Characters of representations for molecular motions

Motion	E	C2	i	σh
Cartesian 3N	0	0	0	0
Translation (x,y,z)	3	-1	-3	1
Rotation (Rx,Ry,Rz)	3	-1	3	-1
Vibration	-6	2	0	0

Decomposition to irreducible representations

Motion	Ag	Bg	Au	Bu	Total
Cartesian 3N	0	0	0	0	0
Translation (x,y,z)	0	0	1	2	3
Rotation (Rx,Ry,Rz)	1	2	0	0	3
Vibration	-1	-2	-1	-2	-6

Molecular parameter

Number of Atoms (N)	0
Number of internal coordinates	-6
Number of independant internal coordinates	-1
Number of vibrational modes	-6

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Force field analysis

Allowed / forbidden vibronational transitions

Operator	Ag	Bg	Au	Bu	Total
Linear (IR)	-1	-2	-1	-2	-3 / -3
Quadratic (Raman)	-1	-2	-1	-2	-3 / -3
IR + Raman	- - - -	- - - -	- - - -	- - - -	0* / 0

* Parity Mutual Exclusion Principle

Characters of force fields
(Symmetric powers of vibration representation)

Force field	E	C2	i	σh
linear	-6	2	0	0
quadratic	15	-1	-3	-3
cubic	-20	-4	0	0
quartic	15	-1	3	3
quintic	-6	2	0	0
sextic	1	1	-1	-1

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

Force field	Ag	Bg	Au	Bu
linear	-1	-2	-1	-2
quadratic	2	4	5	4
cubic	-6	-4	-6	-4
quartic	5	4	2	4
quintic	-1	-2	-1	-2
sextic	0	0	1	0

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_2h

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(Bu)
..1.	BgBg.	..1.	BuBu.
Subtotal: 2 / 2 / 4
Irrep combinations (i,j) with indices: pos(Ag) ≤ i ≤ j ≤ pos(Bu)
Subtotal: 0 / 0 / 6
Total: 2 / 2 / 10

Contributions to nonvanishing cubic force field constants

Irrep combinations (i,i,i) with indices: pos(Ag) ≤ i ≤ pos(Bu)
Subtotal: 0 / 0 / 4
Irrep combinations (i,i,j) (i,j,j) with indices: pos(Ag) ≤ i ≤ j ≤ pos(Bu)
Subtotal: -2 / 0 / 12
Irrep combinations (i,j,k) with indices: pos(Ag) ≤ i ≤ j ≤ k ≤ pos(Bu)
Subtotal: -4 / 0 / 4
Total: -6 / 0 / 20

Contributions to nonvanishing quartic force field constants

Irrep combinations (i,i,i,i) with indices: pos(Ag) ≤ i ≤ pos(Bu)
Subtotal: 0 / 0 / 4
Irrep combinations (i,i,i,j) (i,j,j,j) with indices: pos(Ag) ≤ i ≤ j ≤ pos(Bu)
Subtotal: 0 / 0 / 12
Irrep combinations (i,i,j,j) with indices: pos(Ag) ≤ i ≤ j ≤ pos(Bu)
..1.	BgBgBuBu.
Subtotal: 1 / 1 / 6
Irrep combinations (i,i,j,k) (i,j,j,k) (i,j,k,k) with indices: pos(Ag) ≤ i ≤ j ≤ k ≤ pos(Bu)
Subtotal: 0 / 0 / 12
Irrep combinations (i,j,k,l) with indices: pos(Ag) ≤ i ≤ j ≤ k ≤ l ≤ pos(Bu)
..4.	AgBgAuBu.
Subtotal: 4 / 1 / 1
Total: 5 / 2 / 35

Calculate contributions to

Ag	Bg	Au	Bu

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