Characters of representations for molecular motions

Motion	E	C15	(C15)2	C5	(C15)4	C3	(C5)2	(C15)7	(C15)8	(C5)3	(C3)2	(C15)11	(C5)4	(C15)13	(C15)14	i	(S30)17	(S30)19	(S10)7	(S30)23	(S6)5	(S10)9	(S30)29	S30	S10	S6	(S30)7	(S10)3	(S30)11	(S30)13
Cartesian 3N	0	0.000	0.000	0.000	0.000	0	0.000	0.000	0.000	0.000	0	0.000	0.000	0.000	0.000	0	0.000	0.000	0.000	0.000	0	0.000	0.000	0.000	0.000	0	0.000	0.000	0.000	0.000
Translation (x,y,z)	3	2.827	2.338	1.618	0.791	0	-0.618	-0.956	-0.956	-0.618	0	0.791	1.618	2.338	2.827	-3	-2.827	-2.338	-1.618	-0.791	0	0.618	0.956	0.956	0.618	0	-0.791	-1.618	-2.338	-2.827
Rotation (Rx,Ry,Rz)	3	2.827	2.338	1.618	0.791	0	-0.618	-0.956	-0.956	-0.618	0	0.791	1.618	2.338	2.827	3	2.827	2.338	1.618	0.791	0	-0.618	-0.956	-0.956	-0.618	0	0.791	1.618	2.338	2.827
Vibration	-6	-5.654	-4.677	-3.236	-1.582	0	1.236	1.913	1.913	1.236	0	-1.582	-3.236	-4.677	-5.654	0	0.000	0.000	0.000	-0.000	0	0.000	0.000	0.000	0.000	0	0.000	0.000	0.000	0.000

Decomposition to irreducible representations

Motion	Ag	E1g*	E2g*	E3g*	E4g*	E5g*	E6g*	E7g*	Au	E1u*	E2u*	E3u*	E4u*	E5u*	E6u*	E7u*	Total
Cartesian 3N	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
Translation (x,y,z)	0	0	0	0	0	0	0	0	1	1	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	0	2
Vibration	-1	-1	0	0	0	0	0	0	-1	-1	0	0	0	0	0	0	-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	Ag	E1g*	E2g*	E3g*	E4g*	E5g*	E6g*	E7g*	Au	E1u*	E2u*	E3u*	E4u*	E5u*	E6u*	E7u*	Total
Linear (IR)	-1	-1	0	0	0	0	0	0	-1	-1	0	0	0	0	0	0	-2 / -2
Quadratic (Raman)	-1	-1	0	0	0	0	0	0	-1	-1	0	0	0	0	0	0	-2 / -2
IR + Raman	- - - -	- - - -	- - - -	0	0	0	0	0	- - - -	- - - -	0	0	0	0	0	0	0* / 0

* Parity Mutual Exclusion Principle

Characters of force fields
(Symmetric powers of vibration representation)

Force field	E	C15	(C15)2	C5	(C15)4	C3	(C5)2	(C15)7	(C15)8	(C5)3	(C3)2	(C15)11	(C5)4	(C15)13	(C15)14	i	(S30)17	(S30)19	(S10)7	(S30)23	(S6)5	(S10)9	(S30)29	S30	S10	S6	(S30)7	(S10)3	(S30)11	(S30)13
linear	-6	-5.654	-4.677	-3.236	-1.582	0	1.236	1.913	1.913	1.236	0	-1.582	-3.236	-4.677	-5.654	0	0.000	0.000	0.000	-0.000	0	0.000	0.000	0.000	0.000	0	0.000	0.000	0.000	0.000
quadratic	15	13.647	10.144	5.854	2.207	0	-0.854	-0.998	-0.998	-0.854	0	2.207	5.854	10.144	13.647	-3	-2.338	-0.791	0.618	0.956	0	-1.618	-2.827	-2.827	-1.618	0	0.956	0.618	-0.791	-2.338
cubic	-20	-17.985	-12.935	-7.236	-3.251	-2	-2.764	-3.829	-3.829	-2.764	-2	-3.251	-7.236	-12.935	-17.985	0	-0.000	0.000	0.000	-0.000	0	0.000	0.000	0.000	0.000	0	0.000	0.000	0.000	0.000
quartic	15	13.647	10.144	5.854	2.207	0	-0.854	-0.998	-0.998	-0.854	0	2.207	5.854	10.144	13.647	3	2.338	0.791	-0.618	-0.956	0	1.618	2.827	2.827	1.618	0	-0.956	-0.618	0.791	2.338
quintic	-6	-5.654	-4.677	-3.236	-1.582	0	1.236	1.913	1.913	1.236	0	-1.582	-3.236	-4.677	-5.654	0	0.000	0.000	0.000	0.000	0	0.000	0.000	0.000	0.000	0	-0.000	0.000	0.000	0.000
sextic	1	1.000	1.000	1.000	1.000	1	1.000	1.000	1.000	1.000	1	1.000	1.000	1.000	1.000	-1	-1.000	-1.000	-1.000	-1.000	-1	-1.000	-1.000	-1.000	-1.000	-1	-1.000	-1.000	-1.000	-1.000

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

Force field	Ag	E1g*	E2g*	E3g*	E4g*	E5g*	E6g*	E7g*	Au	E1u*	E2u*	E3u*	E4u*	E5u*	E6u*	E7u*
linear	-1	-1	0	0	0	0	0	0	-1	-1	0	0	0	0	0	0
quadratic	2	2	0	0	0	0	0	0	3	2	1	0	0	0	0	0
cubic	-4	-2	-1	0	0	0	0	0	-4	-2	-1	0	0	0	0	0
quartic	3	2	1	0	0	0	0	0	2	2	0	0	0	0	0	0
quintic	-1	-1	0	0	0	0	0	0	-1	-1	0	0	0	0	0	0
sextic	0	0	0	0	0	0	0	0	1	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 S₃₀

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(Ag) ≤ i ≤ pos(E7u)
..1.	E1gE1g.	..1.	E1uE1u.
Subtotal: 2 / 2 / 16
Irrep combinations (i,j) with indices: pos(Ag) ≤ i ≤ j ≤ pos(E7u)
Subtotal: 0 / 0 / 120
Total: 2 / 2 / 136

Contributions to nonvanishing cubic force field constants

Irrep combinations (i,i,i) with indices: pos(Ag) ≤ i ≤ pos(E7u)
Subtotal: 0 / 0 / 16
Irrep combinations (i,i,j) (i,j,j) with indices: pos(Ag) ≤ i ≤ j ≤ pos(E7u)
Subtotal: -2 / 0 / 240
Irrep combinations (i,j,k) with indices: pos(Ag) ≤ i ≤ j ≤ k ≤ pos(E7u)
Subtotal: -2 / 0 / 560
Total: -4 / 0 / 816

Contributions to nonvanishing quartic force field constants

Irrep combinations (i,i,i,i) with indices: pos(Ag) ≤ i ≤ pos(E7u)
Subtotal: 0 / 0 / 16
Irrep combinations (i,i,i,j) (i,j,j,j) with indices: pos(Ag) ≤ i ≤ j ≤ pos(E7u)
Subtotal: 0 / 0 / 240
Irrep combinations (i,i,j,j) with indices: pos(Ag) ≤ i ≤ j ≤ pos(E7u)
..1.	E1gE1gE1uE1u.
Subtotal: 1 / 1 / 120
Irrep combinations (i,i,j,k) (i,j,j,k) (i,j,k,k) with indices: pos(Ag) ≤ i ≤ j ≤ k ≤ pos(E7u)
Subtotal: 0 / 0 / 1.680
Irrep combinations (i,j,k,l) with indices: pos(Ag) ≤ i ≤ j ≤ k ≤ l ≤ pos(E7u)
..2.	AgE1gAuE1u.
Subtotal: 2 / 1 / 1.820
Total: 3 / 2 / 3.876

Calculate contributions to

Ag	E1g	E2g	E3g	E4g	E5g	E6g	E7g	Au	E1u	E2u	E3u	E4u	E5u	E6u	E7u

Show only nonzero contributions Show all contributions