Characters of representations for molecular motions

Motion	E	C2.(z)	C2.(y)	C2.(x)
Cartesian 3N	66	0	0	-6
Translation (x,y,z)	3	-1	-1	-1
Rotation (Rx,Ry,Rz)	3	-1	-1	-1
Vibration	60	2	2	-4

Decomposition to irreducible representations

Motion	A	B1	B2	B3	Total
Cartesian 3N	15	18	18	15	66
Translation (x,y,z)	0	1	1	1	3
Rotation (Rx,Ry,Rz)	0	1	1	1	3
Vibration	15	16	16	13	60

Molecular parameter

Number of Atoms (N)	22
Number of internal coordinates	60
Number of independant internal coordinates	15
Number of vibrational modes	60

---

Force field analysis

Allowed / forbidden vibronational transitions

Operator	A	B1	B2	B3	Total
Linear (IR)	15	16	16	13	45 / 15
Quadratic (Raman)	15	16	16	13	60 / 0
IR + Raman	- - - -	16	16	13	45 / 0

Characters of force fields
(Symmetric powers of vibration representation)

Force field	E	C2.(z)	C2.(y)	C2.(x)
linear	60	2	2	-4
quadratic	1.830	32	32	38
cubic	37.820	62	62	-132
quartic	595.665	527	527	721
quintic	7.624.512	992	992	-2.240
sextic	82.598.880	5.952	5.952	9.184

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

Force field	A	B1	B2	B3
linear	15	16	16	13
quadratic	483	448	448	451
cubic	9.453	9.488	9.488	9.391
quartic	149.360	148.736	148.736	148.833
quintic	1.906.064	1.906.688	1.906.688	1.905.072
sextic	20.654.992	20.647.424	20.647.424	20.649.040

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 D₂

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(B3)
..120.	AA.	..136.	B1B1.	..136.	B2B2.	..91.	B3B3.
Subtotal: 483 / 4 / 4
Irrep combinations (i,j) with indices: pos(A) ≤ i ≤ j ≤ pos(B3)
Subtotal: 0 / 0 / 6
Total: 483 / 4 / 10

Contributions to nonvanishing cubic force field constants

Irrep combinations (i,i,i) with indices: pos(A) ≤ i ≤ pos(B3)
..680.	AAA.
Subtotal: 680 / 1 / 4
Irrep combinations (i,i,j) (i,j,j) with indices: pos(A) ≤ i ≤ j ≤ pos(B3)
..2.040.	AB1B1.	..2.040.	AB2B2.	..1.365.	AB3B3.
Subtotal: 5.445 / 3 / 12
Irrep combinations (i,j,k) with indices: pos(A) ≤ i ≤ j ≤ k ≤ pos(B3)
..3.328.	B1B2B3.
Subtotal: 3.328 / 1 / 4
Total: 9.453 / 5 / 20

Contributions to nonvanishing quartic force field constants

Irrep combinations (i,i,i,i) with indices: pos(A) ≤ i ≤ pos(B3)
..3.060.	AAAA.	..3.876.	B1B1B1B1.	..3.876.	B2B2B2B2.	..1.820.	B3B3B3B3.
Subtotal: 12.632 / 4 / 4
Irrep combinations (i,i,i,j) (i,j,j,j) with indices: pos(A) ≤ i ≤ j ≤ pos(B3)
Subtotal: 0 / 0 / 12
Irrep combinations (i,i,j,j) with indices: pos(A) ≤ i ≤ j ≤ pos(B3)
..16.320.	AAB1B1.	..16.320.	AAB2B2.	..10.920.	AAB3B3.	..18.496.	B1B1B2B2.	..12.376.	B1B1B3B3.	..12.376.	B2B2B3B3.
Subtotal: 86.808 / 6 / 6
Irrep combinations (i,i,j,k) (i,j,j,k) (i,j,k,k) with indices: pos(A) ≤ i ≤ j ≤ k ≤ pos(B3)
Subtotal: 0 / 0 / 12
Irrep combinations (i,j,k,l) with indices: pos(A) ≤ i ≤ j ≤ k ≤ l ≤ pos(B3)
..49.920.	AB1B2B3.
Subtotal: 49.920 / 1 / 1
Total: 149.360 / 11 / 35

Calculate contributions to

A	B1	B2	B3

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