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2.2 The Schrödinger Equation | 19
real functions than complex functions, we usually take advantage of another property of 
the wave equation. For differential equations of this type, any linear combination of solu-
tions to the equation—sums or differences of the functions, with each multiplied by any 
coefficient—is also a solution to the equation. The combinations usually chosen for the p 
orbitals are the sum and difference of the p orbitals having ml = + 1 and –1, normalized 
by multiplying by the constants 122 and i22 , respectively: 
 !2px =
122(!+1 + !-1) = 12A 3p 3R(r)4 sin u cos f 
 !2py =
i22(!+1 - !-1) = 12A 3p 3R(r)4 sin u sin f 
 TABLE 2.3 Hydrogen Atom Wave Functions: Angular Functions 
 Angular Factors Real Wave Functions 
 Related to Angular Momentum 
 Functions 
of u In Polar Coordinates 
 In Cartesian 
Coordinates Shapes Label 
 l ml " # 
 
z
 
 #"(u, f) #"(x, y, z) 
 0( s ) 0 
 
122p 122 122p 122p 
 
z y
x
 
 s 
 1( p ) 0 
 
122p 262 cos u 12A 3p cos u 12A 3p zr pz 
 +1 
 
122p eif 232 sin u 
 w 
 
 g 12A 3p sin u cos f 12A 3p xr 
 
 
 px 
 -1 
 
122p e- if 232 sin u 12A 3p sin u sin f 12A 3p yr py 
 2( d ) 0 
 
122p 12A52 (3 cos 2 u-1) 14A 5p (3 cos2 u - 1) 14A 5p (2z2 - x2 - y2)r2 
 
 dz2 
 +1 
 
122p eif 2152 cos u sin u 
 w 
 
 g 12A15p cos u sin u cos f 12A15p xzr2 dxz 
 -1 
 
122p e- if 2152 cos u sin u 12A15p cos u sin u sin f 12A15p yzr2 dyz 
 +2 
 
122p e2if 2154 sin2 u 
 w g 14A15p sin2 u cos 2f 14A15p (x
2 - y2)
r2
 
 
 dx2-y2 
 -2 
 
122p e-2if 2154 sin2 u 14A15p sin2 u sin 2f 14A15p xyr2 dxy 
 Source: Hydrogen Atom Wave Functions: Angular Functions, Physical Chemistry, 5th ed.,Gordon Barrow (c) 1988. McGraw-Hill Companies, Inc. 
 NOTE: The relations (eif - e- if)/(2i) = sin f and (eif + e- if)/2 = cos f can be used to convert the exponential imaginary functions to real trigonometric functions, 
combining the two orbitals with ml = { 1 to give two orbitals with sin f and cos f. In a similar fashion, the orbitals with ml = { 2 result in real functions with cos2 f 
and sin2 f. These functions have then been converted to Cartesian form by using the functions x = r sin u cos f, y = r sin u sin f, and z = r cos u. 
20 Chapter 2 | Atomic Structure
 TABLE 2.4 Hydrogen Atom Wave Functions: Radial Functions 
 Radial Functions R ( r ), with s = Z r/a0 
 Orbital n l R ( r ) 
 1s 1 0 R 1s = 2 c Z
a0
d3/2e -s 
 2s 2 0 R 2s = 2 c Z2a0 d3/2(2 - s)e -s/2 
 2p 1 R 2p =
123 c Z2a0 d3/2se -s/2 
 3s 3 0 R 3s =
2
27
c Z
3a0
d3/2(27 - 18s + 2s2)e -s/3 
 3p 1 R 3p =
1
8123 c 2Za0 d3/2(6 - s)s e -s/3 
 3d 2 R 3d =
1
81215 c 2Za0 d3/2s2 e -s/3 
 * We should really call this the d2z2-x2-y2 orbital! 
 The same procedure used on the d orbital functions for ml = { 1 and {2 gives the 
functions in the column headed !"(u, f) in Table 2.3 , which are the familiar d orbitals. 
The dz2 orbital (ml = 0) actually uses the function 2z2 - x2 - y2, which we shorten to z2 
for convenience. * These functions are now real functions, so # = #* and ##* = #2. 
 A more detailed look at the Schrödinger equation shows the mathematical origin of 
atomic orbitals. In three dimensions, # may be expressed in terms of Cartesian coordinates 
( x , y , z ) or in terms of spherical coordinates (r, u, f). Spherical coordinates, as shown in 
 Figure 2.5 , are especially useful in that r represents the distance from the nucleus. The spheri-
cal coordinate u is the angle from the z axis, varying from 0 to p, and f is the angle from 
the x axis, varying from 0 to 2p . Conversion between Cartesian and spherical coordinates 
is carried out with the following expressions: 
 x = r sin u cos f
y = r sin u sin f
z = r cos u 
 In spherical coordinates, the three sides of the volume element are r du, r sin u df, and 
 dr . The product of the three sides is r2 sin u du df dr, equivalent to dx dy dz . The volume 
of the thin shell between r and r + dr is 4pr2 dr, which is the integral over f from 0 to 
 p and over u from 0 to 2p. This integral is useful in describing the electron density as a 
function of distance from the nucleus. 
 # can be factored into a radial component and two angular components. The radial 
function R describes electron density at different distances from the nucleus; the angular 
functions ! and " describe the shape of the orbital and its orientation in space. The two 
angular factors are sometimes combined into one factor, called Y : 
 #(r, u, f) = R(r)!(u)"(f) = R(r)Y(u, f) 
u
Spherical coordinates
Volume element
u
r
r
f
x
y
z
x
y
du
r sin u df
r sin u
df
rdu
dr
f
z
 FIGURE 2.5 Spherical 
 Coordinates and Volume 
Element for a Spherical Shell 
in Spherical Coordinates. 
22 Chapter 2 | Atomic Structure
3p
15R
ad
ia
l f
un
ct
io
n 
a 0
R
(r
)
0 30
-.1
0
.1
.2
23 .3
.4
5 252010
3d
R
ad
ia
l f
un
ct
io
n 
a 0
R
(r
)
0 30
0
.1
.2
.3
23 .4
.5
5 252010
1s
r/a0
15R
ad
ia
l f
un
ct
io
n 
a 0
R
(r
)
0 30
Radial Wave Functions
Radial Probability Functions
0
.4
.8
1.2
23 1.6
2
5 252010
3s
r/a0
15
Pr
ob
ab
ili
ty
 a
0r
2 R
2
Pr
ob
ab
ili
ty
 a
0r
2 R
2
Pr
ob
ab
ili
ty
 a
0r
2 R
2
Pr
ob
ab
ili
ty
 a
0r
2 R
2
Pr
ob
ab
ili
ty
 a
0r
2 R
2
Pr
ob
ab
ili
ty
a 0
r2
R
2
0 30
0
.1
.2
.3
.4
.5
.6
5 252010
0
.1
.2
.3
.4
.5
.6
0
.1
.2
.3
.4
.5
.6
0
.1
.2
.3
.4
.5
.6
0
.1
.2
.3
.4
.5
.6
0
.1
.2
.3
.4
.5
.6
3p
r/a0
150 305 252010
3d
r/a0
150 305 252010
2s
r/a0
150 305 252010
2p
r/a0
150 305 252010
1s
r/a0
150 305 252010
2p
r/a0
15R
ad
ia
l f
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ct
io
n 
a 0
R
(r
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0
.2
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.6
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3s
15R
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a 0
R
(r
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0 30
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0
.1
.2
23 .3
.4
5 252010
2s
r/a0
15
r/a0 r/a0
15
r/a0
R
ad
ia
l f
un
ct
io
n 
a 0
R
(r
)
0 30
-.2
0
.2
.4
23 .6
.8
5 252010
 FIGURE 2.7 Radial Wave 
 Functions and Radial Probability 
Functions. 
22 Chapter 2 | Atomic Structure
3p
15R
ad
ia
l f
un
ct
io
n 
a 0
R
(r
)
0 30
-.1
0
.1
.2
23 .3
.4
5 252010
3d
R
ad
ia
l f
un
ct
io
n 
a 0
R
(r
)
0 30
0
.1
.2
.3
23 .4
.5
5 252010
1s
r/a0
15R
ad
ia
l f
un
ct
io
n 
a 0
R
(r
)
0 30
Radial Wave Functions
Radial Probability Functions
0
.4
.8
1.2
23 1.6
2
5 252010
3s
r/a015
Pr
ob
ab
ili
ty
 a
0r
2 R
2
Pr
ob
ab
ili
ty
 a
0r
2 R
2
Pr
ob
ab
ili
ty
 a
0r
2 R
2
Pr
ob
ab
ili
ty
 a
0r
2 R
2
Pr
ob
ab
ili
ty
 a
0r
2 R
2
Pr
ob
ab
ili
ty
a 0
r2
R
2
0 30
0
.1
.2
.3
.4
.5
.6
5 252010
0
.1
.2
.3
.4
.5
.6
0
.1
.2
.3
.4
.5
.6
0
.1
.2
.3
.4
.5
.6
0
.1
.2
.3
.4
.5
.6
0
.1
.2
.3
.4
.5
.6
3p
r/a0
150 305 252010
3d
r/a0
150 305 252010
2s
r/a0
150 305 252010
2p
r/a0
150 305 252010
1s
r/a0
150 305 252010
2p
r/a0
15R
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a 0
R
(r
)
0 30
0
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1
5 252010
3s
15R
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a 0
R
(r
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0 30
-.1
0
.1
.2
23 .3
.4
5 252010
2s
r/a0
15
r/a0 r/a0
15
r/a0
R
ad
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R
(r
)
0 30
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0
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23 .6
.8
5 252010
 FIGURE 2.7 Radial Wave 
 Functions and Radial Probability 
Functions. 
Black plate (12,1)
The angular part of thewavefunction,Að!;"Þ
Now let us consider the angular parts of the wavefunctions,
Að!;"Þ, for different types of atomic orbitals. These are
independent of the principal quantum number as Table 1.2
illustrates for n ¼ 1 and 2. Moreover, for s orbitals, Að!;"Þ
is independent of the angles ! and " and is of a constant
value. Thus, an s orbital is spherically symmetric about the
nucleus. We noted above that a set of p orbitals is triply
degenerate; by convention they are given the labels px, py
and pz. From Table 1.2, we see that the angular part of the
pz wavefunction is independent of "; the orbital can be
CHEMICAL AND THEORETICAL BACKGROUND
Box 1.4 Notation for 2 and its normalization
Although we use 2 in the text, it should strictly be written
as $ where $ is the complex conjugate of . In the x-
direction, the probability of finding the electron between
the limits x and ðxþ dxÞ is proportional to ðxÞ $ðxÞdx.
In three-dimensional space this is expressed as $ d# in
which we are considering the probability of finding the
electron in a volume element d# . For just the radial part
of the wavefunction, the function is RðrÞR$ðrÞ.
In all of our mathematical manipulations, we must
ensure that the result shows that the electron is somewhere
(i.e. it has not vanished!) and this is done by normalizing the
wavefunction to unity. This means that the probability of
finding the electron somewhere in space is taken to be 1.
Mathematically, the normalization is represented as follows:ð
 2 d# ¼ 1 or more correctly
ð
 $ d# ¼ 1
and this effectively states that the integral (
Ð
) is over all
space (d#) and that the total integral of 2 (or $) must
be unity.
Fig. 1.6 Plots of radial parts of the wavefunction R(r) against r for the 2p, 3p, 4p and 3d atomic orbitals; the nucleus is at r ¼ 0.
Fig. 1.7 Radial distribution functions, 4$r2RðrÞ2, for the 1s, 2s and 3s atomic orbitals of the hydrogen atom.
12 Chapter 1 . Some basic concepts
Black plate (12,1)
The angular part of thewavefunction,Að!;"Þ
Now let us consider the angular parts of the wavefunctions,
Að!;"Þ, for different types of atomic orbitals. These are
independent of the principal quantum number as Table 1.2
illustrates for n ¼ 1 and 2. Moreover, for s orbitals, Að!;"Þ
is independent of the angles ! and " and is of a constant
value. Thus, an s orbital is spherically symmetric about the
nucleus. We noted above that a set of p orbitals is triply
degenerate; by convention they are given the labels px, py
and pz. From Table 1.2, we see that the angular part of the
pz wavefunction is independent of "; the orbital can be
CHEMICAL AND THEORETICAL BACKGROUND
Box 1.4 Notation for 2 and its normalization
Although we use 2 in the text, it should strictly be written
as $ where $ is the complex conjugate of . In the x-
direction, the probability of finding the electron between
the limits x and ðxþ dxÞ is proportional to ðxÞ $ðxÞdx.
In three-dimensional space this is expressed as $ d# in
which we are considering the probability of finding the
electron in a volume element d# . For just the radial part
of the wavefunction, the function is RðrÞR$ðrÞ.
In all of our mathematical manipulations, we must
ensure that the result shows that the electron is somewhere
(i.e. it has not vanished!) and this is done by normalizing the
wavefunction to unity. This means that the probability of
finding the electron somewhere in space is taken to be 1.
Mathematically, the normalization is represented as follows:ð
 2 d# ¼ 1 or more correctly
ð
 $ d# ¼ 1
and this effectively states that the integral (
Ð
) is over all
space (d#) and that the total integral of 2 (or $) must
be unity.
Fig. 1.6 Plots of radial parts of the wavefunction R(r) against r for the 2p, 3p, 4p and 3d atomic orbitals; the nucleus is at r ¼ 0.
Fig. 1.7 Radial distribution functions, 4$r2RðrÞ2, for the 1s, 2s and 3s atomic orbitals of the hydrogen atom.
12 Chapter 1 . Some basic concepts
Black plate (13,1)
represented as two spheres (touching at the origin)†, the
centres of which lie on the z axis. For the px and py orbitals,
Að!;"Þ depends on both the angles ! and "; these orbitals are
similar to pz but are oriented along the x and y axes.
Although we must not lose sight of the fact that wave-
functions are mathematical in origin, most chemists find
such functions hard to visualize and prefer pictorial
representations of orbitals. The boundary surfaces of the s
and three p atomic orbitals are shown in Figure 1.9. The
different colours of the lobes are significant. The boundary
surface of an s orbital has a constant phase, i.e. the amplitude
of the wavefunction associated with the boundary surface of
the s orbital has a constant sign. For a p orbital, there is one
phase change with respect to the boundary surface and this
occurs at a nodal plane as is shown for the pz orbital in
Figure 1.9. The amplitude of a wavefunction may be positive
or negative; this is shown using þ and$ signs, or by shading
the lobes in different colours as in Figure 1.9.
Just as the function 4#r2RðrÞ2 represents the probability of
finding an electron at a distance r from the nucleus, we use a
function dependent upon Að!;"Þ2 to represent the prob-
ability in terms of ! and ". For an s orbital, squaring
Að!;"Þ causes no change in the spherical symmetry, and the
surface boundary for the s atomic orbital shown in Figure
1.10 looks similar to that in Figure 1.9. For the p orbitals
however, going from Að!;"Þ to Að!;"Þ2 has the effect of
elongating the lobes as illustrated in Figure 1.10. Squaring
Að!;"Þ necessarily means that the signs (þ or $) disappear,
but in practice chemists often indicate the amplitude by a
sign or by shading (as in Figure 1.10) because of the impor-
tance of the signs of the wavefunctions with respect to their
overlap during bond formation (see Section 1.13).
Finally, Figure 1.11 shows the boundary surfaces for five
hydrogen-like d orbitals. We shall not consider the mathema-
tical forms of these wavefunctions, but merely represent the
orbitals in the conventional manner. Each d orbital possesses
two nodal planes and as an exercise you should recognize
where these planes lie for each orbital. We consider d orbitals
in more detail in Chapters 19 and 20, and f orbitals in
Chapter 24.
Orbital energies in a hydrogen-like species
Besides providing information about the wavefunctions,
solutions of the Schro¨dinger equation give orbital energies,
E (energy levels), and equation 1.16 shows the dependence
of E on the principal quantum number for hydrogen-like
species.
E ¼ $ k
n2
k ¼ a constant ¼ 1:312& 103 kJmol$1
ð1:16Þ
For each value of n there is only one energy solutionand
for hydrogen-like species, all atomic orbitals with the
same principal quantum number (e.g. 3s, 3p and 3d ) are
degenerate.
Size of orbitals
For a given atom, a series of orbitals with different values of n
but the same values of l and ml (e.g. 1s, 2s, 3s, 4s, . . .) differ in
Fig. 1.8 Radial distribution functions, 4#r2RðrÞ2, for the 3s, 3p and 3d atomic orbitals of the hydrogen atom.
† In order to emphasize that " is a continuous function we have extended
boundary surfaces in representations of orbitals to the nucleus, but for
p orbitals, this is strictly not true if we are considering '95% of the
electronic charge.
Chapter 1 . Atomic orbitals 13
2.2 The Schrödinger Equation | 23
graphs. The Bohr radius, a0 = 52.9 pm, is a common unit in quantum mechanics. It is 
the value of r at the maximum of !2 for a hydrogen 1 s orbital (the most probable distance 
from the hydrogen nucleus for the 1 s electron), and it is also the radius of the n = 1 orbit 
according to the Bohr model. 
 In all the radial probability plots, the electron density, or probability of finding the 
electron, falls off rapidly beyond its maximum as the distance from the nucleus increases. 
It falls off most quickly for the 1 s orbital; by r = 5a0, the probability is approaching zero. 
By contrast, the 3 d orbital has a maximum at r = 9a0 and does not approach zero until 
approximately r = 20a0. All the orbitals, including the s orbitals, have zero probability at 
the center of the nucleus, because 4pr2R2 = 0 at r = 0. The radial probability functions 
are a combination of 4pr2, which increases rapidly with r , and R2, which may have maxima 
and minima, but generally decreases exponentially with r . The product of these two factors 
gives the characteristic probabilities seen in the plots. Because chemical reactions depend 
on the shape and extent of orbitals at large distances from the nucleus, the radial probability 
functions help show which orbitals are most likely to be involved in reactions. 
 Nodal Surfaces
 At large distances from the nucleus, the electron density, or probability of finding the 
electron, falls off rapidly. The 2 s orbital also has a nodal surface , a surface with zero 
electron density, in this case a sphere with r = 2a0 where the probability is zero. Nodes 
appear naturally as a result of the wave nature of the electron. A node is a surface where the 
wave function is zero as it changes sign (as at r = 2a0 in the 2 s orbital); this requires that 
 ! = 0, and the probability of finding the electron at any point on the surface is also zero. 
 If the probability of finding an electron is zero (!2 = 0), ! must also be equal to 
zero. Because 
 ! (r, u, f) = R (r)Y(u, f) 
 in order for ! = 0, either R(r) = 0 or Y(u, f) = 0. We can therefore determine nodal 
surfaces by determining under what conditions R = 0 or Y = 0. 
 Table 2.5 summarizes the nodes for several orbitals. Note that the total number of 
nodes in any orbital is n – 1 if the conical nodes of some d and f orbitals count as two nodes.* 
 * Mathematically, the nodal surface for the dz2 orbital is one surface, but in this instance, it fi ts the pattern better if 
thought of as two nodes. 
 TABLE 2.5 Nodal Surfaces 
 Angular Nodes [Y(u, f) = 0] 
 Examples (number of angular nodes) 
 s orbitals 0 
 p orbitals 1 plane for each orbital 
 d orbitals 2 planes for each orbital except dz2 
 1 conical surface for dz2 
 Radial Nodes [ R ( r )!0] 
 Examples (number of radial nodes) 
 1s 0 2p 0 3d 0 
 2s 1 3p 1 4d 1 
 3s 2 4p 2 5d 2 
pz dx2 - y2
y
y
x x
z
x = -y
x = y
Black plate (19,1)
Fig. 1.12 Radial distribution functions, 4!r2RðrÞ2, for the 1s, 2s and 2p atomic orbitals of the hydrogen atom.
CHEMICAL AND THEORETICAL BACKGROUND
Box 1.6 Effective nuclear charge and Slater’s rules
Slater’s rules
Effective nuclear charges, Zeff, experienced by electrons in
different atomic orbitals may be estimated using Slater’s
rules. These rules are based on experimental data for
electron promotion and ionization energies, and Zeff is
determined from the equation:
Zeff ¼ Z $ S
where Z ¼ nuclear charge, Zeff ¼ effective nuclear charge,
S ¼ screening (or shielding) constant.
Values of S may be estimated as follows:
1. Write out the electronic configuration of the element in
the following order and groupings: (1s), (2s, 2p),
(3s, 3p), (3d ), (4s, 4p), (4d ), (4f ), (5s, 5p) etc.
2. Electrons in any group higher in this sequence than the
electron under consideration contribute nothing to S.
3. Consider a particular electron in an ns or np orbital:
(i) Each of the other electrons in the (ns, np) group
contributes S = 0.35.
(ii) Each of the electrons in the ðn$ 1Þ shell contri-
butes S ¼ 0:85.
(iii) Each of the electrons in the ðn$ 2Þ or lower shells
contributes S ¼ 1:00.
4. Consider a particular electron in an nd or nf orbital:
(i) Each of the other electrons in the (nd, nf ) group
contributes S ¼ 0:35.
(ii) Each of the electrons in a lower group than the one
being considered contributes S ¼ 1:00.
An example of how to apply Slater’s rules
Question: Confirm that the experimentally observed electro-
nic configuration of K, 1s22s22p63s23p64s1, is energetically
more stable than the configuration 1s22s22p63s23p63d1.
For K, Z ¼ 19:
Applying Slater’s rules, the effective nuclear charge
experienced by the 4s electron for the configuration
1s22s22p63s23p64s1 is:
Zeff ¼ Z $ S
¼ 19$ ½ð8& 0:85Þ þ ð10& 1:00Þ(
¼ 2:20
The effective nuclear charge experienced by the 3d electron
for the configuration 1s22s22p63s23p63d1 is:
Zeff ¼ Z $ S
¼ 19$ ð18& 1:00Þ
¼ 1:00
Thus, an electron in the 4s (rather than the 3d) atomic orbital
is under the influence of a greater effective nuclear charge and
in the ground state of potassium, it is the 4s atomic orbital
that is occupied.
Slater versus Clementi and Raimondi values of Zeff
Slater’s rules have been used to estimate ionization energies,
ionic radii and electronegativities. More accurate effective
nuclear charges have been calculated by Clementi and
Raimondi by using self-consistent field (SCF) methods, and
indicate much higher Zeff values for the d electrons.
However, the simplicity of Slater’s approach makes this
an attractive method for ‘back-of-the-envelope’ estimations
of Zeff. "
Chapter 1 . Many-electron atoms 19