Page 190 - PRINCIPLES OF QUANTUM MECHANICS as Applied to Chemistry and Chemical Physics

6.4 Atomic orbitals 181
7=2
1 Z
2
2
j3d 2 2i 2 ÿ1=2 (j3d 2 i‡j3d ÿ2 i) ˆ (x ÿ y )e ÿ( Zr=3a 0 )
x ÿ y
81(2ð) 1=2 a 0
(6:64d)
2 1=2 7=2
Z
j3d xy i ÿi2 ÿ1=2 (j3d 2 iÿj3d ÿ2 i) ˆ xye ÿ( Zr=3a 0 ) (6:64e)
81ð 1=2 a 0

In forming j3d 2 2i and j3d xy i, equations (A.37) and (A.38) were used. Graphs
x ÿ y
of the square of the angular part of these ®ve real functions are shown in Figure
6.3 and the mathematical expressions are listed in Table 6.2.

Radial functions and expectation values
The radial functions R nl (r) for the 1s, 2s, 2p, 3s, 3p, and 3d atomic orbitals are
shown in Figure 6.4. For states with l 6ˆ 0, the radial functions vanish at the
origin. For states with no angular momentum (l ˆ 0), however, the radial
function R n0 (r) has a non-zero value at the origin. The function R nl (r) has
(n ÿ l ÿ 1) nodes between 0 and 1, i.e., the function crosses the r-axis
(n ÿ l ÿ 1) times, not counting the origin.
The probability of ®nding the electron in the hydrogen-like atom, with the
distance r from the nucleus between r and r ‡ dr, with angle è between è and
è ‡ dè, and with the angle j between j and j ‡ dj is

2
2 2
2
jø nlm j dô ˆ [R nl (r)] jY lm (è, j)j r sin è dr dè dj
To ®nd the probability D nl (r)dr that the electron is between r and r ‡ dr
regardless of the direction, we integrate over the angles è and j to obtain

ð 2ð

2
2
2
2
2
D nl (r)dr ˆ r [R nl (r)] dr jY lm (è, j)j sin è dè dj ˆ r [R nl (r)] dr
0 0
(6:65)
Since the spherical harmonics are normalized, the value of the double integral
is unity.
The radial distribution function D nl (r) is the probability density for the
electron being in a spherical shell with inner radius r and outer radius r ‡ dr.
For the 1s, 2s, and 2p states, these functions are