Concise Physical Chemistry

c16 JWBS043-Rogers September 13, 2010 11:28 Printer Name: Yet to Come

262 WAVE MECHANICS OF SIMPLE SYSTEMS

a .529

S(r)

1

81 3 a

1.5

27 18

r

a

2

r

a

2

e

r

3a

R(r) r^2 S(r)^2

0105

0

R(r) 0.01

r

FIGURE 16.9 The radial probability density for an electron in the 3sorbital of hydrogen.

To find the probability density function for the lowest three orbitals, we square

the wave function. At each node the probability density of finding the electron at the

radial distanceris zero. Between the nodes are maxima calledantinodes.Forthe3s

orbital, this leads to Fig. 16.9 in which there are the antinodes at aboutr= 0 .5, 2,

and 7. The radii of maximum probability densities correspond to the “shells” in early

atomic theory.

16.9 ORTHOGONALITY AND OVERLAP

It is not difficult to show that the probability density of the product of thepzorbital

times thepxorbital is zero. Theiroverlapis zero. Unlikesorbitals which are positive

(+) everywhere, the 2porbitals have one angular nodal plane. Therefore they pass

from−to+and back again as the radius vector passes through either 0 orπ.The

porbital has aplane of inversionas one of itssymmetry elements.Theporbitals

have a positivelobeand a negative lobe (Fig. 16.10). If aporbital and ansorbital

+

FIGURE 16.10 The radial node of the 2patomic orbital.