Characters of representations for molecular motions

Motion	E	8C3	6C2	6C4	3C2	i	6S4	8S6	3σh	6σd
Cartesian 3N	36	0	0	4	-4	0	0	0	8	4
Translation (x,y,z)	3	0	-1	1	-1	-3	-1	0	1	1
Rotation (Rx,Ry,Rz)	3	0	-1	1	-1	3	1	0	-1	-1
Vibration	30	0	2	2	-2	0	0	0	8	4

Decomposition to irreducible representations

Motion	A1g	A2g	Eg	T1g	T2g	A1u	A2u	Eu	T1u	T2u	Total
Cartesian 3N	2	0	2	2	2	0	0	0	4	2	14
Translation (x,y,z)	0	0	0	0	0	0	0	0	1	0	1
Rotation (Rx,Ry,Rz)	0	0	0	1	0	0	0	0	0	0	1
Vibration	2	0	2	1	2	0	0	0	3	2	12

Molecular parameter

Number of Atoms (N)	12
Number of internal coordinates	30
Number of independant internal coordinates	2
Number of vibrational modes	12

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Force field analysis

Allowed / forbidden vibronational transitions

Operator	A1g	A2g	Eg	T1g	T2g	A1u	A2u	Eu	T1u	T2u	Total
Linear (IR)	2	0	2	1	2	0	0	0	3	2	3 / 9
Quadratic (Raman)	2	0	2	1	2	0	0	0	3	2	6 / 6
IR + Raman	- - - -	0	- - - -	1	- - - -	0	0	0	- - - -	2	0* / 3

* Parity Mutual Exclusion Principle

Characters of force fields
(Symmetric powers of vibration representation)

Force field	E	8C3	6C2	6C4	3C2	i	6S4	8S6	3σh	6σd
linear	30	0	2	2	-2	0	0	0	8	4
quadratic	465	0	17	1	17	15	-1	0	47	23
cubic	4.960	10	32	0	-32	0	0	0	208	72
quartic	40.920	0	152	8	152	120	8	0	792	256
quintic	278.256	0	272	16	-272	0	0	0	2.640	680
sextic	1.623.160	55	952	8	952	680	-8	5	8.008	1.904

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

Force field	A1g	A2g	Eg	T1g	T2g	A1u	A2u	Eu	T1u	T2u
linear	2	0	2	1	2	0	0	0	3	2
quadratic	19	9	28	21	31	7	8	15	31	29
cubic	129	103	227	286	312	85	95	175	330	320
quartic	967	861	1.828	2.457	2.555	797	823	1.620	2.603	2.577
quintic	6.066	5.824	11.890	17.126	17.360	5.566	5.664	11.230	17.626	17.520
sextic	34.757	34.043	68.770	100.573	101.287	33.252	33.486	66.713	101.967	101.725

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 O_h

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(A1g) ≤ i ≤ pos(T2u)
..3.	A1gA1g.	..3.	EgEg.	..1.	T1gT1g.	..3.	T2gT2g.	..6.	T1uT1u.	..3.	T2uT2u.
Subtotal: 19 / 6 / 10
Irrep combinations (i,j) with indices: pos(A1g) ≤ i ≤ j ≤ pos(T2u)
Subtotal: 0 / 0 / 45
Total: 19 / 6 / 55

Contributions to nonvanishing cubic force field constants

Irrep combinations (i,i,i) with indices: pos(A1g) ≤ i ≤ pos(T2u)
..4.	A1gA1gA1g.	..4.	EgEgEg.	..4.	T2gT2gT2g.
Subtotal: 12 / 3 / 10
Irrep combinations (i,i,j) (i,j,j) with indices: pos(A1g) ≤ i ≤ j ≤ pos(T2u)
..2.	T1gT1gT2g.	..6.	A1gEgEg.	..2.	A1gT1gT1g.	..6.	A1gT2gT2g.	..12.	A1gT1uT1u.	..6.	A1gT2uT2u.	..2.	EgT1gT1g.	..6.	EgT2gT2g.	..12.	EgT1uT1u.	..6.	EgT2uT2u.
..1.	T1gT2gT2g.	..3.	T1gT1uT1u.	..1.	T1gT2uT2u.	..12.	T2gT1uT1u.	..6.	T2gT2uT2u.
Subtotal: 83 / 15 / 90
Irrep combinations (i,j,k) with indices: pos(A1g) ≤ i ≤ j ≤ k ≤ pos(T2u)
..4.	EgT1gT2g.	..12.	EgT1uT2u.	..6.	T1gT1uT2u.	..12.	T2gT1uT2u.
Subtotal: 34 / 4 / 120
Total: 129 / 22 / 220

Contributions to nonvanishing quartic force field constants

Irrep combinations (i,i,i,i) with indices: pos(A1g) ≤ i ≤ pos(T2u)
..5.	A1gA1gA1gA1g.	..6.	EgEgEgEg.	..2.	T1gT1gT1gT1g.	..11.	T2gT2gT2gT2g.	..36.	T1uT1uT1uT1u.	..11.	T2uT2uT2uT2u.
Subtotal: 71 / 6 / 10
Irrep combinations (i,i,i,j) (i,j,j,j) with indices: pos(A1g) ≤ i ≤ j ≤ pos(T2u)
..2.	T1gT1gT1gT2g.	..36.	T1uT1uT1uT2u.	..8.	A1gEgEgEg.	..8.	A1gT2gT2gT2g.	..4.	EgT2gT2gT2g.	..6.	T1gT2gT2gT2g.	..18.	T1uT2uT2uT2u.
Subtotal: 82 / 7 / 90
Irrep combinations (i,i,j,j) with indices: pos(A1g) ≤ i ≤ j ≤ pos(T2u)
..9.	A1gA1gEgEg.	..3.	A1gA1gT1gT1g.	..9.	A1gA1gT2gT2g.	..18.	A1gA1gT1uT1u.	..9.	A1gA1gT2uT2u.	..6.	EgEgT1gT1g.	..18.	EgEgT2gT2g.	..36.	EgEgT1uT1u.	..18.	EgEgT2uT2u.	..9.	T1gT1gT2gT2g.
..18.	T1gT1gT1uT1u.	..9.	T1gT1gT2uT2u.	..57.	T2gT2gT1uT1u.	..28.	T2gT2gT2uT2u.	..57.	T1uT1uT2uT2u.
Subtotal: 304 / 15 / 45
Irrep combinations (i,i,j,k) (i,j,j,k) (i,j,k,k) with indices: pos(A1g) ≤ i ≤ j ≤ k ≤ pos(T2u)
..8.	EgEgT1gT2g.	..24.	EgEgT1uT2u.	..12.	T1gT1gT1uT2u.	..42.	T2gT2gT1uT2u.	..4.	A1gT1gT1gT2g.	..4.	EgT1gT1gT2g.	..4.	A1gEgT1gT1g.	..12.	A1gEgT2gT2g.	..24.	A1gEgT1uT1u.	..12.	A1gEgT2uT2u.
..2.	A1gT1gT2gT2g.	..6.	A1gT1gT1uT1u.	..2.	A1gT1gT2uT2u.	..24.	A1gT2gT1uT1u.	..12.	A1gT2gT2uT2u.	..8.	EgT1gT2gT2g.	..18.	EgT1gT1uT1u.	..8.	EgT1gT2uT2u.	..36.	EgT2gT1uT1u.	..16.	EgT2gT2uT2u.
..30.	T1gT2gT1uT1u.	..14.	T1gT2gT2uT2u.
Subtotal: 322 / 22 / 360
Irrep combinations (i,j,k,l) with indices: pos(A1g) ≤ i ≤ j ≤ k ≤ l ≤ pos(T2u)
..8.	A1gEgT1gT2g.	..24.	A1gEgT1uT2u.	..12.	A1gT1gT1uT2u.	..24.	A1gT2gT1uT2u.	..24.	EgT1gT1uT2u.	..48.	EgT2gT1uT2u.	..48.	T1gT2gT1uT2u.
Subtotal: 188 / 7 / 210
Total: 967 / 57 / 715

Calculate contributions to

A1g	A2g	Eg	T1g	T2g	A1u	A2u	Eu	T1u	T2u

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