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§2.1 Atomic orbitals of hydrogenic atoms
An atomic orbital is a one-electron wave function for an electron in an atom.
According to the QM model, the behavior of a specific electron in an atom can be described by a mathematical expression called a wave function (or “ orbital
”, denoted by the Greek letter ψ), obtained via solution of the Schrodinger equation. A hydrogenic atom or hydrogen-like atom is a one-electron atom or ion of general atomic number Z; examples of hydrogenic atoms are H, He
+
, Li
+2
, O
+7
. A many-electron atom (or polyelectronic atom) is an atom or ion with more than one electron; examples include all neutral atoms other than H. The Schrodinger equation for an electron in a hydrogenic atom can be solved exactly, and providing a set of hydrogenic atomic orbitals to describe the electronic structures of many-electron atoms.
When chemists say that an electron ‘occupies’ a certain orbital, it means that the behavior of the electron is described by the corresponding wave function. We might think of an orbital as looking like a photograph of the electron taken at a slow shutter speed. In such a photo, the orbital would appear as a blurry cloud, indicating the region of space where the electron has been.
For practical purposes we can say that an orbital represents the space where an electron spends 90% to 95% of its time, i.e. the region that describes the probable location of the electron. The probability of finding the electron at a particular point in space is called the electron density or electron distribution , and is related to the square of the orbital wave function.
Quantum numbers and atomic wave functions
(1) quantum numbers
An electron in an orbital with quantum number n has an energy given by
E n
 
32

Z
2
 e
4
2

0
2  2 n
2
, where μ is the reduced mass of the hydrogenic atom,

0
is the vacuum permittivity and
1
4

0

8 .
98755

10
9
J
 m

C
-2
. If we replace the reduced mass by the mass of an electron, 9.10938 × 10
-31
kg, corresponding to a nucleus of infinite mass, the Rydberg constant is

R


32 m
 e e
4
2

2
0
 2

2 .
17987

10

18
J

1.09737

10
5 cm
1
.
* The virial theorem: If the kinetic energy of a particle has the form E
K
= mv x
2
/2, and the potential energy has the form V = kx b
, then its mean potential and kinetic energies are related by 2<E
K
> = b<V>. l , the angular momentum quantum number, is also called the azimuthal quantum number and defines the spatial distribution of the orbital with respect to the nucleus.
An electron in an orbital with quantum number l has an angular momentum of magnitude
 l ( l

1 )

1
2
 . An electron in an orbital with quantum number m l
has a z-component of angular momentum m l
 .
Together, the quantum numbers n , l , m l
define an atomic orbital. The fourth quantum number, m s
, a result of relativistic corrections to the Schrodinger equation, completes the description by accounting for the magnetic moment of the electron.

* In addition to the classical ideas of mass and charge, the electron possess a permanent magnetic moment, μ, which may take one of two values and is generally expressed in terms of Bohr magnetons (μ
B
) viz: μ = -2 m s
μ
B
. Here m s
is called the spin quantum number which may adopt one of two values ±1/2.
* Labels s, p, d, f come from an old classification scheme for atomic spectral lines: sharp, principal, diffuse and fundamental.
(Miessler, et al., “Inorganic Chemistry”, Table2.2)
All the orbitals of a given value of n are said to form a single shell of the atom. The orbitals with the same value of n but different values of l are said to form a subshell of a given shell. Both shells and subshells are generally referred to by letters: n = shell:
1
K
2
L
3
M
4
…
N
… l = 0 subshell: s
1 p
2 d
3 f
4 g
5 h
6 i
7 k
…
…

(Atkins’ Physical Chemistry, 9e, Fig9.8)
An atom’s highest principal quantum number determines the valence shell of the atom, and it is typically only the electrons and orbitals of the valence shell that are involved in bonding.
(2) atomic wave functions
The atomic wave function (or atomic orbital) can be factored into a radial component and two angular components; the two angular components are sometimes combined into one factor, Y :
 n , l , m
 r ,

,
 

R
     

R n , l
 

Y l , m
 
. Both the radial part and the angular part are normalized:

0
 dr R n , l
2  r
2 
1 ,

0
 d
 
0
2
 d

Y l , m
2  sin
 
1 .
The angular wave function Y l , m

(Atkins’ Physical Chemistry, 9e, Table8.2)

(Miessler, et al., “Inorganic Chemistry”, Table2.3)
* Combining the two orbitals with cos
 m l
 
1 gives two orbitals with sin

; similarly, combining the orbitals with m l
 
2
and
results in real functions with cos
2

and sin
2 
. For example,

2 p x

1
2
 

1
 

1


1
2
3


R
   sin
 cos


2 p y
 i
2
 

1
 

1


1
2
3


R
 sin
 sin

* The d z
2
orbital ( m l

0 ) actually use the function 2 z
2  x
2  y
2
, which we shorten to
2 z for convenience.

* In the column “Functions of  ” are the shapes for the 
portion; when the
 portion is included, with values of
 
0 to 2π, the three-dimensional shapes in the far-right column are formed.
* The phase of atomic orbitals (shown above by color blue and clear) is solely a result of the mathematical functions describing the orbitals. One color indicates that the function is positive in this region of space, and the other color indicates that the function is negative. It does not matter which color is defined as positive or negative, only that the two regions are opposite.
The radial wave function R n , l
(Atkins’ Physical Chemistry, 9e, Table9.1)

(Miessler, et al., “Inorganic Chemistry”, Table2.4)
* a = 52.9 pm, the Bohr radius, is the value of r at the maximum of
0

2
for a hydrogen 1s orbital (the most probable distance from the hydrogen nucleus for the 1s electron).
Close to the nucleus the radial wave function is proportional to r l
, and the higher the orbital angular momentum, the less likely its is that the electron will be found there.
Far from the nucleus all radial wave functions approach zero exponentially.

(Atkins’ Physical Chemistry, 9e, Fig9.4)
例 . 寫出氫原子 1s 軌域、 2p z
軌域的數學式子 (wave function) 。

§2.2 Radial distribution functions
The radial distribution function or the radial probability function,
2
R
2 r , describes the probability of finding the electron at a given distance from the nucleus, summed over all angles. 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.
Radial wave functions
(Miessler, et al., “Inorganic Chemistry”, Figure2.7)
Radial probability functions

(Miessler, et al., “Inorganic Chemistry”, Figure2.7)
Each curve above has a maximum value, corresponding to the most probable distance of finding the electron from the nucleus. The value of r to which they correspond characterizes the spatial extent of the wave function, or the radius of the orbital. It is a general result that the radius of hydrogenic atomic orbital, defined as the distance where there is a maximum in the radial distribution function, increases with n (and is close to n
2 a
0
), but depends only slightly on the value of l .
例 . 寫出氫原子各個軌域的 radial distribution function ；當電子位於某一個軌域
時，計算電子距離原子核多遠出現機率最大？找出電子和原子核的平均距離。

§2.3 Nodal surfaces
Nodal surfaces are surfaces with zero electron density (or zero probability of finding the electron); for example, conferring the radial part of the 2s wave function, R
2s
, the
2s orbital has a spherical nodal surface – a sphere of radius 2 a
0
. A node is a surface where the wave function is zero as it changes sign, i.e., a surface separating the positive and negative regions of a wave function.
Angular nodes are planar or conical. There are l angular nodes in any orbital, with the conical surface in the d z
2
orbitals counted as two nodes.
Rewriting the wave functions of 2p orbitals (Z = 1) in terms of the cartesian coordinates,

2 p
X

N
 x
 e
 r
2 a ,

2 p
Y

N
 y
 e
 r
2 a ,

2 p
Z

N
 z
 e
 r
2 a where N represents the normalization constant.
The three functions point along a different geometrical axis, x, y, or z. In the case of
2p z
, for a given value of z the function has the same value for all points located at the same distance, r, from the origin. It is said that this function is cylindrically symmetrical about the z-axis. On the other hand, the values of 2p z
are of opposite sign for two points on the different sides of the xy plane. This function is thus said to be antisymmetrical with respect to the xy plane. Finally the function is identically zero within the xy plane; this plane is a nodal plane of 2p z
.

(Atkins’ Physical Chemistry, 9e, Fig9.15)
(Atkins’ Physical Chemistry, 9e, Fig9.16)
* Radial nodes are also called spherical nodes. The lowest energy orbitals of each classification (1s, 2p, 3d, 4f, etc.) have no radial nodes. The number of radial nodes for a given orbital is equal to n – l –1.

(Atkins’ Physical Chemistry, 9e, Fig9.10)
The total number of nodes in any orbital is n – 1 if the conical node of some d and f orbitals count as two nodes.
Constant electron density surfaces for selected atomic orbitals

(Miessler, et al., “Inorganic Chemistry”, Figure2.8)
例 . 寫出氫原子軌域各有幾個的 nodal surfaces ，以及這些 nodal surfaces 的數學
式子。
Because electrons (around the nucleus, or in an atomic orbital) have wave-like properties, these waves are overlapping in space when the electrons are in the same orbital. However, because the electrons are negatively charged and have particle character also, they tend to repel each other. As a result, their movements are actually correlated. Correlation is the ability of an electron to feel the trajectory of another electron and therefore alter its own course so as to minimize Coulombic repulsions and keep the energy of the system to a minimum.

Exercises ex1. Based on your knowledge of the first few hydrogenic atomic orbitals, deduce general formulas, in terms of n and l , for (i) the number of radial nodes in an atomic orbital, (ii) the number of angular nodes, (iii) the total number of nodes. ex2. The following reaction might occur in the interior of a star:
He
++
+ H

He
+
+ H
+
.
Calculate the electronic energy change, assuming all species be in their ground states.