Characters of representations for molecular motions

Motion	E	2C3	3C'2	i	2S6	3σd
Cartesian 3N	36	0	0	0	0	4
Translation (x,y,z)	3	0	-1	-3	0	1
Rotation (Rx,Ry,Rz)	3	0	-1	3	0	-1
Vibration	30	0	2	0	0	4

Decomposition to irreducible representations

Motion	A1g	A2g	Eg	A1u	A2u	Eu	Total
Cartesian 3N	4	2	6	2	4	6	24
Translation (x,y,z)	0	0	0	0	1	1	2
Rotation (Rx,Ry,Rz)	0	1	1	0	0	0	2
Vibration	4	1	5	2	3	5	20

Molecular parameter

Number of Atoms (N)	12
Number of internal coordinates	30
Number of independant internal coordinates	4
Number of vibrational modes	20

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Force field analysis

Allowed / forbidden vibronational transitions

Operator	A1g	A2g	Eg	A1u	A2u	Eu	Total
Linear (IR)	4	1	5	2	3	5	8 / 12
Quadratic (Raman)	4	1	5	2	3	5	9 / 11
IR + Raman	- - - -	1	- - - -	2	- - - -	- - - -	0* / 3

* Parity Mutual Exclusion Principle

Characters of force fields
(Symmetric powers of vibration representation)

Force field	E	2C3	3C'2	i	2S6	3σd
linear	30	0	2	0	0	4
quadratic	465	0	17	15	0	23
cubic	4.960	10	32	0	0	72
quartic	40.920	0	152	120	0	256
quintic	278.256	0	272	0	0	680
sextic	1.623.160	55	952	680	5	1.904

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

Force field	A1g	A2g	Eg	A1u	A2u	Eu
linear	4	1	5	2	3	5
quadratic	50	30	80	36	39	75
cubic	441	389	825	405	425	825
quartic	3.522	3.318	6.840	3.374	3.426	6.800
quintic	23.426	22.950	46.376	23.086	23.290	46.376
sextic	136.044	134.616	270.630	134.977	135.453	270.405

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 D_3d

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(Eu)
..10.	A1gA1g.	..1.	A2gA2g.	..15.	EgEg.	..3.	A1uA1u.	..6.	A2uA2u.	..15.	EuEu.
Subtotal: 50 / 6 / 6
Irrep combinations (i,j) with indices: pos(A1g) ≤ i ≤ j ≤ pos(Eu)
Subtotal: 0 / 0 / 15
Total: 50 / 6 / 21

Contributions to nonvanishing cubic force field constants

Irrep combinations (i,i,i) with indices: pos(A1g) ≤ i ≤ pos(Eu)
..20.	A1gA1gA1g.	..35.	EgEgEg.
Subtotal: 55 / 2 / 6
Irrep combinations (i,i,j) (i,j,j) with indices: pos(A1g) ≤ i ≤ j ≤ pos(Eu)
..4.	A1gA2gA2g.	..60.	A1gEgEg.	..12.	A1gA1uA1u.	..24.	A1gA2uA2u.	..60.	A1gEuEu.	..10.	A2gEgEg.	..10.	A2gEuEu.	..75.	EgEuEu.
Subtotal: 255 / 8 / 30
Irrep combinations (i,j,k) with indices: pos(A1g) ≤ i ≤ j ≤ k ≤ pos(Eu)
..6.	A2gA1uA2u.	..50.	EgA1uEu.	..75.	EgA2uEu.
Subtotal: 131 / 3 / 20
Total: 441 / 13 / 56

Contributions to nonvanishing quartic force field constants

Irrep combinations (i,i,i,i) with indices: pos(A1g) ≤ i ≤ pos(Eu)
..35.	A1gA1gA1gA1g.	..1.	A2gA2gA2gA2g.	..120.	EgEgEgEg.	..5.	A1uA1uA1uA1u.	..15.	A2uA2uA2uA2u.	..120.	EuEuEuEu.
Subtotal: 296 / 6 / 6
Irrep combinations (i,i,i,j) (i,j,j,j) with indices: pos(A1g) ≤ i ≤ j ≤ pos(Eu)
..140.	A1gEgEgEg.	..35.	A2gEgEgEg.	..70.	A1uEuEuEu.	..105.	A2uEuEuEu.
Subtotal: 350 / 4 / 30
Irrep combinations (i,i,j,j) with indices: pos(A1g) ≤ i ≤ j ≤ pos(Eu)
..10.	A1gA1gA2gA2g.	..150.	A1gA1gEgEg.	..30.	A1gA1gA1uA1u.	..60.	A1gA1gA2uA2u.	..150.	A1gA1gEuEu.	..15.	A2gA2gEgEg.	..3.	A2gA2gA1uA1u.	..6.	A2gA2gA2uA2u.	..15.	A2gA2gEuEu.	..45.	EgEgA1uA1u.
..90.	EgEgA2uA2u.	..550.	EgEgEuEu.	..18.	A1uA1uA2uA2u.	..45.	A1uA1uEuEu.	..90.	A2uA2uEuEu.
Subtotal: 1.277 / 15 / 15
Irrep combinations (i,i,j,k) (i,j,j,k) (i,j,k,k) with indices: pos(A1g) ≤ i ≤ j ≤ k ≤ pos(Eu)
..60.	EgEgA1uA2u.	..150.	EgEgA1uEu.	..225.	EgEgA2uEu.	..40.	A1gA2gEgEg.	..40.	A1gA2gEuEu.	..300.	A1gEgEuEu.	..75.	A2gEgEuEu.	..60.	A1uA2uEuEu.
Subtotal: 950 / 8 / 60
Irrep combinations (i,j,k,l) with indices: pos(A1g) ≤ i ≤ j ≤ k ≤ l ≤ pos(Eu)
..24.	A1gA2gA1uA2u.	..200.	A1gEgA1uEu.	..300.	A1gEgA2uEu.	..50.	A2gEgA1uEu.	..75.	A2gEgA2uEu.
Subtotal: 649 / 5 / 15
Total: 3.522 / 38 / 126

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A1g	A2g	Eg	A1u	A2u	Eu

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