Characters of representations for molecular motions

Motion	E	2C3	3C'2	σh	2S3	3σv
Cartesian 3N	222	0	0	6	0	8
Translation (x,y,z)	3	0	-1	1	-2	1
Rotation (Rx,Ry,Rz)	3	0	-1	-1	2	-1
Vibration	216	0	2	6	0	8

Decomposition to irreducible representations

Motion	A'1	A'2	E'	A''1	A''2	E''	Total
Cartesian 3N	21	17	38	16	20	36	148
Translation (x,y,z)	0	0	1	0	1	0	2
Rotation (Rx,Ry,Rz)	0	1	0	0	0	1	2
Vibration	21	16	37	16	19	35	144

Molecular parameter

Number of Atoms (N)	74
Number of internal coordinates	216
Number of independant internal coordinates	21
Number of vibrational modes	144

---

Force field analysis

Allowed / forbidden vibronational transitions

Operator	A'1	A'2	E'	A''1	A''2	E''	Total
Linear (IR)	21	16	37	16	19	35	56 / 88
Quadratic (Raman)	21	16	37	16	19	35	93 / 51
IR + Raman	- - - -	16	37	16	- - - -	- - - -	37 / 32

Characters of force fields
(Symmetric powers of vibration representation)

Force field	E	2C3	3C'2	σh	2S3	3σv
linear	216	0	2	6	0	8
quadratic	23.436	0	110	126	0	140
cubic	1.703.016	72	218	686	2	952
quartic	93.240.126	0	6.104	7.896	0	9.534
quintic	4.102.565.544	0	11.990	39.522	0	56.952
sextic	151.111.164.204	2.628	227.810	329.042	38	425.516

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	21	16	37	16	19	35
quadratic	2.026	1.901	3.927	1.935	1.950	3.885
cubic	142.280	141.695	283.938	141.689	142.056	283.710
quartic	7.774.578	7.766.759	15.541.337	7.768.495	7.770.210	15.538.705
quintic	341.900.991	341.866.520	683.767.511	341.865.928	341.888.409	683.754.337
sextic	12.592.788.213	12.592.461.550	25.185.248.430	12.592.520.602	12.592.619.455	25.185.138.762

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_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'')
..231.	A'1A'1.	..136.	A'2A'2.	..703.	E'E'.	..136.	A''1A''1.	..190.	A''2A''2.	..630.	E''E''.
Subtotal: 2.026 / 6 / 6
Irrep combinations (i,j) with indices: pos(A'1) ≤ i ≤ j ≤ pos(E'')
Subtotal: 0 / 0 / 15
Total: 2.026 / 6 / 21

Contributions to nonvanishing cubic force field constants

Irrep combinations (i,i,i) with indices: pos(A'1) ≤ i ≤ pos(E'')
..1.771.	A'1A'1A'1.	..9.139.	E'E'E'.
Subtotal: 10.910 / 2 / 6
Irrep combinations (i,i,j) (i,j,j) with indices: pos(A'1) ≤ i ≤ j ≤ pos(E'')
..2.856.	A'1A'2A'2.	..14.763.	A'1E'E'.	..2.856.	A'1A''1A''1.	..3.990.	A'1A''2A''2.	..13.230.	A'1E''E''.	..10.656.	A'2E'E'.	..9.520.	A'2E''E''.	..23.310.	E'E''E''.
Subtotal: 81.181 / 8 / 30
Irrep combinations (i,j,k) with indices: pos(A'1) ≤ i ≤ j ≤ k ≤ pos(E'')
..4.864.	A'2A''1A''2.	..20.720.	E'A''1E''.	..24.605.	E'A''2E''.
Subtotal: 50.189 / 3 / 20
Total: 142.280 / 13 / 56

Contributions to nonvanishing quartic force field constants

Irrep combinations (i,i,i,i) with indices: pos(A'1) ≤ i ≤ pos(E'')
..10.626.	A'1A'1A'1A'1.	..3.876.	A'2A'2A'2A'2.	..247.456.	E'E'E'E'.	..3.876.	A''1A''1A''1A''1.	..7.315.	A''2A''2A''2A''2.	..198.765.	E''E''E''E''.
Subtotal: 471.914 / 6 / 6
Irrep combinations (i,i,i,j) (i,j,j,j) with indices: pos(A'1) ≤ i ≤ j ≤ pos(E'')
..191.919.	A'1E'E'E'.	..146.224.	A'2E'E'E'.	..124.320.	A''1E''E''E''.	..147.630.	A''2E''E''E''.
Subtotal: 610.093 / 4 / 30
Irrep combinations (i,i,j,j) with indices: pos(A'1) ≤ i ≤ j ≤ pos(E'')
..31.416.	A'1A'1A'2A'2.	..162.393.	A'1A'1E'E'.	..31.416.	A'1A'1A''1A''1.	..43.890.	A'1A'1A''2A''2.	..145.530.	A'1A'1E''E''.	..95.608.	A'2A'2E'E'.	..18.496.	A'2A'2A''1A''1.	..25.840.	A'2A'2A''2A''2.	..85.680.	A'2A'2E''E''.	..95.608.	E'E'A''1A''1.
..133.570.	E'E'A''2A''2.	..1.282.050.	E'E'E''E''.	..25.840.	A''1A''1A''2A''2.	..85.680.	A''1A''1E''E''.	..119.700.	A''2A''2E''E''.
Subtotal: 2.382.717 / 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'')
..202.464.	E'E'A''1A''2.	..393.680.	E'E'A''1E''.	..467.495.	E'E'A''2E''.	..223.776.	A'1A'2E'E'.	..199.920.	A'1A'2E''E''.	..489.510.	A'1E'E''E''.	..372.960.	A'2E'E''E''.	..180.880.	A''1A''2E''E''.
Subtotal: 2.530.685 / 8 / 60
Irrep combinations (i,j,k,l) with indices: pos(A'1) ≤ i ≤ j ≤ k ≤ l ≤ pos(E'')
..102.144.	A'1A'2A''1A''2.	..435.120.	A'1E'A''1E''.	..516.705.	A'1E'A''2E''.	..331.520.	A'2E'A''1E''.	..393.680.	A'2E'A''2E''.
Subtotal: 1.779.169 / 5 / 15
Total: 7.774.578 / 38 / 126

Calculate contributions to

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

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