Characters of representations for molecular motions

Motion	E	2C3	3C'2
Cartesian 3N	39	0	-1
Translation (x,y,z)	3	0	-1
Rotation (Rx,Ry,Rz)	3	0	-1
Vibration	33	0	1

Decomposition to irreducible representations

Motion	A1	A2	E	Total
Cartesian 3N	6	7	13	26
Translation (x,y,z)	0	1	1	2
Rotation (Rx,Ry,Rz)	0	1	1	2
Vibration	6	5	11	22

Molecular parameter

Number of Atoms (N)	13
Number of internal coordinates	33
Number of independant internal coordinates	6
Number of vibrational modes	22

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Force field analysis

Allowed / forbidden vibronational transitions

Operator	A1	A2	E	Total
Linear (IR)	6	5	11	16 / 6
Quadratic (Raman)	6	5	11	17 / 5
IR + Raman	- - - -	- - - -	11	11 / 0

Characters of force fields
(Symmetric powers of vibration representation)

Force field	E	2C3	3C'2
linear	33	0	1
quadratic	561	0	17
cubic	6.545	11	17
quartic	58.905	0	153
quintic	435.897	0	153
sextic	2.760.681	66	969

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

Force field	A1	A2	E
linear	6	5	11
quadratic	102	85	187
cubic	1.103	1.086	2.178
quartic	9.894	9.741	19.635
quintic	72.726	72.573	145.299
sextic	460.620	459.651	920.205

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)
..21.	A1A1.	..15.	A2A2.	..66.	EE.
Subtotal: 102 / 3 / 3
Irrep combinations (i,j) with indices: pos(A1) ≤ i ≤ j ≤ pos(E)
Subtotal: 0 / 0 / 3
Total: 102 / 3 / 6

Contributions to nonvanishing cubic force field constants

Irrep combinations (i,i,i) with indices: pos(A1) ≤ i ≤ pos(E)
..56.	A1A1A1.	..286.	EEE.
Subtotal: 342 / 2 / 3
Irrep combinations (i,i,j) (i,j,j) with indices: pos(A1) ≤ i ≤ j ≤ pos(E)
..90.	A1A2A2.	..396.	A1EE.	..275.	A2EE.
Subtotal: 761 / 3 / 6
Irrep combinations (i,j,k) with indices: pos(A1) ≤ i ≤ j ≤ k ≤ pos(E)
Subtotal: 0 / 0 / 1
Total: 1.103 / 5 / 10

Contributions to nonvanishing quartic force field constants

Irrep combinations (i,i,i,i) with indices: pos(A1) ≤ i ≤ pos(E)
..126.	A1A1A1A1.	..70.	A2A2A2A2.	..2.211.	EEEE.
Subtotal: 2.407 / 3 / 3
Irrep combinations (i,i,i,j) (i,j,j,j) with indices: pos(A1) ≤ i ≤ j ≤ pos(E)
..1.716.	A1EEE.	..1.430.	A2EEE.
Subtotal: 3.146 / 2 / 6
Irrep combinations (i,i,j,j) with indices: pos(A1) ≤ i ≤ j ≤ pos(E)
..315.	A1A1A2A2.	..1.386.	A1A1EE.	..990.	A2A2EE.
Subtotal: 2.691 / 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)
..1.650.	A1A2EE.
Subtotal: 1.650 / 1 / 3
Irrep combinations (i,j,k,l) with indices: pos(A1) ≤ i ≤ j ≤ k ≤ l ≤ pos(E)
Subtotal: 0 / 0 / 0
Total: 9.894 / 9 / 15

Calculate contributions to

A1	A2	E

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