Characters of representations for molecular motions

Motion	E	C2
Cartesian 3N	45	-1
Translation (x,y,z)	3	-1
Rotation (Rx,Ry,Rz)	3	-1
Vibration	39	1

Decomposition to irreducible representations

Motion	A	B	Total
Cartesian 3N	22	23	45
Translation (x,y,z)	1	2	3
Rotation (Rx,Ry,Rz)	1	2	3
Vibration	20	19	39

Molecular parameter

Number of Atoms (N)	15
Number of internal coordinates	39
Number of independant internal coordinates	20
Number of vibrational modes	39

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Force field analysis

Allowed / forbidden vibronational transitions

Operator	A	B	Total
Linear (IR)	20	19	39 / 0
Quadratic (Raman)	20	19	39 / 0
IR + Raman	20	19	39 / 0

Characters of force fields
(Symmetric powers of vibration representation)

Force field	E	C2
linear	39	1
quadratic	780	20
cubic	10.660	20
quartic	111.930	210
quintic	962.598	210
sextic	7.059.052	1.540

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

Force field	A	B
linear	20	19
quadratic	400	380
cubic	5.340	5.320
quartic	56.070	55.860
quintic	481.404	481.194
sextic	3.530.296	3.528.756

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 C₂

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(A) ≤ i ≤ pos(B)
..210.	AA.	..190.	BB.
Subtotal: 400 / 2 / 2
Irrep combinations (i,j) with indices: pos(A) ≤ i ≤ j ≤ pos(B)
Subtotal: 0 / 0 / 1
Total: 400 / 2 / 3

Contributions to nonvanishing cubic force field constants

Irrep combinations (i,i,i) with indices: pos(A) ≤ i ≤ pos(B)
..1.540.	AAA.
Subtotal: 1.540 / 1 / 2
Irrep combinations (i,i,j) (i,j,j) with indices: pos(A) ≤ i ≤ j ≤ pos(B)
..3.800.	ABB.
Subtotal: 3.800 / 1 / 2
Irrep combinations (i,j,k) with indices: pos(A) ≤ i ≤ j ≤ k ≤ pos(B)
Subtotal: 0 / 0 / 0
Total: 5.340 / 2 / 4

Contributions to nonvanishing quartic force field constants

Irrep combinations (i,i,i,i) with indices: pos(A) ≤ i ≤ pos(B)
..8.855.	AAAA.	..7.315.	BBBB.
Subtotal: 16.170 / 2 / 2
Irrep combinations (i,i,i,j) (i,j,j,j) with indices: pos(A) ≤ i ≤ j ≤ pos(B)
Subtotal: 0 / 0 / 2
Irrep combinations (i,i,j,j) with indices: pos(A) ≤ i ≤ j ≤ pos(B)
..39.900.	AABB.
Subtotal: 39.900 / 1 / 1
Irrep combinations (i,i,j,k) (i,j,j,k) (i,j,k,k) with indices: pos(A) ≤ i ≤ j ≤ k ≤ pos(B)
Subtotal: 0 / 0 / 0
Irrep combinations (i,j,k,l) with indices: pos(A) ≤ i ≤ j ≤ k ≤ l ≤ pos(B)
Subtotal: 0 / 0 / 0
Total: 56.070 / 3 / 5

Calculate contributions to

A	B

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