Physical Chemistry Third Edition

17.4 The Orbitals of the Hydrogen-Like Atom 745

the origin and atr→∞. TheR 30 (R 3 s) function corresponds to two spherical nodal

surfaces in addition to the nodal surface atr→∞. TheR 31 function vanishes at the

origin and produces a single spherical nodal surface in addition to the nodal surface at

r→∞. TheR 32 function vanishes at the origin but has no nodal surface in addition

to the nodal surface atr→∞.

Exercise 17.10

Sketch rough graphs of the followingRfunctions:

a.R 32 R 3 d

b.R 40 R 4 s

c.R 53

d.R 74

A zero value in theΘfunction produces a nodal surface that is a cone, except that

a node atθ0 andθπcorresponds to a line at thezaxis, and a node atθπ/ 2

corresponds to a nodal cone that is flattened into thexyplane. You can sketch rough

graphs of theΘfunctions from the following pattern: (1) TheΘlmfunction has a

number of nodal cones equal tol−m; (2) theΘl 0 functions are nonzero atθ0 and

θπ; (3) iflis odd,Θl 0 (0)−Θl 0 (π) and iflis even,Θl 0 (0)Θl 0 (π); (3) theΘlm

functions form≥1 vanish atθ0 andθπ. Note that the nodal lineθ0 and

θπis not included in the number of nodal cones. It is part of nodal planes from theΦ

function.

EXAMPLE17.5

Give the number of nodal cones for the followingΘlmfunctions:

a.Θ 00

b.Θ 10

c.Θ 11

d.Θ 20

Solution

a.TheΘ 00 function has no nodal surface and is equal to a nonzero constant.

b.TheΘ 10 function has one nodal cone. Since the cones are generally arranged symmetri-

cally, the nodal cone is flattened into thexyplane.

c.TheΘ 11 function vanishes atθ0 andθπ, but has no nodal cones.

d.TheΘ 20 function has two nodal cones. A graph of it starts at a nonzero constant atθ0,

crosses the axis twice, and has the same value atθπas atθ0.

A zero value in aΦfunction produces a nodal surface that is a vertical half-plane

containing thezaxis. There are always two nodal half-planes that combine to produce

a vertical plane containing thezaxis. These nodal surfaces follow the pattern: (1) The

Φ 0 function has no nodal planes; (2) form≥1 the realΦfunctions have a number

of equally spaced nodal planes equal tom(2mhalf-planes). TheΦmxfunctions are

nonzero atφ0 andφπ, so their nodal planes do not contain thexaxis. TheΦmy

functions are equal to zero atφ0 andφπ, so that one of their nodal planes contains

thexaxis. The real part of the complexΦmfunction is proportional toΦmxand has

the same nodal planes asΦmx. The imaginary part ofΦmis proportional toΦmyand

has the same nodal planes asΦmy. Several graphs ofΘandΦfunctions are shown in