Characters of representations for molecular motions

Motion	E	2C3	3C'2	σh	2S3	3σv
Cartesian 3N	60	0	-2	6	0	8
Translation (x,y,z)	3	0	-1	1	-2	1
Rotation (Rx,Ry,Rz)	3	0	-1	-1	2	-1
Vibration	54	0	0	6	0	8

Decomposition to irreducible representations

Motion	A'1	A'2	E'	A''1	A''2	E''	Total
Cartesian 3N	7	4	11	2	7	9	40
Translation (x,y,z)	0	0	1	0	1	0	2
Rotation (Rx,Ry,Rz)	0	1	0	0	0	1	2
Vibration	7	3	10	2	6	8	36

Molecular parameter

Number of Atoms (N)	20
Number of internal coordinates	54
Number of independant internal coordinates	7
Number of vibrational modes	36

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Force field analysis

Allowed / forbidden vibronational transitions

Operator	A'1	A'2	E'	A''1	A''2	E''	Total
Linear (IR)	7	3	10	2	6	8	16 / 20
Quadratic (Raman)	7	3	10	2	6	8	25 / 11
IR + Raman	- - - -	3	10	2	- - - -	- - - -	10 / 5

Characters of force fields
(Symmetric powers of vibration representation)

Force field	E	2C3	3C'2	σh	2S3	3σv
linear	54	0	0	6	0	8
quadratic	1.485	0	27	45	0	59
cubic	27.720	18	0	200	2	304
quartic	395.010	0	378	930	0	1.434
quintic	4.582.116	0	0	3.396	0	5.760
sextic	45.057.474	171	3.654	12.386	11	21.542

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	7	3	10	2	6	8
quadratic	149	106	255	112	128	240
cubic	2.406	2.254	4.650	2.220	2.372	4.584
quartic	33.448	32.542	65.990	32.576	33.104	65.680
quintic	383.566	380.686	764.252	380.120	383.000	763.120
sextic	3.762.151	3.749.553	7.511.613	3.749.312	3.758.256	7.507.488

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_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'')
..28.	A'1A'1.	..6.	A'2A'2.	..55.	E'E'.	..3.	A''1A''1.	..21.	A''2A''2.	..36.	E''E''.
Subtotal: 149 / 6 / 6
Irrep combinations (i,j) with indices: pos(A'1) ≤ i ≤ j ≤ pos(E'')
Subtotal: 0 / 0 / 15
Total: 149 / 6 / 21

Contributions to nonvanishing cubic force field constants

Irrep combinations (i,i,i) with indices: pos(A'1) ≤ i ≤ pos(E'')
..84.	A'1A'1A'1.	..220.	E'E'E'.
Subtotal: 304 / 2 / 6
Irrep combinations (i,i,j) (i,j,j) with indices: pos(A'1) ≤ i ≤ j ≤ pos(E'')
..42.	A'1A'2A'2.	..385.	A'1E'E'.	..21.	A'1A''1A''1.	..147.	A'1A''2A''2.	..252.	A'1E''E''.	..135.	A'2E'E'.	..84.	A'2E''E''.	..360.	E'E''E''.
Subtotal: 1.426 / 8 / 30
Irrep combinations (i,j,k) with indices: pos(A'1) ≤ i ≤ j ≤ k ≤ pos(E'')
..36.	A'2A''1A''2.	..160.	E'A''1E''.	..480.	E'A''2E''.
Subtotal: 676 / 3 / 20
Total: 2.406 / 13 / 56

Contributions to nonvanishing quartic force field constants

Irrep combinations (i,i,i,i) with indices: pos(A'1) ≤ i ≤ pos(E'')
..210.	A'1A'1A'1A'1.	..15.	A'2A'2A'2A'2.	..1.540.	E'E'E'E'.	..5.	A''1A''1A''1A''1.	..126.	A''2A''2A''2A''2.	..666.	E''E''E''E''.
Subtotal: 2.562 / 6 / 6
Irrep combinations (i,i,i,j) (i,j,j,j) with indices: pos(A'1) ≤ i ≤ j ≤ pos(E'')
..1.540.	A'1E'E'E'.	..660.	A'2E'E'E'.	..240.	A''1E''E''E''.	..720.	A''2E''E''E''.
Subtotal: 3.160 / 4 / 30
Irrep combinations (i,i,j,j) with indices: pos(A'1) ≤ i ≤ j ≤ pos(E'')
..168.	A'1A'1A'2A'2.	..1.540.	A'1A'1E'E'.	..84.	A'1A'1A''1A''1.	..588.	A'1A'1A''2A''2.	..1.008.	A'1A'1E''E''.	..330.	A'2A'2E'E'.	..18.	A'2A'2A''1A''1.	..126.	A'2A'2A''2A''2.	..216.	A'2A'2E''E''.	..165.	E'E'A''1A''1.
..1.155.	E'E'A''2A''2.	..5.220.	E'E'E''E''.	..63.	A''1A''1A''2A''2.	..108.	A''1A''1E''E''.	..756.	A''2A''2E''E''.
Subtotal: 11.545 / 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'')
..540.	E'E'A''1A''2.	..880.	E'E'A''1E''.	..2.640.	E'E'A''2E''.	..945.	A'1A'2E'E'.	..588.	A'1A'2E''E''.	..2.520.	A'1E'E''E''.	..1.080.	A'2E'E''E''.	..336.	A''1A''2E''E''.
Subtotal: 9.529 / 8 / 60
Irrep combinations (i,j,k,l) with indices: pos(A'1) ≤ i ≤ j ≤ k ≤ l ≤ pos(E'')
..252.	A'1A'2A''1A''2.	..1.120.	A'1E'A''1E''.	..3.360.	A'1E'A''2E''.	..480.	A'2E'A''1E''.	..1.440.	A'2E'A''2E''.
Subtotal: 6.652 / 5 / 15
Total: 33.448 / 38 / 126

Calculate contributions to

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

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