Question

a reducible representation for the motion of formic acid (point group Cs) was found to be...

23) A reducible representation for the motion of formic acid (HCOOH; point group is Cs) was found to be: EOh 15 5 Given the character table below, factor this reducible representation into a combination of irreducible representations, following the equation: N= Σ Xirr Xream Cs E On 1 1 x, y, R2 1 -1 |z, R., Ry A" x y², 2², xy lyz, xz

please help unsure how to solve this becauseof the point group!

Answer 1

Formic acid has a Cs point group.

HCOOH	E	σσv
No.of unshifted atoms	5	1
Contribution of each character per atom	3	5
Reducible representations	15	5

Cs A A" On 1 1 x, y, R2 1 -1 2, Rx, Ry x y², 2², xy | yz, xZ

the character table is given in the question

applying the reduction formula

N = 2 XirrXredn

h = order of the group

Ҳ_irr = character of the irreducible representation

Ҳ_red = character of the reducible representation

n = coefficient of each character in the character table

Γ _A' = 1/2 {[15 * 1 * 1] + [5*1*1] } = 10

Γ _A'' = 1/2 {[15 * 1 * 1] + [5*1*(-1)] } = 5

10A' + 5A'' combination of irreducible representation