All of these distances, the most probable
radius (a 0 ), the average value (3/2a 0 ), and the
radius at which a sphere encloses the electron
with a 90% probability (3a 0 ) are measures
of the distribution of the wavefunction
(Figure 12.6). The usefulness of these values
will depend upon what property of the wave-
function needs to be calculated.
p Orbitals
For p orbitals, l=1 and the electron has a
non-zero angular momentum. This
results in the distribution being no longer
radially symmetrical as we saw for the s
orbitals. Let us look at the n=2 levels, of
which there are three orbitals with ml=+1,
0, and −1. The ml=0 orbital is denoted as
p 0 and the tables give the wavefunction as
being:
(12.29)
But in spherical coordinates z=rcosθ, so we can write:
(12.30)
Since the wavefunction can be written as the product of ztimes a radial
exponentially decreasing function, it will have the appearance of a node
at the origin, initially increasing for small values (where the exponential
is approximately a constant), and eventually decreasing exponentially. The
solutions are also seen to be symmetrical about the zaxis.
We can in a similar manner write the two other orbitals using the tables.
Since these orbitals are degenerate we are free to use linear combinations
of the orbitals. The convention is to use the orbitals that give the form
very similar to the term above. The two p orbitals are given by:
pRY (12.31)
ar
±± a==
⎛
⎝
⎜
⎜
⎞
⎠
⎟
⎟
⎛
⎝
⎜
⎜
⎞
12111
03201
46
1
,,/⎠⎠
⎟
⎟
⎡
⎣
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
⎛
⎝
⎜⎜
⎞
⎠
ee−ra/ ⎟⎟ ±i/(^2) sin
12
0 3
8
5
πθ φφ⎡
⎣
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
pz
a
0 era
032(^12)
4
1
2
(^10)
/
= ⎛ /
⎝
⎜⎜
⎞
⎠
⎟⎟
⎡
⎣
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
−
π==zf r()p
ar
a
0 era
0320(^12)
42
(^10)
(c
/
= /
⎛
⎝
⎜
⎜
⎞
⎠
⎟
⎟
⎛
⎝
⎜
⎜
⎞
⎠
⎟
⎟
−
πoos )θ2 Z
CHAPTER 12 THE HYDROGEN ATOM 251
0 1 a 0 2 a 0 3 a 0 4 a 00Maximum probability
for radiusAverage radiusRadius
for 90%
probabilityRadius1.5a 0Radial probability (4π(^2) r
)ψ
*ψ
Figure 12.6The
radial distribution
of the 1s orbital,
showing the most
probable radius,
average value, and
90% probability
value.