Characters of representations for molecular motions

Motion	E	2C3	3σv
Cartesian 3N	60	0	4
Translation (x,y,z)	3	0	1
Rotation (Rx,Ry,Rz)	3	0	-1
Vibration	54	0	4

Decomposition to irreducible representations

Motion	A1	A2	E	Total
Cartesian 3N	12	8	20	40
Translation (x,y,z)	1	0	1	2
Rotation (Rx,Ry,Rz)	0	1	1	2
Vibration	11	7	18	36

Molecular parameter

Number of Atoms (N)	20
Number of internal coordinates	54
Number of independant internal coordinates	11
Number of vibrational modes	36

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Force field analysis

Allowed / forbidden vibronational transitions

Operator	A1	A2	E	Total
Linear (IR)	11	7	18	29 / 7
Quadratic (Raman)	11	7	18	29 / 7
IR + Raman	11	7	18	29 / 7

Characters of force fields
(Symmetric powers of vibration representation)

Force field	E	2C3	3σv
linear	54	0	4
quadratic	1.485	0	35
cubic	27.720	18	120
quartic	395.010	0	610
quintic	4.582.116	0	1.856
sextic	45.057.474	171	7.134

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

Force field	A1	A2	E
linear	11	7	18
quadratic	265	230	495
cubic	4.686	4.566	9.234
quartic	66.140	65.530	131.670
quintic	764.614	762.758	1.527.372
sextic	7.513.203	7.506.069	15.019.101

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_3v

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)
..66.	A1A1.	..28.	A2A2.	..171.	EE.
Subtotal: 265 / 3 / 3
Irrep combinations (i,j) with indices: pos(A1) ≤ i ≤ j ≤ pos(E)
Subtotal: 0 / 0 / 3
Total: 265 / 3 / 6

Contributions to nonvanishing cubic force field constants

Irrep combinations (i,i,i) with indices: pos(A1) ≤ i ≤ pos(E)
..286.	A1A1A1.	..1.140.	EEE.
Subtotal: 1.426 / 2 / 3
Irrep combinations (i,i,j) (i,j,j) with indices: pos(A1) ≤ i ≤ j ≤ pos(E)
..308.	A1A2A2.	..1.881.	A1EE.	..1.071.	A2EE.
Subtotal: 3.260 / 3 / 6
Irrep combinations (i,j,k) with indices: pos(A1) ≤ i ≤ j ≤ k ≤ pos(E)
Subtotal: 0 / 0 / 1
Total: 4.686 / 5 / 10

Contributions to nonvanishing quartic force field constants

Irrep combinations (i,i,i,i) with indices: pos(A1) ≤ i ≤ pos(E)
..1.001.	A1A1A1A1.	..210.	A2A2A2A2.	..14.706.	EEEE.
Subtotal: 15.917 / 3 / 3
Irrep combinations (i,i,i,j) (i,j,j,j) with indices: pos(A1) ≤ i ≤ j ≤ pos(E)
..12.540.	A1EEE.	..7.980.	A2EEE.
Subtotal: 20.520 / 2 / 6
Irrep combinations (i,i,j,j) with indices: pos(A1) ≤ i ≤ j ≤ pos(E)
..1.848.	A1A1A2A2.	..11.286.	A1A1EE.	..4.788.	A2A2EE.
Subtotal: 17.922 / 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)
..11.781.	A1A2EE.
Subtotal: 11.781 / 1 / 3
Irrep combinations (i,j,k,l) with indices: pos(A1) ≤ i ≤ j ≤ k ≤ l ≤ pos(E)
Subtotal: 0 / 0 / 0
Total: 66.140 / 9 / 15

Calculate contributions to

A1	A2	E

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