Characters of representations for molecular motions

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

Decomposition to irreducible representations

Motion	A1g	A2g	B1g	B2g	Eg	A1u	A2u	B1u	B2u	Eu	Total
Cartesian 3N	12	8	9	11	19	7	12	11	8	20	117
Translation (x,y,z)	0	0	0	0	0	0	1	0	0	1	2
Rotation (Rx,Ry,Rz)	0	1	0	0	1	0	0	0	0	0	2
Vibration	12	7	9	11	18	7	11	11	8	19	113

Molecular parameter

Number of Atoms (N)	52
Number of internal coordinates	150
Number of independant internal coordinates	12
Number of vibrational modes	113

---

Force field analysis

Allowed / forbidden vibronational transitions

Operator	A1g	A2g	B1g	B2g	Eg	A1u	A2u	B1u	B2u	Eu	Total
Linear (IR)	12	7	9	11	18	7	11	11	8	19	30 / 83
Quadratic (Raman)	12	7	9	11	18	7	11	11	8	19	50 / 63
IR + Raman	- - - -	7	- - - -	- - - -	- - - -	7	- - - -	11	8	- - - -	0* / 33

* Parity Mutual Exclusion Principle

Characters of force fields
(Symmetric powers of vibration representation)

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

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

Force field	A1g	A2g	B1g	B2g	Eg	A1u	A2u	B1u	B2u	Eu
linear	12	7	9	11	18	7	11	11	8	19
quadratic	774	672	711	733	1.405	690	716	714	691	1.407
cubic	36.138	35.644	35.761	36.023	71.667	35.643	36.061	36.022	35.684	71.745
quartic	1.374.981	1.369.685	1.371.282	1.373.344	2.743.029	1.370.349	1.372.719	1.372.602	1.370.464	2.743.183
quintic	42.262.945	42.238.529	42.243.092	42.258.422	84.496.951	42.238.452	42.259.942	42.258.345	42.240.089	84.500.031
sextic	1.091.556.314	1.091.371.472	1.091.415.477	1.091.512.231	2.182.883.703	1.091.388.210	1.091.496.976	1.091.492.413	1.091.392.733	2.182.889.709

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

---

Contributions to nonvanishing force field constants

pos(X) : Position of irreducible representation (irrep) X in character table of D_4h

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(Eu)
..78.	A1gA1g.	..28.	A2gA2g.	..45.	B1gB1g.	..66.	B2gB2g.	..171.	EgEg.	..28.	A1uA1u.	..66.	A2uA2u.	..66.	B1uB1u.	..36.	B2uB2u.	..190.	EuEu.
Subtotal: 774 / 10 / 10
Irrep combinations (i,j) with indices: pos(A1g) ≤ i ≤ j ≤ pos(Eu)
Subtotal: 0 / 0 / 45
Total: 774 / 10 / 55

Contributions to nonvanishing cubic force field constants

Irrep combinations (i,i,i) with indices: pos(A1g) ≤ i ≤ pos(Eu)
..364.	A1gA1gA1g.
Subtotal: 364 / 1 / 10
Irrep combinations (i,i,j) (i,j,j) with indices: pos(A1g) ≤ i ≤ j ≤ pos(Eu)
..336.	A1gA2gA2g.	..540.	A1gB1gB1g.	..792.	A1gB2gB2g.	..2.052.	A1gEgEg.	..336.	A1gA1uA1u.	..792.	A1gA2uA2u.	..792.	A1gB1uB1u.	..432.	A1gB2uB2u.	..2.280.	A1gEuEu.	..1.071.	A2gEgEg.
..1.197.	A2gEuEu.	..1.539.	B1gEgEg.	..1.710.	B1gEuEu.	..1.881.	B2gEgEg.	..2.090.	B2gEuEu.
Subtotal: 17.840 / 15 / 90
Irrep combinations (i,j,k) with indices: pos(A1g) ≤ i ≤ j ≤ k ≤ pos(Eu)
..693.	A2gB1gB2g.	..539.	A2gA1uA2u.	..616.	A2gB1uB2u.	..693.	B1gA1uB1u.	..792.	B1gA2uB2u.	..616.	B2gA1uB2u.	..1.331.	B2gA2uB1u.	..2.394.	EgA1uEu.	..3.762.	EgA2uEu.	..3.762.	EgB1uEu.
..2.736.	EgB2uEu.
Subtotal: 17.934 / 11 / 120
Total: 36.138 / 27 / 220

Contributions to nonvanishing quartic force field constants

Irrep combinations (i,i,i,i) with indices: pos(A1g) ≤ i ≤ pos(Eu)
..1.365.	A1gA1gA1gA1g.	..210.	A2gA2gA2gA2g.	..495.	B1gB1gB1gB1g.	..1.001.	B2gB2gB2gB2g.	..20.691.	EgEgEgEg.	..210.	A1uA1uA1uA1u.	..1.001.	A2uA2uA2uA2u.	..1.001.	B1uB1uB1uB1u.	..330.	B2uB2uB2uB2u.	..25.460.	EuEuEuEu.
Subtotal: 51.764 / 10 / 10
Irrep combinations (i,i,i,j) (i,j,j,j) with indices: pos(A1g) ≤ i ≤ j ≤ pos(Eu)
Subtotal: 0 / 0 / 90
Irrep combinations (i,i,j,j) with indices: pos(A1g) ≤ i ≤ j ≤ pos(Eu)
..2.184.	A1gA1gA2gA2g.	..3.510.	A1gA1gB1gB1g.	..5.148.	A1gA1gB2gB2g.	..13.338.	A1gA1gEgEg.	..2.184.	A1gA1gA1uA1u.	..5.148.	A1gA1gA2uA2u.	..5.148.	A1gA1gB1uB1u.	..2.808.	A1gA1gB2uB2u.	..14.820.	A1gA1gEuEu.	..1.260.	A2gA2gB1gB1g.
..1.848.	A2gA2gB2gB2g.	..4.788.	A2gA2gEgEg.	..784.	A2gA2gA1uA1u.	..1.848.	A2gA2gA2uA2u.	..1.848.	A2gA2gB1uB1u.	..1.008.	A2gA2gB2uB2u.	..5.320.	A2gA2gEuEu.	..2.970.	B1gB1gB2gB2g.	..7.695.	B1gB1gEgEg.	..1.260.	B1gB1gA1uA1u.
..2.970.	B1gB1gA2uA2u.	..2.970.	B1gB1gB1uB1u.	..1.620.	B1gB1gB2uB2u.	..8.550.	B1gB1gEuEu.	..11.286.	B2gB2gEgEg.	..1.848.	B2gB2gA1uA1u.	..4.356.	B2gB2gA2uA2u.	..4.356.	B2gB2gB1uB1u.	..2.376.	B2gB2gB2uB2u.	..12.540.	B2gB2gEuEu.
..4.788.	EgEgA1uA1u.	..11.286.	EgEgA2uA2u.	..11.286.	EgEgB1uB1u.	..6.156.	EgEgB2uB2u.	..123.633.	EgEgEuEu.	..1.848.	A1uA1uA2uA2u.	..1.848.	A1uA1uB1uB1u.	..1.008.	A1uA1uB2uB2u.	..5.320.	A1uA1uEuEu.	..4.356.	A2uA2uB1uB1u.
..2.376.	A2uA2uB2uB2u.	..12.540.	A2uA2uEuEu.	..2.376.	B1uB1uB2uB2u.	..12.540.	B1uB1uEuEu.	..6.840.	B2uB2uEuEu.
Subtotal: 345.990 / 45 / 45
Irrep combinations (i,i,j,k) (i,j,j,k) (i,j,k,k) with indices: pos(A1g) ≤ i ≤ j ≤ k ≤ pos(Eu)
..11.781.	EgEgA1uA2u.	..13.167.	EgEgA1uB1u.	..9.576.	EgEgA1uB2u.	..20.691.	EgEgA2uB1u.	..15.048.	EgEgA2uB2u.	..13.464.	EgEgB1uB2u.	..12.852.	A1gA2gEgEg.	..14.364.	A1gA2gEuEu.	..18.468.	A1gB1gEgEg.	..20.520.	A1gB1gEuEu.
..22.572.	A1gB2gEgEg.	..25.080.	A1gB2gEuEu.	..10.773.	A2gB1gEgEg.	..11.970.	A2gB1gEuEu.	..13.167.	A2gB2gEgEg.	..14.630.	A2gB2gEuEu.	..15.147.	B1gB2gEgEg.	..16.929.	B1gB2gEuEu.	..13.167.	A1uA2uEuEu.	..14.630.	A1uB1uEuEu.
..10.640.	A1uB2uEuEu.	..22.990.	A2uB1uEuEu.	..16.720.	A2uB2uEuEu.	..15.048.	B1uB2uEuEu.
Subtotal: 373.394 / 24 / 360
Irrep combinations (i,j,k,l) with indices: pos(A1g) ≤ i ≤ j ≤ k ≤ l ≤ pos(Eu)
..8.316.	A1gA2gB1gB2g.	..6.468.	A1gA2gA1uA2u.	..7.392.	A1gA2gB1uB2u.	..8.316.	A1gB1gA1uB1u.	..9.504.	A1gB1gA2uB2u.	..7.392.	A1gB2gA1uB2u.	..15.972.	A1gB2gA2uB1u.	..28.728.	A1gEgA1uEu.	..45.144.	A1gEgA2uEu.	..45.144.	A1gEgB1uEu.
..32.832.	A1gEgB2uEu.	..3.528.	A2gB1gA1uB2u.	..7.623.	A2gB1gA2uB1u.	..5.929.	A2gB2gA1uB1u.	..6.776.	A2gB2gA2uB2u.	..16.758.	A2gEgA1uEu.	..26.334.	A2gEgA2uEu.	..26.334.	A2gEgB1uEu.	..19.152.	A2gEgB2uEu.	..7.623.	B1gB2gA1uA2u.
..8.712.	B1gB2gB1uB2u.	..21.546.	B1gEgA1uEu.	..33.858.	B1gEgA2uEu.	..33.858.	B1gEgB1uEu.	..24.624.	B1gEgB2uEu.	..26.334.	B2gEgA1uEu.	..41.382.	B2gEgA2uEu.	..41.382.	B2gEgB1uEu.	..30.096.	B2gEgB2uEu.	..6.776.	A1uA2uB1uB2u.
Subtotal: 603.833 / 30 / 210
Total: 1.374.981 / 109 / 715

Calculate contributions to

A1g	A2g	B1g	B2g	Eg	A1u	A2u	B1u	B2u	Eu

Show only nonzero contributions Show all contributions
Up to quartic force fieldUp to quintic force fieldUp to sextic force field