Characters of representations for molecular motions

Motion	E	2C5	2(C5)2	5C'2	σh	2S5	2(S5)3	5σv
Cartesian 3N	210	0.000	-0.000	0	10	-0.000	-0.000	4
Translation (x,y,z)	3	1.618	-0.618	-1	1	-0.382	-2.618	1
Rotation (Rx,Ry,Rz)	3	1.618	-0.618	-1	-1	0.382	2.618	-1
Vibration	204	-3.236	1.236	2	10	-0.000	-0.000	4

Decomposition to irreducible representations

Motion	A'1	A'2	E'1	E'2	A''1	A''2	E''1	E''2	Total
Cartesian 3N	12	10	22	22	9	11	20	20	126
Translation (x,y,z)	0	0	1	0	0	1	0	0	2
Rotation (Rx,Ry,Rz)	0	1	0	0	0	0	1	0	2
Vibration	12	9	21	22	9	10	19	20	122

Molecular parameter

Number of Atoms (N)	70
Number of internal coordinates	204
Number of independant internal coordinates	12
Number of vibrational modes	122

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Force field analysis

Allowed / forbidden vibronational transitions

Operator	A'1	A'2	E'1	E'2	A''1	A''2	E''1	E''2	Total
Linear (IR)	12	9	21	22	9	10	19	20	31 / 91
Quadratic (Raman)	12	9	21	22	9	10	19	20	53 / 69
IR + Raman	- - - -	9	- - - -	- - - -	9	- - - -	- - - -	20	0 / 38

Characters of force fields
(Symmetric powers of vibration representation)

Force field	E	2C5	2(C5)2	5C'2	σh	2S5	2(S5)3	5σv
linear	204	-3.236	1.236	2	10	-0.000	-0.000	4
quadratic	20.910	5.854	-0.854	104	152	0.618	-1.618	110
cubic	1.435.820	-7.236	-2.764	206	1.190	-0.000	0.000	420
quartic	74.303.685	5.854	-0.854	5.459	10.803	-0.618	1.618	6.085
quintic	3.091.033.296	38.764	43.236	10.712	70.872	2.000	2.000	22.256
sextic	107.670.993.144	-134.915	52.915	192.816	492.624	-1.000	-1.000	225.784

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

Force field	A'1	A'2	E'1	E'2	A''1	A''2	E''1	E''2
linear	12	9	21	22	9	10	19	20
quadratic	1.107	1.000	2.107	2.105	1.037	1.040	2.076	2.075
cubic	72.006	71.693	143.701	143.702	71.677	71.784	143.463	143.464
quartic	3.718.611	3.712.839	7.431.449	7.431.448	3.714.488	3.714.801	7.429.289	7.429.287
quintic	154.563.459	154.546.975	309.110.412	309.110.413	154.545.243	154.551.015	309.096.238	309.096.239
sextic	5.383.678.930	5.383.469.630	10.767.148.560	10.767.148.602	5.383.516.776	5.383.533.260	10.767.050.035	10.767.050.077

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_5h

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''2)
..78.	A'1A'1.	..45.	A'2A'2.	..231.	E'1E'1.	..253.	E'2E'2.	..45.	A''1A''1.	..55.	A''2A''2.	..190.	E''1E''1.	..210.	E''2E''2.
Subtotal: 1.107 / 8 / 8
Irrep combinations (i,j) with indices: pos(A'1) ≤ i ≤ j ≤ pos(E''2)
Subtotal: 0 / 0 / 28
Total: 1.107 / 8 / 36

Contributions to nonvanishing cubic force field constants

Irrep combinations (i,i,i) with indices: pos(A'1) ≤ i ≤ pos(E''2)
..364.	A'1A'1A'1.
Subtotal: 364 / 1 / 8
Irrep combinations (i,i,j) (i,j,j) with indices: pos(A'1) ≤ i ≤ j ≤ pos(E''2)
..5.082.	E'1E'1E'2.	..540.	A'1A'2A'2.	..2.772.	A'1E'1E'1.	..3.036.	A'1E'2E'2.	..540.	A'1A''1A''1.	..660.	A'1A''2A''2.	..2.280.	A'1E''1E''1.	..2.520.	A'1E''2E''2.	..1.890.	A'2E'1E'1.	..2.079.	A'2E'2E'2.
..1.539.	A'2E''1E''1.	..1.710.	A'2E''2E''2.	..5.313.	E'1E'2E'2.	..4.410.	E'1E''2E''2.	..4.180.	E'2E''1E''1.
Subtotal: 38.551 / 15 / 56
Irrep combinations (i,j,k) with indices: pos(A'1) ≤ i ≤ j ≤ k ≤ pos(E''2)
..810.	A'2A''1A''2.	..3.591.	E'1A''1E''1.	..3.990.	E'1A''2E''1.	..7.980.	E'1E''1E''2.	..3.960.	E'2A''1E''2.	..4.400.	E'2A''2E''2.	..8.360.	E'2E''1E''2.
Subtotal: 33.091 / 7 / 56
Total: 72.006 / 23 / 120

Contributions to nonvanishing quartic force field constants

Irrep combinations (i,i,i,i) with indices: pos(A'1) ≤ i ≤ pos(E''2)
..1.365.	A'1A'1A'1A'1.	..495.	A'2A'2A'2A'2.	..26.796.	E'1E'1E'1E'1.	..32.131.	E'2E'2E'2E'2.	..495.	A''1A''1A''1A''1.	..715.	A''2A''2A''2A''2.	..18.145.	E''1E''1E''1E''1.	..22.155.	E''2E''2E''2E''2.
Subtotal: 102.297 / 8 / 8
Irrep combinations (i,i,i,j) (i,j,j,j) with indices: pos(A'1) ≤ i ≤ j ≤ pos(E''2)
..38.962.	E'1E'1E'1E'2.	..26.600.	E''1E''1E''1E''2.	..42.504.	E'1E'2E'2E'2.	..29.260.	E''1E''2E''2E''2.
Subtotal: 137.326 / 4 / 56
Irrep combinations (i,i,j,j) with indices: pos(A'1) ≤ i ≤ j ≤ pos(E''2)
..3.510.	A'1A'1A'2A'2.	..18.018.	A'1A'1E'1E'1.	..19.734.	A'1A'1E'2E'2.	..3.510.	A'1A'1A''1A''1.	..4.290.	A'1A'1A''2A''2.	..14.820.	A'1A'1E''1E''1.	..16.380.	A'1A'1E''2E''2.	..10.395.	A'2A'2E'1E'1.	..11.385.	A'2A'2E'2E'2.	..2.025.	A'2A'2A''1A''1.
..2.475.	A'2A'2A''2A''2.	..8.550.	A'2A'2E''1E''1.	..9.450.	A'2A'2E''2E''2.	..106.953.	E'1E'1E'2E'2.	..10.395.	E'1E'1A''1A''1.	..12.705.	E'1E'1A''2A''2.	..123.690.	E'1E'1E''1E''1.	..88.410.	E'1E'1E''2E''2.	..11.385.	E'2E'2A''1A''1.	..13.915.	E'2E'2A''2A''2.
..87.571.	E'2E'2E''1E''1.	..150.150.	E'2E'2E''2E''2.	..2.475.	A''1A''1A''2A''2.	..8.550.	A''1A''1E''1E''1.	..9.450.	A''1A''1E''2E''2.	..10.450.	A''2A''2E''1E''1.	..11.550.	A''2A''2E''2E''2.	..72.390.	E''1E''1E''2E''2.
Subtotal: 844.581 / 28 / 28
Irrep combinations (i,i,j,k) (i,j,j,k) (i,j,k,k) with indices: pos(A'1) ≤ i ≤ j ≤ k ≤ pos(E''2)
..18.900.	E'1E'1A''1A''2.	..41.580.	E'1E'1A''1E''2.	..46.200.	E'1E'1A''2E''2.	..87.780.	E'1E'1E''1E''2.	..20.790.	E'2E'2A''1A''2.	..43.263.	E'2E'2A''1E''1.	..48.070.	E'2E'2A''2E''1.	..96.140.	E'2E'2E''1E''2.	..60.984.	A'1E'1E'1E'2.	..45.738.	A'2E'1E'1E'2.
..34.200.	A''1E''1E''1E''2.	..38.000.	A''2E''1E''1E''2.	..22.680.	A'1A'2E'1E'1.	..24.948.	A'1A'2E'2E'2.	..18.468.	A'1A'2E''1E''1.	..20.520.	A'1A'2E''2E''2.	..63.756.	A'1E'1E'2E'2.	..52.920.	A'1E'1E''2E''2.	..50.160.	A'1E'2E''1E''1.	..47.817.	A'2E'1E'2E'2.
..39.690.	A'2E'1E''2E''2.	..37.620.	A'2E'2E''1E''1.	..87.780.	E'1E'2E''1E''1.	..97.020.	E'1E'2E''2E''2.	..15.390.	A''1A''2E''1E''1.	..17.100.	A''1A''2E''2E''2.	..35.910.	A''1E''1E''2E''2.	..39.900.	A''2E''1E''2E''2.
Subtotal: 1.253.324 / 28 / 168
Irrep combinations (i,j,k,l) with indices: pos(A'1) ≤ i ≤ j ≤ k ≤ l ≤ pos(E''2)
..9.720.	A'1A'2A''1A''2.	..43.092.	A'1E'1A''1E''1.	..47.880.	A'1E'1A''2E''1.	..95.760.	A'1E'1E''1E''2.	..47.520.	A'1E'2A''1E''2.	..52.800.	A'1E'2A''2E''2.	..100.320.	A'1E'2E''1E''2.	..32.319.	A'2E'1A''1E''1.	..35.910.	A'2E'1A''2E''1.	..71.820.	A'2E'1E''1E''2.
..35.640.	A'2E'2A''1E''2.	..39.600.	A'2E'2A''2E''2.	..75.240.	A'2E'2E''1E''2.	..79.002.	E'1E'2A''1E''1.	..83.160.	E'1E'2A''1E''2.	..87.780.	E'1E'2A''2E''1.	..92.400.	E'1E'2A''2E''2.	..351.120.	E'1E'2E''1E''2.
Subtotal: 1.381.083 / 18 / 70
Total: 3.718.611 / 86 / 330

Calculate contributions to

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

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