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ATOMIC
STRUCTURE
Written by: B.K Chaudary
Assistant professor
PGGC Sector 11
Chandigarh
B.K Chaudhary
Department of Chemistry PGGC 11 CHD
1
Heisenberg Uncertainty Principle

In the world of very small particles, one cannot measure any
property of a particle without interacting with it in some way

This introduces an unavoidable uncertainty into the result

One can never measure all the
properties exactly
B.K Chaudhary
Department of Chemistry PGGC 11 CHD
Werner Heisenberg (1901-1976)2
Measuring the position and
momentum of an electron

Shine light on electron and detect reflected
light using a microscope
Minimum uncertainty in position
is given by the wavelength of the
light
So to determine the position
accurately, it is necessary to use
light with a short wavelength
B.K Chaudhary
Department of Chemistry PGGC 11 CHD
3
Measuring the position and momentum
of an electron (cont’d)

By Planck’s law E = hc/λ, a photon with a short wavelength has a
large energy

Thus, it would impart a large ‘kick’ to the electron

But to determine its momentum accurately,
electron must only be given a small kick

This means using light of long wavelength!
B.K Chaudhary
Department of Chemistry PGGC 11 CHD
4
Fundamental Trade Off …

Use light with short wavelength:
– accurate measurement of position but not
momentum

Use light with long wavelength
– accurate measurement of momentum but not
position
B.K Chaudhary
Department of Chemistry PGGC 11 CHD
5
Planck’s Distribution
•
Energies are limited to discrete value
– Quantization of energy
E  nh
•
, n  0,1,2,...
Max Planck
Planck’s distribution
dE  d

8hc
5 (e hc / kT  1)
• At high frequencies approaches the Rayleigh-Jeans
law
(e hc / kT  1)  (1 
hc
hc
 ....)  1 
kT
kT
• The Planck’s distribution also follows StefanBoltzmann’s Las
B.K Chaudhary
Department of Chemistry PGGC 11 CHD
6
Wave-Particle Duality
-The particle character of wave
•
Particle character of electromagnetic radiation
– Observation :
• Energies of electromagnetic radiation of frequency v
can only have E = 0, h, v 2hv, …
(corresponds to particles n= 0, 1, 2, … with energy = hv)
– Particles of electromagnetic radiation : Photon
– Discrete spectra from atoms and molecules can be explained
as generating a photon of energy hn .
– ∆E = hv
B.K Chaudhary
Department of Chemistry PGGC 11 CHD
7
•
Wave-Particle Duality
-The particle character of wave
Photoelectric effect
– Ejection of electrons from metals when
they are exposed to UV radiation
– Experimental characteristic
• No electrons are ejected, regardless
of the intensity of radiation, unless UV
its frequency exceeds a threshold
value characteristic of the metal.
• The kinetic energy of ejected
electrons increases linearly with the
frequency of the incident radiation
but is independent of the intensity of
the radiation .
• Even at low light intensities, electrons
are ejected immediately if the
frequency is above threshold.
B.K Chaudhary
Department of Chemistry PGGC 11 CHD
electrons
Metal
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Wave-Particle Duality
-The particle character of wave
•
Photoelectric effect
– Observations suggests ;
• Collision of particle – like projectile that carries energy
• Kinetic energy of electron = hν - Φ
Φ : work function (characteristic of the meltal)
energy required to remove a electron from the metal
to infinity
• For the electron ejection , hν > Φ required.
• In case hν < Φ , no ejection of electrons
B.K Chaudhary
Department of Chemistry PGGC 11 CHD
9
Wave-Particle Duality
-The particle character of wave
• Photoelectric effect
B.K Chaudhary
Department of Chemistry PGGC 11 CHD
10
Wave-Particle Duality
-The wave character of particles
• Diffraction of electron beam from metal
surface
– Davison and Germer (1925)
– Diffraction is characteristic property of
wave
– Particles (electrons) have wave like
properties !
– From interference pattern, we can get
structural information of a surface
LEED (Low Energy Electron Diffraction)
B.K Chaudhary
Department of Chemistry PGGC 11 CHD
11
Wave Particle Duality
•
De Brogile Relation
(1924)
– Any particle traveling with a linear
Matter wave:
= mvlength
= h/l
momentum
p haspwave
– Macroscopic bodies have high
momenta (large p)
 small wave length
 wave like properties are not observed
B.K Chaudhary
Department of Chemistry PGGC 11 CHD
12
Schrödinger equation
• 1926, Erwin Schrödinger (Austria)
– Describe a particle with wave function
– Wave function has full information about the
particle
Time independent Schrödinger equation
for a particle in one dimension
B.K Chaudhary
Department of Chemistry PGGC 11 CHD
13
Schrodinger Wave Equation
In 1926 Schrodinger wrote an equation that
described both the particle and wave nature of the eWave function (Y) describes:
1. energy of e- with a given Y
2. probability of finding e- in a volume of space
Schrodinger’s equation can only be solved exactly
for the hydrogen atom. Must approximate its
solution for multi-electron systems.
B.K Chaudhary
Department of Chemistry PGGC 11 CHD
14
Schrodinger Equation
General form
HY = E Y
H= T + V
: Hamiltonian
operator
B.K Chaudhary
Department of Chemistry PGGC 11 CHD
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The Schrodinger equation:
Kinetic
energy
+
Potential
energy
=
Total
energy
For a given U(x),
• what are the possible (x)?
• What are the corresponding E?
B.K Chaudhary
Department of Chemistry PGGC 11 CHD
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For a free particle, U(x) = 0, so
 (x)  Ae
ikx
Where k = 2
= anything real
2
2
k
E
2m
= any value from
0 to infinity

The free particle can be found anywhere, with
equal probability
B.K Chaudhary
Department of Chemistry PGGC 11 CHD
17
Normalization

When ψ is a solution, so is Nψ

We can always find a normalization const. such that the
proportionality of Born becomes equality
N 2  * dx  1
*

  dx  1
*
*


dxdydz



  d  1
Normalization const. are
already contained in wave
function
B.K Chaudhary
Department of Chemistry PGGC 11 CHD
18
Quantization

Energy of a particle is
quantized
 Acceptable energy can be found
by solving Schrödinger equation
 There are certain limitation in
energies of particles
B.K Chaudhary
Department of Chemistry PGGC 11 CHD
19
The information in a wavefunction

Simple case
– One dimensional motion, V=0
 2 d 2

 E
2
2m dx
Solution
  Aeikx  Be ikx
k 2 2
E
2m
B.K Chaudhary
Department of Chemistry PGGC 11 CHD
20
Probability Density
B=0
  Ae
ikx
  A
2
2
A=0
  Be
 ikx
 B
2
2
A=B
  2 Acos kx
2
  4 A cos
kx
B.K Chaudhary
2
Department of Chemistry PGGC 11 CHD
nodes
21
Eigenvalues and eigenfucntions

Eigenvalue equation
(Operator)(function) = (constant factor)*(same function)
̂  
Operator
Eigenfunction
Eigenvalue
Solution : Wave function
Allowed energy (quantization)
(operator correspond ing to observable )  (value of observable ) 
B.K Chaudhary
Department of Chemistry PGGC 11 CHD
22
Quantum Mechanics and
Atomic Orbitals

The first orbital of all elements is spherical.

Other orbitals have a characteristic shape and position as
described by 4 quantum numbers: n,l,ml,ms. All are integers
except ms

Principal Quantum Number (n): an integer from 1... Total

Angular quantum number (l). (permitted values l = 0 to
n1): the subshell shape.
# e in a shell = n2.
– Common usage for l = 0, 1, 2, 3, 4, and use s, p, d, f, g,...
respectively.
– Subshell described as 1s, 2s, 2p, etc.
B.K Chaudhary
Department of Chemistry PGGC 11 CHD
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– Magnetic quantum number,ml, (allowed l
to +l ) directionality of an l subshell orbital.
 Total number of possible orbitals is 2l+1.
 E.g. s and p subshells have 1 & 3 orbitals,
respectively.

Spin quantum number,ms (allowed values 1/2).
Due to induced magnetic fields from rotating electrons.

Pauli exclusion principle: no two electrons in an atom
can have the same four quantum numbers.
B.K Chaudhary
Department of Chemistry PGGC 11 CHD
24
Permissible Quantum States
B.K Chaudhary
Department of Chemistry PGGC 11 CHD
25
Orbital energies of the hydrogen atom.
B.K Chaudhary
Department of Chemistry PGGC 11 CHD
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Shapes of orbitals (electron
probability clouds)
s orbitals are spherical (1).
 p orbitals are dumbbell shaped (3).
 d orbitals have four lobes (5).
 f orbitals are very complex (7).

B.K Chaudhary
Department of Chemistry PGGC 11 CHD
27
Orbital Energies of Multielectron
Atoms
All elements have the same number of
orbitals (s,p, d, and etc.).
 In hydrogen these orbitals all have the
same energy.
 In other elements there are slight
orbital energy differences as a result of
the presence of other electrons in the
atom.
 The presence of more than one electron
changes the energy of the electron

B.K Chaudhary
Department of Chemistry PGGC 11 CHD
28
Shape of 1s Orbital
B.K Chaudhary
Department of Chemistry PGGC 11 CHD
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Shape of 2p Orbital
B.K Chaudhary
Department of Chemistry PGGC 11 CHD
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Shape of 3d Orbitals
B.K Chaudhary
Department of Chemistry PGGC 11 CHD
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Elements
and
Their
Electronic
Configurations
B.K Chaudhary
Department of Chemistry PGGC 11 CHD
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Electron Configuration
The arrangement of electrons in an
atom in the ground state.
 Need to learn some simple rules or
principles.

B.K Chaudhary
Department of Chemistry PGGC 11 CHD
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Rules are…
 Aufbau
principle
 Pauli’s exclusion principle
 Hund’s Rule
B.K Chaudhary
Department of Chemistry PGGC 11 CHD
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Aufbau Principle
German for building up.
 An electron occupies the lowestenergy orbital that can receive it.
 In Hydrogen, the electron goes into
the 1s orbital because it’s the lowest
energy orbital.

B.K Chaudhary
Department of Chemistry PGGC 11 CHD
35
A general rule -they arrange
themselves to
have the lowest
possible energy.
Ground State
Configuration
B.K Chaudhary
Department of Chemistry PGGC 11 CHD
36
Pauli Exclusion Principle
No two electrons in the same atom can
have the same set of four quantum
numbers.
 Each electron in the same atom has a
unique set of quantum numbers.

B.K Chaudhary
Department of Chemistry PGGC 11 CHD
37
Hund’s Rule
Equivalent orbitals of equal
energy are each occupied
by one electron before any
one orbital is occupied by a
second electron.
B.K Chaudhary
Department of Chemistry PGGC 11 CHD
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Hund’s Rule (cont.)
All electrons in singly
occupied orbitals have
the same spin.
B.K Chaudhary
Department of Chemistry PGGC 11 CHD
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Writing Electron Configurations
B.K Chaudhary
Department of Chemistry PGGC 11 CHD
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Standard Notation
of Fluorine
Number of electrons
in the sub level 2,2,5
2
1s
2
2s
5
2p
Sublevels
B.K Chaudhary
Department of Chemistry PGGC 11 CHD
41