Characters of representations for molecular motions

Motion	E	S4	C2	(S4)3
Cartesian 3N	36	0	-4	0
Translation (x,y,z)	3	-1	-1	-1
Rotation (Rx,Ry,Rz)	3	1	-1	1
Vibration	30	0	-2	0

Decomposition to irreducible representations

Motion	A	B	E*	Total
Cartesian 3N	8	8	10	26
Translation (x,y,z)	0	1	1	2
Rotation (Rx,Ry,Rz)	1	0	1	2
Vibration	7	7	8	22

Molecular parameter

Number of Atoms (N)	12
Number of internal coordinates	30
Number of independant internal coordinates	7
Number of vibrational modes	22

---

Force field analysis

Allowed / forbidden vibronational transitions

Operator	A	B	E*	Total
Linear (IR)	7	7	8	15 / 7
Quadratic (Raman)	7	7	8	22 / 0
IR + Raman	- - - -	7	8	15 / 0

Characters of force fields
(Symmetric powers of vibration representation)

Force field	E	S4	C2	(S4)3
linear	30	0	-2	0
quadratic	465	-1	17	-1
cubic	4.960	0	-32	0
quartic	40.920	8	152	8
quintic	278.256	0	-272	0
sextic	1.623.160	-8	952	-8

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

Force field	A	B	E*
linear	7	7	8
quadratic	120	121	112
cubic	1.232	1.232	1.248
quartic	10.272	10.264	10.192
quintic	69.496	69.496	69.632
sextic	406.024	406.032	405.552

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 S₄

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(E)
..28.	AA.	..28.	BB.	..64.	EE.
Subtotal: 120 / 3 / 3
Irrep combinations (i,j) with indices: pos(A) ≤ i ≤ j ≤ pos(E)
Subtotal: 0 / 0 / 3
Total: 120 / 3 / 6

Contributions to nonvanishing cubic force field constants

Irrep combinations (i,i,i) with indices: pos(A) ≤ i ≤ pos(E)
..84.	AAA.
Subtotal: 84 / 1 / 3
Irrep combinations (i,i,j) (i,j,j) with indices: pos(A) ≤ i ≤ j ≤ pos(E)
..196.	ABB.	..448.	AEE.	..504.	BEE.
Subtotal: 1.148 / 3 / 6
Irrep combinations (i,j,k) with indices: pos(A) ≤ i ≤ j ≤ k ≤ pos(E)
Subtotal: 0 / 0 / 1
Total: 1.232 / 4 / 10

Contributions to nonvanishing quartic force field constants

Irrep combinations (i,i,i,i) with indices: pos(A) ≤ i ≤ pos(E)
..210.	AAAA.	..210.	BBBB.	..1.956.	EEEE.
Subtotal: 2.376 / 3 / 3
Irrep combinations (i,i,i,j) (i,j,j,j) with indices: pos(A) ≤ i ≤ j ≤ pos(E)
Subtotal: 0 / 0 / 6
Irrep combinations (i,i,j,j) with indices: pos(A) ≤ i ≤ j ≤ pos(E)
..784.	AABB.	..1.792.	AAEE.	..1.792.	BBEE.
Subtotal: 4.368 / 3 / 3
Irrep combinations (i,i,j,k) (i,j,j,k) (i,j,k,k) with indices: pos(A) ≤ i ≤ j ≤ k ≤ pos(E)
..3.528.	ABEE.
Subtotal: 3.528 / 1 / 3
Irrep combinations (i,j,k,l) with indices: pos(A) ≤ i ≤ j ≤ k ≤ l ≤ pos(E)
Subtotal: 0 / 0 / 0
Total: 10.272 / 7 / 15

Calculate contributions to

A	B	E

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