## CHM 501

29. Find the irreducible representations of the C π orbitals in . Draw sketches of each of the orbitals and energy order them. Show the electron occupation of the orbitals.

The molecule belongs to the C2v point group. Use the p orbitals perpendicular to the page as the basis set. The z axis is along the C-Cl bond and let the x axis be in the plane of the paper. Let p1 be on the C connected to Cl and number clockwise around the ring.

 C2v E C2 σv(xz) σv'(yz) a1 1 1 1 1 a2 1 1 –1 –1 b1 1 –1 1 –1 b2 1 –1 –1 1 p orbitals 3 –1 –3 1 R(p1) p1 –p1 –p1 p1 R(p2) p2 –p3 –p2 p3

n(a1) = [(1)(1)(3) + (1)(1)(–1) + (1)(1)(–3) + (1)(1)(1)]/4 = 0

n(a2) = [(1)(1)(3) + (1)(1)(–1) + (1)(–1)(–3) + (1)(–1)(1)]/4 = 1

n(b1) = [(1)(1)(3) + (1)(–1)(–1) + (1)(1)(–3) + (1)(–1)(1)]/4 = 0

n(b2) = [(1)(1)(3) + (1)(–1)(–1) + (1)(–1)(–3) + (1)(1)(1)]/4 = 2

a2 + 2b2

From the results of the symmetry operations on each orbital, p1 is never changed with any other orbital, so its representation is found by simply inspecting the characters (1, –1, –1, 1) and is one of the b2 orbitals.

Using the projection operators gives:

P(a2 p2) = (1)p2 + (1)(–p3) + (–1)(–p2) + (–1)(p3) = 2[p2 – p3]

Ψ(a2) = 2½[p2 – p3]

P(b2 p2) = (1)p2 + (–1)(–p3) + (–1)(–p2) + (1)(p3) = 2[p2 + p3]

Ψ(b2) = 2½[p2 + p3]