Characters of representations for molecular motions

Motion	E	2C3	3σv
Cartesian 3N	18	0	4
Translation (x,y,z)	3	0	1
Rotation (Rx,Ry,Rz)	3	0	-1
Vibration	12	0	4

Decomposition to irreducible representations

Motion	A1	A2	E	Total
Cartesian 3N	5	1	6	12
Translation (x,y,z)	1	0	1	2
Rotation (Rx,Ry,Rz)	0	1	1	2
Vibration	4	0	4	8

Molecular parameter

Number of Atoms (N)	6
Number of internal coordinates	12
Number of independant internal coordinates	4
Number of vibrational modes	8

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Force field analysis

Allowed / forbidden vibronational transitions

Operator	A1	A2	E	Total
Linear (IR)	4	0	4	8 / 0
Quadratic (Raman)	4	0	4	8 / 0
IR + Raman	4	0	4	8 / 0

Characters of force fields
(Symmetric powers of vibration representation)

Force field	E	2C3	3σv
linear	12	0	4
quadratic	78	0	14
cubic	364	4	36
quartic	1.365	0	85
quintic	4.368	0	176
sextic	12.376	10	344

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

Force field	A1	A2	E
linear	4	0	4
quadratic	20	6	26
cubic	80	44	120
quartic	270	185	455
quintic	816	640	1.456
sextic	2.238	1.894	4.122

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 C_3v

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(E)
..10.	A1A1.	..10.	EE.
Subtotal: 20 / 2 / 3
Irrep combinations (i,j) with indices: pos(A1) ≤ i ≤ j ≤ pos(E)
Subtotal: 0 / 0 / 3
Total: 20 / 2 / 6

Contributions to nonvanishing cubic force field constants

Irrep combinations (i,i,i) with indices: pos(A1) ≤ i ≤ pos(E)
..20.	A1A1A1.	..20.	EEE.
Subtotal: 40 / 2 / 3
Irrep combinations (i,i,j) (i,j,j) with indices: pos(A1) ≤ i ≤ j ≤ pos(E)
..40.	A1EE.
Subtotal: 40 / 1 / 6
Irrep combinations (i,j,k) with indices: pos(A1) ≤ i ≤ j ≤ k ≤ pos(E)
Subtotal: 0 / 0 / 1
Total: 80 / 3 / 10

Contributions to nonvanishing quartic force field constants

Irrep combinations (i,i,i,i) with indices: pos(A1) ≤ i ≤ pos(E)
..35.	A1A1A1A1.	..55.	EEEE.
Subtotal: 90 / 2 / 3
Irrep combinations (i,i,i,j) (i,j,j,j) with indices: pos(A1) ≤ i ≤ j ≤ pos(E)
..80.	A1EEE.
Subtotal: 80 / 1 / 6
Irrep combinations (i,i,j,j) with indices: pos(A1) ≤ i ≤ j ≤ pos(E)
..100.	A1A1EE.
Subtotal: 100 / 1 / 3
Irrep combinations (i,i,j,k) (i,j,j,k) (i,j,k,k) with indices: pos(A1) ≤ i ≤ j ≤ k ≤ pos(E)
Subtotal: 0 / 0 / 3
Irrep combinations (i,j,k,l) with indices: pos(A1) ≤ i ≤ j ≤ k ≤ l ≤ pos(E)
Subtotal: 0 / 0 / 0
Total: 270 / 4 / 15

Calculate contributions to

A1	A2	E

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