Characters of representations for molecular motions

Motion	E	2C3	3C'2	σh	2S3	3σv
Cartesian 3N	102	0	0	0	0	14
Translation (x,y,z)	3	0	-1	1	-2	1
Rotation (Rx,Ry,Rz)	3	0	-1	-1	2	-1
Vibration	96	0	2	0	0	14

Decomposition to irreducible representations

Motion	A'1	A'2	E'	A''1	A''2	E''	Total
Cartesian 3N	12	5	17	5	12	17	68
Translation (x,y,z)	0	0	1	0	1	0	2
Rotation (Rx,Ry,Rz)	0	1	0	0	0	1	2
Vibration	12	4	16	5	11	16	64

Molecular parameter

Number of Atoms (N)	34
Number of internal coordinates	96
Number of independant internal coordinates	12
Number of vibrational modes	64

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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)	12	4	16	5	11	16	27 / 37
Quadratic (Raman)	12	4	16	5	11	16	44 / 20
IR + Raman	- - - -	4	16	5	- - - -	- - - -	16 / 9

Characters of force fields
(Symmetric powers of vibration representation)

Force field	E	2C3	3C'2	σh	2S3	3σv
linear	96	0	2	0	0	14
quadratic	4.656	0	50	48	0	146
cubic	152.096	32	98	0	0	1.134
quartic	3.764.376	0	1.274	1.176	0	7.546
quintic	75.287.520	0	2.450	0	0	43.582
sextic	1.267.339.920	528	22.050	19.600	16	227.458

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	12	4	16	5	11	16
quadratic	441	343	784	360	408	768
cubic	12.988	12.372	25.344	12.421	12.939	25.344
quartic	316.001	311.591	627.592	312.032	315.168	627.200
quintic	6.285.468	6.262.452	12.547.920	6.263.677	6.284.243	12.547.920
sextic	105.675.761	105.551.007	211.226.496	105.558.760	105.661.464	211.219.968

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'')
..78.	A'1A'1.	..10.	A'2A'2.	..136.	E'E'.	..15.	A''1A''1.	..66.	A''2A''2.	..136.	E''E''.
Subtotal: 441 / 6 / 6
Irrep combinations (i,j) with indices: pos(A'1) ≤ i ≤ j ≤ pos(E'')
Subtotal: 0 / 0 / 15
Total: 441 / 6 / 21

Contributions to nonvanishing cubic force field constants

Irrep combinations (i,i,i) with indices: pos(A'1) ≤ i ≤ pos(E'')
..364.	A'1A'1A'1.	..816.	E'E'E'.
Subtotal: 1.180 / 2 / 6
Irrep combinations (i,i,j) (i,j,j) with indices: pos(A'1) ≤ i ≤ j ≤ pos(E'')
..120.	A'1A'2A'2.	..1.632.	A'1E'E'.	..180.	A'1A''1A''1.	..792.	A'1A''2A''2.	..1.632.	A'1E''E''.	..480.	A'2E'E'.	..480.	A'2E''E''.	..2.176.	E'E''E''.
Subtotal: 7.492 / 8 / 30
Irrep combinations (i,j,k) with indices: pos(A'1) ≤ i ≤ j ≤ k ≤ pos(E'')
..220.	A'2A''1A''2.	..1.280.	E'A''1E''.	..2.816.	E'A''2E''.
Subtotal: 4.316 / 3 / 20
Total: 12.988 / 13 / 56

Contributions to nonvanishing quartic force field constants

Irrep combinations (i,i,i,i) with indices: pos(A'1) ≤ i ≤ pos(E'')
..1.365.	A'1A'1A'1A'1.	..35.	A'2A'2A'2A'2.	..9.316.	E'E'E'E'.	..70.	A''1A''1A''1A''1.	..1.001.	A''2A''2A''2A''2.	..9.316.	E''E''E''E''.
Subtotal: 21.103 / 6 / 6
Irrep combinations (i,i,i,j) (i,j,j,j) with indices: pos(A'1) ≤ i ≤ j ≤ pos(E'')
..9.792.	A'1E'E'E'.	..3.264.	A'2E'E'E'.	..4.080.	A''1E''E''E''.	..8.976.	A''2E''E''E''.
Subtotal: 26.112 / 4 / 30
Irrep combinations (i,i,j,j) with indices: pos(A'1) ≤ i ≤ j ≤ pos(E'')
..780.	A'1A'1A'2A'2.	..10.608.	A'1A'1E'E'.	..1.170.	A'1A'1A''1A''1.	..5.148.	A'1A'1A''2A''2.	..10.608.	A'1A'1E''E''.	..1.360.	A'2A'2E'E'.	..150.	A'2A'2A''1A''1.	..660.	A'2A'2A''2A''2.	..1.360.	A'2A'2E''E''.	..2.040.	E'E'A''1A''1.
..8.976.	E'E'A''2A''2.	..51.392.	E'E'E''E''.	..990.	A''1A''1A''2A''2.	..2.040.	A''1A''1E''E''.	..8.976.	A''2A''2E''E''.
Subtotal: 106.258 / 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'')
..6.600.	E'E'A''1A''2.	..10.880.	E'E'A''1E''.	..23.936.	E'E'A''2E''.	..5.760.	A'1A'2E'E'.	..5.760.	A'1A'2E''E''.	..26.112.	A'1E'E''E''.	..8.704.	A'2E'E''E''.	..6.600.	A''1A''2E''E''.
Subtotal: 94.352 / 8 / 60
Irrep combinations (i,j,k,l) with indices: pos(A'1) ≤ i ≤ j ≤ k ≤ l ≤ pos(E'')
..2.640.	A'1A'2A''1A''2.	..15.360.	A'1E'A''1E''.	..33.792.	A'1E'A''2E''.	..5.120.	A'2E'A''1E''.	..11.264.	A'2E'A''2E''.
Subtotal: 68.176 / 5 / 15
Total: 316.001 / 38 / 126

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

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