Characters of representations for molecular motions

Motion	E	C2	i	σh
Cartesian 3N	36	-4	0	8
Translation (x,y,z)	3	-1	-3	1
Rotation (Rx,Ry,Rz)	3	-1	3	-1
Vibration	30	-2	0	8

Decomposition to irreducible representations

Motion	Ag	Bg	Au	Bu	Total
Cartesian 3N	10	8	6	12	36
Translation (x,y,z)	0	0	1	2	3
Rotation (Rx,Ry,Rz)	1	2	0	0	3
Vibration	9	6	5	10	30

Molecular parameter

Number of Atoms (N)	12
Number of internal coordinates	30
Number of independant internal coordinates	9
Number of vibrational modes	30

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Force field analysis

Allowed / forbidden vibronational transitions

Operator	Ag	Bg	Au	Bu	Total
Linear (IR)	9	6	5	10	15 / 15
Quadratic (Raman)	9	6	5	10	15 / 15
IR + Raman	- - - -	- - - -	- - - -	- - - -	0* / 0

* Parity Mutual Exclusion Principle

Characters of force fields
(Symmetric powers of vibration representation)

Force field	E	C2	i	σh
linear	30	-2	0	8
quadratic	465	17	15	47
cubic	4.960	-32	0	208
quartic	40.920	152	120	792
quintic	278.256	-272	0	2.640
sextic	1.623.160	952	680	8.008

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

Force field	Ag	Bg	Au	Bu
linear	9	6	5	10
quadratic	136	104	105	120
cubic	1.284	1.196	1.180	1.300
quartic	10.496	10.024	10.040	10.360
quintic	70.156	68.972	68.836	70.292
sextic	408.200	403.720	403.856	407.384

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)
..45.	AgAg.	..21.	BgBg.	..15.	AuAu.	..55.	BuBu.
Subtotal: 136 / 4 / 4
Irrep combinations (i,j) with indices: pos(Ag) ≤ i ≤ j ≤ pos(Bu)
Subtotal: 0 / 0 / 6
Total: 136 / 4 / 10

Contributions to nonvanishing cubic force field constants

Irrep combinations (i,i,i) with indices: pos(Ag) ≤ i ≤ pos(Bu)
..165.	AgAgAg.
Subtotal: 165 / 1 / 4
Irrep combinations (i,i,j) (i,j,j) with indices: pos(Ag) ≤ i ≤ j ≤ pos(Bu)
..189.	AgBgBg.	..135.	AgAuAu.	..495.	AgBuBu.
Subtotal: 819 / 3 / 12
Irrep combinations (i,j,k) with indices: pos(Ag) ≤ i ≤ j ≤ k ≤ pos(Bu)
..300.	BgAuBu.
Subtotal: 300 / 1 / 4
Total: 1.284 / 5 / 20

Contributions to nonvanishing quartic force field constants

Irrep combinations (i,i,i,i) with indices: pos(Ag) ≤ i ≤ pos(Bu)
..495.	AgAgAgAg.	..126.	BgBgBgBg.	..70.	AuAuAuAu.	..715.	BuBuBuBu.
Subtotal: 1.406 / 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)
..945.	AgAgBgBg.	..675.	AgAgAuAu.	..2.475.	AgAgBuBu.	..315.	BgBgAuAu.	..1.155.	BgBgBuBu.	..825.	AuAuBuBu.
Subtotal: 6.390 / 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)
..2.700.	AgBgAuBu.
Subtotal: 2.700 / 1 / 1
Total: 10.496 / 11 / 35

Calculate contributions to

Ag	Bg	Au	Bu

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