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Chapter 4
many-electron atoms
Periodic system and shell structure
The empirically determined physical and chemical
properties of the atoms and their dependence on the
atomic number Z lead to the discovery of the periodic
system of the elements. It is one of the goals of atomic
physics to understand the ordering and the properties of
the chemical elements in periodic system. These are
explainable starting from the electronic structures of the
atoms. With some important experimental facts, for
example, alkali atoms and x-ray spectra, one-electron
spectra and many-electron spectra, it leads to the concept
of the shell structure of atoms.
 The periodic system of the elements;
 Schrödinger equation for a many-electron
problem — Hartree-Fock technique;
 shell structures of the atoms;
The periodic system of the elements
Atomic volumes and ionisation energies as function of the position in the
periodic system of the elements. Particularly noticeable are the (relatively)
large atomic volumes of the alkali metal atoms and the large ionisation
energies of the noble gas atoms.
Properties of elements in periodic table
The halogens
The transition metals
Lanthanides
Actinides
The alkaline earth metals
The Alkali metals
The noble gases
periodicity
With increasing atomic number, the electron configuration of the
atoms display a periodic variation. Because of this the elements
show periodic variations of both physical and chemical behavior.
Atomic Radius: the atomic radii increases top to bottom and right
to left in the periodic table.
 The size of the electron cloud increases as n increases.
 look down the periodic table, the size of atoms in each group is
going to increase.
 look across the periodic table, all the atoms in each group have
the same n. However, for each element, the positive charge on the
nucleus increases by one proton, the outer electron cloud is pulled
in a little tighter. they tend to decrease in size from left to right
across a period of the table.
periodicity
Ionization Energy: The energy needed to remove the most loosely
held electron from an atom.
 Ionization energies are periodic.
 In any column or group, there is a gradual decrease in ionization
energy as the atomic number increases. Metals typically have a low
ionization energy. Nonmetals typically have a high ionization energy.
Electron Affinity: The attraction of an atom for an electron.
 Metals have low electron affinities while nonmetals have high
electron affinities.
 The general trend as you go down a column is a decreasing tendancy
to gain electrons.
 As you go across a row there is also a trend for a greater attraction for
electrons.
Shell structures and periodic
properties of the elements
Properties of elements in Group 1A, the alkali metals
 The alkali metals are the elements Li, Na, K, Rb, Cs.
 These elements have relatively low melting points. Cesium melts at body
temperature, 37o Celsius.
 They are soft enough that they can be cut with a knife.
 They react violently with water.
Properties of elements in Group 8A, the noble gases
• The noble gases, He, Ne, Ar, Kr, Xe, Ra, are nonmetals.
• The noble gases are very unreactive. They were once called the inert gases.
• The noble gases are so unreactive they are used to protect materials from reaction.
Welding is done using argon atmosphere to prevent oxygen in the air from reacting
with the hot welded metals.
Shell structures and periodic
properties of the elements
Properties of elements in Group 7A, the halogens
 The halogens, F, Cl, Br, I, At, are nonmetals.
 Halogens have lower melting points than the alkali metals. Fluorine and
chlorine are gases at room temperature
 The halogens are corrosive and so reactive that in nature they are only found
combined with other elements
Properties of elements in Group 2A, the alkaline earth
metals
• The alkali metals are the elements Be, Mg, Ca, Sr, Ba, Rd.
• These elements have higher melting points than the alkali metals.
• They are harder and tougher. Magnesium is used to fabricate car parts like
wheels.
• Calcium reacts with water. Magnesium clearly doesn't react with water.
Schrödinger equation for a many-electron
problem — Hartree-Fock technique;
Strictly speaking, in order to understand the electronic
structures of any atoms, and in order to define the
quantum numbers of an electron in the atom, one would
have to solve the Schrödinger equation for a manyparticle problem, namely for all the electrons in the atom---many body problem---no analytical solution.
The problem is solved approximately using the HartreeFock technique, which is based on the model of
independent particles.
Hartree-Fock technique
The basic idea is that instead of trying to calculate the
interactions of N-1 electrons with Nth electron, one
replaces the Coulomb attraction of the nucleus for the
Nth electron by an effective potential. One then
calculates the eigenstates and the eigenvalues of the
Nth electron in this potential field. These one-particle
functions are often referred to in the literature as
orbitals.
In the actual problem, the Coulomb interaction among the
electrons is taken into account. The energy of the Coulomb
interaction between the pair of electrons j, k is given by
e2/(40rjk) (rjk is the distance between the electrons), so the
Halmitonian is:
1
e2
H  H j  
 H j  
2 j  k 4 0 rjk
j 1
j  k 4 0 rjk
j 1
N
e2
N
The next step is to solve the Schrödinger equation:
H  Et 
There is no exact solution for the many-body problem, it has to
be reduced to a one-electron problem.
One-particle problem
As we know from the electrostatics, there is an interaction energy
between a charge at position r and the charge distribution given by
other electrons in the many-electron atom, which is the effective
potential V(r):
2
V (r ) 
1
4 0
e (rj )
 r r
j
d j 
1
4 0

e Q ( R j )
r  rj
d j
Where  is the charge distribution,  is the wave-function, the
integral covers the total volume.
Then we must solve the Schrödinger equation in which both the
Coulomb potential of the nucleus and the interaction energy with
all the other electrons. If the chosen electron has the subscript k,
and the coordinate Rk:
 2 2
 1
Ze2
0
1




V
(
r
)

(
R
)

E

k
k  k
k
k ( Rk )

4 0 rk
 2m0

Hartree-Fock method
The Hartree-Fock method needs a trial functions (0). The
superscript (0) means that we use a given (or guessed)
wavefunction to start the whole procedure. The trial wavefunction
is belong to a given potential V(0). The next step is to solve the
Schrödinger equation with given potential V(0), and we have the
new wavefunction (1). (1) again is belong to a given potential
V(1). Then repeat the same process… Seen schematically, we have
the calculation procedure:
 (0)  V (0)   (1)  V (1)   ( 2)  V ( 2)     ( j )  
The  was achieve by a series of iteratively process, and the
calculations were done by computers. We will not involve in
detailed calculations. The results are interesting for the structures of
many-electron atoms.
Orbitals and quantum numbers
Orbitals: the one-electron function calculated by Hartree-Fock
method is referred to in the literature as orbitals, which are the
eigenstates and eigenvalues of the Nth electron in the manyelectron atom.
Similar to one electron atom, H, it needs quantum numbers to
characterise the atomic structures:
The principal quantum number: n = 1, 2, 3, 4…;
The orbital quantum number: l = 0, 1, 2, 3, …, n-1;
The magnetic quantum number: ml = 0, ±1, ±2, …, ±l;
The magnetic spin quantum number: ms = ±1/2.
The electronic configuration
The electronic configuration: a particular states of the energy
levels or terms of an atoms by electrons is called electronic
configuration of the atom in that state – in this case, the ground
state.
The questions for the electronic configuration:
 What are the possible electronic configurations in the atoms;
 Which are particularly stable;
 How the electrons of an atom are distributed among the possible
quantum number.
Pauli principle
Pauli principle (Pauli 1925): the electronic states of an atom
can only be occupied in such a way that no two electron have
exactly the same set of quantum numbers.
The electrons must therefore differ in at least one quantum
number. A set of quantum numbers: n, l, ml, ms.
A state of an atom, the states with the lowest energies are
naturally the occupied ones.
Shell structures
Shell: the group of the electrons with the same principal
quantum number n. The shell is also refer to energy level
or term.
n = 1, 2, 3, 4, 5, … refers to K, L, M, N, O shells, …
Electron in K shell is much closer to the nucleus than L
shell.
Subshell: the group of the electrons with the same
principal quantum number n and the same orbital
quantum number l.
l = 0, 1, 2, 3, 4, … refers to S, P, D, F, G, … subshells, …
subshells
Shell structures
level or shell number of
subshells
type of
subshells
Maximun number
of electrons in
shell 2n2
n=1
1
S
2
n=2
2
S and P
8
n=3
3
S, P, D
18
n=4
4
S, P, D, F
32
n=5
5
S, P, D, F, G
-
n=6
6
S, P, D, F, G, H
-
n=7
7
S, P, D, F, G, H, I
-
the number of electrons (112) in the biggest atom known can be
accommodated by using only the common s, p, d, and f sublevels. The
subshells g, h, i etc. are never needed.
Distance between nucleus
and an electron
The principal quantum number "n" is very useful. It tells
the relative energy for an electron in an atom and it also
indicates the average distance between the nucleus and
an electron. The higher the value for "n" the greater the
distance between the nucleus and the electron. An
electron in the first level, n = 1, is closer to the nucleus on
average than an electron in the n = 4 shell
Subshells s, p, d and f
The shape for all "s" orbitals is spherical with the center of the
sphere at the nucleus. The size of the sphere increases for
increasing quantum numbers. The 1s has a smaller diameter
than the 2s and so forth.
Subshells s, p, d and f
The p orbitals have a double lobe shape with the point of
contact of the lobes at the center of the nucleus. The three
different p orbitals are each aligned along one of the three
coordinate axes x, y, or z
Number of orbitals in a shell
orbital type
S
p
d
f
maximum number of
orbitals in a subshell
1
3
5
7
number of electrons that
can fit into orbital type
2
6
10
14
The maximum number of electrons with a
given principal quantum number which can be
bound to an atom:
For a given n there are n different values for l;
For every value of l there are 2l+1 different values of ml;
For each pair of l and ml there are two different values of ms;
Thus for each pair of numbers n and l, there are at most 2(2l+1)
electrons.
The maximum number of electrons in a shell with a given
value of n is:
n 1
2
2
(
2
l

1
)

2
n

l 0
It is the possible electronic configuration of many-electron atoms.
The shell structure of the atomic
energy levels and their ordering,
for the last added electron and
for the inner electrons
What is the electronic
configuration for the elements?
He(2)
B(5)
Ni(28)
(1s)2
(1s)2(2s)2(2p)1
(1s)2(2s)2(2p)6(3s)2(3p)6(4s)2(3d)8
Orbital diagram
Electronic configurations
Na(11)
Mg(12)
Al(13)
Si(14)
P(15)
S(16)
Cl(17)
Ar(18)
2
2
6
1
1s 2s 2p 3s
2
2
6
2
1s 2s 2p 3s
1s22s22p63s23p1
1s22s22p63s23p2
1s22s22p63s23p3
2
2
6
2
4
1s 2s 2p 3s 3p
2
2
6
2
5
1s 2s 2p 3s 3p
1s22s22p63s23p6
The orbital diagrams
Many-electron atoms
Examples of the filling of the electron configurations
Element
He
Li
Be
O
Cl
K
Electrons in Element
2
3
4
8
17
19
Electron Configuration
1s2
1s22s1
1s22s2
1s22s22p4
1s22s22p63s23p5
1s22s22p63s23p64s1
Closed shell
A closed shell or noble gas configuration occurs whenever the next
electron to be added would occupy the s state of the next higher
principal quantum number n, and it is not necessary that all the
states belong to the lower principal quantum numbers be filled.
The electrons in the closed shells are more closer to the nucleus
and are more strongly bound.
The angular momentum and magnetic moments add up to zero, so
that the closed shells are spherically symmetrical and is especially
stable.
The closed shells are particularly stable electronic configurations.
2He (1s)2
(1s)2(2s)2(2p)6
18Ar (3s)2(3p)6
10Ne
36Kr
(4s)2(3d)10(4p)6
54Xe (5s)2(4d)10(5p)6
86Rn (6s)2(4f)14(5d)10(6p)6
Electron configuration and the periodic table
The transition metals
The noble
gases
The halogens
The Alkali metals
The alkaline earth metals
Lanthanides
Actinides
•The left-most columns include the alkali metals and the alkaline earth metals. In these
elements the valence s orbitals are being filled
•On the right hand side, the right-most block of six elements are those in which the valence p
orbitals are being filled These two groups comprise the main-group elements
•In the middle is a block of ten columns that contain transition metals. These are elements in
which d orbitals are being filled
•Below this group are two rows with 14 columns. These are commonly referred to the f-block
metals. In these columns the f orbitals are being filled
homework
19.1, 19.4, 19.6, 19.7