BioPHYSICAL chemistry

So, the total angular momentum is quantized by land the projection of
the angular momentum along the zdirection is quantized by ml.

Orbitals

The solutions to Schrödinger’s equation consist of the product of a radial
term and an angular term:

ψn,l,ml(r,θ,φ) =Rn,l(r)Ylml(θ,φ) (12.11)

where each solution is uniquely defined by the three indices, n, l, ml, that
are restricted to be n=1, 2, 3,..., l=0, 1, 2,..., n−1, and ml=l,
l−1, l−2,..., −l, where nis the principal quantum number arising
from the quantization of energy as:

(12.12)

lis the angular-momentum quantum number arising from quantization

of the angular momentum with magnitude , and mlarises from

the quantization of the zcomponent of angular momentum mlZ. These
solutions are degenerate in energy; that is, they are dependent only upon
nand not lor ml, as seen on the energy diagram (Figure 12.1).
All of the orbitals are directly related to these wave-
functions. Let us look first at the properties of the l= 0
and l=1 wavefunctions that correspond to the s and p
orbitals (see Figure 12.2).

s Orbitals

An s orbital is one with l=0. As a result the angular
component is a constant and the dependence is strictly
radial. For the 1s orbital, the wavefunction is:

(12.13)

The higher orbitals all have the same general appear-
ance as there is an exponential dependence multiplied
by a polynomial term. For example, the 2s orbital is:

ψ (12.14)

(^200) π 03 0
(^12)
22

1

4

=− 2 /^0

⎛

⎝

⎜⎜

⎞

⎠

⎟⎟ −

a

r

a

era

ψ

(^100) π 03
(^1) / 0
= −
a
era
ll()+ 1 Z

E

hcR

n

R

me

n hc

H

H

=− = e

2

4

0

where 8 ε 23

CHAPTER 12 THE HYDROGEN ATOM 247

 s

3

2

1

n

pdf

3 s 3 p

2 s 2 p

[1]

[1]

1 s

[1]

[3]

[3] [5]

3 d

Energy

Figure 12.1An energy diagram for

the solutions of the hydrogen atom.