Characters of representations for molecular motions

Motion	E	C28	C14	(C28)3	C7	(C28)5	(C14)3	C4	(C7)2	(C28)9	(C14)5	(C28)11	(C7)3	(C28)13	C2	(C28)15	(C7)4	(C28)17	(C14)9	(C28)19	(C7)5	(C4)3	(C14)11	(C28)23	(C7)6	(C28)25	(C14)13	(C28)27
Cartesian 3N	0	0.000	0.000	0.000	0.000	0.000	0.000	0	0.000	0.000	0.000	0.000	0.000	0.000	0	0.000	0.000	0.000	0.000	0.000	0.000	0	0.000	0.000	0.000	0.000	0.000	0.000
Translation (x,y,z)	3	2.950	2.802	2.564	2.247	1.868	1.445	1	0.555	0.132	-0.247	-0.564	-0.802	-0.950	-1	-0.950	-0.802	-0.564	-0.247	0.132	0.555	1	1.445	1.868	2.247	2.564	2.802	2.950
Rotation (Rx,Ry,Rz)	3	2.950	2.802	2.564	2.247	1.868	1.445	1	0.555	0.132	-0.247	-0.564	-0.802	-0.950	-1	-0.950	-0.802	-0.564	-0.247	0.132	0.555	1	1.445	1.868	2.247	2.564	2.802	2.950
Vibration	-6	-5.900	-5.604	-5.127	-4.494	-3.736	-2.890	-2	-1.110	-0.264	0.494	1.127	1.604	1.900	2	1.900	1.604	1.127	0.494	-0.264	-1.110	-2	-2.890	-3.736	-4.494	-5.127	-5.604	-5.900

Decomposition to irreducible representations

Motion	A	B	E1*	E2*	E3*	E4*	E5*	E6*	E7*	E8*	E9*	E10*	E11*	E12*	E13*	Total
Cartesian 3N	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
Translation (x,y,z)	1	0	1	0	0	0	0	0	0	0	0	0	0	0	0	2
Rotation (Rx,Ry,Rz)	1	0	1	0	0	0	0	0	0	0	0	0	0	0	0	2
Vibration	-2	0	-2	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	B	E1*	E2*	E3*	E4*	E5*	E6*	E7*	E8*	E9*	E10*	E11*	E12*	E13*	Total
Linear (IR)	-2	0	-2	0	0	0	0	0	0	0	0	0	0	0	0	-4 / 0
Quadratic (Raman)	-2	0	-2	0	0	0	0	0	0	0	0	0	0	0	0	-4 / 0
IR + Raman	-2	0	-2	- - - -	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	C28	C14	(C28)3	C7	(C28)5	(C14)3	C4	(C7)2	(C28)9	(C14)5	(C28)11	(C7)3	(C28)13	C2	(C28)15	(C7)4	(C28)17	(C14)9	(C28)19	(C7)5	(C4)3	(C14)11	(C28)23	(C7)6	(C28)25	(C14)13	(C28)27
linear	-6	-5.900	-5.604	-5.127	-4.494	-3.736	-2.890	-2	-1.110	-0.264	0.494	1.127	1.604	1.900	2	1.900	1.604	1.127	0.494	-0.264	-1.110	-2	-2.890	-3.736	-4.494	-5.127	-5.604	-5.900
quadratic	15	14.601	13.455	11.700	9.543	7.224	4.978	3	1.418	0.282	-0.433	-0.810	-0.961	-0.997	-1	-0.997	-0.961	-0.810	-0.433	0.282	1.418	3	4.978	7.224	9.543	11.700	13.455	14.601
cubic	-20	-19.403	-17.702	-15.145	-12.098	-8.977	-6.176	-4	-2.616	-2.035	-2.122	-2.635	-3.286	-3.804	-4	-3.804	-3.286	-2.635	-2.122	-2.035	-2.616	-4	-6.176	-8.977	-12.098	-15.145	-17.702	-19.403
quartic	15	14.601	13.455	11.700	9.543	7.224	4.978	3	1.418	0.282	-0.433	-0.810	-0.961	-0.997	-1	-0.997	-0.961	-0.810	-0.433	0.282	1.418	3	4.978	7.224	9.543	11.700	13.455	14.601
quintic	-6	-5.900	-5.604	-5.127	-4.494	-3.736	-2.890	-2	-1.110	-0.264	0.494	1.127	1.604	1.900	2	1.900	1.604	1.127	0.494	-0.264	-1.110	-2	-2.890	-3.736	-4.494	-5.127	-5.604	-5.900
sextic	1	1.000	1.000	1.000	1.000	1.000	1.000	1	1.000	1.000	1.000	1.000	1.000	1.000	1	1.000	1.000	1.000	1.000	1.000	1.000	1	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	B	E1*	E2*	E3*	E4*	E5*	E6*	E7*	E8*	E9*	E10*	E11*	E12*	E13*
linear	-2	0	-2	0	0	0	0	0	0	0	0	0	0	0	0
quadratic	5	0	4	1	0	0	0	0	0	0	0	0	0	0	0
cubic	-8	0	-4	-2	0	0	0	0	0	0	0	0	0	0	0
quartic	5	0	4	1	0	0	0	0	0	0	0	0	0	0	0
quintic	-2	0	-2	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 C₂₈

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(E13)
..1.	AA.	..4.	E1E1.
Subtotal: 5 / 2 / 15
Irrep combinations (i,j) with indices: pos(A) ≤ i ≤ j ≤ pos(E13)
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(E13)
Subtotal: 0 / 0 / 15
Irrep combinations (i,i,j) (i,j,j) with indices: pos(A) ≤ i ≤ j ≤ pos(E13)
Subtotal: -8 / 0 / 210
Irrep combinations (i,j,k) with indices: pos(A) ≤ i ≤ j ≤ k ≤ pos(E13)
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(E13)
..1.	E1E1E1E1.
Subtotal: 1 / 1 / 15
Irrep combinations (i,i,i,j) (i,j,j,j) with indices: pos(A) ≤ i ≤ j ≤ pos(E13)
Subtotal: 0 / 0 / 210
Irrep combinations (i,i,j,j) with indices: pos(A) ≤ i ≤ j ≤ pos(E13)
..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(E13)
Subtotal: 0 / 0 / 1.365
Irrep combinations (i,j,k,l) with indices: pos(A) ≤ i ≤ j ≤ k ≤ l ≤ pos(E13)
Subtotal: 0 / 0 / 1.365
Total: 5 / 2 / 3.060

Calculate contributions to

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

Show only nonzero contributions Show all contributions