Characters of representations for molecular motions

Motion	E	8C3	6C2	6C4	3C2	i	6S4	8S6	3σh	6σd
Cartesian 3N	156	0	-2	0	0	0	0	0	4	14
Translation (x,y,z)	3	0	-1	1	-1	-3	-1	0	1	1
Rotation (Rx,Ry,Rz)	3	0	-1	1	-1	3	1	0	-1	-1
Vibration	150	0	0	-2	2	0	0	0	4	14

Decomposition to irreducible representations

Motion	A1g	A2g	Eg	T1g	T2g	A1u	A2u	Eu	T1u	T2u	Total
Cartesian 3N	5	2	7	8	11	1	5	6	12	8	65
Translation (x,y,z)	0	0	0	0	0	0	0	0	1	0	1
Rotation (Rx,Ry,Rz)	0	0	0	1	0	0	0	0	0	0	1
Vibration	5	2	7	7	11	1	5	6	11	8	63

Molecular parameter

Number of Atoms (N)	52
Number of internal coordinates	150
Number of independant internal coordinates	5
Number of vibrational modes	63

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Force field analysis

Allowed / forbidden vibronational transitions

Operator	A1g	A2g	Eg	T1g	T2g	A1u	A2u	Eu	T1u	T2u	Total
Linear (IR)	5	2	7	7	11	1	5	6	11	8	11 / 52
Quadratic (Raman)	5	2	7	7	11	1	5	6	11	8	23 / 40
IR + Raman	- - - -	2	- - - -	7	- - - -	1	5	6	- - - -	8	0* / 29

* Parity Mutual Exclusion Principle

Characters of force fields
(Symmetric powers of vibration representation)

Force field	E	8C3	6C2	6C4	3C2	i	6S4	8S6	3σh	6σd
linear	150	0	0	-2	2	0	0	0	4	14
quadratic	11.325	0	75	3	77	75	1	0	83	173
cubic	573.800	50	0	-4	152	0	0	0	312	1.512
quartic	21.947.850	0	2.850	42	3.002	2.850	38	0	3.466	11.866
quintic	675.993.780	0	0	-80	5.852	0	0	0	12.320	79.492
sextic	17.463.172.650	1.275	73.150	118	79.002	73.150	38	25	97.174	490.042

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

Force field	A1g	A2g	Eg	T1g	T2g	A1u	A2u	Eu	T1u	T2u
linear	5	2	7	7	11	1	5	6	11	8
quadratic	279	216	495	672	733	222	246	468	716	691
cubic	12.180	11.803	23.958	35.644	36.023	11.763	12.142	23.880	36.061	35.684
quartic	459.560	455.861	915.421	1.369.685	1.373.344	456.032	458.285	914.317	1.372.719	1.370.464
quintic	14.094.266	14.074.413	28.168.679	42.238.529	42.258.422	14.072.853	14.092.746	28.165.599	42.259.942	42.240.089
sextic	363.899.267	363.758.430	727.657.047	1.091.371.472	1.091.512.231	363.761.544	363.865.747	727.626.666	1.091.496.976	1.091.392.733

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 O_h

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(T2u)
..15.	A1gA1g.	..3.	A2gA2g.	..28.	EgEg.	..28.	T1gT1g.	..66.	T2gT2g.	..1.	A1uA1u.	..15.	A2uA2u.	..21.	EuEu.	..66.	T1uT1u.	..36.	T2uT2u.
Subtotal: 279 / 10 / 10
Irrep combinations (i,j) with indices: pos(A1g) ≤ i ≤ j ≤ pos(T2u)
Subtotal: 0 / 0 / 45
Total: 279 / 10 / 55

Contributions to nonvanishing cubic force field constants

Irrep combinations (i,i,i) with indices: pos(A1g) ≤ i ≤ pos(T2u)
..35.	A1gA1gA1g.	..84.	EgEgEg.	..35.	T1gT1gT1g.	..286.	T2gT2gT2g.
Subtotal: 440 / 4 / 10
Irrep combinations (i,i,j) (i,j,j) with indices: pos(A1g) ≤ i ≤ j ≤ pos(T2u)
..308.	T1gT1gT2g.	..15.	A1gA2gA2g.	..140.	A1gEgEg.	..140.	A1gT1gT1g.	..330.	A1gT2gT2g.	..5.	A1gA1uA1u.	..75.	A1gA2uA2u.	..105.	A1gEuEu.	..330.	A1gT1uT1u.	..180.	A1gT2uT2u.
..42.	A2gEgEg.	..30.	A2gEuEu.	..196.	EgT1gT1g.	..462.	EgT2gT2g.	..147.	EgEuEu.	..462.	EgT1uT1u.	..252.	EgT2uT2u.	..385.	T1gT2gT2g.	..385.	T1gT1uT1u.	..196.	T1gT2uT2u.
..726.	T2gT1uT1u.	..396.	T2gT2uT2u.
Subtotal: 5.307 / 22 / 90
Irrep combinations (i,j,k) with indices: pos(A1g) ≤ i ≤ j ≤ k ≤ pos(T2u)
..154.	A2gT1gT2g.	..10.	A2gA1uA2u.	..176.	A2gT1uT2u.	..539.	EgT1gT2g.	..42.	EgA1uEu.	..210.	EgA2uEu.	..616.	EgT1uT2u.	..77.	T1gA1uT1u.	..280.	T1gA2uT2u.	..462.	T1gEuT1u.
..336.	T1gEuT2u.	..616.	T1gT1uT2u.	..88.	T2gA1uT2u.	..605.	T2gA2uT1u.	..726.	T2gEuT1u.	..528.	T2gEuT2u.	..968.	T2gT1uT2u.
Subtotal: 6.433 / 17 / 120
Total: 12.180 / 43 / 220

Contributions to nonvanishing quartic force field constants

Irrep combinations (i,i,i,i) with indices: pos(A1g) ≤ i ≤ pos(T2u)
..70.	A1gA1gA1gA1g.	..5.	A2gA2gA2gA2g.	..406.	EgEgEgEg.	..616.	T1gT1gT1gT1g.	..3.212.	T2gT2gT2gT2g.	..1.	A1uA1uA1uA1u.	..70.	A2uA2uA2uA2u.	..231.	EuEuEuEu.	..3.212.	T1uT1uT1uT1u.	..996.	T2uT2uT2uT2u.
Subtotal: 8.819 / 10 / 10
Irrep combinations (i,i,i,j) (i,j,j,j) with indices: pos(A1g) ≤ i ≤ j ≤ pos(T2u)
..2.156.	T1gT1gT1gT2g.	..5.808.	T1uT1uT1uT2u.	..420.	A1gEgEgEg.	..175.	A1gT1gT1gT1g.	..1.430.	A1gT2gT2gT2g.	..168.	A2gEgEgEg.	..168.	A2gT1gT1gT1g.	..330.	A2gT2gT2gT2g.	..784.	EgT1gT1gT1g.	..3.080.	EgT2gT2gT2g.
..5.082.	T1gT2gT2gT2g.	..56.	A1uEuEuEu.	..165.	A1uT1uT1uT1u.	..120.	A1uT2uT2uT2u.	..280.	A2uEuEuEu.	..1.430.	A2uT1uT1uT1u.	..280.	A2uT2uT2uT2u.	..2.640.	EuT1uT1uT1u.	..1.008.	EuT2uT2uT2u.	..3.168.	T1uT2uT2uT2u.
Subtotal: 28.748 / 20 / 90
Irrep combinations (i,i,j,j) with indices: pos(A1g) ≤ i ≤ j ≤ pos(T2u)
..45.	A1gA1gA2gA2g.	..420.	A1gA1gEgEg.	..420.	A1gA1gT1gT1g.	..990.	A1gA1gT2gT2g.	..15.	A1gA1gA1uA1u.	..225.	A1gA1gA2uA2u.	..315.	A1gA1gEuEu.	..990.	A1gA1gT1uT1u.	..540.	A1gA1gT2uT2u.	..84.	A2gA2gEgEg.
..84.	A2gA2gT1gT1g.	..198.	A2gA2gT2gT2g.	..3.	A2gA2gA1uA1u.	..45.	A2gA2gA2uA2u.	..63.	A2gA2gEuEu.	..198.	A2gA2gT1uT1u.	..108.	A2gA2gT2uT2u.	..1.568.	EgEgT1gT1g.	..3.696.	EgEgT2gT2g.	..28.	EgEgA1uA1u.
..420.	EgEgA2uA2u.	..1.491.	EgEgEuEu.	..3.696.	EgEgT1uT1u.	..2.016.	EgEgT2uT2u.	..6.699.	T1gT1gT2gT2g.	..28.	T1gT1gA1uA1u.	..420.	T1gT1gA2uA2u.	..1.176.	T1gT1gEuEu.	..6.699.	T1gT1gT1uT1u.	..3.612.	T1gT1gT2uT2u.
..66.	T2gT2gA1uA1u.	..990.	T2gT2gA2uA2u.	..2.772.	T2gT2gEuEu.	..16.093.	T2gT2gT1uT1u.	..8.668.	T2gT2gT2uT2u.	..15.	A1uA1uA2uA2u.	..21.	A1uA1uEuEu.	..66.	A1uA1uT1uT1u.	..36.	A1uA1uT2uT2u.	..315.	A2uA2uEuEu.
..990.	A2uA2uT1uT1u.	..540.	A2uA2uT2uT2u.	..2.772.	EuEuT1uT1u.	..1.512.	EuEuT2uT2u.	..8.668.	T1uT1uT2uT2u.
Subtotal: 79.816 / 45 / 45
Irrep combinations (i,i,j,k) (i,j,j,k) (i,j,k,k) with indices: pos(A1g) ≤ i ≤ j ≤ k ≤ pos(T2u)
..3.773.	EgEgT1gT2g.	..105.	EgEgA1uA2u.	..168.	EgEgA1uEu.	..840.	EgEgA2uEu.	..4.312.	EgEgT1uT2u.	..168.	T1gT1gA1uEu.	..231.	T1gT1gA1uT1u.	..224.	T1gT1gA1uT2u.	..840.	T1gT1gA2uEu.	..1.540.	T1gT1gA2uT1u.
..840.	T1gT1gA2uT2u.	..3.234.	T1gT1gEuT1u.	..2.352.	T1gT1gEuT2u.	..6.776.	T1gT1gT1uT2u.	..396.	T2gT2gA1uEu.	..605.	T2gT2gA1uT1u.	..528.	T2gT2gA1uT2u.	..1.980.	T2gT2gA2uEu.	..3.630.	T2gT2gA2uT1u.	..2.200.	T2gT2gA2uT2u.
..7.986.	T2gT2gEuT1u.	..5.808.	T2gT2gEuT2u.	..16.456.	T2gT2gT1uT2u.	..3.168.	EuEuT1uT2u.	..1.540.	A1gT1gT1gT2g.	..462.	A2gT1gT1gT2g.	..3.773.	EgT1gT1gT2g.	..528.	A1uT1uT1uT2u.	..2.200.	A2uT1uT1uT2u.	..5.808.	EuT1uT1uT2u.
..210.	A1gA2gEgEg.	..150.	A1gA2gEuEu.	..980.	A1gEgT1gT1g.	..2.310.	A1gEgT2gT2g.	..735.	A1gEgEuEu.	..2.310.	A1gEgT1uT1u.	..1.260.	A1gEgT2uT2u.	..1.925.	A1gT1gT2gT2g.	..1.925.	A1gT1gT1uT1u.	..980.	A1gT1gT2uT2u.
..3.630.	A1gT2gT1uT1u.	..1.980.	A1gT2gT2uT2u.	..392.	A2gEgT1gT1g.	..924.	A2gEgT2gT2g.	..294.	A2gEgEuEu.	..924.	A2gEgT1uT1u.	..504.	A2gEgT2uT2u.	..924.	A2gT1gT2gT2g.	..924.	A2gT1gT1uT1u.	..504.	A2gT1gT2uT2u.
..1.210.	A2gT2gT1uT1u.	..616.	A2gT2gT2uT2u.	..5.929.	EgT1gT2gT2g.	..5.929.	EgT1gT1uT1u.	..3.136.	EgT1gT2uT2u.	..9.317.	EgT2gT1uT1u.	..4.928.	EgT2gT2uT2u.	..2.772.	T1gT2gEuEu.	..14.399.	T1gT2gT1uT1u.	..7.700.	T1gT2gT2uT2u.
..75.	A1uA2uEuEu.	..396.	A1uEuT1uT1u.	..216.	A1uEuT2uT2u.	..308.	A1uT1uT2uT2u.	..1.980.	A2uEuT1uT1u.	..1.080.	A2uEuT2uT2u.	..1.980.	A2uT1uT2uT2u.	..4.224.	EuT1uT2uT2u.
Subtotal: 172.451 / 68 / 360
Irrep combinations (i,j,k,l) with indices: pos(A1g) ≤ i ≤ j ≤ k ≤ l ≤ pos(T2u)
..770.	A1gA2gT1gT2g.	..50.	A1gA2gA1uA2u.	..880.	A1gA2gT1uT2u.	..2.695.	A1gEgT1gT2g.	..210.	A1gEgA1uEu.	..1.050.	A1gEgA2uEu.	..3.080.	A1gEgT1uT2u.	..385.	A1gT1gA1uT1u.	..1.400.	A1gT1gA2uT2u.	..2.310.	A1gT1gEuT1u.
..1.680.	A1gT1gEuT2u.	..3.080.	A1gT1gT1uT2u.	..440.	A1gT2gA1uT2u.	..3.025.	A1gT2gA2uT1u.	..3.630.	A1gT2gEuT1u.	..2.640.	A1gT2gEuT2u.	..4.840.	A1gT2gT1uT2u.	..1.078.	A2gEgT1gT2g.	..84.	A2gEgA1uEu.	..420.	A2gEgA2uEu.
..1.232.	A2gEgT1uT2u.	..112.	A2gT1gA1uT2u.	..770.	A2gT1gA2uT1u.	..924.	A2gT1gEuT1u.	..672.	A2gT1gEuT2u.	..1.232.	A2gT1gT1uT2u.	..242.	A2gT2gA1uT1u.	..880.	A2gT2gA2uT2u.	..1.452.	A2gT2gEuT1u.	..1.056.	A2gT2gEuT2u.
..1.936.	A2gT2gT1uT2u.	..539.	EgT1gA1uT1u.	..392.	EgT1gA1uT2u.	..2.695.	EgT1gA2uT1u.	..1.960.	EgT1gA2uT2u.	..6.468.	EgT1gEuT1u.	..4.704.	EgT1gEuT2u.	..8.624.	EgT1gT1uT2u.	..847.	EgT2gA1uT1u.	..616.	EgT2gA1uT2u.
..4.235.	EgT2gA2uT1u.	..3.080.	EgT2gA2uT2u.	..10.164.	EgT2gEuT1u.	..7.392.	EgT2gEuT2u.	..13.552.	EgT2gT1uT2u.	..385.	T1gT2gA1uA2u.	..462.	T1gT2gA1uEu.	..847.	T1gT2gA1uT1u.	..616.	T1gT2gA1uT2u.	..2.310.	T1gT2gA2uEu.
..4.235.	T1gT2gA2uT1u.	..3.080.	T1gT2gA2uT2u.	..10.164.	T1gT2gEuT1u.	..7.392.	T1gT2gEuT2u.	..27.104.	T1gT2gT1uT2u.	..440.	A1uA2uT1uT2u.	..528.	A1uEuT1uT2u.	..2.640.	A2uEuT1uT2u.
Subtotal: 169.726 / 58 / 210
Total: 459.560 / 201 / 715

Calculate contributions to

A1g	A2g	Eg	T1g	T2g	A1u	A2u	Eu	T1u	T2u

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