BioPHYSICAL chemistry

We can plot the radial dependence of these wavefunctions and see that the

higher orbitals have a more complex distribution with nodes appearing.

For example, the 2s orbital has nodes at:

(12.15)

Since the solutions are non-zero for all values of r(excluding the nodes),

the electron has a finite probability of being located anywhere in space.

However, the probability is clearly highest near the nucleus. Let’s calculate

these probabilities using the wavefunctions just as we did for the particle

in a box and harmonic oscillator. The probability of finding an electron at

20 2

0

−= = 0

r

a

so ra

248 PART 2 QUANTUM MECHANICS AND SPECTROSCOPY

0.8

0.2

0

0.6

0.4

0.2

0 5 10 15

Ra

3/2 0

r/a 0

(a) 2.0 (b)

0.5

1.5

1 s

1.0 2 s

0

0123

Ra

3/2 0

r/a 0

(c)0.4

0.1

0

0.3

0.2

0.1

0

Ra

3/2 0

r/a 0

(d)0.15

0.10

0.05

0

0 5 10 15

Ra

3/2 0

r/a 0

(e)0.10

0

0.05

0.05

0

Ra

3/2 0

r/a 0

(f)0.05

0.03

0.02

0.01

0.04

0

0

5

5 5

10

10 10

15

15 15

20

20 20

Ra

3/2 0

r/a 0

3 s

2 p^3 p

3 d

Figure 12.2The radial wavefunctions for some solutions: (a–c) the first three s orbitals, (d,e) the
first two p orbitals, and (f) the first d orbital.