Characters of representations for molecular motions

Motion	E	C'2	i	σd
Cartesian 3N	0	0	0	0
Translation (x,y,z)	1	-1	-1	1
Rotation (Rx,Ry,Rz)	0	0	0	0
Vibration	-1	1	1	-1

Decomposition to irreducible representations

Motion	A1g	A2g	A1u	A2u	Total
Cartesian 3N	0	0	0	0	0
Translation (x,y,z)	0	0	0	1	1
Rotation (Rx,Ry,Rz)	0	0	0	0	0
Vibration	0	0	0	-1	-1

Molecular parameter

Number of Atoms (N)	0
Number of internal coordinates	-1
Number of independant internal coordinates	0
Number of vibrational modes	-1

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Force field analysis

Allowed / forbidden vibronational transitions

Operator	A1g	A2g	A1u	A2u	Total
Linear (IR)	0	0	0	-1	-1 / 0
Quadratic (Raman)	0	0	0	-1	0 / -1
IR + Raman	- - - -	0	0	- - - -	0* / 0

* Parity Mutual Exclusion Principle

Characters of force fields
(Symmetric powers of vibration representation)

Force field	E	C'2	i	σd
linear	-1	1	1	-1
quadratic	0	0	0	0
cubic	0	0	0	0
quartic	0	0	0	0
quintic	0	0	0	0
sextic	0	0	0	0

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

Force field	A1g	A2g	A1u	A2u
linear	0	0	0	-1
quadratic	0	0	0	0
cubic	0	0	0	0
quartic	0	0	0	0
quintic	0	0	0	0
sextic	0	0	0	0

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 D_1d

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(A1g) ≤ i ≤ pos(A2u)
Subtotal: 0 / 0 / 4
Irrep combinations (i,j) with indices: pos(A1g) ≤ i ≤ j ≤ pos(A2u)
Subtotal: 0 / 0 / 6
Total: 0 / 0 / 10

Contributions to nonvanishing cubic force field constants

Irrep combinations (i,i,i) with indices: pos(A1g) ≤ i ≤ pos(A2u)
Subtotal: 0 / 0 / 4
Irrep combinations (i,i,j) (i,j,j) with indices: pos(A1g) ≤ i ≤ j ≤ pos(A2u)
Subtotal: 0 / 0 / 12
Irrep combinations (i,j,k) with indices: pos(A1g) ≤ i ≤ j ≤ k ≤ pos(A2u)
Subtotal: 0 / 0 / 4
Total: 0 / 0 / 20

Contributions to nonvanishing quartic force field constants

Irrep combinations (i,i,i,i) with indices: pos(A1g) ≤ i ≤ pos(A2u)
Subtotal: 0 / 0 / 4
Irrep combinations (i,i,i,j) (i,j,j,j) with indices: pos(A1g) ≤ i ≤ j ≤ pos(A2u)
Subtotal: 0 / 0 / 12
Irrep combinations (i,i,j,j) with indices: pos(A1g) ≤ i ≤ j ≤ pos(A2u)
Subtotal: 0 / 0 / 6
Irrep combinations (i,i,j,k) (i,j,j,k) (i,j,k,k) with indices: pos(A1g) ≤ i ≤ j ≤ k ≤ pos(A2u)
Subtotal: 0 / 0 / 12
Irrep combinations (i,j,k,l) with indices: pos(A1g) ≤ i ≤ j ≤ k ≤ l ≤ pos(A2u)
Subtotal: 0 / 0 / 1
Total: 0 / 0 / 35

Calculate contributions to

A1g	A2g	A1u	A2u

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