Characters of representations for molecular motions

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

Decomposition to irreducible representations

Motion	A'	A''	Total
Cartesian 3N	12	9	21
Translation (x,y,z)	2	1	3
Rotation (Rx,Ry,Rz)	1	2	3
Vibration	9	6	15

Molecular parameter

Number of Atoms (N)	7
Number of internal coordinates	15
Number of independant internal coordinates	9
Number of vibrational modes	15

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Force field analysis

Allowed / forbidden vibronational transitions

Operator	A'	A''	Total
Linear (IR)	9	6	15 / 0
Quadratic (Raman)	9	6	15 / 0
IR + Raman	9	6	15 / 0

Characters of force fields
(Symmetric powers of vibration representation)

Force field	E	σh
linear	15	3
quadratic	120	12
cubic	680	28
quartic	3.060	72
quintic	11.628	144
sextic	38.760	300

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

Force field	A'	A''
linear	9	6
quadratic	66	54
cubic	354	326
quartic	1.566	1.494
quintic	5.886	5.742
sextic	19.530	19.230

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_s

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(A'')
..45.	A'A'.	..21.	A''A''.
Subtotal: 66 / 2 / 2
Irrep combinations (i,j) with indices: pos(A') ≤ i ≤ j ≤ pos(A'')
Subtotal: 0 / 0 / 1
Total: 66 / 2 / 3

Contributions to nonvanishing cubic force field constants

Irrep combinations (i,i,i) with indices: pos(A') ≤ i ≤ pos(A'')
..165.	A'A'A'.
Subtotal: 165 / 1 / 2
Irrep combinations (i,i,j) (i,j,j) with indices: pos(A') ≤ i ≤ j ≤ pos(A'')
..189.	A'A''A''.
Subtotal: 189 / 1 / 2
Irrep combinations (i,j,k) with indices: pos(A') ≤ i ≤ j ≤ k ≤ pos(A'')
Subtotal: 0 / 0 / 0
Total: 354 / 2 / 4

Contributions to nonvanishing quartic force field constants

Irrep combinations (i,i,i,i) with indices: pos(A') ≤ i ≤ pos(A'')
..495.	A'A'A'A'.	..126.	A''A''A''A''.
Subtotal: 621 / 2 / 2
Irrep combinations (i,i,i,j) (i,j,j,j) with indices: pos(A') ≤ i ≤ j ≤ pos(A'')
Subtotal: 0 / 0 / 2
Irrep combinations (i,i,j,j) with indices: pos(A') ≤ i ≤ j ≤ pos(A'')
..945.	A'A'A''A''.
Subtotal: 945 / 1 / 1
Irrep combinations (i,i,j,k) (i,j,j,k) (i,j,k,k) with indices: pos(A') ≤ i ≤ j ≤ k ≤ pos(A'')
Subtotal: 0 / 0 / 0
Irrep combinations (i,j,k,l) with indices: pos(A') ≤ i ≤ j ≤ k ≤ l ≤ pos(A'')
Subtotal: 0 / 0 / 0
Total: 1.566 / 3 / 5

Calculate contributions to

A'	A''

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