BioPHYSICAL chemistry

This reduces to:

(12.32)

Using the relationships

x=rsinθcosφ

y=rsinθsinφ (12.33)

e±iφ=cosφ±isinφ

it is easy to show that the p orbitals can be rewritten as

(12.34)

(12.35)

These combinations have the zcomponent of angular momentum can-

celing and have the same basic form as for p 0. These versions of the

orbitals yield the conventional representation shown

in Figure 12.7.

d Orbitals

The d orbitals represent the l=2 orbitals and arise when

nis at least 3; then=3 shell contains one 3s orbital,

three 3p orbitals, and five 3d orbitals. The electrons in

the 3d orbitals have an angular momentum with

mlequal to −2, −1, 0, +1, and +2. As was found for the

p orbitals, the wavefunctions with opposite values of

mlcan be combined in pairs to give rise to conventional

standing orbitals (Figure 12.8), expressed as:

dxy=xy f(r) (12.36)

dyz=yz f(r)

dzx=zx f(r)

dx (^2) −y 2 =
dz 2 =

1

23

()() 3 zrfr^22 −

1

2

()()xyfr^22 −

6 Z

p

i

y=+∝()pp yr+−

2

11

pppxrx= ()

−

+−−∝

1

2

11

p

a

± = re±i

⎛

⎝

⎜

⎜

⎞

⎠

⎟

(^10) ⎟
52
1
8

1

sin

/

5

π

θ φ

252 PART 2 QUANTUM MECHANICS AND SPECTROSCOPY

z

pz py

px

y

x

Figure 12.7A representation of p
orbitals of the hydrogen atom. A nodal
plane separates the two lobes of each
orbital.