Characters of representations for molecular motions

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

Decomposition to irreducible representations

Motion	A'	A''	Total
Cartesian 3N	24	21	45
Translation (x,y,z)	2	1	3
Rotation (Rx,Ry,Rz)	1	2	3
Vibration	21	18	39

Molecular parameter

Number of Atoms (N)	15
Number of internal coordinates	39
Number of independant internal coordinates	21
Number of vibrational modes	39

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Force field analysis

Allowed / forbidden vibronational transitions

Operator	A'	A''	Total
Linear (IR)	21	18	39 / 0
Quadratic (Raman)	21	18	39 / 0
IR + Raman	21	18	39 / 0

Characters of force fields
(Symmetric powers of vibration representation)

Force field	E	σh
linear	39	3
quadratic	780	24
cubic	10.660	64
quartic	111.930	294
quintic	962.598	714
sextic	7.059.052	2.464

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

Force field	A'	A''
linear	21	18
quadratic	402	378
cubic	5.362	5.298
quartic	56.112	55.818
quintic	481.656	480.942
sextic	3.530.758	3.528.294

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'')
..231.	A'A'.	..171.	A''A''.
Subtotal: 402 / 2 / 2
Irrep combinations (i,j) with indices: pos(A') ≤ i ≤ j ≤ pos(A'')
Subtotal: 0 / 0 / 1
Total: 402 / 2 / 3

Contributions to nonvanishing cubic force field constants

Irrep combinations (i,i,i) with indices: pos(A') ≤ i ≤ pos(A'')
..1.771.	A'A'A'.
Subtotal: 1.771 / 1 / 2
Irrep combinations (i,i,j) (i,j,j) with indices: pos(A') ≤ i ≤ j ≤ pos(A'')
..3.591.	A'A''A''.
Subtotal: 3.591 / 1 / 2
Irrep combinations (i,j,k) with indices: pos(A') ≤ i ≤ j ≤ k ≤ pos(A'')
Subtotal: 0 / 0 / 0
Total: 5.362 / 2 / 4

Contributions to nonvanishing quartic force field constants

Irrep combinations (i,i,i,i) with indices: pos(A') ≤ i ≤ pos(A'')
..10.626.	A'A'A'A'.	..5.985.	A''A''A''A''.
Subtotal: 16.611 / 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'')
..39.501.	A'A'A''A''.
Subtotal: 39.501 / 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: 56.112 / 3 / 5

Calculate contributions to

A'	A''

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