Characters of representations for molecular motions

Motion	E	2C3	3C'2	σh	2S3	3σv
Cartesian 3N	90	0	0	30	0	0
Translation (x,y,z)	3	0	-1	1	-2	1
Rotation (Rx,Ry,Rz)	3	0	-1	-1	2	-1
Vibration	84	0	2	30	0	0

Decomposition to irreducible representations

Motion	A'1	A'2	E'	A''1	A''2	E''	Total
Cartesian 3N	10	10	20	5	5	10	60
Translation (x,y,z)	0	0	1	0	1	0	2
Rotation (Rx,Ry,Rz)	0	1	0	0	0	1	2
Vibration	10	9	19	5	4	9	56

Molecular parameter

Number of Atoms (N)	30
Number of internal coordinates	84
Number of independant internal coordinates	10
Number of vibrational modes	56

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Force field analysis

Allowed / forbidden vibronational transitions

Operator	A'1	A'2	E'	A''1	A''2	E''	Total
Linear (IR)	10	9	19	5	4	9	23 / 33
Quadratic (Raman)	10	9	19	5	4	9	38 / 18
IR + Raman	- - - -	9	19	5	- - - -	- - - -	19 / 14

Characters of force fields
(Symmetric powers of vibration representation)

Force field	E	2C3	3C'2	σh	2S3	3σv
linear	84	0	2	30	0	0
quadratic	3.570	0	44	492	0	42
cubic	102.340	28	86	5.770	10	0
quartic	2.225.895	0	989	53.853	0	903
quintic	39.175.752	0	1.892	423.516	0	0
sextic	581.106.988	406	15.136	2.907.424	64	13.244

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

Force field	A'1	A'2	E'	A''1	A''2	E''
linear	10	9	19	5	4	9
quadratic	360	317	677	257	256	513
cubic	9.037	8.994	18.012	8.072	8.029	16.092
quartic	190.452	189.506	379.958	181.025	180.982	362.007
quintic	3.300.412	3.299.466	6.599.878	3.229.826	3.228.880	6.458.706
sextic	48.675.041	48.660.851	97.335.657	48.183.827	48.182.881	96.366.537

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 D_3h

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(A'1) ≤ i ≤ pos(E'')
..55.	A'1A'1.	..45.	A'2A'2.	..190.	E'E'.	..15.	A''1A''1.	..10.	A''2A''2.	..45.	E''E''.
Subtotal: 360 / 6 / 6
Irrep combinations (i,j) with indices: pos(A'1) ≤ i ≤ j ≤ pos(E'')
Subtotal: 0 / 0 / 15
Total: 360 / 6 / 21

Contributions to nonvanishing cubic force field constants

Irrep combinations (i,i,i) with indices: pos(A'1) ≤ i ≤ pos(E'')
..220.	A'1A'1A'1.	..1.330.	E'E'E'.
Subtotal: 1.550 / 2 / 6
Irrep combinations (i,i,j) (i,j,j) with indices: pos(A'1) ≤ i ≤ j ≤ pos(E'')
..450.	A'1A'2A'2.	..1.900.	A'1E'E'.	..150.	A'1A''1A''1.	..100.	A'1A''2A''2.	..450.	A'1E''E''.	..1.539.	A'2E'E'.	..324.	A'2E''E''.	..855.	E'E''E''.
Subtotal: 5.768 / 8 / 30
Irrep combinations (i,j,k) with indices: pos(A'1) ≤ i ≤ j ≤ k ≤ pos(E'')
..180.	A'2A''1A''2.	..855.	E'A''1E''.	..684.	E'A''2E''.
Subtotal: 1.719 / 3 / 20
Total: 9.037 / 13 / 56

Contributions to nonvanishing quartic force field constants

Irrep combinations (i,i,i,i) with indices: pos(A'1) ≤ i ≤ pos(E'')
..715.	A'1A'1A'1A'1.	..495.	A'2A'2A'2A'2.	..18.145.	E'E'E'E'.	..70.	A''1A''1A''1A''1.	..35.	A''2A''2A''2A''2.	..1.035.	E''E''E''E''.
Subtotal: 20.495 / 6 / 6
Irrep combinations (i,i,i,j) (i,j,j,j) with indices: pos(A'1) ≤ i ≤ j ≤ pos(E'')
..13.300.	A'1E'E'E'.	..11.970.	A'2E'E'E'.	..825.	A''1E''E''E''.	..660.	A''2E''E''E''.
Subtotal: 26.755 / 4 / 30
Irrep combinations (i,i,j,j) with indices: pos(A'1) ≤ i ≤ j ≤ pos(E'')
..2.475.	A'1A'1A'2A'2.	..10.450.	A'1A'1E'E'.	..825.	A'1A'1A''1A''1.	..550.	A'1A'1A''2A''2.	..2.475.	A'1A'1E''E''.	..8.550.	A'2A'2E'E'.	..675.	A'2A'2A''1A''1.	..450.	A'2A'2A''2A''2.	..2.025.	A'2A'2E''E''.	..2.850.	E'E'A''1A''1.
..1.900.	E'E'A''2A''2.	..23.256.	E'E'E''E''.	..150.	A''1A''1A''2A''2.	..675.	A''1A''1E''E''.	..450.	A''2A''2E''E''.
Subtotal: 57.756 / 15 / 15
Irrep combinations (i,i,j,k) (i,j,j,k) (i,j,k,k) with indices: pos(A'1) ≤ i ≤ j ≤ k ≤ pos(E'')
..3.420.	E'E'A''1A''2.	..8.550.	E'E'A''1E''.	..6.840.	E'E'A''2E''.	..15.390.	A'1A'2E'E'.	..3.240.	A'1A'2E''E''.	..8.550.	A'1E'E''E''.	..7.695.	A'2E'E''E''.	..720.	A''1A''2E''E''.
Subtotal: 54.405 / 8 / 60
Irrep combinations (i,j,k,l) with indices: pos(A'1) ≤ i ≤ j ≤ k ≤ l ≤ pos(E'')
..1.800.	A'1A'2A''1A''2.	..8.550.	A'1E'A''1E''.	..6.840.	A'1E'A''2E''.	..7.695.	A'2E'A''1E''.	..6.156.	A'2E'A''2E''.
Subtotal: 31.041 / 5 / 15
Total: 190.452 / 38 / 126

Calculate contributions to

A'1	A'2	E'	A''1	A''2	E''

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