Physical Chemistry Third Edition

854 20 The Electronic States of Diatomic Molecules

This orbital is orthogonal toψ 2 sp(1).

EXAMPLE20.9

Using the fact that the 2sand 2pzorbitals are normalized and orthogonal to each other, show

thatc 1 andc 2 both equal

√

1 /2 to normalize the hybrid orbitals.

Solution

∫

ψ∗ 2 sp(1)ψ 2 sp(1)dq 1

c 12

∫

[−ψ 2 s+ψ 2 pz][−ψ 2 s+ψ 2 pz]dq 1

c 12

∫

ψ^22 sdq− 2 c^21

∫

ψ 2 sψ 2 pzdq+c^21

∫

ψ^22 pzdqc^21 (1− 2 × 0 +1) 2 c^21  1

c 1 √^12

∫

ψ∗ 2 sp(2)ψ 2 sp(2)dq 1

c 22

∫

[−ψ 2 s−ψ 2 pz]^2 dq 1

c^22

∫

ψ 22 sdq+ 2 c^22

∫

ψ 2 pzψ 2 sdq+c^22

∫

ψ 22 pzdqc^22 (1+ 0 +1) 2 c^22  1

c 2 √^1

2

Exercise 20.15

Show that the 2sp(1) and 2sp(2) hybrid orbitals are orthogonal to each other.

Figure 20.15 shows cross sections of the orbital regions of the 2sand 2pzorbitals

and of the 2sp(1) and 2sp(2) hybrid orbitals. In the 2sp(1) hybrid orbital the 2sand

the 2pzorbitals add in the direction of the positivezaxis and partially cancel in the

direction of the negativezaxis. We say that this hybrid orbital is “directional,” with

its orbital region extending in the positivezdirection. The orbital region of the 2sp(2)

hybrid orbital extends in the opposite direction from that of the 2sp(1) orbital.

Using these hybrid orbitals as part of the basis set, we can approximate the two

occupied LCAOMOs as linear combinations of no more than two atomic orbitals:

ψ 1 σψ 1 sLi (20.4-8a)

ψ 2 σc 2 sp(1)Liψ 2 sp(1)Li+c 1 sHψ 1 sH− 0. 47 ψ 2 sp(1)Li− 0. 88 ψ 1 sH (20.4-8b)

In the second expression for the 2σorbital the values of the coefficients were cho-

sen to maintain approximately the same relative weights of the atomic orbitals as in

the Hartree–Fock–Roothaan orbital. Figure 20.16 shows the orbital region of the 2σ

LCAOMO and shows that it is a bonding orbital with overlap between the nuclei. It