Characters of representations for molecular motions

Motion	E	2C3	3σv
Cartesian 3N	48	0	8
Translation (x,y,z)	3	0	1
Rotation (Rx,Ry,Rz)	3	0	-1
Vibration	42	0	8

Decomposition to irreducible representations

Motion	A1	A2	E	Total
Cartesian 3N	12	4	16	32
Translation (x,y,z)	1	0	1	2
Rotation (Rx,Ry,Rz)	0	1	1	2
Vibration	11	3	14	28

Molecular parameter

Number of Atoms (N)	16
Number of internal coordinates	42
Number of independant internal coordinates	11
Number of vibrational modes	28

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Force field analysis

Allowed / forbidden vibronational transitions

Operator	A1	A2	E	Total
Linear (IR)	11	3	14	25 / 3
Quadratic (Raman)	11	3	14	25 / 3
IR + Raman	11	3	14	25 / 3

Characters of force fields
(Symmetric powers of vibration representation)

Force field	E	2C3	3σv
linear	42	0	8
quadratic	903	0	53
cubic	13.244	14	256
quartic	148.995	0	1.095
quintic	1.370.754	0	4.056
sextic	10.737.573	105	13.803

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

Force field	A1	A2	E
linear	11	3	14
quadratic	177	124	301
cubic	2.340	2.084	4.410
quartic	25.380	24.285	49.665
quintic	230.487	226.431	456.918
sextic	1.796.532	1.782.729	3.579.156

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.	..6.	A2A2.	..105.	EE.
Subtotal: 177 / 3 / 3
Irrep combinations (i,j) with indices: pos(A1) ≤ i ≤ j ≤ pos(E)
Subtotal: 0 / 0 / 3
Total: 177 / 3 / 6

Contributions to nonvanishing cubic force field constants

Irrep combinations (i,i,i) with indices: pos(A1) ≤ i ≤ pos(E)
..286.	A1A1A1.	..560.	EEE.
Subtotal: 846 / 2 / 3
Irrep combinations (i,i,j) (i,j,j) with indices: pos(A1) ≤ i ≤ j ≤ pos(E)
..66.	A1A2A2.	..1.155.	A1EE.	..273.	A2EE.
Subtotal: 1.494 / 3 / 6
Irrep combinations (i,j,k) with indices: pos(A1) ≤ i ≤ j ≤ k ≤ pos(E)
Subtotal: 0 / 0 / 1
Total: 2.340 / 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.	..15.	A2A2A2A2.	..5.565.	EEEE.
Subtotal: 6.581 / 3 / 3
Irrep combinations (i,i,i,j) (i,j,j,j) with indices: pos(A1) ≤ i ≤ j ≤ pos(E)
..6.160.	A1EEE.	..1.680.	A2EEE.
Subtotal: 7.840 / 2 / 6
Irrep combinations (i,i,j,j) with indices: pos(A1) ≤ i ≤ j ≤ pos(E)
..396.	A1A1A2A2.	..6.930.	A1A1EE.	..630.	A2A2EE.
Subtotal: 7.956 / 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)
..3.003.	A1A2EE.
Subtotal: 3.003 / 1 / 3
Irrep combinations (i,j,k,l) with indices: pos(A1) ≤ i ≤ j ≤ k ≤ l ≤ pos(E)
Subtotal: 0 / 0 / 0
Total: 25.380 / 9 / 15

Calculate contributions to

A1	A2	E

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