Characters of representations for molecular motions

Motion	E	C3	(C3)2	i	(S6)5	S6
Cartesian 3N	21	0	0	-3	0	0
Translation (x,y,z)	3	0	0	-3	0	0
Rotation (Rx,Ry,Rz)	3	0	0	3	0	0
Vibration	15	0	0	-3	0	0

Decomposition to irreducible representations

Motion	Ag	Eg*	Au	Eu*	Total
Cartesian 3N	3	3	4	4	14
Translation (x,y,z)	0	0	1	1	2
Rotation (Rx,Ry,Rz)	1	1	0	0	2
Vibration	2	2	3	3	10

Molecular parameter

Number of Atoms (N)	7
Number of internal coordinates	15
Number of independant internal coordinates	2
Number of vibrational modes	10

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Force field analysis

Allowed / forbidden vibronational transitions

Operator	Ag	Eg*	Au	Eu*	Total
Linear (IR)	2	2	3	3	6 / 4
Quadratic (Raman)	2	2	3	3	4 / 6
IR + Raman	- - - -	- - - -	- - - -	- - - -	0* / 0

* Parity Mutual Exclusion Principle

Characters of force fields
(Symmetric powers of vibration representation)

Force field	E	C3	(C3)2	i	(S6)5	S6
linear	15	0	0	-3	0	0
quadratic	120	0	0	12	0	0
cubic	680	5	5	-28	-1	-1
quartic	3.060	0	0	72	0	0
quintic	11.628	0	0	-144	0	0
sextic	38.760	15	15	300	3	3

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

Force field	Ag	Eg*	Au	Eu*
linear	2	2	3	3
quadratic	22	22	18	18
cubic	110	108	120	117
quartic	522	522	498	498
quintic	1.914	1.914	1.962	1.962
sextic	6.516	6.507	6.414	6.408

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 S₆

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(Eu)
..3.	AgAg.	..4.	EgEg.	..6.	AuAu.	..9.	EuEu.
Subtotal: 22 / 4 / 4
Irrep combinations (i,j) with indices: pos(Ag) ≤ i ≤ j ≤ pos(Eu)
Subtotal: 0 / 0 / 6
Total: 22 / 4 / 10

Contributions to nonvanishing cubic force field constants

Irrep combinations (i,i,i) with indices: pos(Ag) ≤ i ≤ pos(Eu)
..4.	AgAgAg.	..8.	EgEgEg.
Subtotal: 12 / 2 / 4
Irrep combinations (i,i,j) (i,j,j) with indices: pos(Ag) ≤ i ≤ j ≤ pos(Eu)
..8.	AgEgEg.	..12.	AgAuAu.	..18.	AgEuEu.	..24.	EgEuEu.
Subtotal: 62 / 4 / 12
Irrep combinations (i,j,k) with indices: pos(Ag) ≤ i ≤ j ≤ k ≤ pos(Eu)
..36.	EgAuEu.
Subtotal: 36 / 1 / 4
Total: 110 / 7 / 20

Contributions to nonvanishing quartic force field constants

Irrep combinations (i,i,i,i) with indices: pos(Ag) ≤ i ≤ pos(Eu)
..5.	AgAgAgAg.	..9.	EgEgEgEg.	..15.	AuAuAuAu.	..36.	EuEuEuEu.
Subtotal: 65 / 4 / 4
Irrep combinations (i,i,i,j) (i,j,j,j) with indices: pos(Ag) ≤ i ≤ j ≤ pos(Eu)
..16.	AgEgEgEg.	..60.	AuEuEuEu.
Subtotal: 76 / 2 / 12
Irrep combinations (i,i,j,j) with indices: pos(Ag) ≤ i ≤ j ≤ pos(Eu)
..12.	AgAgEgEg.	..18.	AgAgAuAu.	..27.	AgAgEuEu.	..24.	EgEgAuAu.	..72.	EgEgEuEu.	..54.	AuAuEuEu.
Subtotal: 207 / 6 / 6
Irrep combinations (i,i,j,k) (i,j,j,k) (i,j,k,k) with indices: pos(Ag) ≤ i ≤ j ≤ k ≤ pos(Eu)
..54.	EgEgAuEu.	..48.	AgEgEuEu.
Subtotal: 102 / 2 / 12
Irrep combinations (i,j,k,l) with indices: pos(Ag) ≤ i ≤ j ≤ k ≤ l ≤ pos(Eu)
..72.	AgEgAuEu.
Subtotal: 72 / 1 / 1
Total: 522 / 15 / 35

Calculate contributions to

Ag	Eg	Au	Eu

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