Physical Chemistry Third Edition

744 17 The Electronic States of Atoms. I. The Hydrogen Atom

2 0.2

2 0.4

0

0.2

0.0

0.4

1 s

0.8

0.6

1.0

2468101214

(a)

r/a

(b)

2 0.2

2 0.4

0

0.2

0.0

0.4

0.8

0.6

1.0

2468101214

2 p

2 s

(c)

2 0.2

0

r/a

0.0

0.2

0.4

2 4 6 8 10 12 14

3 s

3 p

3 d

r/a

R

(r

)^3

3/2a

R

(r

)^3

3/2a

R

(r)

3

a

3/2

Figure 17.6 Radial Factors for Hydrogen-Like Energy Eigenfunctions.(a)n=1.

(b)n= 2. (c)n=3.Asnis increased withlfixed, the number of spherical nodal sur-

faces increases. Aslis increased withnfixed, the number of spherical nodal surfaces

decreases.

If theRfactor in a wave function vanishes at some value ofr, this corresponds to a

spherical nodal surface. Figure 17.6 shows graphs of theRfunctions for the first three

shells. It is possible to construct rough graphs of theRfunctions without referring

to the formulas, using the following pattern: (1) thes(l0) radial factors (Rn 0 or

Rns) are nonzero at the origin but all of the otherRnlfunctions vanish at the origin;

(2) A curve representing a givenRnlfunction crosses the axisn−l−1 times and

must approach the axis for large values ofr, corresponding ton−lnodal spheres

including the one atr→∞. A node at the origin does not count in this number of

nodal spheres. For example,R 1 svanishes only atr→∞. The curve representing the

R 20 (R 2 s) function crosses the axis once, corresponding to one spherical nodal surface

in addition to the nodal surface atr→∞. TheR 21 (R 2 p) function vanishes only at