Characters of representations for molecular motions

Motion	E	2C6	2C3	C2	3C'2	3C''2	i	2S3	2S6	σh	3σd	3σv
Cartesian 3N	216	0	0	0	-8	0	0	0	0	72	0	8
Translation (x,y,z)	3	2	0	-1	-1	-1	-3	-2	0	1	1	1
Rotation (Rx,Ry,Rz)	3	2	0	-1	-1	-1	3	2	0	-1	-1	-1
Vibration	210	-4	0	2	-6	2	0	0	0	72	0	8

Decomposition to irreducible representations

Motion	A1g	A2g	B1g	B2g	E1g	E2g	A1u	A2u	B1u	B2u	E1u	E2u	Total
Cartesian 3N	12	12	4	8	12	24	4	8	12	12	24	12	144
Translation (x,y,z)	0	0	0	0	0	0	0	1	0	0	1	0	2
Rotation (Rx,Ry,Rz)	0	1	0	0	1	0	0	0	0	0	0	0	2
Vibration	12	11	4	8	11	24	4	7	12	12	23	12	140

Molecular parameter

Number of Atoms (N)	72
Number of internal coordinates	210
Number of independant internal coordinates	12
Number of vibrational modes	140

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Force field analysis

Allowed / forbidden vibronational transitions

Operator	A1g	A2g	B1g	B2g	E1g	E2g	A1u	A2u	B1u	B2u	E1u	E2u	Total
Linear (IR)	12	11	4	8	11	24	4	7	12	12	23	12	30 / 110
Quadratic (Raman)	12	11	4	8	11	24	4	7	12	12	23	12	47 / 93
IR + Raman	- - - -	11	4	8	- - - -	- - - -	4	- - - -	12	12	- - - -	12	0* / 63

* Parity Mutual Exclusion Principle

Characters of force fields
(Symmetric powers of vibration representation)

Force field	E	2C6	2C3	C2	3C'2	3C''2	i	2S3	2S6	σh	3σd	3σv
linear	210	-4	0	2	-6	2	0	0	0	72	0	8
quadratic	22.155	8	0	107	123	107	105	0	0	2.697	105	137
cubic	1.565.620	-10	70	212	-668	212	0	24	0	69.792	0	928
quartic	83.369.265	8	0	5.777	7.521	5.777	5.565	0	0	1.399.197	5.565	9.117
quintic	3.568.204.542	-4	0	11.342	-37.482	11.342	0	0	0	23.121.576	0	54.120
sextic	127.860.662.755	37	2.485	209.827	305.731	209.827	198.485	323	35	327.363.605	198.485	397.333

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

Force field	A1g	A2g	B1g	B2g	E1g	E2g	A1u	A2u	B1u	B2u	E1u	E2u
linear	12	11	4	8	11	24	4	7	12	12	23	12
quadratic	1.104	986	808	812	1.622	2.088	810	813	1.032	1.020	2.054	1.621
cubic	68.217	68.099	62.096	62.548	124.630	136.295	62.165	62.511	68.148	68.136	136.258	124.667
quartic	3.535.990	3.528.995	3.415.184	3.415.636	6.830.822	7.064.983	3.415.256	3.415.602	3.532.208	3.530.884	7.063.094	6.830.856
quintic	149.642.558	149.635.563	147.698.450	147.724.186	295.422.635	299.278.122	147.702.230	147.722.295	149.638.778	149.637.454	299.276.231	295.424.526
sextic	5.341.323.940	5.341.046.096	5.313.874.304	5.313.900.040	10.627.773.804	10.682.369.316	5.313.878.085	5.313.898.150	5.341.187.824	5.341.114.136	10.682.301.276	10.627.775.694

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_6h

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(A1g) ≤ i ≤ pos(E2u)
..78.	A1gA1g.	..66.	A2gA2g.	..10.	B1gB1g.	..36.	B2gB2g.	..66.	E1gE1g.	..300.	E2gE2g.	..10.	A1uA1u.	..28.	A2uA2u.	..78.	B1uB1u.	..78.	B2uB2u.
..276.	E1uE1u.	..78.	E2uE2u.
Subtotal: 1.104 / 12 / 12
Irrep combinations (i,j) with indices: pos(A1g) ≤ i ≤ j ≤ pos(E2u)
Subtotal: 0 / 0 / 66
Total: 1.104 / 12 / 78

Contributions to nonvanishing cubic force field constants

Irrep combinations (i,i,i) with indices: pos(A1g) ≤ i ≤ pos(E2u)
..364.	A1gA1gA1g.	..2.600.	E2gE2gE2g.
Subtotal: 2.964 / 2 / 12
Irrep combinations (i,i,j) (i,j,j) with indices: pos(A1g) ≤ i ≤ j ≤ pos(E2u)
..1.584.	E1gE1gE2g.	..792.	A1gA2gA2g.	..120.	A1gB1gB1g.	..432.	A1gB2gB2g.	..792.	A1gE1gE1g.	..3.600.	A1gE2gE2g.	..120.	A1gA1uA1u.	..336.	A1gA2uA2u.	..936.	A1gB1uB1u.	..936.	A1gB2uB2u.
..3.312.	A1gE1uE1u.	..936.	A1gE2uE2u.	..605.	A2gE1gE1g.	..3.036.	A2gE2gE2g.	..2.783.	A2gE1uE1u.	..726.	A2gE2uE2u.	..6.624.	E2gE1uE1u.	..1.872.	E2gE2uE2u.
Subtotal: 29.542 / 18 / 132
Irrep combinations (i,j,k) with indices: pos(A1g) ≤ i ≤ j ≤ k ≤ pos(E2u)
..352.	A2gB1gB2g.	..308.	A2gA1uA2u.	..1.584.	A2gB1uB2u.	..1.056.	B1gE1gE2g.	..192.	B1gA1uB1u.	..336.	B1gA2uB2u.	..1.104.	B1gE1uE2u.	..2.112.	B2gE1gE2g.	..384.	B2gA1uB2u.	..672.	B2gA2uB1u.
..2.208.	B2gE1uE2u.	..1.012.	E1gA1uE1u.	..1.771.	E1gA2uE1u.	..1.584.	E1gB1uE2u.	..1.584.	E1gB2uE2u.	..3.036.	E1gE1uE2u.	..1.152.	E2gA1uE2u.	..2.016.	E2gA2uE2u.	..6.624.	E2gB1uE1u.	..6.624.	E2gB2uE1u.
Subtotal: 35.711 / 20 / 220
Total: 68.217 / 40 / 364

Contributions to nonvanishing quartic force field constants

Irrep combinations (i,i,i,i) with indices: pos(A1g) ≤ i ≤ pos(E2u)
..1.365.	A1gA1gA1gA1g.	..1.001.	A2gA2gA2gA2g.	..35.	B1gB1gB1gB1g.	..330.	B2gB2gB2gB2g.	..2.211.	E1gE1gE1gE1g.	..45.150.	E2gE2gE2gE2g.	..35.	A1uA1uA1uA1u.	..210.	A2uA2uA2uA2u.	..1.365.	B1uB1uB1uB1u.	..1.365.	B2uB2uB2uB2u.
..38.226.	E1uE1uE1uE1u.	..3.081.	E2uE2uE2uE2u.
Subtotal: 94.374 / 12 / 12
Irrep combinations (i,i,i,j) (i,j,j,j) with indices: pos(A1g) ≤ i ≤ j ≤ pos(E2u)
..31.200.	A1gE2gE2gE2g.	..28.600.	A2gE2gE2gE2g.	..1.144.	B1gE1gE1gE1g.	..2.288.	B2gE1gE1gE1g.	..1.456.	A1uE2uE2uE2u.	..2.548.	A2uE2uE2uE2u.	..27.600.	B1uE1uE1uE1u.	..27.600.	B2uE1uE1uE1u.
Subtotal: 122.436 / 8 / 132
Irrep combinations (i,i,j,j) with indices: pos(A1g) ≤ i ≤ j ≤ pos(E2u)
..5.148.	A1gA1gA2gA2g.	..780.	A1gA1gB1gB1g.	..2.808.	A1gA1gB2gB2g.	..5.148.	A1gA1gE1gE1g.	..23.400.	A1gA1gE2gE2g.	..780.	A1gA1gA1uA1u.	..2.184.	A1gA1gA2uA2u.	..6.084.	A1gA1gB1uB1u.	..6.084.	A1gA1gB2uB2u.	..21.528.	A1gA1gE1uE1u.
..6.084.	A1gA1gE2uE2u.	..660.	A2gA2gB1gB1g.	..2.376.	A2gA2gB2gB2g.	..4.356.	A2gA2gE1gE1g.	..19.800.	A2gA2gE2gE2g.	..660.	A2gA2gA1uA1u.	..1.848.	A2gA2gA2uA2u.	..5.148.	A2gA2gB1uB1u.	..5.148.	A2gA2gB2uB2u.	..18.216.	A2gA2gE1uE1u.
..5.148.	A2gA2gE2uE2u.	..360.	B1gB1gB2gB2g.	..660.	B1gB1gE1gE1g.	..3.000.	B1gB1gE2gE2g.	..100.	B1gB1gA1uA1u.	..280.	B1gB1gA2uA2u.	..780.	B1gB1gB1uB1u.	..780.	B1gB1gB2uB2u.	..2.760.	B1gB1gE1uE1u.	..780.	B1gB1gE2uE2u.
..2.376.	B2gB2gE1gE1g.	..10.800.	B2gB2gE2gE2g.	..360.	B2gB2gA1uA1u.	..1.008.	B2gB2gA2uA2u.	..2.808.	B2gB2gB1uB1u.	..2.808.	B2gB2gB2uB2u.	..9.936.	B2gB2gE1uE1u.	..2.808.	B2gB2gE2uE2u.	..54.780.	E1gE1gE2gE2g.	..660.	E1gE1gA1uA1u.
..1.848.	E1gE1gA2uA2u.	..5.148.	E1gE1gB1uB1u.	..5.148.	E1gE1gB2uB2u.	..50.347.	E1gE1gE1uE1u.	..13.926.	E1gE1gE2uE2u.	..3.000.	E2gE2gA1uA1u.	..8.400.	E2gE2gA2uA2u.	..23.400.	E2gE2gB1uB1u.	..23.400.	E2gE2gB2uB2u.	..235.428.	E2gE2gE1uE1u.
..65.016.	E2gE2gE2uE2u.	..280.	A1uA1uA2uA2u.	..780.	A1uA1uB1uB1u.	..780.	A1uA1uB2uB2u.	..2.760.	A1uA1uE1uE1u.	..780.	A1uA1uE2uE2u.	..2.184.	A2uA2uB1uB1u.	..2.184.	A2uA2uB2uB2u.	..7.728.	A2uA2uE1uE1u.	..2.184.	A2uA2uE2uE2u.
..6.084.	B1uB1uB2uB2u.	..21.528.	B1uB1uE1uE1u.	..6.084.	B1uB1uE2uE2u.	..21.528.	B2uB2uE1uE1u.	..6.084.	B2uB2uE2uE2u.	..59.754.	E1uE1uE2uE2u.
Subtotal: 817.015 / 66 / 66
Irrep combinations (i,i,j,k) (i,j,j,k) (i,j,k,k) with indices: pos(A1g) ≤ i ≤ j ≤ k ≤ pos(E2u)
..1.540.	E1gE1gA1uA2u.	..3.168.	E1gE1gA1uE2u.	..5.544.	E1gE1gA2uE2u.	..7.920.	E1gE1gB1uB2u.	..18.216.	E1gE1gB1uE1u.	..18.216.	E1gE1gB2uE1u.	..7.728.	E2gE2gA1uA2u.	..14.400.	E2gE2gA1uE2u.	..25.200.	E2gE2gA2uE2u.	..39.744.	E2gE2gB1uB2u.
..82.800.	E2gE2gB1uE1u.	..82.800.	E2gE2gB2uE1u.	..19.008.	A1gE1gE1gE2g.	..17.424.	A2gE1gE1gE2g.	..13.248.	A1uE1uE1uE2u.	..23.184.	A2uE1uE1uE2u.	..7.260.	A1gA2gE1gE1g.	..36.432.	A1gA2gE2gE2g.	..33.396.	A1gA2gE1uE1u.	..8.712.	A1gA2gE2uE2u.
..79.488.	A1gE2gE1uE1u.	..22.464.	A1gE2gE2uE2u.	..72.864.	A2gE2gE1uE1u.	..20.592.	A2gE2gE2uE2u.	..1.760.	B1gB2gE1gE1g.	..8.832.	B1gB2gE2gE2g.	..8.096.	B1gB2gE1uE1u.	..2.112.	B1gB2gE2uE2u.	..13.200.	B1gE1gE2gE2g.	..12.144.	B1gE1gE1uE1u.
..3.432.	B1gE1gE2uE2u.	..26.400.	B2gE1gE2gE2g.	..24.288.	B2gE1gE1uE1u.	..6.864.	B2gE1gE2uE2u.	..7.084.	A1uA2uE1uE1u.	..1.848.	A1uA2uE2uE2u.	..36.432.	B1uB2uE1uE1u.	..9.504.	B1uB2uE2uE2u.	..21.528.	B1uE1uE2uE2u.	..21.528.	B2uE1uE2uE2u.
Subtotal: 866.400 / 40 / 660
Irrep combinations (i,j,k,l) with indices: pos(A1g) ≤ i ≤ j ≤ k ≤ l ≤ pos(E2u)
..4.224.	A1gA2gB1gB2g.	..3.696.	A1gA2gA1uA2u.	..19.008.	A1gA2gB1uB2u.	..12.672.	A1gB1gE1gE2g.	..2.304.	A1gB1gA1uB1u.	..4.032.	A1gB1gA2uB2u.	..13.248.	A1gB1gE1uE2u.	..25.344.	A1gB2gE1gE2g.	..4.608.	A1gB2gA1uB2u.	..8.064.	A1gB2gA2uB1u.
..26.496.	A1gB2gE1uE2u.	..12.144.	A1gE1gA1uE1u.	..21.252.	A1gE1gA2uE1u.	..19.008.	A1gE1gB1uE2u.	..19.008.	A1gE1gB2uE2u.	..36.432.	A1gE1gE1uE2u.	..13.824.	A1gE2gA1uE2u.	..24.192.	A1gE2gA2uE2u.	..79.488.	A1gE2gB1uE1u.	..79.488.	A1gE2gB2uE1u.
..11.616.	A2gB1gE1gE2g.	..2.112.	A2gB1gA1uB2u.	..3.696.	A2gB1gA2uB1u.	..12.144.	A2gB1gE1uE2u.	..23.232.	A2gB2gE1gE2g.	..4.224.	A2gB2gA1uB1u.	..7.392.	A2gB2gA2uB2u.	..24.288.	A2gB2gE1uE2u.	..11.132.	A2gE1gA1uE1u.	..19.481.	A2gE1gA2uE1u.
..17.424.	A2gE1gB1uE2u.	..17.424.	A2gE1gB2uE2u.	..33.396.	A2gE1gE1uE2u.	..12.672.	A2gE2gA1uE2u.	..22.176.	A2gE2gA2uE2u.	..72.864.	A2gE2gB1uE1u.	..72.864.	A2gE2gB2uE1u.	..896.	B1gB2gA1uA2u.	..4.608.	B1gB2gB1uB2u.	..2.112.	B1gE1gA1uE2u.
..3.696.	B1gE1gA2uE2u.	..12.144.	B1gE1gB1uE1u.	..12.144.	B1gE1gB2uE1u.	..8.832.	B1gE2gA1uE1u.	..15.456.	B1gE2gA2uE1u.	..13.824.	B1gE2gB1uE2u.	..13.824.	B1gE2gB2uE2u.	..26.496.	B1gE2gE1uE2u.	..4.224.	B2gE1gA1uE2u.	..7.392.	B2gE1gA2uE2u.
..24.288.	B2gE1gB1uE1u.	..24.288.	B2gE1gB2uE1u.	..17.664.	B2gE2gA1uE1u.	..30.912.	B2gE2gA2uE1u.	..27.648.	B2gE2gB1uE2u.	..27.648.	B2gE2gB2uE2u.	..52.992.	B2gE2gE1uE2u.	..12.672.	E1gE2gA1uB1u.	..12.672.	E1gE2gA1uB2u.	..24.288.	E1gE2gA1uE1u.
..22.176.	E1gE2gA2uB1u.	..22.176.	E1gE2gA2uB2u.	..42.504.	E1gE2gA2uE1u.	..38.016.	E1gE2gB1uE2u.	..38.016.	E1gE2gB2uE2u.	..218.592.	E1gE2gE1uE2u.	..4.032.	A1uA2uB1uB2u.	..13.248.	A1uB1uE1uE2u.	..13.248.	A1uB2uE1uE2u.	..23.184.	A2uB1uE1uE2u.
..23.184.	A2uB2uE1uE2u.
Subtotal: 1.635.765 / 71 / 495
Total: 3.535.990 / 197 / 1.365

Calculate contributions to

A1g	A2g	B1g	B2g	E1g	E2g	A1u	A2u	B1u	B2u	E1u	E2u

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