Characters of representations for molecular motions

Motion	E	C2	i	σh
Cartesian 3N	42	0	0	6
Translation (x,y,z)	3	-1	-3	1
Rotation (Rx,Ry,Rz)	3	-1	3	-1
Vibration	36	2	0	6

Decomposition to irreducible representations

Motion	Ag	Bg	Au	Bu	Total
Cartesian 3N	12	9	9	12	42
Translation (x,y,z)	0	0	1	2	3
Rotation (Rx,Ry,Rz)	1	2	0	0	3
Vibration	11	7	8	10	36

Molecular parameter

Number of Atoms (N)	14
Number of internal coordinates	36
Number of independant internal coordinates	11
Number of vibrational modes	36

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Force field analysis

Allowed / forbidden vibronational transitions

Operator	Ag	Bg	Au	Bu	Total
Linear (IR)	11	7	8	10	18 / 18
Quadratic (Raman)	11	7	8	10	18 / 18
IR + Raman	- - - -	- - - -	- - - -	- - - -	0* / 0

* Parity Mutual Exclusion Principle

Characters of force fields
(Symmetric powers of vibration representation)

Force field	E	C2	i	σh
linear	36	2	0	6
quadratic	666	20	18	36
cubic	8.436	38	0	146
quartic	82.251	209	171	561
quintic	658.008	380	0	1.812
sextic	4.496.388	1.520	1.140	5.552

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

Force field	Ag	Bg	Au	Bu
linear	11	7	8	10
quadratic	185	157	158	166
cubic	2.155	2.063	2.082	2.136
quartic	20.798	20.413	20.432	20.608
quintic	165.050	163.954	164.144	164.860
sextic	1.126.150	1.122.614	1.122.804	1.124.820

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)
..66.	AgAg.	..28.	BgBg.	..36.	AuAu.	..55.	BuBu.
Subtotal: 185 / 4 / 4
Irrep combinations (i,j) with indices: pos(Ag) ≤ i ≤ j ≤ pos(Bu)
Subtotal: 0 / 0 / 6
Total: 185 / 4 / 10

Contributions to nonvanishing cubic force field constants

Irrep combinations (i,i,i) with indices: pos(Ag) ≤ i ≤ pos(Bu)
..286.	AgAgAg.
Subtotal: 286 / 1 / 4
Irrep combinations (i,i,j) (i,j,j) with indices: pos(Ag) ≤ i ≤ j ≤ pos(Bu)
..308.	AgBgBg.	..396.	AgAuAu.	..605.	AgBuBu.
Subtotal: 1.309 / 3 / 12
Irrep combinations (i,j,k) with indices: pos(Ag) ≤ i ≤ j ≤ k ≤ pos(Bu)
..560.	BgAuBu.
Subtotal: 560 / 1 / 4
Total: 2.155 / 5 / 20

Contributions to nonvanishing quartic force field constants

Irrep combinations (i,i,i,i) with indices: pos(Ag) ≤ i ≤ pos(Bu)
..1.001.	AgAgAgAg.	..210.	BgBgBgBg.	..330.	AuAuAuAu.	..715.	BuBuBuBu.
Subtotal: 2.256 / 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)
..1.848.	AgAgBgBg.	..2.376.	AgAgAuAu.	..3.630.	AgAgBuBu.	..1.008.	BgBgAuAu.	..1.540.	BgBgBuBu.	..1.980.	AuAuBuBu.
Subtotal: 12.382 / 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)
..6.160.	AgBgAuBu.
Subtotal: 6.160 / 1 / 1
Total: 20.798 / 11 / 35

Calculate contributions to

Ag	Bg	Au	Bu

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