# Principles of Quantum Mechanics as Applied to Chemistry and Chemical Physics

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```orbitals are given in Table 6.2.
d orbitals
The five wave functions for n  3, l  2 are
j3d0i  j320i  1
81

6ð
p Z
a0
\ufffd \ufffd7=2
r2eÿ( Zr=3a0)(3 cos2 \u141ÿ 1) (6:63a)
j3d\ufffd1i  j32\ufffd 1i  1
81

ð
p Z
a0
\ufffd \ufffd7=2
r2eÿ( Zr=3a0) sin \u141 cos\u141 e\ufffdij (6:63b)
j3d\ufffd2i  j32\ufffd 2i  1
162

ð
p Z
a0
\ufffd \ufffd7=2
r2eÿ( Zr=3a0) sin2\u141 e\ufffdi2j (6:63c)
The orbital j3d0i is real. Substitution of equation (5.29c) into (6.63a) and a
change in notation for the subscript give
6.4 Atomic orbitals 179
j3dz2i 
1
81

6ð
p Z
a0
\ufffd \ufffd7=2
(3z2 ÿ r2)eÿ( Zr=3a0) (6:64a)
From the four complex orbitals j3d1i, j3dÿ1i, j3d2i, and j3dÿ2i, we construct
four equivalent real orbitals by the relations
j3dxzi \ufffd 2ÿ1=2(j3d1i  j3dÿ1i)  2
1=2
81ð1=2
Z
a0
\ufffd \ufffd7=2
xzeÿ( Zr=3a0) (6:64b)
j3d yzi \ufffd ÿi2ÿ1=2(j3d1i ÿ j3dÿ1i)  2
1=2
81ð1=2
Z
a0
\ufffd \ufffd7=2
yzeÿ( Zr=3a0) (6:64c)
z
1
2
2pz
any axis \u2019
z-axis
12 12
any axis \u2019
x-axis
any axis \u2019
y-axis
2px 2py
x y
Figure 6.2 Polar graphs of the hydrogen 2p atomic orbitals. Regions of positive and
negative values of the orbitals are indicated by  and ÿ signs, respectively. The
distance of the curve from the origin is proportional to the square of the angular part
of the atomic orbital.
180 The hydrogen atom
j3dx2ÿ y2i \ufffd 2ÿ1=2(j3d2i  j3dÿ2i) 
1
81(2ð)1=2
Z
a0
\ufffd \ufffd7=2
(x2 ÿ y2)eÿ( Zr=3a0)
(6:64d)
j3dxyi \ufffd ÿi2ÿ1=2(j3d2i ÿ j3dÿ2i)  2
1=2
81ð1=2
Z
a0
\ufffd \ufffd7=2
xyeÿ( Zr=3a0) (6:64e)
In forming j3dx2ÿ y2i and j3dxyi, equations (A.37) and (A.38) were used. Graphs
of the square of the angular part of these five real functions are shown in Figure
6.3 and the mathematical expressions are listed in Table 6.2.
The radial functions Rnl(r) for the 1s, 2s, 2p, 3s, 3p, and 3d atomic orbitals are
shown in Figure 6.4. For states with l 6 0, the radial functions vanish at the
origin. For states with no angular momentum (l  0), however, the radial
function Rn0(r) has a non-zero value at the origin. The function Rnl(r) has
(nÿ l ÿ 1) nodes between 0 and 1, i.e., the function crosses the r-axis
(nÿ l ÿ 1) times, not counting the origin.
The probability of finding the electron in the hydrogen-like atom, with the
distance r from the nucleus between r and r  dr, with angle \u141 between \u141 and
\u141 d\u141, and with the angle j between j and j dj is
j\u142nlmj2 dô  [Rnl(r)]2jYlm(\u141, j)j2 r2 sin\u141 dr d\u141 dj
To find the probability Dnl(r) dr that the electron is between r and r  dr
regardless of the direction, we integrate over the angles \u141 and j to obtain
Dnl(r) dr  r2[Rnl(r)]2 dr
ð
0
2ð
0
jYlm(\u141, j)j2 sin\u141 d\u141 dj  r2[Rnl(r)]2 dr
(6:65)
Since the spherical harmonics are normalized, the value of the double integral
is unity.
The radial distribution function Dnl(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
6.4 Atomic orbitals 181
2 1
1 2
z
x
3dxz
2 1
1 2
z
y
3dyz
2 1
1 2
y
x
3dxy
Figure 6.3 Polar graphs of the hydrogen 3d atomic orbitals. Regions of positive and
negative values of the orbitals are indicated by  and ÿ signs, respectively. The
distance of the curve from the origin is proportional to the square of the angular part
of the atomic orbital.
2
2
1 1
y
x
3dx22y2
z
1
1
3dz2
any axis \u2019
z-axis
2 2
182 The hydrogen atom
Figure 6.4 The radial functions Rnl(r) for the hydrogen-like atom.
0 1 2 3 4 5
r/aì0
0.5
1
1.5
2aì
3/2R10
0 2 4 6 8 10
20.2
0
0.2
0.4
0.6
0.8
l 5 0
l 5 1
r/aì
aì
3/2R2l
0 2 4 6 8 10 12 14 16 18 20
r/aì
aì
3/2R3l
20.1
0.0
0.1
0.2
0.3
0.4
l 5 0
l 5 1
l 5 2
6.4 Atomic orbitals 183
D10(r)  4 Z
a0
\ufffd \ufffd3
r2eÿ2 Zr=a0
D20(r)  1
8
Z
a0
\ufffd \ufffd3
r2 2ÿ Zr
a0
\ufffd \ufffd2
eÿZr=a0 (6:66)
D21(r)  1
24
Z
a0
\ufffd \ufffd5
r4eÿZr=a0
distribution functions for the 1s, 2s, 2p, 3s, 3p, and 3d states are shown in
Figure 6.5.
The most probable value rmp of r for the 1s state is found by setting the
derivative of D10(r) equal to zero
dD10(r)
dr
 8 Z
a0
\ufffd \ufffd3
r 1ÿ Zr
a0
\ufffd \ufffd
eÿ2 Zr=a0  0
which gives
rmp  a0=Z (6:67)
Thus, for the hydrogen atom (Z  1) the most probable distance of the electron
from the nucleus is equal to the radius of the first Bohr orbit.
The radial distribution functions may be used to calculate expectation values
of functions of the radial variable r. For example, the average distance of the
electron from the nucleus for the 1s state is given by
hri1s 
1
0
rD10(r) dr  4 Z
a0
\ufffd \ufffd31
0
r3eÿ2 Zr=a0 dr  3a0
2Z
(6:68)
where equations (A.26) and (A.28) were used to evaluate the integral. By the
same method, we find
hri2s  6a0
Z
, hri2p  5a0
Z
The expectation values of powers and inverse powers of r for any arbitrary
state of the hydrogen-like atom are defined by
hrkinl 
1
0
rkDnl(r) dr 
1
0
rk[Rnl(r)]
2 r2 dr (6:69)
In Appendix H we show that these expectation values obey the recurrence
relation
k  1
n2
hrkinl ÿ (2k  1) a0
Z
hr kÿ1inl  k l(l  1) 1ÿ k
2
4
\ufffd \ufffd
a20
Z2
hr kÿ2inl  0
(6:70)
184 The hydrogen atom
Figure 6.5 The radial distribution functions Dnl(r) for the hydrogen-like atom.
0 2 4 6 8 10 12 14 16 18 20
0.00
0.10
0.20
0.30
0.40
0.50
0.60a0D10
r/a0
0 2 4 6 8 10 12 14 16 18 20
0.00
0.10
0.05
0.10
0.15
0.20a0D2l
r/a0
D21
D20
0 2 4 6 8 10 12 14 16 18 20
0.00
0.02
0.04
0.06
0.08
0.10
0.12a0D3l
r/a0
D30
D31
D32
6.4 Atomic orbitals 185
For k  0, equation (6.70) gives
hrÿ1inl  Z
n2a0
(6:71)
For k  1, equation (6.70) gives
2
n2
hrinl ÿ 3a0
Z
 l(l  1) a
2
0
Z2
hrÿ1inl  0
or
hrinl  a0
2Z
[3n2 ÿ l(l  1)] (6:72)
For k  2, equation (6.70) gives
3
n2
hr2inl ÿ 5a0
Z
hrinl  2[l(l  1)ÿ 34]
a20
Z2
 0
or
hr2inl  n
2a20
2Z2
[5n2 ÿ 3l(l  1) 1] (6:73)
For higher values of k, equation (6.70) leads to hr3inl, hr4inl, . . .
For k  ÿ1, equation (6.70) relates hrÿ3inl to hrÿ2inl
hrÿ3inl  Z
l(l  1)a0 hr
ÿ2inl (6:74)
For k  ÿ2, ÿ3, . . . , equation (6.70) gives successively hrÿ4inl, hrÿ5inl, . . .
expressed in terms of hrÿ2inl.
Although the expectation value hrÿ2inl cannot be obtained from equation
(6.70), it can be evaluated by regarding the azimuthal quantum number l as the
parameter in the Hellmann\u2013Feynman theorem (equation (3.71)). Thus, we
have
@En
@ l
 @ H^ l
@ l
\ufffd \ufffd
(6:75)
where the Hamiltonian operator H^ l is given by equation (6.18) and the energy
levels En by equation (6.57). The derivative @ H^ l=@ l is just
@ H^ l
@ l
 &quot;
2
2ìr2
(2l  1) (6:76)
In the derivation of (6.57), the quantum number n is shown to be the value of l
plus a positive integer. Accordingly, we have @n=@ l  1 and
@En
@ l
 ÿ Z
2e92
2a0
@
@ l
nÿ2  ÿ Z
2e92
2a0
@n
@ l
@
@n
nÿ2  Z
2&quot;2
ìa20
nÿ3 (6:77)
where aì  &quot;2=ìe92 has been replaced by a0  &quot;2=mee92. Substitution of
equations (6.76) and (6.77) into (6.75) gives the desired result
186 The hydrogen atom
hrÿ2inl  Z
2
n3(l  1
2
)a20
(6:78)
Expression (6.71) for the expectation value of rÿ1 may be used to calculate
the average potential energy of the electron in the state jnlmi. The potential
energy V (r) is given by equation (6.13). Its expectation value is
hV inl  ÿZe92hrÿ1inl  ÿ Z
2e92
a0 n2
(6:79)
The result depends only on the principal quantum```