Characters of representations for molecular motions

Motion	E	8C3	3C2	6S4	6σd
Cartesian 3N	12	0	0	0	2
Translation (x,y,z)	3	0	-1	-1	1
Rotation (Rx,Ry,Rz)	3	0	-1	1	-1
Vibration	6	0	2	0	2

Decomposition to irreducible representations

Motion	A1	A2	E	T1	T2	Total
Cartesian 3N	1	0	1	1	2	5
Translation (x,y,z)	0	0	0	0	1	1
Rotation (Rx,Ry,Rz)	0	0	0	1	0	1
Vibration	1	0	1	0	1	3

Molecular parameter

Number of Atoms (N)	4
Number of internal coordinates	6
Number of independant internal coordinates	1
Number of vibrational modes	3

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Force field analysis

Allowed / forbidden vibronational transitions

Operator	A1	A2	E	T1	T2	Total
Linear (IR)	1	0	1	0	1	1 / 2
Quadratic (Raman)	1	0	1	0	1	3 / 0
IR + Raman	- - - -	0	- - - -	0	1	1 / 0

Characters of force fields
(Symmetric powers of vibration representation)

Force field	E	8C3	3C2	6S4	6σd
linear	6	0	2	0	2
quadratic	21	0	5	1	5
cubic	56	2	8	0	8
quartic	126	0	14	2	14
quintic	252	0	20	0	20
sextic	462	3	30	2	30

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

Force field	A1	A2	E	T1	T2
linear	1	0	1	0	1
quadratic	3	0	3	1	3
cubic	6	2	6	4	8
quartic	11	3	14	11	17
quintic	18	8	26	24	34
sextic	32	16	45	47	61

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 T_d

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(A1) ≤ i ≤ pos(T2)
..1.	A1A1.	..1.	EE.	..1.	T2T2.
Subtotal: 3 / 3 / 5
Irrep combinations (i,j) with indices: pos(A1) ≤ i ≤ j ≤ pos(T2)
Subtotal: 0 / 0 / 10
Total: 3 / 3 / 15

Contributions to nonvanishing cubic force field constants

Irrep combinations (i,i,i) with indices: pos(A1) ≤ i ≤ pos(T2)
..1.	A1A1A1.	..1.	EEE.	..1.	T2T2T2.
Subtotal: 3 / 3 / 5
Irrep combinations (i,i,j) (i,j,j) with indices: pos(A1) ≤ i ≤ j ≤ pos(T2)
..1.	A1EE.	..1.	A1T2T2.	..1.	ET2T2.
Subtotal: 3 / 3 / 20
Irrep combinations (i,j,k) with indices: pos(A1) ≤ i ≤ j ≤ k ≤ pos(T2)
Subtotal: 0 / 0 / 10
Total: 6 / 6 / 35

Contributions to nonvanishing quartic force field constants

Irrep combinations (i,i,i,i) with indices: pos(A1) ≤ i ≤ pos(T2)
..1.	A1A1A1A1.	..1.	EEEE.	..2.	T2T2T2T2.
Subtotal: 4 / 3 / 5
Irrep combinations (i,i,i,j) (i,j,j,j) with indices: pos(A1) ≤ i ≤ j ≤ pos(T2)
..1.	A1EEE.	..1.	A1T2T2T2.
Subtotal: 2 / 2 / 20
Irrep combinations (i,i,j,j) with indices: pos(A1) ≤ i ≤ j ≤ pos(T2)
..1.	A1A1EE.	..1.	A1A1T2T2.	..2.	EET2T2.
Subtotal: 4 / 3 / 10
Irrep combinations (i,i,j,k) (i,j,j,k) (i,j,k,k) with indices: pos(A1) ≤ i ≤ j ≤ k ≤ pos(T2)
..1.	A1ET2T2.
Subtotal: 1 / 1 / 30
Irrep combinations (i,j,k,l) with indices: pos(A1) ≤ i ≤ j ≤ k ≤ l ≤ pos(T2)
Subtotal: 0 / 0 / 5
Total: 11 / 9 / 70

Calculate contributions to

A1	A2	E	T1	T2

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