Atomic Structure and Periodic
Trends
Atomic Structure and Periodic
Trends
 Lectures:
week1: W9 am; week2: W9 am (ICL) & 11 am (DP), F 9 am
(DP); week 3: 9 am (ICL)
 Books:
• Inorganic Chemistry by Shriver and Atkins
• Physical Chemistry by P.W.Atkins , J. De Paula
• Essential Trends in Inorganic Chemistry D Mingos
• Introduction to Quantum Theory and Atomic Structure
by P. A. Cox
 Other resources:
– Web-pages: http://timmel.chem.ox.ac.uk/lectures/
– Ritchie/Titmuss, Quantum Theory of Atoms and
Molecules, Hilary Term
Why study atomic electronic
structure?



All of chemistry (+biochemistry etc.) ultimately boils
down to molecular electronic structure.
Reason: electronic structure governs bonding and
thus molecular structure and reactivity
Before it is possible to understand molecular
electronic structure we need to introduce a number of
concepts which are more easily demonstrated in
atoms.
atomic structure  molecular structure  chemistry
The Periodic Table
Mendeleyev,
Dmitri Ivanovich
(1834-1907)
Mendeleyev was the first chemist to understand that all elements are related
members of a single ordered system. From his table he predicted the properties of
elements then unknown, of which three (gallium, scandium, and germanium) were
discovered in his lifetime.
This course......
will introduce new concepts gradually
starting with the “simplest”:
H-Atom Energy levels, Wavefunctions, Born

Interpretation, Orbitals
Many electron atoms Effects of other electrons,
Penetration, Quantum Defect
The Aufbau Principle Electronic Configuration of
atoms and their ions
Trends in the PT
Ionisation Energy, Electron Affinity,
Size of atoms and ions
The H-Atom
1
H
1.008
The Hydrogen Atom
Revision

+
The simplest possible system and the basis
for all others:
Electron orbits the proton
v under the influence
of the Coulomb Force:
r
-
F
H-Atom: consider the energy
v
ETotal =
E KE  v
1
2
2
c.m.
2
E PE
+
e

4π 0 r
r
2
E Total
e
 v 
4π 0 r
1
2
2
-
Energy Levels?
Consider 3 approaches:
 Classical
 Bohr Model (old quantum theory)
 Full Quantum: Schrödinger Equation

2
E Total
e
 v 
4π 0 r
1
2
2

Classically:

There is no theoretical restriction at all on v or r
(there are infinitely many combinations with the
same energy).
Hence the energy can take any value - it is
.
As a result transitions should be possible
everywhere across the electromagnetic spectrum.
So lets see the spectrum...



The H-atom Emission Spectrum
+
lens
lens
H2
-
prism
InfraUltra-violet
red visible
XUV
Principles of Quantum Mechanics
Quantization
Energy levels
Revision
The Rydberg Formula
In 1890 Rydberg showed that the frequencies
of all transitions could be fit by a single equation:
 1
1
v  cRy 2  2 
 n1 n 2 
Bohr Theory (old quantum)
Bohr explained the observed frequencies by
restricting the allowed orbits the electron
could occupy to particular circular orbits(by
quantizing the angular momentum).
 His theory gives energy levels:

hc Ry
e
E   2 where Ry  2 3
n
8ε 0 h c
4
4
3
2
1
n
The problem with Bohr Theory
Bohr theory works extremely well for the Hatom.
 However;
 it provides no explanation for the
quantization of energy -it just happens to fit
the observed spectrum
But, more seriously:
 it just doesn’t work at all for any other atoms

Quantum mechanical Principles
and the Solution of the
Schrödinger Equation
Principles of Quantum Mechanics
Quantization
Classical mechanics
m 2
E v
2
is continuous
Quantum mechanics
Energy levels
Principles of Quantum Mechanics
It’s all about probability
In classical mechanics
Position of object specified
In quantum mechanics
Only
of object at a particular location
Principles of Quantum Mechanics
e-
How do we describe the
electrons in atoms?
You know: Electrons can be described as
(characterised by mass, momentum, position…)
p = h/l
De Broglie
However: Electrons can also be described as
(characterised by wavelength, frequency, amplitude)
and show properties such as interference,
diffraction
h = Planck’s constant = 6.626 10-34 Js
The Davisson Germer Experiment
Proving the wave
properties of electrons
(matter!)
Intensity variation in
diffracted beam shows
constructive and
destructive interference
of wave
Principles of Quantum Mechanics
The Wavefunction
Important
In quantum mechanics, an electron, just like any other
particle, is described by a
Y(position, time)
Contains all information there is to
know about the particle
More on than in Hilary Term
The Results of Quantum Mechanics
Schrödinger equation:
  2 2

  V(r)  Y  EY

 2μ




where   2  2  2 ,
x
y z
2
2
2
2
Y is the wavefunction,
V(r) the potential energy and
E the total energy
Spherical Polar Coordinates

Instead of Cartesian (x,y,z) the maths
works out easier if we use a different
coordinate system:
z 
r
x

y
x = r sin cos
y = r sin sin
z = r cos
(takes advantage of
the spherical symmetry
of the system)
More on than in Hilary Term
So Schrödinger’s Equation
becomes.....
  2 2

  V(r) Y  EY

 2μ

2
as before
e
V(r)  
with
4ππ0 r
2
1

1 2
2
But now  
r 2 
2
r r
r
2
1

1 

2
where
 

sin 
2
2
sin  
sin  

....which can be solved exactly for the Hatom with the solutions called orbitals, more
specifically, atomic orbitals.
We separate the wavefunction into 2 parts:
 a radial part R(r) and
 an angular part Y(,),
Important

such that Y=

The solution introduces 3 quantum
numbers:
....which can be solved exactly for the Hatom with the solutions called orbitals, more
specifically, atomic orbitals.
We separate the wavefunction into 2 parts:
 a radial part R(r) and
 an angular part Y(,),
Important

such that Y=R(r)Y(,)

The solution introduces 3 quantum numbers:
The quantum numbers;
quantum numbers arise in the solution;
 R(r) gives rise to:
the principal quantum number, n
Important


Y(,) yields:
the orbital angular momentum quantum
number, l and the magnetic quantum number,
ml
i.e., Y=Rn,l(r)Yl,m(,)
Important
The values of n, l, & ml
n = 1, 2, 3, 4, .......
l = 0, 1, 2, 3, ......(n-1)
ml = -l, -l+1, -l+2,..0,..., l-1, l
You are familiar with these.....
n is the integer number associated with
an orbital
 Different l values have different names
arising from early spectroscopy
e.g.,
l =0 is labelled
l =1 is labelled
l =2 is labelled
l =3 is labelled
etc...
Important

Important
Looked at another way.....
n =1 l=0 ml=0
1s orbital (1 of)
n =2 l=0 ml=0
2s orbital (1 of)
l=1 ml=-1, 0, 1
2p orbitals (3 of)
n =3 l=0 ml=0
3s orbital (1 of)
l=1 ml=-1, 0, 1
3p orbitals (3 of)
l=2 ml=-2,-1,0,1,2 3d orbitals (5 of)
etc. etc. etc.
Hence we begin to see the structure behind
the periodic table emerge.
Exercise 1;
Work out, and name, all the possible
orbitals with principal
quantum number n=5.
How many orbitals have n=5?
The Radial Wavefunctions

The radial wavefunctions for H-atom are
the set of Laguerre functions in terms of n, l
a0 (=0.05292nm)
is the
-the most probable
orbital radius of
an H-atom 1s
electron.
R(r)
Important
1s
R(r)
2p
2s
R(r)
3s
3p
3d
Revisit: The Born Interpretation
YY* 
The square of the wavefunction,
at a point is proportional to the
of finding the particle at that point.
Y
2
Y* is the complex conjugate of Y.
In 1D: If the amplitude of the wavefunction of a particle at
some point x is Y(x) then the probability of finding the
particle between x and x +dx is proportional to
Y(x)Y*(x)dx
 Y(x) dx
2
1s
2
R(r)
R(r)
1s
2p
2p
2s
2s
3p
3s
3p
3d
3d
3s
Radial Wavefunctions and the
Born Interpretation
At long distances from the nucleus, all wavefunctions decay
to
.
Some wavefunctions are zero at the nucleus (namely, all but
the l=0 (s) orbitals). For these orbitals, the electron has a zero
probability of being found
.
Some orbitals have nodes, ie, the wave function passes
through zero; There are
such radial nodes for each orbital.
In 3D: If the amplitude of the wavefunction of a particle
at some point (x,y,z) is Y(x,y,z) then the probability of
finding the particle between x and x+dx, y+dy and z+dz,
ie, in a volume dV = dx dy dz is given by
Y(x,y,z) dV
2
z
It follows therefore that
Y(x,y,z)
2
dz
dx
is a probability density.
In spherical coordinates
2
Y(r,,f)
dy
y
x
More important to know probability of finding
electron at a given distance from nucleus!
…..in a shell of
z
dz
dx
x
dy
y
Cartesian coordinates
not very useful to describe orbitals!
dA=dxdy
dV=dxdydz
Surface element
Volume element
dA = r2sindfd
dV = r2sindfddr
The Surface area of a sphere is hence:
2 

2
r
 sin  df d
0 0
2

0
0
 r 2  sin  d  df
 r 2 ( cos   cos 0)(2  0)
 4 r 2
For spherically symmetric orbitals:
the radial distribution function is defined as
P(r)=
2
4r Y(r)
2
and P(r)dr is the probability of finding the
electron in a shell of radius r and thickness dr
Construction of the radial
distribution function
For spherically symmetrical orbitals
P(r)
Radial distribution function P(r)
Important
Born
interpretation
prob = Y2dt
Hence plot
P(r) = r2R(r)2
P(r)=4r2Y2
(for spherical
symmetry)
Pn(r) has
nodes
R(r)
So what do we learn?
1s
R(r)2
2s
2p
r
r
The 2p orbital is on average closer
to the nucleus, but note the 2s
orbital’s high probability of being
P(r)
1s
2s
2p
r
P(r)
3d vs 4s
4s
3d
The Angular Wavefunction
The Ylm(,) angular part
of the solution form a set
of functions called the
n.b. These are generally
imaginary functions -we use
real linear combinations to
picture them.
The Shapes of Wavefunctions
(Orbitals)
Shapes arise from combination of radial R(r)
and angular parts Ylm(,) of the
wavefunction.
 Usually represented as boundary surfaces
which include
of the probability
density. The electron does not undergo
planetary style circular orbits.
 These are the familiar spherical s-orbitals and
dumbell-shaped p-orbitals, etc...
Important

Electron densities representations
The s-orbitals:
R2
R2
*
R(r)
R(r)
r
Important
+
+
+
Signs of corresponding
wave functions
Please note that R and R2 are arbitrarily scaled.
-
of corresponding wave
functions (white areas)
Boundary Model
Boundary model of an s orbital within which there is 90%
probability of finding the electron
The p-orbitals: boundary surfaces
- actually imaginary functions but linear
combinations give the familiar dumbells
Important
Different shades denote
different signs of the
wavefunction
One
Nodal
plane
px
py
pz
The d-orbitals : boundary surfaces
z
z
z
y
dxz
x
dxy
z
y
dz2
Again, different shades denote different
signs of the wavefunction
dyz
x
Important
x
y
y
x
2 2
dx-y
The d-orbitals : boundary surfaces
Two Nodal planes which split orbital into 4 lobes, orbitals lie in a plane
perpendicular to the two nodal panes and point between the axes
z
z
z
y
x
ddzx
xzxz
y
y
x
ddxyxy
x
ddyzyz
Two
Nodal
planes
Again, different shades denote different
signs of the wavefunction
The d-orbitals : boundary surfaces
z
y
x
2 2
Two Nodal planes which
split orbital into 4 lobes,
orbitals lie in xy-plane,
pointing along the x and y
axes, nodal planes at 45o to
xz- and yz-planes
2
Cylindrical symmetry, two
angular nodes which take
the form of cones at 54.7o
and 125.3o to the z-axis.
54.7o
125.3
o
The energies of orbitals
Important

In the case of the H-atom (& only the Hatom) the energy is determined exclusively
by the principal quantum number, n:
μe
where Ry is the Rydberg constant R y  2 3
8ε 0 h c
Ry for H-atom = 109 677 cm-1
(= Ionization energy, 13.6eV)
4
H-atom Energy
Levels
Important
i) All levels with the same n
i.e., E(3s)=E(3p)=E(3d)
ii) All energies are negative
(because the electron is bound)
iii) n= by definition has
energy zero, hence E- E1
= ionization energy (13.6 eV)
= Ry (109 677 cm-1)
The Ionization Energy
Corresponds to the total removal of an
electron (i.e., transition to n=)
 Since in the ground state H-atom n=1 is the
highest occupied atomic orbital, the
ionization energy is given by:
Important

 1 1
Ionization Energy   R y hc     R y hc
  1
See later for Koopman’s theorem
Other Atoms
A) Other single electron
atoms/ions (H-like atoms)
The ions He+, Li2+, Be3+, B4+,... etc. are said to
be
with the H-atom (same
electron configuration).
 Again ns, np, nd etc. levels are degenerate.
 However the attractive Coulomb force is much
bigger due to the Z+charge of the nuclei.
 The energy levels are given by a similar
expression to that for the H-atom with the
2
inclusion of a
:
Z R y hc
En  
n2

Orbitals contract with increasing Z
Note that
exponential function
decays faster as Z
Orbitals contract with increasing Z
Y2s
Z=1
Z=2
Z=3
P(r)2s
Z=1
Z=2
Z=3
H-like atoms/ions : Summary






The orbitals contract with increasing Z
The effect of the Z2 term is to increase the energy level
spacing.
E.g., The energy level spacings in the He+ spectrum are
approx. 4 times those in the H-atom whilst in Be3+ they are
16 times larger (Z=4).
This is only approximate because of the slight mass
dependence of the Rydberg constant.
Ry(H) = 109 677cm-1
Ry(mass) = 109 737 cm-1
Compare ionisation energies for IE(H) = 13.6eV, IE(He+) =
54.4eV, IE(Li2+)=122.4eV
B) Multi-electron Atoms
-
2+
Helium atom

Schrödinger equation cannot be solved
analytically anymore (apart from He)
Need to develop an approximate picture for
multi-electron atoms
The orbital approximation in
quantum mechanics
Y(r1, r2, …) = y(r1) y(r2)….
Total wavefunction of
many electron atom
Each electron is occupying
its individual orbital with
nuclear charge modified to
take account of all other
electrons’ presence
(repulsion!)
The orbital approximation in words
Make
multi electron atom look like a one
electron atom
Assumes every electron to be on its own
experiencing an effective nuclear charge, Zeff
Then the orbitals for the electrons take the
form of those in hydrogen but their energies
and sizes are modified by simply using an
effective nuclear charge
-
2+
One electron atom
Two-electron atom
-
Zeff
Orbital Approximation
The Concept of Electron Spin




The solution of the Schrödinger equation above
accounts very well for the structure of the H-atom
spectrum and, as we will see, for other atoms too.
However under very high resolution “fine
structure” is observed in the transitions which is
not explained using this approach.
Dirac extended wave mechanics to include Special
relativity and an extra coordinate - time.
Solution of the Dirac equation yields a new
intrinsic angular momentum -
The Electron Spin
An electron has
 The projection of this spin is also quantized
(by analogy with orbital angular
momentum; l and ml) such that the spin
projection quantum number, ms=½, (spin up
 or a) and ms=-½, (spin down  or b)
 This is the final theoretical plank behind the
structure of the periodic table.
Important

Important
Pauli Exclusion Principle



no two electrons in the same atom can have
the same 4 quantum numbers n, l, ml, ms
(as all electrons necessarily have the same s=1/2)
Hence, no individual orbital may be occupied by more
than 2 electrons
Electrons occupying the same orbital must be “paired up”.
The Pauli Principle and Symmetry
The Pauli Exclusion Principle applies to any
pair of identical
(spin quantum
number s = half integer) – protons, electrons,
neutrons have s = ½ and are fermions.
BUT, any number of
(s = integer)
can occupy the same orbital  12C (s = 0) and
the photon (s = 1)are bosons
Now, as we proceed from H to
other atoms, we need to consider
1.
The Pauli Exclusion Principle
2. The coulombic repulsions between
the electrons
-
2+
Two-electron atom
B) Atoms with one outer electron



Easiest: alkali metals Li, Na, K, Rb but also Be+,
Mg+, Ag (4d105s1) as only one outer electron.
But still, the situation is no longer as simple as the
H-atom due to the inner electrons - the “core”;
The core electrons will
tend to
the outer
electron from the full
nuclear charge.
Shielding and Penetration
Important




If an electron is always outside the core it
experiences only a net charge of nucleus and
core.
If, however, the electron spends much of its
time close to the nucleus (within the core) it
will experience a larger nuclear attraction and
have a lower energy (more tightly bound).
Hence the energy of the outer electron depends
on how much it
the core region.
This in turn depends on the type (s, p, d, f etc.)
of orbital it is in.
Important
Recall the radial distribution
functions.....
nucleus

An e- in the 3s orbital spends more time close to
the nucleus than an electron in 3p and is thus more
tightly bound (lower energy).
Ramifications
The energy of a given quantum state is now
no longer simply a function of its principal
quantum number but also of its penetration
into the core region which depends on the
orbital shape (and thus l).
 i.e., E=En,l
 In general the energies of sub-shells of the
same principal quantum number n lie in the
order
Important

Zeff – the effective nuclear charge
Important

To account for the effects of penetration and shielding
we use
an
effective nuclear charge Zeff such that
2
E nl  
Z eff Ry hc
n
2

( Z   ) 2 Ry hc
n2
where  is the shielding parameter and Z is the charge of
the nucleus.
Zeff is
a function of n and l as electrons in different
shells and subshells approach nucleus to different
extents
Trends in Zeff
The Grotrian diagram for Na
Important
Note:
•Different l levels have
different energy.
•The H-atom levels are
marked on the RHS
H-Atom
energy
levels
•Note more rapid
stabilisation of 4s with
respect to 3d due to
Radial Distribution Function
3d vs 4s
4s
3d
The Aufbau Principle and the
Structure of the Periodic Table
Electron Configurations
To
obtain a ground state configuration for an
atom we apply the Pauli exclusion and the
Aufbau principle which states that electrons are
added to orbitals in increasing order of energy.
n
3d
3p
l
l = 2 (d)
n =3
l = 1 (p)
3s
l = 0 (s)
2p
l = 1 (p)
2s
1s
n =2
l = 0 (s)
n =1
l = 0 (s)
# of orbitals Possible
# of e
2l+1
Principles of how to build up electron
configurations
3d
4s
3p
3s
2p

The Aufbau Principle “The buildingup”principle
When establishing the
ground state
configuration of an
atom start at the
energetic bottom and
work your way up
2s
1s
NB: The energy ordering of the
orbitals changes
with the number of electrons.
Principles of how to build up electron
configurations
(1) The Pauli Exclusion Principle - No two
electrons in one atom may have the same set of
four quantum numbers (that is they must differ in
one or more of n, l, ml , ms)
Differ in
Differ in
2p
2p
2s
1s
2s
1s
Differ in
Differ in
2p
2p
2s
1s
2s
1s
Remember, when two electrons share
one orbital, their magnetic spin
quantum numbers must be
up
down
Paired electron spins
Principles of how to build up
electron configurations
(2) HUND’s Rule - When electrons occupy orbitals of the
same energy, the lowest energy state corresponds to the
configuration with the greatest number of unpaired electrons
Energetically favored
2p
2p
2s
1s
2s
1s
E(Paired)
E(Unpaired)
Maximizing the number of parallel spins The exchange interaction
Quantum mechanical in origin
 Arguments based on the fact that total
wavefunction has to be
with respect to exchange of the electrons
(Pauli)
 Nothing to do with the fact that electrons
are charged!
 Result is that each electron pair with
parallel spins leads to a lowering of the
electronic energy of the atom

Now, let’s start remembering
(1)
The Pauli Exclusion Principle (only two
anitparallel spins in one orbital)
(2) Hund’s Rule (parallel spin configuration of lower
energy for degenerate orbitals)
(3) Aufbau Principle (from energetic bottom to
energetic top)
1H
Configuration 1s1
Can lose one electron to
from stable ion H+
Can form single bond
H2O, H2
2He
Configuration 1s2
Full Shell
Inert
Noble Gas
Unlikely to form bonds
or ions
3 (Pauli)
Li
–
Lithium
–
remember:
can’t
have
s
3
•Configuration 1s22s1
•Easily ionised to Li+ (1s2)
•Alkali Metal
•Under standard
conditions: lightest metal
and least dense element
4Be- Beryllium
2 2
•Configuration 1s 2s
•Be2+ (1s2) stable ion
•Alkaline Earth Metal
•Toxic: replaces Magnesium from
Magnesium activated enzymes due to
stronger coordination ability
5B
- Boron
Number of Electrons: 5
Config 1s22s2 2p1
Forms stable covalently bonded molecular networks
Mainly tervalent
Lewis acidity of many of its compounds and multicentre bonding
Chemistry highly diverse and complex
6C
- Carbon
Config: 1s22s2 2p2
Diamond, graphite, amorphous, fullerenes (e.g., C60)
Forms (usually) four bonds (tetravalent), see CH4
Non-metallic
Carbon's unique characteristic of bonding to itself is
responsible for complex molecules composed of long
chains of carbon atoms, the skeleton of life
7N
- Nitrogen
Config 1s22s2 2p3
In compounds typically forms three bonds
(trivalent), see NH3, N2
Non-metal
N2: •Gaseous, odourless, tasteless
•78% of air is N2
•Very inert at room temperature
(and below) due to strong triple
bond
8O
Config 1s22s2 2p4
member of chalcogen group
Normally considered divalent but other oxidations states vary widely
hugely electronegative (see later)
O2: Colourless, odourless, highly reactive gas,
Created biologically from CO2 by green
photosynthesizing plants
Paramagnetic (attracted by a magnetic field)
Exercise 3;
Name the electron configurations of the
F and Ne atoms. Which is more stable
and why?
9F
•Config
•Member of the halogen group
•Most reactive of all elements
•Forms F readily
•highly electronegative!
•Does not exist in nature in the elemental state at all because of high
reactivity
10Ne
- Neon
Config
Full Outer shell
Noble Gas
Unlikely to form bonds or ions
And now the cycle repeats itself….
s-block
p-block
s1 s2
p1 p2 p3 p4 p5 p6
1
2
3
4
5
6
7
Remember:
Li: 1s22s1
Ne: 1s22s2 2p6
Na: 1s22s2 2p63s1
The Third Period
Na:
1s22s22p63s1
cf. Li (second period) 1s22s1
Mg:
1s22s22p63s2
cf. Be (second period) 1s22s2
Al:
1s22s22p63s2 3p1
cf. B (second period) 1s22s22p1
Si:
1s22s22p63s2 3p2
cf. C (second period) 1s22s22p2
P:
1s22s22p63s2 3p3
cf. N (second period) 1s22s22p3
S:
1s22s22p63s2 3p4
cf. O (second period) 1s22s22p4
Cl:
1s22s22p63s2 3p5
cf. F (second period) 1s22s22p5
Ar:
1s22s22p63s2 3p6
cf. Ne (second period) 1s22s22p6
And what now?
Here, again, is the picture in the H-atom
Energy
4s
3s
3p
The 3d electron is more firmly bound than the 4s
because of its lower principle quantum number
3d
Remember: Grotrian diagram for Na
•Note more rapid
stabilisation of 4s with
respect to 3d due to
difference in penetration!
H-Atom
energy
levels
Remember:
Radial Distribution Function
3d vs 4s
4s
3d
4s
through the inner shells to
some extent and from N on all the way to Ca, it is more
stable (stronger bound) than 3d
The energy of individual (singly
occupied orbitals)
Hence, from N
Energy
3d
4s
3p
3s
4s is lower in energy than 3d
Ca
K
3d
4s
3d
4s
3p
3p
3s
Ar
1s22s22p6
3s
Ar
1s22s22p6
And now we start filling the d-block!
1s22s22p63s23p64s23d1
1s22s22p63s23p64s23d2
Careful though – the actual energy ordering is now:
4s
3d
4s
3d
3p
3p
3s
Ar
1s22s22p6
3s
Ar
1s22s22p6
Sc
1s22s22p63s23p64s23d1
Order in which they were filled
1s22s22p63s23p63d14s2
Energy Ordering
Ti
1s22s22p63s23p64s23d2
Order in which they were filled
1s22s22p63s23p6
Energy Ordering
And all
the way
to Zn
Important for formation of ions!
Ti
1s22s22p63s23p63d24s2
Ti3+
1s22s22p63s23p63d24s2
Ti3+
1s22s22p63s23p63d1
Ar
Sc
Ti
V
Cr
Mn
Fe
Co
Ni
Cu
Zn
1s22s22p63s23p63d14s2
1s22s22p63s23p63d24s2
1s22s22p63s23p63d34s2
1s22s22p63s23p6
1s22s22p63s23p63d54s2
1s22s22p63s23p63d64s2
1s22s22p63s23p63d74s2
1s22s22p63s23p63d84s2
1s22s22p63s23p6
1s22s22p63s23p63d104s2
Now, we should really understand the
Structure of the periodic table
s-block
d-block
p-block
s1 s2 d1 d2 d3 d4 d5 d6 d7 d8 d9 d10 p1 p2 p3 p4 p5 p6
1
2
3
4
5
6
7
4f
5f
Electron Configuration of Ba
1s22s22p63s23p64s23d104p65s24d105p66s2
1
2
3
4
5
6
7
4f
5f
The Lanthanides
TheActinides
Alkali
Metals (s1)
1s
Halogens
(p5)
Alkaline
Earth Metals (s2)
Noble
Gases
2p
2s
Transition Metals
3s
(d1-10)
4s
Inner Transition Metals
5s (f1-14)
3p
3d
4p
4d
5p
6s
4f
5d
6p
7s
5f
6d
7p
The Periodic Table of the Elements
1s
2s
2p
3s
3p
4s
3d
4p
5s
4d
5p
6s
5d
6p
7s
6d
7p
4f
5f
Periodic Trends
Periodic Trends
1) Effective Nuclear Charge - Zeff
2s – 3s: Zeff greater for 3s probably due to actual higher overall charge
Periodic Trends – (Zeff/n)2
Increase across periods unchanged
(but note that relative slopes are different)for (Zeff/n)2)
BUT
Zeff
H(1) 1
Li(2s) 1.26
Na(3s) 1.85
(Zeff/n)2
1
0.40
0.38
Periodic Trends
2) The Ionization Energies (I) within the Periodic Table
Ionization is the process that removes an electron from the neutral
gas phase atom, eg,
1st IE M(g)
M+(g) + e(g)
I1 = E(M+, g) – E(M, g)
2nd IE M+(g)
M2+(g) + e(g)
I2 = E(M2+, g) – E(M+, g)
These reactions require energy (endothermic).
Li(g)
Li+(g) + e(g)
I1 = 520 kJ/mol
Li+( g)
Li2+(g) + e(g)
I2 = 7300 kJ/mol
Determining Ionisation Energies
a) Electron Impact
+V
e accelerated
through potential
M
M(g) + e(g)
M2+(g) + 2 e(g)
e ejected from M
if Eh high enough
I = Eh - Eelkin
b) Photo Electron Spectroscopy
Light
(Eh)
M
Trends in Ionisation Energies
General increase across periods
But kinks?
Let’s recall….
Rcall: The orbital energies
2
E nl  
Z eff Ry hc
n
2

( Z   ) 2 Ry hc
n2
Koopman’s Theorm
I   orbital energy
Together with our trend in (Zeff/n)2 across the
period, that explains the general trend beautifully
But the kinks?
1st Ionization Energies in the 1st Period
H
He
Li
Be
B
C
N
O
F
Ne
Recall: Maximizing the number of parallel
spins - The exchange interaction
Quantum mechanical in origin
 Arguments based on the fact that total
wavefunction has to be antiparallel with
respect to exchange of the electrons (Pauli)
 Nothing to do with the fact that electrons
are charged!
 Result is that each electron pair with
parallel spins leads to a lowering of the
electronic energy of the atom

Space for extra Notes
Moving on through the Periodic Table
Ionisation Energy /eV
20
18
Group
17 F
16
14
Cl
Group 13 B
12
Br
I
As
10
Al
8
Ga
In
Tl
6
2
3
4
5
6
n
Moving on through the Periodic Table
IE(eV)
14
12
10
8
6
4
2
0
H
Cu Ag
Li
Na
Au
K Rb Cs
n
1
3
5
Note large increase
in IE from K to Cu (10 extra units
of nuclear charge badly shielded by d electrons and again from
Rb to Ag). Between Cs and Au the 4f shell is filled giving a total
increase of 24 units of nuclear charge!
More on Ionization Energies
Ionization Potentials tend to
for the successively
heavier elements within a period as the number of protons in the nucleus
increases and electrons are successively added to the same shell (Zeff
increases at constant n).
However, some irregularities (penetration of p vs s and due to exchange
energy contributions) occur.
Ionization Potentials tend to
for the successively
heavier elements in a group in the periodic table (as Zeff increases but n
also increases and does so faster) .
Note transition metal and lanthanide contraction affect these trends.
The trends in second IE are similar but shifted by one atomic number:
The second IEs are
than the first IE for that element.
Also, note particularly large increases as we start to take electrons from
inner shells, e.g., the first, second and third ionisation energies of
beryllium are: 899 kJ mol-1, 1756 kJ mol-1 and 14846 kJ mol-1.
More Periodic Trends
3) Electron Affinities (EAs) within the Periodic Table
The amount of energy needed to remove an electron from a negative
ion = amount of energy released when a neutral atom in its ground
state gains an electron.
X -(g)
X(g) + e
Cl- (g)
Cl(g) + e
E = 348 kJ/mol
A positive electron affinity tells us that X -(g) has a lower
(more favorable) energy than the neutral atom, X(g).
Together with IEs, EAs tell us about chemical bonding: if M has a
low ionization energy an X a high EA, then it is likely that
M + X
M+ + X -
MX will be ionic
3) The Electron Affinities
F-
1s22s2 2p6
C
F
1s22s2 2p5
C-
1s22s2 2p2 1s22s2 2p3
Note: EA always less than IE due to extra electron repulsion on adding an electron!
4) Atomic Radii
3s1
2s1
3s23p6
2s22p6
Again, the Lanthanide Contraction
Nb(Z=41) and Ta(Z=73) have identical atomic radii
4) Atomic Radii
a) Atomic radii generally decrease moving from left to right within the periods
(nuclear charge keeps on increasing but electrons are added to the same shell),
eg, going from Li (1s2 2s1) 157pm to F (1s2 2s22p6) 64pm; for both n = 2
b) Atomic radii generally increase down the group with increasing atomic
number as electrons are occupying more and more distant electron shells, eg,
going from Li (1s2 2s1) 157pm to Cs (1s2 2s22p63s23p64s23d104p65s24d105p66s2)
272pm
c) There is a large
increase
as electrons go into next shell (like
between He and Li or Ne to Na)
d) All anions are larger than their parent atoms and all cations are smaller,
compare Be2+(27 pm) and Be (112pm), I(206pm) and I(133) – please note that
ionic radius depends on coordination number of ion
e) Ionic radii generally
decrease
on the same ion (Tl+, 164pm > Tl3+, 88pm)
with increasing positive charge
5) Electronegativity
•The electronegativity of an atom is a measure of its power when in
chemical combination to attract electrons to itself
•With few exceptions, electronegativity increases
across the periodic table and decreases down a group,
•F is far more electronegative than I
•F is far more electronegative than Li
Appendices
1) Revisit: The Born Interpretation
The wave function is normalised so that:
Y
d
t

1

2
where the integration is over all space accessible to the electron. This
expression simply shows that the probability of finding the electron
somewhere must be 1 (100%).
2) Radial Wavefunctions
3) Volume Element in spherical coordinates
z
rsindf

dr
The radius of the latitude is
rd
rsin
r
r
y
f
x

df
Remember that the arc length, s, is given by s = r a (with a in radians)
 r2sindfd
The Surface
Element follows hence as dA = rsindfrd
The Volume
Element follows hence as dV = rsindfrddr  r2sindfddr
4) Pauli Principle
When the labels of any two identical fermions are
exchanged, the total wavefunction changes sign.
fermion
Y
(2,1) = 
fermion
Y
(1,2)
When the labels of any two identical bosons are
exchanged the total wavefunction retains the
same sign.
Yboson (2,1) = Yboson (1,2)
Two particles (fermions)
Total wave function of
particles 1 and 2
Total Spin wave
function of particles 1
and 2
Y(1,2) =y(1)y(2) (1,2)
Space wave functions of particles 1
and 2 residing in the same orbital
(characterized by the same n, l, ml)
exchange
Now exchanging labels
Y(1,2) =y(1)y(2) (1,2)
Y(2,1) =y(2)y(1) (2,1)
And in y that is easy:
y(1)y(2)  y(2)y(1)
As it is just a product and a x b = b x a!
But in the spin wave functions (2,1)?
(1,2)  a(1)a(2)
(1,2)  b(1)b(2)
(1,2)  1/2[a(1)b(2)  b(1)a(2)]

(1,2)  1/2[a(1)b(2) b(1)a(2)]
+
But in the spin wave functions (2,1)?
ex
symmetric
(1,2)  a(1)a(2)  a(2)a(1)  (2,1)
ex
(1,2)  b(1)b(2)  b(2)b(1)  (2,1)
(1,2)  1/2[a(1)b(2)  b(1)a(2)]
symmetric
-
ex
1/2[a(2)b(1)  b(2)a(1)]
 1/2[b(2)a(1)  a(2)b(1)]
 1/2[a(1)b(2)  b(1)a(2)]   (2,1)
antisymmetric
In analogy:
(1,2)  1/2[a(1)b(2)  b(1)a(2)]
+
ex
1/2[a(2)b(1)  b(2)a(1)]
 1/2[b(2)a(1)  a(2)b(1)]
 1/2[a(1)b(2)  b(1)a(2)]   (2,1)
symmetric
Hence, the only allowed overall
wavefunction is:
Y(1,2) =y(1)y(2) [a(1)b(2)  b(1)a(2)]/2
With an antiparallel
arrangement of spins
5) Slater’s Rules
Approximate method for estimating
the effective nuclear charge
Zeff = Z - S
Where Z is the actual nuclear charge
and S is a shielding constant.
Computing S
1. Divide orbitals into groups
(1s) (2s2p) (3s, 3p) (3d) (4s, 4p) (4d) (4f)
Note s and p with same n grouped together
2. There is no contribution to S from electron to the right of the one
being considered.
3. A contribution is added to S for each electron in the same group as
the one being considered – except in the (1s) group where the
contribution is 0.30.
3. If the electron being considered is in an ns or np orbital, the
electrons in the next lowest shell (n-1) each contribute 0.85 to S. Those
electrons in lower shells ((n-2) and lower) contribute 1.00 to S.
4. If the electron being considered is an nd or nf orbital, all electrons
below it in energy contribute 1.00 to S.
Example
For P:
Zeff = 15 – 4 x 0.35 – 8 x 0.85 – 2 x 1.00 = 4.8
Trends right, actual
values bad
s and p orbitals treated
the same – huge
differences for the
orbitals in terms of
penetration!!!!!