Characters of representations for molecular motions

Motion	E	i
Cartesian 3N	39	-3
Translation (x,y,z)	3	-3
Rotation (Rx,Ry,Rz)	3	3
Vibration	33	-3

Decomposition to irreducible representations

Motion	Ag	Au	Total
Cartesian 3N	18	21	39
Translation (x,y,z)	0	3	3
Rotation (Rx,Ry,Rz)	3	0	3
Vibration	15	18	33

Molecular parameter

Number of Atoms (N)	13
Number of internal coordinates	33
Number of independant internal coordinates	15
Number of vibrational modes	33

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Force field analysis

Allowed / forbidden vibronational transitions

Operator	Ag	Au	Total
Linear (IR)	15	18	18 / 15
Quadratic (Raman)	15	18	15 / 18
IR + Raman	- - - -	- - - -	0* / 0

* Parity Mutual Exclusion Principle

Characters of force fields
(Symmetric powers of vibration representation)

Force field	E	i
linear	33	-3
quadratic	561	21
cubic	6.545	-55
quartic	58.905	225
quintic	435.897	-531
sextic	2.760.681	1.653

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

Force field	Ag	Au
linear	15	18
quadratic	291	270
cubic	3.245	3.300
quartic	29.565	29.340
quintic	217.683	218.214
sextic	1.381.167	1.379.514

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.	..171.	AuAu.
Subtotal: 291 / 2 / 2
Irrep combinations (i,j) with indices: pos(Ag) ≤ i ≤ j ≤ pos(Au)
Subtotal: 0 / 0 / 1
Total: 291 / 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)
..2.565.	AgAuAu.
Subtotal: 2.565 / 1 / 2
Irrep combinations (i,j,k) with indices: pos(Ag) ≤ i ≤ j ≤ k ≤ pos(Au)
Subtotal: 0 / 0 / 0
Total: 3.245 / 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.	..5.985.	AuAuAuAu.
Subtotal: 9.045 / 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)
..20.520.	AgAgAuAu.
Subtotal: 20.520 / 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: 29.565 / 3 / 5

Calculate contributions to

Ag	Au

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