Characters of representations for molecular motions

Motion	E	C2	i	σh
Cartesian 3N	24	0	0	8
Translation (x,y,z)	3	-1	-3	1
Rotation (Rx,Ry,Rz)	3	-1	3	-1
Vibration	18	2	0	8

Decomposition to irreducible representations

Motion	Ag	Bg	Au	Bu	Total
Cartesian 3N	8	4	4	8	24
Translation (x,y,z)	0	0	1	2	3
Rotation (Rx,Ry,Rz)	1	2	0	0	3
Vibration	7	2	3	6	18

Molecular parameter

Number of Atoms (N)	8
Number of internal coordinates	18
Number of independant internal coordinates	7
Number of vibrational modes	18

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Force field analysis

Allowed / forbidden vibronational transitions

Operator	Ag	Bg	Au	Bu	Total
Linear (IR)	7	2	3	6	9 / 9
Quadratic (Raman)	7	2	3	6	9 / 9
IR + Raman	- - - -	- - - -	- - - -	- - - -	0* / 0

* Parity Mutual Exclusion Principle

Characters of force fields
(Symmetric powers of vibration representation)

Force field	E	C2	i	σh
linear	18	2	0	8
quadratic	171	11	9	41
cubic	1.140	20	0	160
quartic	5.985	65	45	525
quintic	26.334	110	0	1.512
sextic	100.947	275	165	3.941

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

Force field	Ag	Bg	Au	Bu
linear	7	2	3	6
quadratic	58	32	33	48
cubic	330	240	250	320
quartic	1.655	1.360	1.370	1.600
quintic	6.989	6.178	6.233	6.934
sextic	26.332	24.224	24.279	26.112

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)
..28.	AgAg.	..3.	BgBg.	..6.	AuAu.	..21.	BuBu.
Subtotal: 58 / 4 / 4
Irrep combinations (i,j) with indices: pos(Ag) ≤ i ≤ j ≤ pos(Bu)
Subtotal: 0 / 0 / 6
Total: 58 / 4 / 10

Contributions to nonvanishing cubic force field constants

Irrep combinations (i,i,i) with indices: pos(Ag) ≤ i ≤ pos(Bu)
..84.	AgAgAg.
Subtotal: 84 / 1 / 4
Irrep combinations (i,i,j) (i,j,j) with indices: pos(Ag) ≤ i ≤ j ≤ pos(Bu)
..21.	AgBgBg.	..42.	AgAuAu.	..147.	AgBuBu.
Subtotal: 210 / 3 / 12
Irrep combinations (i,j,k) with indices: pos(Ag) ≤ i ≤ j ≤ k ≤ pos(Bu)
..36.	BgAuBu.
Subtotal: 36 / 1 / 4
Total: 330 / 5 / 20

Contributions to nonvanishing quartic force field constants

Irrep combinations (i,i,i,i) with indices: pos(Ag) ≤ i ≤ pos(Bu)
..210.	AgAgAgAg.	..5.	BgBgBgBg.	..15.	AuAuAuAu.	..126.	BuBuBuBu.
Subtotal: 356 / 4 / 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)
..84.	AgAgBgBg.	..168.	AgAgAuAu.	..588.	AgAgBuBu.	..18.	BgBgAuAu.	..63.	BgBgBuBu.	..126.	AuAuBuBu.
Subtotal: 1.047 / 6 / 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)
..252.	AgBgAuBu.
Subtotal: 252 / 1 / 1
Total: 1.655 / 11 / 35

Calculate contributions to

Ag	Bg	Au	Bu

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