Characters of representations for molecular motions

Motion	E	2C3	3σv
Cartesian 3N	21	0	3
Translation (x,y,z)	3	0	1
Rotation (Rx,Ry,Rz)	3	0	-1
Vibration	15	0	3

Decomposition to irreducible representations

Motion	A1	A2	E	Total
Cartesian 3N	5	2	7	14
Translation (x,y,z)	1	0	1	2
Rotation (Rx,Ry,Rz)	0	1	1	2
Vibration	4	1	5	10

Molecular parameter

Number of Atoms (N)	7
Number of internal coordinates	15
Number of independant internal coordinates	4
Number of vibrational modes	10

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Force field analysis

Allowed / forbidden vibronational transitions

Operator	A1	A2	E	Total
Linear (IR)	4	1	5	9 / 1
Quadratic (Raman)	4	1	5	9 / 1
IR + Raman	4	1	5	9 / 1

Characters of force fields
(Symmetric powers of vibration representation)

Force field	E	2C3	3σv
linear	15	0	3
quadratic	120	0	12
cubic	680	5	28
quartic	3.060	0	72
quintic	11.628	0	144
sextic	38.760	15	300

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

Force field	A1	A2	E
linear	4	1	5
quadratic	26	14	40
cubic	129	101	225
quartic	546	474	1.020
quintic	2.010	1.866	3.876
sextic	6.615	6.315	12.915

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.	..1.	A2A2.	..15.	EE.
Subtotal: 26 / 3 / 3
Irrep combinations (i,j) with indices: pos(A1) ≤ i ≤ j ≤ pos(E)
Subtotal: 0 / 0 / 3
Total: 26 / 3 / 6

Contributions to nonvanishing cubic force field constants

Irrep combinations (i,i,i) with indices: pos(A1) ≤ i ≤ pos(E)
..20.	A1A1A1.	..35.	EEE.
Subtotal: 55 / 2 / 3
Irrep combinations (i,i,j) (i,j,j) with indices: pos(A1) ≤ i ≤ j ≤ pos(E)
..4.	A1A2A2.	..60.	A1EE.	..10.	A2EE.
Subtotal: 74 / 3 / 6
Irrep combinations (i,j,k) with indices: pos(A1) ≤ i ≤ j ≤ k ≤ pos(E)
Subtotal: 0 / 0 / 1
Total: 129 / 5 / 10

Contributions to nonvanishing quartic force field constants

Irrep combinations (i,i,i,i) with indices: pos(A1) ≤ i ≤ pos(E)
..35.	A1A1A1A1.	..1.	A2A2A2A2.	..120.	EEEE.
Subtotal: 156 / 3 / 3
Irrep combinations (i,i,i,j) (i,j,j,j) with indices: pos(A1) ≤ i ≤ j ≤ pos(E)
..140.	A1EEE.	..35.	A2EEE.
Subtotal: 175 / 2 / 6
Irrep combinations (i,i,j,j) with indices: pos(A1) ≤ i ≤ j ≤ pos(E)
..10.	A1A1A2A2.	..150.	A1A1EE.	..15.	A2A2EE.
Subtotal: 175 / 3 / 3
Irrep combinations (i,i,j,k) (i,j,j,k) (i,j,k,k) with indices: pos(A1) ≤ i ≤ j ≤ k ≤ pos(E)
..40.	A1A2EE.
Subtotal: 40 / 1 / 3
Irrep combinations (i,j,k,l) with indices: pos(A1) ≤ i ≤ j ≤ k ≤ l ≤ pos(E)
Subtotal: 0 / 0 / 0
Total: 546 / 9 / 15

Calculate contributions to

A1	A2	E

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