Characters of representations for molecular motions

Motion	E	S28	C14	(S28)3	C7	(S28)5	(C14)3	S4	(C7)2	(S28)9	(C14)5	(S28)11	(C7)3	(S28)13	C2	(S28)15	(C7)4	(S28)17	(C14)9	(S28)19	(C7)5	(S4)3	(C14)11	(S28)23	(C7)6	(S28)25	(C14)13	(S28)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	0.950	2.802	0.564	2.247	-0.132	1.445	-1	0.555	-1.868	-0.247	-2.564	-0.802	-2.950	-1	-2.950	-0.802	-2.564	-0.247	-1.868	0.555	-1	1.445	-0.132	2.247	0.564	2.802	0.950
Rotation (Rx,Ry,Rz)	3	-0.950	2.802	-0.564	2.247	0.132	1.445	1	0.555	1.868	-0.247	2.564	-0.802	2.950	-1	2.950	-0.802	2.564	-0.247	1.868	0.555	1	1.445	0.132	2.247	-0.564	2.802	-0.950
Vibration	-6	0.000	-5.604	-0.000	-4.494	0.000	-2.890	0	-1.110	0.000	0.494	0.000	1.604	-0.000	2	-0.000	1.604	-0.000	0.494	-0.000	-1.110	0	-2.890	0.000	-4.494	0.000	-5.604	0.000

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)	0	1	1	0	0	0	0	0	0	0	0	0	0	0	0	2
Rotation (Rx,Ry,Rz)	1	0	0	0	0	0	0	0	0	0	0	0	0	0	1	2
Vibration	-1	-1	-1	0	0	0	0	0	0	0	0	0	0	0	-1	-4

Molecular parameter

Number of Atoms (N)	0
Number of internal coordinates	-6
Number of independant internal coordinates	-1
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)	-1	-1	-1	0	0	0	0	0	0	0	0	0	0	0	-1	-2 / -2
Quadratic (Raman)	-1	-1	-1	0	0	0	0	0	0	0	0	0	0	0	-1	-2 / -2
IR + Raman	- - - -	- - - -	- - - -	- - - -	0	0	0	0	0	0	0	0	0	0	- - - -	0 / 0

Characters of force fields
(Symmetric powers of vibration representation)

Force field	E	S28	C14	(S28)3	C7	(S28)5	(C14)3	S4	(C7)2	(S28)9	(C14)5	(S28)11	(C7)3	(S28)13	C2	(S28)15	(C7)4	(S28)17	(C14)9	(S28)19	(C7)5	(S4)3	(C14)11	(S28)23	(C7)6	(S28)25	(C14)13	(S28)27
linear	-6	0.000	-5.604	-0.000	-4.494	0.000	-2.890	0	-1.110	0.000	0.494	0.000	1.604	-0.000	2	-0.000	1.604	-0.000	0.494	-0.000	-1.110	0	-2.890	0.000	-4.494	0.000	-5.604	0.000
quadratic	15	-2.802	13.455	-1.445	9.543	0.247	4.978	1	1.418	0.247	-0.433	-1.445	-0.961	-2.802	-1	-2.802	-0.961	-1.445	-0.433	0.247	1.418	1	4.978	0.247	9.543	-1.445	13.455	-2.802
cubic	-20	-0.000	-17.702	0.000	-12.098	-0.000	-6.176	0	-2.616	0.000	-2.122	0.000	-3.286	0.000	-4	0.000	-3.286	0.000	-2.122	-0.000	-2.616	0	-6.176	0.000	-12.098	-0.000	-17.702	-0.000
quartic	15	2.802	13.455	1.445	9.543	-0.247	4.978	-1	1.418	-0.247	-0.433	1.445	-0.961	2.802	-1	2.802	-0.961	1.445	-0.433	-0.247	1.418	-1	4.978	-0.247	9.543	1.445	13.455	2.802
quintic	-6	0.000	-5.604	-0.000	-4.494	0.000	-2.890	0	-1.110	-0.000	0.494	-0.000	1.604	-0.000	2	-0.000	1.604	-0.000	0.494	0.000	-1.110	0	-2.890	-0.000	-4.494	0.000	-5.604	0.000
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	-1	-1	-1	0	0	0	0	0	0	0	0	0	0	0	-1
quadratic	2	3	2	0	0	0	0	0	0	0	0	0	0	1	2
cubic	-4	-4	-2	-1	0	0	0	0	0	0	0	0	0	-1	-2
quartic	3	2	2	1	0	0	0	0	0	0	0	0	0	0	2
quintic	-1	-1	-1	0	0	0	0	0	0	0	0	0	0	0	-1
sextic	0	1	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 S₂₈

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.	E1E1.	..1.	E13E13.
Subtotal: 2 / 2 / 15
Irrep combinations (i,j) with indices: pos(A) ≤ i ≤ j ≤ pos(E13)
Subtotal: 0 / 0 / 105
Total: 2 / 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: -2 / 0 / 210
Irrep combinations (i,j,k) with indices: pos(A) ≤ i ≤ j ≤ k ≤ pos(E13)
Subtotal: -2 / 0 / 455
Total: -4 / 0 / 680

Contributions to nonvanishing quartic force field constants

Irrep combinations (i,i,i,i) with indices: pos(A) ≤ i ≤ pos(E13)
Subtotal: 0 / 0 / 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)
..1.	E1E1E13E13.
Subtotal: 1 / 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)
..2.	ABE1E13.
Subtotal: 2 / 1 / 1.365
Total: 3 / 2 / 3.060

Calculate contributions to

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

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