Characters of representations for molecular motions

Motion	E	2C11	2(C11)2	2(C11)3	2(C11)4	2(C11)5	11C'2
Cartesian 3N	0	0.000	0.000	0.000	0.000	0.000	0
Translation (x,y,z)	3	2.683	1.831	0.715	-0.310	-0.919	-1
Rotation (Rx,Ry,Rz)	3	2.683	1.831	0.715	-0.310	-0.919	-1
Vibration	-6	-5.365	-3.662	-1.431	0.619	1.838	2

Decomposition to irreducible representations

Motion	A1	A2	E1	E2	E3	E4	E5	Total
Cartesian 3N	0	0	0	0	0	0	0	0
Translation (x,y,z)	0	1	1	0	0	0	0	2
Rotation (Rx,Ry,Rz)	0	1	1	0	0	0	0	2
Vibration	0	-2	-2	0	0	0	0	-4

Molecular parameter

Number of Atoms (N)	0
Number of internal coordinates	-6
Number of independant internal coordinates	0
Number of vibrational modes	-4

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Force field analysis

Allowed / forbidden vibronational transitions

Operator	A1	A2	E1	E2	E3	E4	E5	Total
Linear (IR)	0	-2	-2	0	0	0	0	-4 / 0
Quadratic (Raman)	0	-2	-2	0	0	0	0	-2 / -2
IR + Raman	- - - -	- - - -	-2	- - - -	0	0	0	-2 / 0

Characters of force fields
(Symmetric powers of vibration representation)

Force field	E	2C11	2(C11)2	2(C11)3	2(C11)4	2(C11)5	11C'2
linear	-6	-5.365	-3.662	-1.431	0.619	1.838	2
quadratic	15	12.561	7.014	1.942	-0.524	-0.993	-1
cubic	-20	-16.392	-8.704	-3.024	-2.192	-3.689	-4
quartic	15	12.561	7.014	1.942	-0.524	-0.993	-1
quintic	-6	-5.365	-3.662	-1.431	0.619	1.838	2
sextic	1	1.000	1.000	1.000	1.000	1.000	1

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

Force field	A1	A2	E1	E2	E3	E4	E5
linear	0	-2	-2	0	0	0	0
quadratic	2	3	4	1	0	0	0
cubic	-6	-2	-4	-2	0	0	0
quartic	2	3	4	1	0	0	0
quintic	0	-2	-2	0	0	0	0
sextic	1	0	0	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 D₁₁

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(E5)
..1.	A2A2.	..1.	E1E1.
Subtotal: 2 / 2 / 7
Irrep combinations (i,j) with indices: pos(A1) ≤ i ≤ j ≤ pos(E5)
Subtotal: 0 / 0 / 21
Total: 2 / 2 / 28

Contributions to nonvanishing cubic force field constants

Irrep combinations (i,i,i) with indices: pos(A1) ≤ i ≤ pos(E5)
Subtotal: 0 / 0 / 7
Irrep combinations (i,i,j) (i,j,j) with indices: pos(A1) ≤ i ≤ j ≤ pos(E5)
Subtotal: -6 / 0 / 42
Irrep combinations (i,j,k) with indices: pos(A1) ≤ i ≤ j ≤ k ≤ pos(E5)
Subtotal: 0 / 0 / 35
Total: -6 / 0 / 84

Contributions to nonvanishing quartic force field constants

Irrep combinations (i,i,i,i) with indices: pos(A1) ≤ i ≤ pos(E5)
..1.	E1E1E1E1.
Subtotal: 1 / 1 / 7
Irrep combinations (i,i,i,j) (i,j,j,j) with indices: pos(A1) ≤ i ≤ j ≤ pos(E5)
Subtotal: 0 / 0 / 42
Irrep combinations (i,i,j,j) with indices: pos(A1) ≤ i ≤ j ≤ pos(E5)
..1.	A2A2E1E1.
Subtotal: 1 / 1 / 21
Irrep combinations (i,i,j,k) (i,j,j,k) (i,j,k,k) with indices: pos(A1) ≤ i ≤ j ≤ k ≤ pos(E5)
Subtotal: 0 / 0 / 105
Irrep combinations (i,j,k,l) with indices: pos(A1) ≤ i ≤ j ≤ k ≤ l ≤ pos(E5)
Subtotal: 0 / 0 / 35
Total: 2 / 2 / 210

Calculate contributions to

A1	A2	E1	E2	E3	E4	E5

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