Characters of representations for molecular motions

Motion	E	2C3	3C'2
Cartesian 3N	24	0	0
Translation (x,y,z)	3	0	-1
Rotation (Rx,Ry,Rz)	3	0	-1
Vibration	18	0	2

Decomposition to irreducible representations

Motion	A1	A2	E	Total
Cartesian 3N	4	4	8	16
Translation (x,y,z)	0	1	1	2
Rotation (Rx,Ry,Rz)	0	1	1	2
Vibration	4	2	6	12

Molecular parameter

Number of Atoms (N)	8
Number of internal coordinates	18
Number of independant internal coordinates	4
Number of vibrational modes	12

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Force field analysis

Allowed / forbidden vibronational transitions

Operator	A1	A2	E	Total
Linear (IR)	4	2	6	8 / 4
Quadratic (Raman)	4	2	6	10 / 2
IR + Raman	- - - -	- - - -	6	6 / 0

Characters of force fields
(Symmetric powers of vibration representation)

Force field	E	2C3	3C'2
linear	18	0	2
quadratic	171	0	11
cubic	1.140	6	20
quartic	5.985	0	65
quintic	26.334	0	110
sextic	100.947	21	275

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

Force field	A1	A2	E
linear	4	2	6
quadratic	34	23	57
cubic	202	182	378
quartic	1.030	965	1.995
quintic	4.444	4.334	8.778
sextic	16.969	16.694	33.642

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 D₃

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(A1) ≤ i ≤ pos(E)
..10.	A1A1.	..3.	A2A2.	..21.	EE.
Subtotal: 34 / 3 / 3
Irrep combinations (i,j) with indices: pos(A1) ≤ i ≤ j ≤ pos(E)
Subtotal: 0 / 0 / 3
Total: 34 / 3 / 6

Contributions to nonvanishing cubic force field constants

Irrep combinations (i,i,i) with indices: pos(A1) ≤ i ≤ pos(E)
..20.	A1A1A1.	..56.	EEE.
Subtotal: 76 / 2 / 3
Irrep combinations (i,i,j) (i,j,j) with indices: pos(A1) ≤ i ≤ j ≤ pos(E)
..12.	A1A2A2.	..84.	A1EE.	..30.	A2EE.
Subtotal: 126 / 3 / 6
Irrep combinations (i,j,k) with indices: pos(A1) ≤ i ≤ j ≤ k ≤ pos(E)
Subtotal: 0 / 0 / 1
Total: 202 / 5 / 10

Contributions to nonvanishing quartic force field constants

Irrep combinations (i,i,i,i) with indices: pos(A1) ≤ i ≤ pos(E)
..35.	A1A1A1A1.	..5.	A2A2A2A2.	..231.	EEEE.
Subtotal: 271 / 3 / 3
Irrep combinations (i,i,i,j) (i,j,j,j) with indices: pos(A1) ≤ i ≤ j ≤ pos(E)
..224.	A1EEE.	..112.	A2EEE.
Subtotal: 336 / 2 / 6
Irrep combinations (i,i,j,j) with indices: pos(A1) ≤ i ≤ j ≤ pos(E)
..30.	A1A1A2A2.	..210.	A1A1EE.	..63.	A2A2EE.
Subtotal: 303 / 3 / 3
Irrep combinations (i,i,j,k) (i,j,j,k) (i,j,k,k) with indices: pos(A1) ≤ i ≤ j ≤ k ≤ pos(E)
..120.	A1A2EE.
Subtotal: 120 / 1 / 3
Irrep combinations (i,j,k,l) with indices: pos(A1) ≤ i ≤ j ≤ k ≤ l ≤ pos(E)
Subtotal: 0 / 0 / 0
Total: 1.030 / 9 / 15

Calculate contributions to

A1	A2	E

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