition to irreducible representations

Motion	Ag	Au	Total
Cartesian 3N	12	12	24
Translation (x,y,z)	0	3	3
Rotation (Rx,Ry,Rz)	3	0	3
Vibration	9	9	18

Molecular parameter

Number of Atoms (N)	8
Number of internal coordinates	18
Number of independant internal coordinates	9
Number of vibrational modes	18

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Force field analysis

Allowed / forbidden vibronational transitions

Operator	Ag	Au	Total
Linear (IR)	9	9	9 / 9
Quadratic (Raman)	9	9	9 / 9
IR + Raman	- - - -	- - - -	0* / 0

* Parity Mutual Exclusion Principle

Characters of force fields
(Symmetric powers of vibration representation)

Force field	E	i
linear	18	0
quadratic	171	9
cubic	1.140	0
quartic	5.985	45
quintic	26.334	0
sextic	100.947	165

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

Force field	Ag	Au
linear	9	9
quadratic	90	81
cubic	570	570
quartic	3.015	2.970
quintic	13.167	13.167
sextic	50.556	50.391

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)
..45.	AgAg.	..45.	AuAu.
Subtotal: 90 / 2 / 2
Irrep combinations (i,j) with indices: pos(Ag) ≤ i ≤ j ≤ pos(Au)
Subtotal: 0 / 0 / 1
Total: 90 / 2 / 3

Contributions to nonvanishing cubic force field constants

Irrep combinations (i,i,i) with indices: pos(Ag) ≤ i ≤ pos(Au)
..165.	AgAgAg.
Subtotal: 165 / 1 / 2
Irrep combinations (i,i,j) (i,j,j) with indices: pos(Ag) ≤ i ≤ j ≤ pos(Au)
..405.	AgAuAu.
Subtotal: 405 / 1 / 2
Irrep combinations (i,j,k) with indices: pos(Ag) ≤ i ≤ j ≤ k ≤ pos(Au)
Subtotal: 0 / 0 / 0
Total: 570 / 2 / 4

Contributions to nonvanishing quartic force field constants

Irrep combinations (i,i,i,i) with indices: pos(Ag) ≤ i ≤ pos(Au)
..495.	AgAgAgAg.	..495.	AuAuAuAu.
Subtotal: 990 / 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)
..2.025.	AgAgAuAu.
Subtotal: 2.025 / 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: 3.015 / 3 / 5

Calculate contributions to

Ag	Au

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