Physical Chemistry Third Edition

746 17 The Electronic States of Atoms. I. The Hydrogen Atom

Figure 17.7. It is possible to draw rough graphs of otherΘandΦfunctions from the

patterns of their nodes.

Exercise 17.11

Draw approximate graphs of the following functions:

a.Φ 3 y

b.Φ 4 x

c.Θ 32

d.Θ 40

e.Θ 53

The total number of nodal surfaces for any hydrogen-like orbital is

Total number of nodal surfacesn−l+l−m+mn (17.4-2)

which includes the nodal sphere atr→∞. The pattern of nodal surfaces is summa-

rized in Table 17.4. It is possible to draw rough sketches of the nodal surfaces for

any hydrogen-like orbital from the patterns of the nodes of the three factors.

Table 17.4 Nodal Surface for the Hydrogen-Like Orbitals

Rnl n−lnodal spheres includingr→∞

nonzero at origin only ifl 0

Θlm l−mnodal cones not including zeros atθ0 andθπ

nonzero atθ0 andθπonly ifm 0

Φmx mvertical nodal planes, none containing thexaxis

Φmy mvertical nodal planes, including one containing thexaxis

Φm real part: same asΦmx

imaginary part: same asΦmy

Q 00

0

Q 11

0

Q 10

0

F 0

0 2



Re(F 1 )

0 2



Im(F 1 )

0 2



F 1 x

0 2



F 1 y

0 2



Figure 17.7 SomeΘandΦFunctions.