Characters of representations for molecular motions

Motion	E	i
Cartesian 3N	36	0
Translation (x,y,z)	3	-3
Rotation (Rx,Ry,Rz)	3	3
Vibration	30	0

Decomposition to irreducible representations

Motion	Ag	Au	Total
Cartesian 3N	18	18	36
Translation (x,y,z)	0	3	3
Rotation (Rx,Ry,Rz)	3	0	3
Vibration	15	15	30

Molecular parameter

Number of Atoms (N)	12
Number of internal coordinates	30
Number of independant internal coordinates	15
Number of vibrational modes	30

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Force field analysis

Allowed / forbidden vibronational transitions

Operator	Ag	Au	Total
Linear (IR)	15	15	15 / 15
Quadratic (Raman)	15	15	15 / 15
IR + Raman	- - - -	- - - -	0* / 0

* Parity Mutual Exclusion Principle

Characters of force fields
(Symmetric powers of vibration representation)

Force field	E	i
linear	30	0
quadratic	465	15
cubic	4.960	0
quartic	40.920	120
quintic	278.256	0
sextic	1.623.160	680

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

Force field	Ag	Au
linear	15	15
quadratic	240	225
cubic	2.480	2.480
quartic	20.520	20.400
quintic	139.128	139.128
sextic	811.920	811.240

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)
..120.	AgAg.	..120.	AuAu.
Subtotal: 240 / 2 / 2
Irrep combinations (i,j) with indices: pos(Ag) ≤ i ≤ j ≤ pos(Au)
Subtotal: 0 / 0 / 1
Total: 240 / 2 / 3

Contributions to nonvanishing cubic force field constants

Irrep combinations (i,i,i) with indices: pos(Ag) ≤ i ≤ pos(Au)
..680.	AgAgAg.
Subtotal: 680 / 1 / 2
Irrep combinations (i,i,j) (i,j,j) with indices: pos(Ag) ≤ i ≤ j ≤ pos(Au)
..1.800.	AgAuAu.
Subtotal: 1.800 / 1 / 2
Irrep combinations (i,j,k) with indices: pos(Ag) ≤ i ≤ j ≤ k ≤ pos(Au)
Subtotal: 0 / 0 / 0
Total: 2.480 / 2 / 4

Contributions to nonvanishing quartic force field constants

Irrep combinations (i,i,i,i) with indices: pos(Ag) ≤ i ≤ pos(Au)
..3.060.	AgAgAgAg.	..3.060.	AuAuAuAu.
Subtotal: 6.120 / 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)
..14.400.	AgAgAuAu.
Subtotal: 14.400 / 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: 20.520 / 3 / 5

Calculate contributions to

Ag	Au

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