Characters of representations for molecular motions

Motion	E	2C3	3C'2
Cartesian 3N	48	0	0
Translation (x,y,z)	3	0	-1
Rotation (Rx,Ry,Rz)	3	0	-1
Vibration	42	0	2

Decomposition to irreducible representations

Motion	A1	A2	E	Total
Cartesian 3N	8	8	16	32
Translation (x,y,z)	0	1	1	2
Rotation (Rx,Ry,Rz)	0	1	1	2
Vibration	8	6	14	28

Molecular parameter

Number of Atoms (N)	16
Number of internal coordinates	42
Number of independant internal coordinates	8
Number of vibrational modes	28

---

Force field analysis

Allowed / forbidden vibronational transitions

Operator	A1	A2	E	Total
Linear (IR)	8	6	14	20 / 8
Quadratic (Raman)	8	6	14	22 / 6
IR + Raman	- - - -	- - - -	14	14 / 0

Characters of force fields
(Symmetric powers of vibration representation)

Force field	E	2C3	3C'2
linear	42	0	2
quadratic	903	0	23
cubic	13.244	14	44
quartic	148.995	0	275
quintic	1.370.754	0	506
sextic	10.737.573	105	2.277

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

Force field	A1	A2	E
linear	8	6	14
quadratic	162	139	301
cubic	2.234	2.190	4.410
quartic	24.970	24.695	49.665
quintic	228.712	228.206	456.918
sextic	1.790.769	1.788.492	3.579.156

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 D₃

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)
..36.	A1A1.	..21.	A2A2.	..105.	EE.
Subtotal: 162 / 3 / 3
Irrep combinations (i,j) with indices: pos(A1) ≤ i ≤ j ≤ pos(E)
Subtotal: 0 / 0 / 3
Total: 162 / 3 / 6

Contributions to nonvanishing cubic force field constants

Irrep combinations (i,i,i) with indices: pos(A1) ≤ i ≤ pos(E)
..120.	A1A1A1.	..560.	EEE.
Subtotal: 680 / 2 / 3
Irrep combinations (i,i,j) (i,j,j) with indices: pos(A1) ≤ i ≤ j ≤ pos(E)
..168.	A1A2A2.	..840.	A1EE.	..546.	A2EE.
Subtotal: 1.554 / 3 / 6
Irrep combinations (i,j,k) with indices: pos(A1) ≤ i ≤ j ≤ k ≤ pos(E)
Subtotal: 0 / 0 / 1
Total: 2.234 / 5 / 10

Contributions to nonvanishing quartic force field constants

Irrep combinations (i,i,i,i) with indices: pos(A1) ≤ i ≤ pos(E)
..330.	A1A1A1A1.	..126.	A2A2A2A2.	..5.565.	EEEE.
Subtotal: 6.021 / 3 / 3
Irrep combinations (i,i,i,j) (i,j,j,j) with indices: pos(A1) ≤ i ≤ j ≤ pos(E)
..4.480.	A1EEE.	..3.360.	A2EEE.
Subtotal: 7.840 / 2 / 6
Irrep combinations (i,i,j,j) with indices: pos(A1) ≤ i ≤ j ≤ pos(E)
..756.	A1A1A2A2.	..3.780.	A1A1EE.	..2.205.	A2A2EE.
Subtotal: 6.741 / 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)
..4.368.	A1A2EE.
Subtotal: 4.368 / 1 / 3
Irrep combinations (i,j,k,l) with indices: pos(A1) ≤ i ≤ j ≤ k ≤ l ≤ pos(E)
Subtotal: 0 / 0 / 0
Total: 24.970 / 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