Characters of representations for molecular motions

Motion	E	2C3	3σv
Cartesian 3N	36	0	4
Translation (x,y,z)	3	0	1
Rotation (Rx,Ry,Rz)	3	0	-1
Vibration	30	0	4

Decomposition to irreducible representations

Motion	A1	A2	E	Total
Cartesian 3N	8	4	12	24
Translation (x,y,z)	1	0	1	2
Rotation (Rx,Ry,Rz)	0	1	1	2
Vibration	7	3	10	20

Molecular parameter

Number of Atoms (N)	12
Number of internal coordinates	30
Number of independant internal coordinates	7
Number of vibrational modes	20

---

Force field analysis

Allowed / forbidden vibronational transitions

Operator	A1	A2	E	Total
Linear (IR)	7	3	10	17 / 3
Quadratic (Raman)	7	3	10	17 / 3
IR + Raman	7	3	10	17 / 3

Characters of force fields
(Symmetric powers of vibration representation)

Force field	E	2C3	3σv
linear	30	0	4
quadratic	465	0	23
cubic	4.960	10	72
quartic	40.920	0	256
quintic	278.256	0	680
sextic	1.623.160	55	1.904

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

Force field	A1	A2	E
linear	7	3	10
quadratic	89	66	155
cubic	866	794	1.650
quartic	6.948	6.692	13.640
quintic	46.716	46.036	92.752
sextic	271.497	269.593	541.035

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_3v

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(E)
..28.	A1A1.	..6.	A2A2.	..55.	EE.
Subtotal: 89 / 3 / 3
Irrep combinations (i,j) with indices: pos(A1) ≤ i ≤ j ≤ pos(E)
Subtotal: 0 / 0 / 3
Total: 89 / 3 / 6

Contributions to nonvanishing cubic force field constants

Irrep combinations (i,i,i) with indices: pos(A1) ≤ i ≤ pos(E)
..84.	A1A1A1.	..220.	EEE.
Subtotal: 304 / 2 / 3
Irrep combinations (i,i,j) (i,j,j) with indices: pos(A1) ≤ i ≤ j ≤ pos(E)
..42.	A1A2A2.	..385.	A1EE.	..135.	A2EE.
Subtotal: 562 / 3 / 6
Irrep combinations (i,j,k) with indices: pos(A1) ≤ i ≤ j ≤ k ≤ pos(E)
Subtotal: 0 / 0 / 1
Total: 866 / 5 / 10

Contributions to nonvanishing quartic force field constants

Irrep combinations (i,i,i,i) with indices: pos(A1) ≤ i ≤ pos(E)
..210.	A1A1A1A1.	..15.	A2A2A2A2.	..1.540.	EEEE.
Subtotal: 1.765 / 3 / 3
Irrep combinations (i,i,i,j) (i,j,j,j) with indices: pos(A1) ≤ i ≤ j ≤ pos(E)
..1.540.	A1EEE.	..660.	A2EEE.
Subtotal: 2.200 / 2 / 6
Irrep combinations (i,i,j,j) with indices: pos(A1) ≤ i ≤ j ≤ pos(E)
..168.	A1A1A2A2.	..1.540.	A1A1EE.	..330.	A2A2EE.
Subtotal: 2.038 / 3 / 3
Irrep combinations (i,i,j,k) (i,j,j,k) (i,j,k,k) with indices: pos(A1) ≤ i ≤ j ≤ k ≤ pos(E)
..945.	A1A2EE.
Subtotal: 945 / 1 / 3
Irrep combinations (i,j,k,l) with indices: pos(A1) ≤ i ≤ j ≤ k ≤ l ≤ pos(E)
Subtotal: 0 / 0 / 0
Total: 6.948 / 9 / 15

Calculate contributions to

A1	A2	E

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