Characters of representations for molecular motions

Motion	E	C2	i	σh
Cartesian 3N	48	0	0	16
Translation (x,y,z)	3	-1	-3	1
Rotation (Rx,Ry,Rz)	3	-1	3	-1
Vibration	42	2	0	16

Decomposition to irreducible representations

Motion	Ag	Bg	Au	Bu	Total
Cartesian 3N	16	8	8	16	48
Translation (x,y,z)	0	0	1	2	3
Rotation (Rx,Ry,Rz)	1	2	0	0	3
Vibration	15	6	7	14	42

Molecular parameter

Number of Atoms (N)	16
Number of internal coordinates	42
Number of independant internal coordinates	15
Number of vibrational modes	42

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Force field analysis

Allowed / forbidden vibronational transitions

Operator	Ag	Bg	Au	Bu	Total
Linear (IR)	15	6	7	14	21 / 21
Quadratic (Raman)	15	6	7	14	21 / 21
IR + Raman	- - - -	- - - -	- - - -	- - - -	0* / 0

* Parity Mutual Exclusion Principle

Characters of force fields
(Symmetric powers of vibration representation)

Force field	E	C2	i	σh
linear	42	2	0	16
quadratic	903	23	21	149
cubic	13.244	44	0	1.024
quartic	148.995	275	231	5.735
quintic	1.370.754	506	0	27.568
sextic	10.737.573	2.277	1.771	117.483

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

Force field	Ag	Bg	Au	Bu
linear	15	6	7	14
quadratic	274	188	189	252
cubic	3.578	3.044	3.066	3.556
quartic	38.809	35.804	35.826	38.556
quintic	349.707	335.670	335.923	349.454
sextic	2.714.776	2.654.896	2.655.149	2.712.752

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)
..120.	AgAg.	..21.	BgBg.	..28.	AuAu.	..105.	BuBu.
Subtotal: 274 / 4 / 4
Irrep combinations (i,j) with indices: pos(Ag) ≤ i ≤ j ≤ pos(Bu)
Subtotal: 0 / 0 / 6
Total: 274 / 4 / 10

Contributions to nonvanishing cubic force field constants

Irrep combinations (i,i,i) with indices: pos(Ag) ≤ i ≤ pos(Bu)
..680.	AgAgAg.
Subtotal: 680 / 1 / 4
Irrep combinations (i,i,j) (i,j,j) with indices: pos(Ag) ≤ i ≤ j ≤ pos(Bu)
..315.	AgBgBg.	..420.	AgAuAu.	..1.575.	AgBuBu.
Subtotal: 2.310 / 3 / 12
Irrep combinations (i,j,k) with indices: pos(Ag) ≤ i ≤ j ≤ k ≤ pos(Bu)
..588.	BgAuBu.
Subtotal: 588 / 1 / 4
Total: 3.578 / 5 / 20

Contributions to nonvanishing quartic force field constants

Irrep combinations (i,i,i,i) with indices: pos(Ag) ≤ i ≤ pos(Bu)
..3.060.	AgAgAgAg.	..126.	BgBgBgBg.	..210.	AuAuAuAu.	..2.380.	BuBuBuBu.
Subtotal: 5.776 / 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)
..2.520.	AgAgBgBg.	..3.360.	AgAgAuAu.	..12.600.	AgAgBuBu.	..588.	BgBgAuAu.	..2.205.	BgBgBuBu.	..2.940.	AuAuBuBu.
Subtotal: 24.213 / 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)
..8.820.	AgBgAuBu.
Subtotal: 8.820 / 1 / 1
Total: 38.809 / 11 / 35

Calculate contributions to

Ag	Bg	Au	Bu

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