Characters of representations for molecular motions

Motion	E	C2.(z)	σv.(xz)	σd.(yz)
Cartesian 3N	54	0	6	0
Translation (x,y,z)	3	-1	1	1
Rotation (Rx,Ry,Rz)	3	-1	-1	-1
Vibration	48	2	6	0

Decomposition to irreducible representations

Motion	A1	A2	B1	B2	Total
Cartesian 3N	15	12	15	12	54
Translation (x,y,z)	1	0	1	1	3
Rotation (Rx,Ry,Rz)	0	1	1	1	3
Vibration	14	11	13	10	48

Molecular parameter

Number of Atoms (N)	18
Number of internal coordinates	48
Number of independant internal coordinates	14
Number of vibrational modes	48

---

Force field analysis

Allowed / forbidden vibronational transitions

Operator	A1	A2	B1	B2	Total
Linear (IR)	14	11	13	10	37 / 11
Quadratic (Raman)	14	11	13	10	48 / 0
IR + Raman	14	- - - -	13	10	37 / 0

Characters of force fields
(Symmetric powers of vibration representation)

Force field	E	C2.(z)	σv.(xz)	σd.(yz)
linear	48	2	6	0
quadratic	1.176	26	42	24
cubic	19.600	50	182	0
quartic	249.900	350	798	300
quintic	2.598.960	650	2.814	0
sextic	22.957.480	3.250	9.730	2.600

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

Force field	A1	A2	B1	B2
linear	14	11	13	10
quadratic	317	284	292	283
cubic	4.958	4.867	4.933	4.842
quartic	62.837	62.288	62.512	62.263
quintic	650.606	649.199	650.281	648.874
sextic	5.743.265	5.737.100	5.740.340	5.736.775

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_2v

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(B2)
..105.	A1A1.	..66.	A2A2.	..91.	B1B1.	..55.	B2B2.
Subtotal: 317 / 4 / 4
Irrep combinations (i,j) with indices: pos(A1) ≤ i ≤ j ≤ pos(B2)
Subtotal: 0 / 0 / 6
Total: 317 / 4 / 10

Contributions to nonvanishing cubic force field constants

Irrep combinations (i,i,i) with indices: pos(A1) ≤ i ≤ pos(B2)
..560.	A1A1A1.
Subtotal: 560 / 1 / 4
Irrep combinations (i,i,j) (i,j,j) with indices: pos(A1) ≤ i ≤ j ≤ pos(B2)
..924.	A1A2A2.	..1.274.	A1B1B1.	..770.	A1B2B2.
Subtotal: 2.968 / 3 / 12
Irrep combinations (i,j,k) with indices: pos(A1) ≤ i ≤ j ≤ k ≤ pos(B2)
..1.430.	A2B1B2.
Subtotal: 1.430 / 1 / 4
Total: 4.958 / 5 / 20

Contributions to nonvanishing quartic force field constants

Irrep combinations (i,i,i,i) with indices: pos(A1) ≤ i ≤ pos(B2)
..2.380.	A1A1A1A1.	..1.001.	A2A2A2A2.	..1.820.	B1B1B1B1.	..715.	B2B2B2B2.
Subtotal: 5.916 / 4 / 4
Irrep combinations (i,i,i,j) (i,j,j,j) with indices: pos(A1) ≤ i ≤ j ≤ pos(B2)
Subtotal: 0 / 0 / 12
Irrep combinations (i,i,j,j) with indices: pos(A1) ≤ i ≤ j ≤ pos(B2)
..6.930.	A1A1A2A2.	..9.555.	A1A1B1B1.	..5.775.	A1A1B2B2.	..6.006.	A2A2B1B1.	..3.630.	A2A2B2B2.	..5.005.	B1B1B2B2.
Subtotal: 36.901 / 6 / 6
Irrep combinations (i,i,j,k) (i,j,j,k) (i,j,k,k) with indices: pos(A1) ≤ i ≤ j ≤ k ≤ pos(B2)
Subtotal: 0 / 0 / 12
Irrep combinations (i,j,k,l) with indices: pos(A1) ≤ i ≤ j ≤ k ≤ l ≤ pos(B2)
..20.020.	A1A2B1B2.
Subtotal: 20.020 / 1 / 1
Total: 62.837 / 11 / 35

Calculate contributions to

A1	A2	B1	B2

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