Characters of representations for molecular motions

Motion	E	σh
Cartesian 3N	39	9
Translation (x,y,z)	3	1
Rotation (Rx,Ry,Rz)	3	-1
Vibration	33	9

Decomposition to irreducible representations

Motion	A'	A''	Total
Cartesian 3N	24	15	39
Translation (x,y,z)	2	1	3
Rotation (Rx,Ry,Rz)	1	2	3
Vibration	21	12	33

Molecular parameter

Number of Atoms (N)	13
Number of internal coordinates	33
Number of independant internal coordinates	21
Number of vibrational modes	33

---

Force field analysis

Allowed / forbidden vibronational transitions

Operator	A'	A''	Total
Linear (IR)	21	12	33 / 0
Quadratic (Raman)	21	12	33 / 0
IR + Raman	21	12	33 / 0

Characters of force fields
(Symmetric powers of vibration representation)

Force field	E	σh
linear	33	9
quadratic	561	57
cubic	6.545	273
quartic	58.905	1.113
quintic	435.897	3.969
sextic	2.760.681	12.817

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

Force field	A'	A''
linear	21	12
quadratic	309	252
cubic	3.409	3.136
quartic	30.009	28.896
quintic	219.933	215.964
sextic	1.386.749	1.373.932

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

---

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'.	..78.	A''A''.
Subtotal: 309 / 2 / 2
Irrep combinations (i,j) with indices: pos(A') ≤ i ≤ j ≤ pos(A'')
Subtotal: 0 / 0 / 1
Total: 309 / 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'')
..1.638.	A'A''A''.
Subtotal: 1.638 / 1 / 2
Irrep combinations (i,j,k) with indices: pos(A') ≤ i ≤ j ≤ k ≤ pos(A'')
Subtotal: 0 / 0 / 0
Total: 3.409 / 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'.	..1.365.	A''A''A''A''.
Subtotal: 11.991 / 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'')
..18.018.	A'A'A''A''.
Subtotal: 18.018 / 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: 30.009 / 3 / 5

Calculate contributions to

A'	A''

Show only nonzero contributions Show all contributions
Up to quartic force fieldUp to quintic force fieldUp to sextic force field