CHM 501 Lecture

Group Theoretical assessment of the energies of orbitals

Consider all trans butadiene. Find the relative orbital energies of the orbitals and label them with their irreducible representations.

Point Group: C_2h

C2h	E	C2	i	h
ag	1	1	1	1	Rz	x2, y2, z2, xy
bg	1	–1	1	–1	Rx, Ry	xz, yz
au	1	1	–1	–1	z
bu	1	–1	–1	1	x, y

 

Find the transformation properties of the p orbitals that form the orbitals:

	E	C2	i	h
p orbitals	4	0	0	–4

This gives 2b_g + 2a_u

Now use projection operators to find the composition of each molecular orbital:

	E	C2	i	h
p1	p1	p4	–p4	–p1
p2	p2	p3	–p3	–p2

P(b_g-1) = [(1)p₁ + (–1)p₄ + (1)(–p₄) + (–1)(–p₁)] = 2(p₁ – p₄)

(b_g-1) = 2^–½(p₁–p₄)

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

(b_g-2) = 2^–½(p₂ – p₃)

P(a_u-1) = [(1)p₁ + (1)p₄ + (–1)(–p₄) + (–1)(–p₁)] = 2(p₁ + p₄)

(a_u-1) = 2^–½(p₁ + p₄)

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

(a_u-2) = 2^–½(p₂ + p₃)

Now draw the MO diagram

Ozone