Characters of representations for molecular motions

Motion	E	C29	(C29)2	(C29)3	(C29)4	(C29)5	(C29)6	(C29)7	(C29)8	(C29)9	(C29)10	(C29)11	(C29)12	(C29)13	(C29)14	(C29)15	(C29)16	(C29)17	(C29)18	(C29)19	(C29)20	(C29)21	(C29)22	(C29)23	(C29)24	(C29)25	(C29)26	(C29)27	(C29)28
Cartesian 3N	0	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000
Translation (x,y,z)	3	2.953	2.815	2.592	2.295	1.937	1.535	1.108	0.676	0.260	-0.122	-0.452	-0.714	-0.895	-0.988	-0.988	-0.895	-0.714	-0.452	-0.122	0.260	0.676	1.108	1.535	1.937	2.295	2.592	2.815	2.953
Rotation (Rx,Ry,Rz)	3	2.953	2.815	2.592	2.295	1.937	1.535	1.108	0.676	0.260	-0.122	-0.452	-0.714	-0.895	-0.988	-0.988	-0.895	-0.714	-0.452	-0.122	0.260	0.676	1.108	1.535	1.937	2.295	2.592	2.815	2.953
Vibration	-6	-5.906	-5.630	-5.184	-4.590	-3.874	-3.070	-2.217	-1.353	-0.519	0.245	0.904	1.427	1.791	1.977	1.977	1.791	1.427	0.904	0.245	-0.519	-1.353	-2.217	-3.070	-3.874	-4.590	-5.184	-5.630	-5.906

Decomposition to irreducible representations

Motion	A	E1*	E2*	E3*	E4*	E5*	E6*	E7*	E8*	E9*	E10*	E11*	E12*	E13*	E14*	Total
Cartesian 3N	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
Translation (x,y,z)	1	1	0	0	0	0	0	0	0	0	0	0	0	0	0	2
Rotation (Rx,Ry,Rz)	1	1	0	0	0	0	0	0	0	0	0	0	0	0	0	2
Vibration	-2	-2	0	0	0	0	0	0	0	0	0	0	0	0	0	-4

Molecular parameter

Number of Atoms (N)	0
Number of internal coordinates	-6
Number of independant internal coordinates	-2
Number of vibrational modes	-4

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Force field analysis

Allowed / forbidden vibronational transitions

Operator	A	E1*	E2*	E3*	E4*	E5*	E6*	E7*	E8*	E9*	E10*	E11*	E12*	E13*	E14*	Total
Linear (IR)	-2	-2	0	0	0	0	0	0	0	0	0	0	0	0	0	-4 / 0
Quadratic (Raman)	-2	-2	0	0	0	0	0	0	0	0	0	0	0	0	0	-4 / 0
IR + Raman	-2	-2	- - - -	0	0	0	0	0	0	0	0	0	0	0	0	-4 / 0

Characters of force fields
(Symmetric powers of vibration representation)

Force field	E	C29	(C29)2	(C29)3	(C29)4	(C29)5	(C29)6	(C29)7	(C29)8	(C29)9	(C29)10	(C29)11	(C29)12	(C29)13	(C29)14	(C29)15	(C29)16	(C29)17	(C29)18	(C29)19	(C29)20	(C29)21	(C29)22	(C29)23	(C29)24	(C29)25	(C29)26	(C29)27	(C29)28
linear	-6	-5.906	-5.630	-5.184	-4.590	-3.874	-3.070	-2.217	-1.353	-0.519	0.245	0.904	1.427	1.791	1.977	1.977	1.791	1.427	0.904	0.245	-0.519	-1.353	-2.217	-3.070	-3.874	-4.590	-5.184	-5.630	-5.906
quadratic	15	14.628	13.555	11.904	9.856	7.625	5.427	3.445	1.810	0.587	-0.230	-0.700	-0.918	-0.989	-1.000	-1.000	-0.989	-0.918	-0.700	-0.230	0.587	1.810	3.445	5.427	7.625	9.856	11.904	13.555	14.628
cubic	-20	-19.443	-17.850	-15.439	-12.532	-9.503	-6.713	-4.457	-2.915	-2.135	-2.030	-2.409	-3.019	-3.603	-3.953	-3.953	-3.603	-3.019	-2.409	-2.030	-2.135	-2.915	-4.457	-6.713	-9.503	-12.532	-15.439	-17.850	-19.443
quartic	15	14.628	13.555	11.904	9.856	7.625	5.427	3.445	1.810	0.587	-0.230	-0.700	-0.918	-0.989	-1.000	-1.000	-0.989	-0.918	-0.700	-0.230	0.587	1.810	3.445	5.427	7.625	9.856	11.904	13.555	14.628
quintic	-6	-5.906	-5.630	-5.184	-4.590	-3.874	-3.070	-2.217	-1.353	-0.519	0.245	0.904	1.427	1.791	1.977	1.977	1.791	1.427	0.904	0.245	-0.519	-1.353	-2.217	-3.070	-3.874	-4.590	-5.184	-5.630	-5.906
sextic	1	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000

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

Force field	A	E1*	E2*	E3*	E4*	E5*	E6*	E7*	E8*	E9*	E10*	E11*	E12*	E13*	E14*
linear	-2	-2	0	0	0	0	0	0	0	0	0	0	0	0	0
quadratic	5	4	1	0	0	0	0	0	0	0	0	0	0	0	0
cubic	-8	-4	-2	0	0	0	0	0	0	0	0	0	0	0	0
quartic	5	4	1	0	0	0	0	0	0	0	0	0	0	0	0
quintic	-2	-2	0	0	0	0	0	0	0	0	0	0	0	0	0
sextic	1	0	0	0	0	0	0	0	0	0	0	0	0	0	0

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 C₂₉

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) ≤ i ≤ pos(E14)
..1.	AA.	..4.	E1E1.
Subtotal: 5 / 2 / 15
Irrep combinations (i,j) with indices: pos(A) ≤ i ≤ j ≤ pos(E14)
Subtotal: 0 / 0 / 105
Total: 5 / 2 / 120

Contributions to nonvanishing cubic force field constants

Irrep combinations (i,i,i) with indices: pos(A) ≤ i ≤ pos(E14)
Subtotal: 0 / 0 / 15
Irrep combinations (i,i,j) (i,j,j) with indices: pos(A) ≤ i ≤ j ≤ pos(E14)
Subtotal: -8 / 0 / 210
Irrep combinations (i,j,k) with indices: pos(A) ≤ i ≤ j ≤ k ≤ pos(E14)
Subtotal: 0 / 0 / 455
Total: -8 / 0 / 680

Contributions to nonvanishing quartic force field constants

Irrep combinations (i,i,i,i) with indices: pos(A) ≤ i ≤ pos(E14)
..1.	E1E1E1E1.
Subtotal: 1 / 1 / 15
Irrep combinations (i,i,i,j) (i,j,j,j) with indices: pos(A) ≤ i ≤ j ≤ pos(E14)
Subtotal: 0 / 0 / 210
Irrep combinations (i,i,j,j) with indices: pos(A) ≤ i ≤ j ≤ pos(E14)
..4.	AAE1E1.
Subtotal: 4 / 1 / 105
Irrep combinations (i,i,j,k) (i,j,j,k) (i,j,k,k) with indices: pos(A) ≤ i ≤ j ≤ k ≤ pos(E14)
Subtotal: 0 / 0 / 1.365
Irrep combinations (i,j,k,l) with indices: pos(A) ≤ i ≤ j ≤ k ≤ l ≤ pos(E14)
Subtotal: 0 / 0 / 1.365
Total: 5 / 2 / 3.060

Calculate contributions to

A	E1	E2	E3	E4	E5	E6	E7	E8	E9	E10	E11	E12	E13	E14

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