Characters of representations for molecular motions

Motion	E	C2.(z)	σv.(xz)	σd.(yz)
Cartesian 3N	18	0	6	0
Translation (x,y,z)	3	-1	1	1
Rotation (Rx,Ry,Rz)	3	-1	-1	-1
Vibration	12	2	6	0

Decomposition to irreducible representations

Motion	A1	A2	B1	B2	Total
Cartesian 3N	6	3	6	3	18
Translation (x,y,z)	1	0	1	1	3
Rotation (Rx,Ry,Rz)	0	1	1	1	3
Vibration	5	2	4	1	12

Molecular parameter

Number of Atoms (N)	6
Number of internal coordinates	12
Number of independant internal coordinates	5
Number of vibrational modes	12

---

Force field analysis

Allowed / forbidden vibronational transitions

Operator	A1	A2	B1	B2	Total
Linear (IR)	5	2	4	1	10 / 2
Quadratic (Raman)	5	2	4	1	12 / 0
IR + Raman	5	- - - -	4	1	10 / 0

Characters of force fields
(Symmetric powers of vibration representation)

Force field	E	C2.(z)	σv.(xz)	σd.(yz)
linear	12	2	6	0
quadratic	78	8	24	6
cubic	364	14	74	0
quartic	1.365	35	195	21
quintic	4.368	56	456	0
sextic	12.376	112	976	56

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

Force field	A1	A2	B1	B2
linear	5	2	4	1
quadratic	29	14	22	13
cubic	113	76	106	69
quartic	404	296	376	289
quintic	1.220	992	1.192	964
sextic	3.380	2.864	3.296	2.836

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)
..15.	A1A1.	..3.	A2A2.	..10.	B1B1.	..1.	B2B2.
Subtotal: 29 / 4 / 4
Irrep combinations (i,j) with indices: pos(A1) ≤ i ≤ j ≤ pos(B2)
Subtotal: 0 / 0 / 6
Total: 29 / 4 / 10

Contributions to nonvanishing cubic force field constants

Irrep combinations (i,i,i) with indices: pos(A1) ≤ i ≤ pos(B2)
..35.	A1A1A1.
Subtotal: 35 / 1 / 4
Irrep combinations (i,i,j) (i,j,j) with indices: pos(A1) ≤ i ≤ j ≤ pos(B2)
..15.	A1A2A2.	..50.	A1B1B1.	..5.	A1B2B2.
Subtotal: 70 / 3 / 12
Irrep combinations (i,j,k) with indices: pos(A1) ≤ i ≤ j ≤ k ≤ pos(B2)
..8.	A2B1B2.
Subtotal: 8 / 1 / 4
Total: 113 / 5 / 20

Contributions to nonvanishing quartic force field constants

Irrep combinations (i,i,i,i) with indices: pos(A1) ≤ i ≤ pos(B2)
..70.	A1A1A1A1.	..5.	A2A2A2A2.	..35.	B1B1B1B1.	..1.	B2B2B2B2.
Subtotal: 111 / 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)
..45.	A1A1A2A2.	..150.	A1A1B1B1.	..15.	A1A1B2B2.	..30.	A2A2B1B1.	..3.	A2A2B2B2.	..10.	B1B1B2B2.
Subtotal: 253 / 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)
..40.	A1A2B1B2.
Subtotal: 40 / 1 / 1
Total: 404 / 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