Gnp ch103-lecture notes

CH103
Physical Chemistry: Introduction to Bonding
G. Naresh Patwari
Room No. 215; Department of Chemistry
2576 7182 naresh@chem.iitb.ac.in

Physical Chemistry –I.N. Levine
Physical Chemistry – P.W. Atkins
Physical Chemistry: A Molecular Approach – McQuarrie and Simon
Websites:
http://www.chem.iitb.ac.in/~naresh/courses.html
www.chem.iitb.ac.in/academics/menu.php
IITB-Moodle http://moodle.iitb.ac.in
http://ocw.mit.edu/OcwWeb/web/courses/courses/index.htm#Chemistry
http://education.jimmyr.com/Berkeley_Chemistry_Courses_23_2008.php
Recommended Texts (Physical Chemistry)

What Do You Get to LEARN?
Why Chemistry?
Classical Mechanics Doesn't Work all the time!
Is there an alternative? QUANTUM MECHANICS
Origin of Quantization & Schrodinger Equation
Applications of Quantum Mechanics to Chemistry
Atomic Structure; Chemical Bonding; Molecular
Structure

Why should Chemistry interest you?
Chemistry plays major role in
1. Daily use materials: Plastics, LCD displays
2. Medicine: Aspirin, Vitamin supplements
3. Energy: Li-ion Batteries, Photovoltaics
4. Atmospheric Science Green-house gasses, Ozone depletion
5. Biotechnology Insulin, Botox
6. Molecular electronics Transport junctions, DNA wires
Haber Process
Haber Process
The Haber process remains
largest chemical and economic
venture. Sustains third of
worlds population
Transport Junctions
Quantum theory is necessary for the understanding and the
development of chemical processes and molecular devices
LCD Display

Classical Mechanics
Newton's Laws of Motion
1. Every object in a state of uniform motion tends to remain in
that state of motion unless an external force is applied to it.
2. The relationship between an object's mass m, its acceleration
a, and the applied force F is F = ma. The direction of the force
vector is the same as the direction of the acceleration vector
3. For every action there is an equal and opposite reaction.

Black-Body Radiation; Beginnings of Quantum Theory
Rayleigh-Jeans law was based
on equipartitioning of energy
Planck’s hypothesis
The permitted values of
energies are integral
multiples of frequencies
E = nhν = nhc/λn = 0,1,2,…
Value of ‘h’ (6.626 x 10-34 J s)
was determined by fitting the
experimental curve to the
Planck’s radiation law
kT
4
8π
ρ
λ
=
Planck’s
radiation law
( )hc
kT
hc
e5
8
1λ
π
ρ
λ
=
−
Towards
Ultraviolet Catastrophe
Hot objects glow
Planck did not believe in the
quantum theory and
struggled to avoid quantum
theory and make its influence
as small as possible
λ =
b
T
max

Heat Capacities of Solids
Dulong – Petit Law
The molar heat capacity of all solids have nearly same value of ~25 kJ
Element Gram heat
capacity
J deg-1 g-1
Atomic
weight
Molar heat
capacity
J deg-1 mol-
1
Bi 0.120 212.8 25.64
Au 0.125 198.9 24.79
Pt 0.133 188.6 25.04
Sn 0.215 117.6 25.30
Zn 0.388 64.5 25.01
Ga 0.382 64.5 24.60
Cu 0.397 63.31 25.14
Ni 0.433 59.0 25.56
Fe 0.460 54.27 24.98
Ca 0.627 39.36 24.67
S 0.787 32.19 25.30
1
,
3 3
3 25
A
m
V m
V
Um N kT RT
U
C R kJmol
T
−
= =
∂ 
= = ≈ ∂ 

Einstein formula
Einstein considered the oscillations of
atoms in the crystal about its
equilibrium position with a single
frequency ‘ν’ and invoked the Planck’s
hypothesis that these vibrations have
quantized energies nhνννν
2
2
2
, 3 ;
1
E
E
T
E
V m E
T
e h
C R
T ke
θ
θ
θ ν
θ
 
   = =     − 
3
1
A
m h
kT
N h
U
e
ν
ν
=
−

Debye formula
Averaging of all the frequencies νD
3 4
, 2
3 ;
( 1)
D x
TD D
V m Dxo
hx e
C R dx
T e k
θ
θ ν
θ
 
= = 
− 
∫

Rutherford Model of Atom
Alpha particles were (He2+) bombarded on a 0.00004 cm (few hundreds of
atoms) thick gold foil and most of the alpha particles were not deflected

Rutherford Model of Atom
Positive Charge
Negatively Charged
Particles
Thompson’s model of atom is incorrect.
Cannot explain Rutherford’s
experimental results
Planetary model of atoms with
central positively charged nucleus
and electrons going around
Classical electrodynamics predicts that
such an arrangement emits radiation
continuously and is unstable

Atomic Spectra
Balmer Series
410.1 nm
434.0 nm
486.1 nm
656.2 nm 2 1
R
n n
R 1 9678 x 1 nm
2 2
1 2
1 1 1
.0 0
λ ∞
− −
∞
 
= − 
 
=
“RH is the most accurately measured fundamental physical constant”
The Rydberg-Ritz Combination
Principle states that the spectral lines
of any element include frequencies
that are either the sum or the
difference of the frequencies of two
other lines.

Bohr Phenomenological Model of Atom
Electrons rotate in circular orbits around a central (massive) nucleus, and
obeys the laws of classical mechanics.
Allowed orbits are those for which the electron’s angular momentum
equals an integral multiple of h/2π i.e. mevr = nh/2π
Energy of H-atom can only take certain discrete values: “Stationary States”
The Atom in a stationary state does not emit electromagnetic radiation
When an atom makes a transition from one stationary state of energy Ea to
another of energy Eb, it emits or absorbs a photon of light: Ea – Eb = hv

Energy expression
Bohr Model of Atom
Angular momentum quantized
n=1,2,3,...
2
(2 )
π
π λ
=
=
nh
mvr
r n
4
2 2 2
0
1
.
8ε
= − e
n
m e
E
h n
Spectral lines
4
2 2 2 2
1 1
, 1, 2, 3,...
8
ν
ε
 
∆ = − = =  
 
e
i f
i f
m e
E h n n
h n n
Explains Rydberg formula
Ionization potential of H atom 13.6 eV
4
2 1
2 2
1.09678 x 10 nm
8ε
− −
∞ = =em e
R
h

Bohr Model of Atom
The Bohr model is a primitive model of the hydrogen atom.
As a theory, it can be derived as a first-order approximation
of the hydrogen atom using the broader and much more
accurate quantum mechanics

Photoelectric Effect: Wave –Particle Duality
Experimental Observations
Increasing the intensity of the light increased the number of
photoelectrons, but not their maximum kinetic energy!
Red light will not cause the ejection of electrons, no matter
what the intensity!
Weak violet light will eject only a few electrons! But their
maximum kinetic energies are greater than those for very
intense light of longer (red) wavelengths
Electromagnetic Radiation
Wave energy is related to Intensity, I ∝ E2
0
and is independent of ω
0 ( )ω= −E E Sin kx t

Photoelectric Effect: Wave –Particle Duality
Einstein borrowed Planck’s idea that ∆E=hν and
proposed that radiation itself existed as small packets of
energy (Quanta)now known as PHOTONS
Energy is frequency dependent
φ = Energy required to remove electron from surface
ν=PE h
21
2
φ φ= = + = +P ME hv KE mv

Diffraction of Electrons : Wave –Particle Duality
Davisson-Germer Experiment
A beam of electrons is directed onto
the surface of a nickel crystal.
Electrons are scattered, and are
detected by means of a detector that
can be rotated through an angle θ.
When the Bragg condition
mλ = 2dsinθ was satisfied (d is the
distance between the nickel atom,
and m an integer) constructive
interference produced peaks of high
intensity

G. P. Thomson Experiment
Electrons from an electron source
were accelerated towards a positive
electrode into which was drilled a
small hole. The resulting narrow
beam of electrons was directed
towards a thin film of nickel. The
lattice of nickel atoms acted as a
diffraction grating, producing a
typical diffraction pattern on a
screen

de Broglie Hypothesis: Mater waves
Since Nature likes symmetry,
Particles also should have wave-like nature
De Broglie wavelength
λ = =
h h
p mv
Electron moving @ 106 m/s
-34
10
-31 6
6.6x10 J s
7 10
9.1x10 Kg 1x10 m/s
λ −
= = = ×
×
h
m
mv
He-atom scattering
Diffraction pattern of He atoms at the speed
2347 m s-1 on a silicon nitride transmission
grating with 1000 lines per millimeter.
Calculated de Broglie wavelength 42.5x10-12 m
de Broglie wavelength too small for
macroscopic objects

The wavelength of the electrons was calculated, and found to be in
close agreement with that expected from the De Broglie equation

Wave –Particle Duality
Light can be Waves or Particles. NEWTON was RIGHT!
Electron (matter) can be Particles or Waves
Electrons and Photons show both wave and particle nature
“WAVICLE”
Best suited to be called a form of “Energy”

Bohr – de Broglie Atom
Constructive Interference of the electron-waves
can result in stationary states (Bohr orbits)
If wavelength don’t match, there can not be any
energy level (state)
Bohr condition &
De Broglie wavelength
2 n=1,2,3,...
n=1,2,3,...
2
π λ
λ
π
=
=
=
r n
h
mv
nh
mvr
Electrons in atoms behave as
standing waves

Uncertainty Principle
Uncertainty principle
.
4π
∆ ∆ ≥x
h
x p

Schrodinger’s philosophy
PARTICLES can be WAVES
and WAVES can be PARTICLES
New theory is required to explain the behavior of electrons, atoms and
molecules
Should be Probabilistic, not deterministic (non-Newtonian) in nature
Wavelike equation for describing sub/atomic systems

PARTICLES can be WAVES
and WAVES can be PARTICLES
A concoction of
2
21
2 2
Wave is Particle
2
Particle is Wave
p
E T V mv V V
m
E h
h
p k
ν ω
π
λ
= + = + = +
= =
= =
let me start with
classical wave equation

Do I need to know any Math?
Algebra
Trigonometry
Differentiation
Integration
Differential equations
[ ]1 1 2 2 1 1 2 2( ) ( ) ( ) ( )+ = +A c f x c f x c Af x c Af x
( ) ( ) ikx
Sin kx Cos kx e
2 2
2 2
∂ ∂
∂ ∂
d d
dx dx x x
( )∫ ∫
b
ikx
a
e dx f x dx
2 2
2 2
( ) ( ) ( ) ( )
( ) ( )+ +
∂ ∂ ∂ ∂
+ + + =
∂ ∂∂ ∂
f f f f
m
x y x y
f x nf y k
x yx y

Remember!
∂ Ψ ∂ Ψ
=
∂ ∂
Ψ =
 
Ψ = = − 
 
= =
= =
⋅ − ⋅ 
= − = 
 
2 2
2 2 2
( , ) 1 ( , )
Classical Wave Equation
( , ) Amplitude
( , ) ; Where 2 is the phase
2
2
i
x t x t
x c t
x t
x
x t Ce t
E h
h
p k
x x p E t
t
α
α π ν
λ
ν ω
π
λ
α π ν
λ

ix t E
iCe i x t i x t
t t t
( , )
( , ) ( , )α α α∂Ψ ∂ ∂ − 
= ⋅ = ⋅Ψ ⋅ = ⋅Ψ ⋅ ∂ ∂ ∂  
x t
E x t
i t
( , )
( , )
− ∂Ψ
= ⋅Ψ
∂
i x p E t
x t Ce( , ) andα
α
⋅ − ⋅
Ψ = =

∂Ψ
= ⋅Ψ
∂
( , )
( , )x
x t
p x t
i x
i x p E t
x t Ce( , ) andα
α
⋅ − ⋅
Ψ = =
α α α∂Ψ ∂ ∂  
= ⋅ = ⋅Ψ ⋅ = ⋅Ψ ⋅ ∂ ∂ ∂  
( , )
( , ) ( , )i xpx t
iCe i x t i x t
x x x

ix t E
iCe i x t i x t
t t t
( , )
( , ) ( , )α α α∂Ψ ∂ ∂ − 
= ⋅ = ⋅Ψ ⋅ = ⋅Ψ ⋅ ∂ ∂ ∂  
x t
E x t
i t
( , )
( , )
− ∂Ψ
= ⋅Ψ
∂
i xpx t
iCe i x t i x t
x x x
( , )
( , ) ( , )α α α∂Ψ ∂ ∂  
= ⋅ = ⋅Ψ ⋅ = ⋅Ψ ⋅ ∂ ∂ ∂  
x
x t
p x t
i x
( , )
( , )
∂Ψ
= ⋅Ψ
∂
i x p E t
x t Ce( , ) andα
α
⋅ − ⋅
Ψ = =

− ∂ ∂ ∂ ∂
= = = − =
∂ ∂ ∂ ∂
Operatorsxi E i p
i t t i x x
Operators
x
x t x t
E x t p x t
i t i x
( , ) ( , )
( , ) ( , )
− ∂Ψ ∂Ψ
= ⋅Ψ = ⋅Ψ
∂ ∂
Operator
A symbol that tells you to do something to whatever follows it
Operators can be real or complex,
Operators can also be represented as matrices
xx t E x t x t p x t
i t i x
( , ) ( , ) ( , ) ( , )
− ∂ ∂
Ψ = ⋅Ψ Ψ = ⋅Ψ
∂ ∂

Operators and Eigenvalues
Operator operating on a function results in re-generating
the same function multiplied by a number
The function f(x) is eigenfunction of operator Â and a its
eigenvalue
( )( ) α=f x Sin x
( )( ) α α= ⋅
d
f x Cos x
dx
( ) ( )
2
2 2
2
( ) ( )α α α α α= ⋅ = − ⋅ = − ⋅  
d d
f x Cos x Sin x f x
dx dx
is an eigenfunction of
operator and is its
eigenvalue
( )αSin x
2
2
d
dx
2
α−
( ) ( ) Eigen Value EquationA f x a f x⋅ = ⋅

The mathematical description of quantum mechanics is
built upon the concept of an operator
The values which come up as result of an experiment are
the eigenvalues of the self-adjoint linear operator.
The average value of the observable corresponding to
operator Â is
The state of a system is completely specified by the
wavefunction Ψ(x,y,z,t) which evolves according to
time-dependent Schrodinger equation
Laws of Quantum Mechanics
ˆ* υ= Ψ Ψ∫a A d

Probability Distribution and Expectation Values
Classical mechanics uses probability theory to obtain
relationships for systems composed of larger number of
particles
For a probability distribution function P(x) the average
value is given by
2 2
1 1
: ( ) and ( )
= =
= =∑ ∑
n n
j j j j j j
j j
Mean x x P x x x P x

Let us consider Maxwell distribution of speeds
The mean speed is calculated by taking the product of
each speed with the fraction of molecules with that
particular speed and summing up all the products.
However, when the distribution of speeds is continuous,
summation is replaced with an integral
RT
v vf v dv
M
1
2
0
8
( )
π
∞  
= =  
 
∫
Mv
RT
M
f v v e
RT
2
3
2
2 2
( ) 4
2
π
π
− 
=  
 
Probability Distribution and Expectation Values

Born Interpretation
In the classical wave equation Ψ(x,t) is the
Amplitude and |Ψ(x,t)|2 is the Intensity
The state of a quantum mechanical system is completely
specified by a wavefunction Ψ(x,t) ,which can be
complex
All possible information can be derived from Ψ(x,t)
From the analogy of classical wave equation, Intensity is
replaced by Probability. The probability is proportional
to the square of the of the wavefunction |Ψ(x,t)|2 ,
known as probability density P(x)

Born Interpretation
P x x t x t x t
2
( ) ( , ) ( , ) ( , )∗
= Ψ = Ψ ⋅Ψ
Probability density
Probability
a a a aP x x x dx x t dx x t x t dx
2
( ) ( , ) ( , ) ( , )∗
≤ ≤ + = Ψ = Ψ ⋅Ψ
Probability in 3-dimensions
*
2
P( , , )
( , , , '). ( , , , ')
( , , , ') τ
≤ ≤ + ≤ ≤ + ≤ ≤ +
= Ψ Ψ
= Ψ
a a a a a a
a a a a a a
a a a
x x x dx y y y dy z z z dz
x y z t x y z t dxdydz
x y z t d

Normalization of Wavefunction
∞∞
x
Ψ
∞∞
x
Ψ
Unacceptable wavefunction
Since Ψ*Ψdτ is the probability, the total
probability of finding the particle
somewhere in space has to be unity
If integration diverges, i.e. ∞: Ψ can not
be normalized, and therefore is NOT an
acceptable wave function. However, a
constant value C ≠ 1 is perfectly
acceptable.
*
*
( , , ). ( , , )
1τ
Ψ Ψ
= Ψ Ψ = Ψ Ψ =
∫∫∫
∫
all space
all space
x y z x y z dxdydz
d
Ψ must vanish at ±∞, or more appropriately at the boundaries
and Ψ must be finite

xx
x
x
x x
d d
p mv p i
i dx dx
p
T
m
2
Position,
Momentum,
Kinetic Energy,
2
= = = −
= x
yx z
d
T
m dx
pp p
T T
m m m m x y z
V x V x
2 2
2
22 2 2 2 2 2
2 2 2
2
Kinetic Energy, +
2 2 2 2
Potential Energy, ( ) ( )
−
=
 − ∂ ∂ ∂
= + = + + 
∂ ∂ ∂ 
Classical Variable QM Operator
The mathematical description of QM mechanics is built
upon the concept of an operator

The values which come up as result of an experiment are
the eigenvalues of the self-adjoint linear operator
In any measurement of observable associated with
operator Â, the only values that will be ever observed are
the eigenvalues an, which satisfy the eigenvalue equation:
Ψn are the eigenfunctions of the system and an are
corresponding eigenvalues
If the system is in state Ψk , a measurement on the
system will yield an eigenvalue ak
⋅Ψ = ⋅Ψn n nA a

2
2 2
2
2
2 2
2
If ( ) ( )
( ) ( )
( ) ( ) ( )
If ( )
( )
( ) ( )
x
x
x
x Sin cx
d
x c Cos cx
dx
d
x c Sin cx c x
dx
x e
d
x e
dx
d
x e x
dx
α
α
α
α
α α
Ψ =
Ψ = ⋅
Ψ = − ⋅ = − ⋅Ψ
Ψ =
Ψ = ⋅
Ψ = ⋅ = ⋅Ψ
Only real eigenvalues will be observed, which will specify
a number corresponding to the classical variable
There may be, and typically are,
many eigenfunctions for the same
QM operator!

All the eigenfunctions of Quantum Mechanical
operators are “Orthogonal”
*
( ) ( ) 0 forψ ψ ψ ψ
+∞
−∞
= = ≠∫ m n m nx x dx m n

The average value of the observable corresponding to
operator Â is ˆ* υ= Ψ Ψ∫a A d
From classical correspondence we can define average
values for a distribution function P(x)
<a> corresponds to the average value of a classical
physical quantity or observable , and Â represents the
corresponding Quantum mechanical operator
2 2
( ) and ( )
∞ ∞
−∞ −∞
= ⋅ = ⋅∫ ∫x xP x dx x x P x dx
2 * ˆ. ( ) = .
+∞ +∞
−∞ −∞
= Ψ ≈ Ψ Ψ = Ψ Ψ∫ ∫ ∫all space
a A P x dx A dx A dx A

Time-dependent Schrodinger equation
where
Time evolution of the wavefunction is related to the
total energy of the system/particle
2
2
( , ) ( , ) ( , )
2
 ∂
Ψ = − ∇ + Ψ ∂  
xi x t V x t x t
t m
2
2
( , )
2
= − ∇ +xH V x t
m
2
2
2
∂
∇ =
∂
x
x
The wavefunction Ψ(x,y,z,t) of a system evolves
according to time-dependent Schrodinger equation

Operators
xi E i p
i t t i x x
− ∂ ∂ ∂ ∂
= = = −
∂ ∂ ∂ ∂
xx t E x t x t p x t
i t i x
( , ) ( , ) ( , ) ( , )
− ∂ ∂
Ψ = ⋅Ψ Ψ = ⋅Ψ
∂ ∂
Total energy operator is also known as Hamiltonian
E V x H
m x
2
2
( )
2
− ∂
= + =
∂
xp
E T V V x
m
i
x
E V x V x
m m x
2
2
2
2
( )
2
( ) ( )
2 2
= + = +
∂ 
−  − ∂∂ = + = +
∂

Schrodinger Equation
i x t H x t V x x t
t m
2
2
( , ) ( , ) ( ) ( , )
2
 ∂ −
Ψ = ⋅Ψ = ∇ + Ψ 
∂  
In 3-dimensions
i x y z t V x y z H x y z t
t m
2
2
( , , , ) ( , , ) ( , , , )
2
 ∂ −
Ψ = ∇ + Ψ = ⋅Ψ 
∂  
x y z
2 2 2
2
2 2 2
where
∂ ∂ ∂
∇ = + +
∂ ∂ ∂
i x t H x t
t
( , ) ( , )
∂
Ψ = ⋅Ψ
∂
E and Ĥ can be
interchangeably used
i x t E x t
t
( , ) ( , )
∂
Ψ = ⋅Ψ
∂

i x t H x t V x x t
t m
2
2
( , ) ( , ) ( ) ( , )
2
 ∂ −
Ψ = ⋅Ψ = ∇ + Ψ 
∂  
H x y z t i x y z t H V x y z
t m
2
2
( , , , ) ( , , , ) ; ( , , )
2
∂ −
⋅Ψ = Ψ = ∇ +
∂
x y z t x y z t( , , , ) ( , , ) ( )ψ φ ψ φΨ = ⋅ ⇒ Ψ = ⋅
Schrodinger equation in 3-dimensions
H i
t
∂
⋅Ψ = Ψ
∂
H i
t
( ) ( )ψ φ ψ φ
∂
⋅ = ⋅
∂

H i
t
H
t
( ) ( )
operates only on ψ and operates only on
ψ φ ψ φ
φ
∂
⋅ = ⋅
∂
∂
∂
H i
t
φ ψ ψ φ
∂ 
⋅ =  ∂ 
H
i
t
Divide by
1
ψ φ
ψ
φ
ψ φ
⋅
∂ 
=  ∂ 
LHS is a function of co-ordinates and RHS is function of
time. If these two have to be equal then both functions
must be equal to constant, say W

H
i W
t
1ψ
φ
ψ φ
∂ 
= = ∂ 
H
W H W
i W i W
t t
1
ψ
ψ ψ
ψ
φ φ φ
φ
⋅
= =
∂ ∂ 
= = ∂ ∂ 
The solution of the differential equation
iWt
i W t e
t
is ( )φ φ φ
−∂
= =
∂
Separation of variables

iWt
t e( )φ
−
=
iWt iWt
e e e
2 0
1φ φ φ
−
∗
= ⋅ = ⋅ = =
The probability distribution function
is independent of time
2 2 2 2 2
ψ φ ψ φ ψΨ = ⋅ = ⋅ =
is the time independent Schrodinger Equation
represents Stationary States of the system
H Wψ ψ=

In classical mechanics Ĥ represents total energy
We can therefore write
H W H Easψ ψ ψ ψ= =
H Eψ ψ=
V x x E x
m x
2 2
2
( ) ( ) ( )
2
ψ ψ
 ∂
− + = ⋅ 
∂ 
Schrodinger equation is an eigen-value equation
There can be many solutions ψn(x) each corresponding
to different energy En

In 3-dimensions the Schrodinger equation is
V x y z x y z E x y z
m x y z
2 2 2 2
2 2 2
( , , ) ( , , ) ( , , )
2
ψ ψ
  ∂ ∂ ∂
− + + + = ⋅  
∂ ∂ ∂  
For ‘n’ particle system the Schrodinger equation in 3-
dimensions is
ψ ψ
ψ ψ
=
− − −
  ∂ ∂ ∂
− + + + = ⋅  
∂ ∂ ∂   
⇐
∑
2 2 2 2
2 2 2
1
1 2 3 1 1 2 3 1 1 2 3 1
( , , )
2
( , , ,... , , , , ,... , , , , ,... , )
n
i i i i
n n n n n n
V x y z E
m x y z
x x x x x y y y y y z z z z z

ψ ψ
 ∂ ∂ ∂
= − + + 
∂ ∂ ∂ 
 ∂ ∂ ∂
− + + 
∂ ∂ ∂ 
 ∂ ∂ ∂
− + +  ∂ ∂ ∂ 
 ∂ ∂ ∂
− + +  ∂ ∂ ∂ 
+ + + + + +
⇐
2 2 2 2
2 2 2
1 1 1 1
2 2 2 2
2 2 2
2 2 2 2
2 2 2 2
2 2 2
3 3 3 3
2 2 2 2
2 2 2
4 4 4 4
12 13 14 23 24 34
1 2 3
2
2
2
2
( , , ,
H
m x y z
m x y z
m x y z
m x y z
V V V V V V
x x x x4 1 2 3 4 1 2 3 4, , , , , , , , )y y y y z z z z
( )1 1 1 1, ,m x y z
( )3 3 3 3, ,m x y z
( )2 2 2 2, ,m x y z
( )4 4 4 4, ,m x y z

Restrictions on wavefunction
ψ must be a solution of the Schrodinger equation
ψ must be normalizable: ψ must be finite and 0 at
boundaries/ ±∞
Ψ must be a continuous function of x,y,z
dΨ/dq must be must be continuous in x,y,z
Ψ must be single-valued
Ψ must be quadratically-intergrable
(square of the wavefunction should be integrable)

Unacceptable because
ψ is not continuous
ψ is not single-valued
dψ/dq is not continuous
ψ goes to infinity

Because of these restrictions, solutions of the
Schrodinger equations do not in general exist for
arbitrary values of energy
In other words, a particle may possess only certain
energies otherwise its wavefunction would be
Unacceptable
The energy of a particle is quantized

Quantization?
The function f(x) = x2 can take any
values
If we impose arbitrary condition that
f(x) can only be multiples of three,
then values if x are restricted.
Quantization!
Physically meaningful boundary
conditions lead to quantization ☺

Not deterministic: Can not precisely determine many
parameters in the system, but Ψ can provide all the
information (spatio-temporal) of a system.
Only average values and probabilities can be obtained for
classical variables, now in new form of “operators”.
Total energy is conserved, but quantization of energy
levels come spontaneously from restriction on wave
function or boundary condition
Final outputs tally very well with experimental results,
and does not violate Classical mechanics for large value
of mass.
Essence of Quantum Mechanics

Quantum Mechanics
Examples of Exactly Solvable Systems
1. Free Particle
2. Particle in a Square-Well Potential
3. Hydrogen Atom

Time-independent Schrodinger equation
Free Particle
H Eψ ψ=
V x x E x
m x
2 2
2
( ) ( ) ( )
2
ψ ψ
 ∂
− + = ⋅ 
∂ 
For a free particle V(x)=0
There are no external forces acting
x E x
m x
2 2
2
( ) ( )
2
ψ ψ
∂
− = ⋅
∂

Free Particle
( ) ( )
( )
x A kx B kx
x A kx B kx k A kx B kx
dx dx
x k A kx B kx k x
dx
2
2 2
2
( ) sin cos
( ) sin cos cos sin
( ) sin cos ( )
ψ
ψ
ψ ψ
= +
∂ ∂
= + = −
∂
= − + = −
x E x
m x
2 2
2
( ) ( )
2
ψ ψ
∂
− = ⋅
∂
m
x
Second-order linear differential equation
Let us assume
Trial Solutionx A kx B kx( ) sin cosψ = +

Free Particle
x E x
m x
2 2
2
( ) ( )
2
ψ ψ
∂
− = ⋅
∂
k mE
k x E x E k
m m
2 2 2
2 2
( ) ( )
2 2
ψ ψ= ⋅ ⇒ = ⇒ = ±
m
x

There are no restrictions on k
E can have any value
Energies of free particles are continuous
Free Particle
x E x
m x
2 2
2
( ) ( )
2
ψ ψ
∂
− = ⋅
∂
k mE
k x E x E k
m m
2 2 2
2 2
( ) ( )
2 2
ψ ψ= ⋅ ⇒ = ⇒ = ±
mE mE
x A x B x
2 2
( ) sin cosψ = +
k
E
m
2 2
2
=
No Quantization All energies are allowed
m
x
de Broglie wave

x
V x x L
x L
0
( ) 0 0
∞ <

= ≤ ≤
∞ >
x V x x E x
m x
2 2
2
( ) ( ) ( ) ( )
2
ψ ψ ψ
∂
− + = ⋅
∂
For regions in the space x < 0 and x > L ⇒ V = ∞
( )
m
x V E x x
x
2
2 2
2
( ) ( ) ( )ψ ψ ψ
∂
= − ⋅ = ∞⋅
∂
Normalization condition not satisfied ⇒
x x L( 0) 0 and ( ) 0ψ ψ< = > =
Particle in 1-D Square-Well Potential

x V x x E x
m x
2 2
2
( ) ( ) ( ) ( )
2
ψ ψ ψ
∂
− + = ⋅
∂
For regions in the space 0 ≤ x ≤ L ⇒ V = 0
x E x
m x
2 2
2
( ) ( )
2
ψ ψ
∂
− = ⋅
∂
This equation is similar to free particle Schrodinger
However, boundary conditions are present
Let is assume
Trial Solution
Energy
x A kx B kx( ) sin cosψ = +
k
E
m
2 2
2
=

x A kx B kx( ) sin cosψ = +
Boundary Condition x x0 ( ) 0ψ= ⇒ =
Boundary Condition
x A kx( ) sin cos0 1ψ = =
x L L( ) 0ψ= ⇒ =
L A kL A kL( ) 0 sin 0 0 or sin 0ψ = ⇒ = ⇒ = =
But the wavefunction ψ(x) CANNOT be ZERO everywhere
kL kL nsin 0 n=1,2,3,4...π= ⇒ =
Wavefunction is x A kx( ) sinψ =

k n
E k
m L
2 2
and
2
π
= =
n
n n h
E
mL mL
2 2 2 2 2
2 2
n=1,2,3,4...
2 8
π
= =
Energy is no longer continues but has
discrete values; Quantization of energy
Energy separation increases with
increasing values of n
The lowest allowed energy level is for n=1
has a non zero value ⇒ Zero Point EnergyE
mL
2 2
1 2
2
π
=

( )f f
f i f i
n h n h h
h E E E n n
mL mL mL
2 2 2 2 2
2 2
2 2 2
-
8 8 8
ν = ∆ = − = = −
Larger the box, smaller the energy of hν
Particle in 1-D Square-Well Potential: Spectroscopy

Wavefunction
Normalization
n
x A kx A x
L
( ) sin sin
π
ψ = =
L L n
x x dx A x dx
L
2 2
0 0
( ) ( ) sin 1
π
ψ ψ∗
⋅ ⋅ = ⋅ =∫ ∫
n
A x x
L L L
2 2
( ) sin
π
ψ= =
Homework
Evaluate the above integral

Wavefunction
n
x x
L L
2
( ) sin
π
ψ =
n=1,3.. (odd)
Symmetric
(even function)
n=2,4.. (even)
Anti-Symmetric
(odd function)
Number of Nodes
(zero crossings) = n-1
Particle in 1-D Square-Well Potential: Spectroscopy

Expectation values
ψ ψ
π π
π
∗
= ⋅ ⋅ ⋅
= ⋅ ⋅ ⋅
= ⋅ ⋅
=
∫
∫
∫
0
2
0
2 2
sin sin
2
sin
2
L
L
x x dx
n n
x x x dx
L L L L
n
x x dx
L L
L
Homework
Verify!

Expectation values
Homework
Verify!
ψ ψ
π π
π π π
∗ ∂ 
= ⋅ − ⋅ ⋅ ∂ 
∂
= − ⋅ ⋅ ⋅
∂
−
= ⋅ ⋅
=
∫
∫
∫
0
2 0
2 2
sin sin
2
sin cos
0
x
L
L
p i dx
x
n n
i x x dx
L L x L L
i n n n
x x dx
L L L

Hamiltonian
∂ ∂
= − − = +
∂ ∂
2 2
2 2
2 2
x yH H H
m x m y
ψ ψ⋅ = ⋅( , ) ( , )nH x y E x y
ψ ψ ψ= ⋅
Let us assume that
( , ) ( ) ( )x y x y

( )H x y H x y( , ) ( ) ( )ψ ψ ψ⋅ = ⋅ ⋅
( ) ( )x yE E x y( , )ψ= + ⋅
( )x yH H x y( ) ( )ψ ψ = + ⋅
 
x yy H x x H y( ) ( ) ( ) ( )ψ ψ ψ ψ= ⋅ ⋅ + ⋅ ⋅
x yy E x x E y( ) ( ) ( ) ( )ψ ψ ψ ψ= ⋅ ⋅ + ⋅ ⋅
x yE x y E x y( ) ( ) ( ) ( )ψ ψ ψ ψ= ⋅ ⋅ + ⋅ ⋅
( ) ( )x yE E x y( ) ( )ψ ψ= + ⋅ ⋅

Hamiltonian
∂ ∂
= − − = +
∂ ∂
2 2
2 2
2 2
x yH H H
m x m y
ψ is a product of the eigenfunctions of the parts of Ĥ
E is sum of the eigenvalues of the parts of Ĥ
ψ ψ⋅ = ⋅( , ) ( , )nH x y E x y
ψ ψ ψ= ⋅( , ) ( ) ( )x y x y
x y x yn n n n nE E E E,= = +

x y x yn n n n
yx
x y
yx
x y
x y
E E E
n hn h
mL mL
nnh
n n
m L L
,
2 22 2
2 2
222
2 2
8 8
, 1,2,3,4...
8
= +
= +
 
= + =  
 
x x y y
x yx y
x y x y
n n
x y
L L L L
n n
x y
L LL L
( , ) ( ) ( )
2 2
sin sin
2
sin sin
ψ ψ ψ
π π
π π
= ⋅
= ⋅
= ⋅
V=0
Lx
Ly

( )
x y x yn n n n
yx
x y x y
E E E
n hn h
mL mL
h
n n n n
mL
,
2 22 2
2 2
2
2 2
2
8 8
, 1,2,3,4...
8
= +
= +
= + =
x y x y
n n
x y
L L L L
n n
x y
L L L
( , ) ( ) ( )
2 2
sin sin
2
sin sin
ψ ψ ψ
π π
π π
= ⋅
= ⋅
= ⋅
V=0
Lx
Ly
Square Box
⇒ Lx = Ly = L

x y
L L L
h
E E E
mL
1,2 1 2
2
1,2 1 2 2
2 2
sin sin
5
8
π π
ψ ψ ψ= ⋅ = ⋅
= + =
V=0
Lx
Ly
x y
L L L
h
E E E
mL
2,1 2 1
2
2,1 2 1 2
2 2
sin sin
5
8
π π
ψ ψ ψ= ⋅ = ⋅
= + =
⇒ are degenerate wavefunctionsE E1,2 2,1= 1,2 2,1andψ ψ
Square Box
⇒ Lx = Ly = L

V=0
Lx
Ly
V=0
Lx
Ly
(1,1)
(2,1) (1,2)
(2,2)
(3,1) (1,3)
(1,1)
(2,1)
(3,1)
(1,2)
(2,2)
(3,2)
(1,3)
Particle in 2-D Square-Well Potential – Symmetry

Number of nodes = nx+ny-2
Particle in a 2-D Well – Wavefunctions

Particle in a 3D-Box
yx z
x x y y z z
x y z x y z
nn n
x y z
L L L L L L
( , , ) ( ) ( ) ( )
2 2 2
sin sin sin
ψ ψ ψ ψ
ππ π
= ⋅ ⋅
= ⋅ ⋅
x y z x y zn n n n n n
yx z
x y z
x y z
E E E E
n hn h n h
n n n
mL mL mL
, ,
2 22 2 2 2
2 2 2
, , 1,2,3,4...
8 8 8
= + +
= + + =

Agrees well with the experimental
value of 258 nm
Particle in a box is a good model
Particle in a Box – Application in Chemistry
Hexatriene is a linear molecule of length 7.3 Å
It absorbs at 258 nm
Use particle in a box model to explain the results.
Six π electron fill
lower three levels
( )
( )
λ
λ
∆ = − = − =
= − ≈
2
2 2
2
2
2 2
8
8
251nm
f i f i
f i
h hc
E E E n n
mL
mL c
n n
h

Increase in bridge length increase the
emission wavelength.
Predicts correct trend and gets the
wavelength almost right.
Particle in a box is a good model
Particle in a Box – Application in Chemistry
Electronic spectra of conjugated molecules
λ
λ
= ⇒ ∝
2
2
2
8
hc h
L
mL
Β-carotene is orange because of 11
conjugated double bonds

Particle in a Box – Application in nano-science
Band gap changes due to
confinement, and so does
the color of emitted light
Quantum Dots have a huge
application in chemistry,
biology, and materials science
for photoemission imaging
purpose, as well as light
harvesting/energy science

What have we learnt?
Formulate a correct Hamiltonian
(energy) Operator H
Solve TISE HΨΨΨΨ=E ΨΨΨΨ
by separation of variables and
intelligent trial wavefunction
Impose boundary conditions for
eigenfunctions and obtain
Quantum numbers
Eigenstatesor Wavefunctions:
Should be “well behaved” -
Normalization of Wavefunction
Probabilities and Expectation Values

Hydrogen Atom
πε
= − ∇ − ∇ −
22 2
2 2
0
1
2 2 4
N e
N e
N e eN
Z Z e
H
m m r
N e N eH T T V -= + +
(xe,ye,ze)
= − + − + −2 2 2
( ) ( ) ( )eN e N e N e Nr x x y y z z
(xN,yN,zN)
N e
N N N e e ex y z x y z
2 2 2 2 2 2
2 2
2 2 2 2 2 2
∂ ∂ ∂ ∂ ∂ ∂
∇ = + + ∇ = + +
∂ ∂ ∂ ∂ ∂ ∂
Two particle central-force problem
Completely solvable – a rare example!

Hydrogen Atom
 
− ∇ − ∇ − Ψ = ⋅Ψ 
 
2 2 2
2 2
2 2
N e Total Total Total
N e eN
QZe
E
m m r
Total N N N e e ex y z x y z( , , , , , )Ψ = Ψ
πε
= − ∇ − ∇ −
22 2
2 2
0
1
2 2 4
N e
N e
N e eN
Z Z e
H
m m r
πε
= − ∇ − ∇ −
= = =
2 2 2
2 2
0
2 2
1
with 1 and
4
N e
N e eN
N e
QZe
H
m m r
Z Z Z Q

Hydrogen Atom: Relative Frame of Reference
Separation of Ĥ into Center of Mass and Internal co-ordinates
x
z
y
-re
R
rN
me(xe,ye,ze)
CM
reN
MN
(xN,yN,zN)
( )
( )
( )
= −
= −
= −
= + +
= + +
= = −
= + +
2 2 2
2 2 2
2 2 2
e N
e N
e N
e e e e
N N N N
eN e N
x x x
y y y
z z z
r x y z
r x y z
r r r r
x y z
 
− ∇ − ∇ − Ψ = ⋅Ψ 
 
2 2 2
2 2
2 2
N e eN
QZe
E
m m r

Separation of Ĥ into Center of Mass and Internal co-ordinates
x
z
y
-re
R
rN
me(xe,ye,ze)
CM
reN
MN
(xN,yN,zN)
+
=
+
+
=
+
+
=
+
+
=
+
e e N n
e N
e e N n
e N
e e N n
e N
e e N N
e N
m x m x
X
m m
m y m y
Y
m m
m z m z
Z
m m
m r m r
R
m m
 
− ∇ − ∇ − Ψ = ⋅Ψ 
 
2 2 2
2 2
2 2
N e eN
QZe
E
m m r
( )
( )
( )
= −
= −
= −
= + +
= + +
= = −
= + +
2 2 2
2 2 2
2 2 2
e N
e N
e N
e e e e
N N N N
eN e N
x x x
y y y
z z z
r x y z
r x y z
r r r r
x y z

µ
µ
 
− ∇ − ∇ − Ψ = ⋅Ψ 
 
= + =
+
2 2 2
2 2
2 2
where and
R r Total Total Total
e N
e N
e N
QZe
E
M r
m m
M m m
m m
⇓
 
− ∇ − ∇ − Ψ = ⋅Ψ 
 
2 2 2
2 2
2 2
N e eN
QZe
E
m m r
Checkout Appendix-1

Hydrogen Atom: Separation to Relative Frame
Hydrogen atom has two particles the nucleus and
electron with co-ordinates xN,yN,zN and xe,ye,ze
The potential energy between the two is function of
relative co-ordinates x=xe-xN, y=ye-yN, z=ze-zN
= + +
= − = − = −
= + +
+ + +
= = =
+ + +
, ,
, ,
e N e N e N
e e N n e e N n e e N n
e N e N e N
r ix jy kz
x x x y y y z z z
R iX jY kZ
m x m x m y m y m z m z
X Y Z
m m m m m m
x
z
y
-re
R
rN
me(xe,ye,ze)
CM
reN
MN
(xN,yN,zN)
Appendix-1

+
=
+
= = −
= −
+
= −
+
e e N N
e N
eN e N
N
e
e N
e
N
e N
m r m r
R
m m
r r r r
m
r R r
m m
m
r R r
m m
x
z
y
-re
R
rN
me(xe,ye,ze)
CM
reN
MN
(xN,yN,zN)
Appendix-1

( )
µ µ
= +
   
= − ⋅ −   
+ +   
   
+ − ⋅ −   
+ +   
 
= + +  
+ 
= + = + =
+
2 2
2 2
2 2
1 1
2 2
1
2
1
2
1 1
2 2
1 1
where and
2 2
e e N N
N N
e
e N e N
e e
e
e N e N
e N
e N
e N
e N
e N
e N
T m r m r
m m
T m R r R r
m m m m
m m
m R r R r
m m m m
m m
T m m R r
m m
m m
T M R r M m m
m m
=
=
=
=
e
e
N
N
dr
r
dt
dr
r
dt
dr
r
dt
dR
R
dt
Appendix-1

µ
µ
= +
= +
2 2
2 2
1 1
2 2
2 2
R r
T M R r
p p
T
M
In the above equation the first term represent the
kinetic energy of the center of mass (CM) motion and
second term represents the kinetic energy of the relative
motion of electron and
µ
µ
⋅
= + −
⋅
= − ∇ − ∇ −
2 2
2 2
2 2
2 2
2 2
N eR r
N e
R r
Z Zp p
H
M r
Z Z
H
M r
Appendix-1

Free particle!
Kinetic energy of the atom
Hydrogen Atom: Separation of CM motion
χ χ χ
 
= − ∇ = 
 
2
2
2
N N R N N NH E
M
=
2 2
2
N
k
E
M
χ ψΨ = ⋅Total N e= +N eH H H = +Total N eE E E
µ
 
− ∇ − ∇ − Ψ = ⋅Ψ 
 
2 2 2
2 2
2 2
R r Total Total Total
QZe
E
M r
µ
− ∇ = − ∇ − =
2 2 2
2 2
2 2
N eR r
QZe
H H
M r

Hydrogen Atom: Electronic Hamiltonian
( )
ψ ψ
µ
ψ
 ∂ ∂ ∂
− + + − 
∂ ∂ ∂  + +
= ⋅
2 2 2 2 2
2 2 2
2 2 2
( , , ) ( , , )
2
( , , )
e e
e e
QZe
x y z x y z
x y z x y z
E x y z
Not possible to separate out into three different co-ordinates.
Need a new co-ordinate system
r
ψ ψ ψ
µ
ψ ψ
 
⋅ = − ∇ − = ⋅ 
 
⇒
2 2
2
2
( , , )
e e r e e e
e e
QZe
H E
r
x y z

Spherical Polar Co-ordinates
θ φ
θ φ
θ
=
=
=
sin cos
sin sin
cos
x r
y r
z r
( )
θ
φ
−
= + +
 
=  
 
 
= −  
 
2 2 2
1
cos
tan 1
r x y z
z
r
y
x
τ θ θ φ= ⋅ ⋅ = ⋅ ⋅ ⋅ ⋅2
sind dx dy dz r dr d d
‘r’ ranges from 0 to ∞
‘θ’ ranges from 0 to π
‘φ’ ranges from 0 to 2π

Spherical Polar Co-ordinates
ψ ψ θ φ ψ⇒ ⇐( , , ) ( , , )e e er x y z
θ
θ θ θ θ φ
 ∂ ∂ ∂
+ + 
∂ ∂ ∂ 
∂ ∂ ∂ ∂ ∂   
= + +   ∂ ∂ ∂ ∂ ∂   
2 2 2
2 2 2
2
2
2 2 2 2 2
1 1 1
sin
sin sin
f
x y z
f f f
r
r r r r r
ψ ψ ψ
θ
µ θ θ θ θ φ
ψ ψ
 ∂ ∂ ∂∂ ∂   
− + +    ∂ ∂ ∂ ∂ ∂    
− =
22
2
2 2 2 2 2
2
1 1 1
sin
2 sin sin
e e e
e e e
r
r r r r r
QZe
E
r

ψ ψ ψ
θ
µ θ θ θ θ φ
ψ ψ
 ∂ ∂ ∂∂ ∂   
− + +    ∂ ∂ ∂ ∂ ∂    
− =
22
2
2 2 2 2 2
2
1 1 1
sin
2 sin sin
e e e
e e e
r
r r r r r
QZe
E
r
µ− 2
2
2
Multiply with
r
ψ ψ ψ
θ
θ θ θ θ φ
µ µ
ψ ψ
∂ ∂ ∂∂ ∂   
+ +   ∂ ∂ ∂ ∂ ∂   
+ + =
2
2
2 2
2 2
2 2
1 1
sin
sin sin
2 2
0
e e e
e e e
r
r r
rQZe r
E

( )ψ θ φ θ φ
ψ
⇒ ⋅Θ ⋅Φ
⇒ ⋅Θ⋅Φ
( , , ) ( ) ( )e
e
r R r
R
θ
θ θ θ θ φ
µ µ
∂ ∂ ⋅Θ⋅Φ ∂ ∂ ⋅Θ⋅Φ ∂ ⋅Θ⋅Φ   
+ +   ∂ ∂ ∂ ∂ ∂   
+ ⋅Θ⋅Φ + ⋅Θ⋅Φ =
2
2
2 2
2 2
2 2
( ) 1 ( ) 1 ( )
sin
sin sin
2 2
( ) ( ) 0e
R R R
r
r r
rQZe r
R E R
ψ ψ ψ
θ
θ θ θ θ φ
µ µ
ψ ψ
∂ ∂ ∂∂ ∂   
+ +   ∂ ∂ ∂ ∂ ∂   
+ + =
2
2
2 2
2 2
2 2
1 1
sin
sin sin
2 2
0
e e e
e e e
r
r r
rQZe r
E

θ
θ θ θ θ φ
µ µ
∂ ∂ ⋅Θ⋅Φ ∂ ∂ ⋅Θ⋅Φ ∂ ⋅Θ⋅Φ   
+ +   ∂ ∂ ∂ ∂ ∂   
+ ⋅Θ⋅Φ + ⋅Θ⋅Φ =
2
2
2 2
2 2
2 2
( ) 1 ( ) 1 ( )
sin
sin sin
2 2
( ) ( ) 0e
R R R
r
r r
rQZe r
R E R
θ
θ θ θ θ φ
µ µ
∂ ∂ ∂ ∂Θ ∂ Φ   
Θ⋅Φ + ⋅Φ + ⋅Θ   ∂ ∂ ∂ ∂ ∂   
+ ⋅Θ⋅Φ + ⋅Θ⋅Φ =
2
2
2 2
2 2
2 2
1 1
( ) ( ) sin ( )
sin sin
2 2
( ) ( ) 0e
R
r R R
r r
rQZe r
R E R
Rearrange

∂ ∂ ∂ ∂Θ ∂ Φ   
Θ⋅Φ + ⋅Φ + ⋅Θ   ∂ ∂ ∂ ∂ ∂   
+ ⋅Θ⋅Φ + ⋅Θ⋅Φ =
2
2
2 2
2 2
2 2
1 1
( ) ( ) sin ( )
sin sin
2 2
( ) ( ) 0e
R
r R R
r r
rQZe r
R E R
θ
θ θ θ θ φ
µ µ
⋅Θ⋅Φ
1
Multiply with
R
∂ ∂ ∂ ∂Θ ∂ Φ   
+ +   ∂ ∂ Θ ∂ ∂ Φ ∂   
+ + =
2
2
2 2
2 2
2 2
1 1 1 1 1
sin
sin sin
2 2
0e
R
r
R r r
rQZe r
E
θ
θ θ θ θ φ
µ µ

∂ ∂ ∂ ∂Θ ∂ Φ   
+ +   ∂ ∂ Θ ∂ ∂ Φ ∂   
+ + =
2
2
2 2
2 2
2 2
1 1 1 1 1
sin
sin sin
2 2
0e
R
r
R r r
rQZe r
E
θ
θ θ θ θ φ
µ µ
Rearrange
∂ ∂ 
+ + + ∂ ∂ 
 ∂ ∂Θ ∂ Φ 
= − +  Θ ∂ ∂ Φ ∂  
2 2
2
2 2
2
2 2
1 2 2
1 1 1 1
sin
sin sin
e
R rQZe r
r E
R r r
µ µ
θ
θ θ θ θ φ
LHS = f(r)=f(θ ,φ) =RHS
⇒ f(r)=f(θ ,φ) =constant=β

∂ ∂ ∂ ∂Θ ∂ Φ   
+ +   ∂ ∂ Θ ∂ ∂ Φ ∂   
+ + =
2
2
2 2
2 2
2 2
1 1 1 1 1
sin
sin sin
2 2
0e
R
r
R r r
rQZe r
E
θ
θ θ θ θ φ
µ µ
Rearrange
∂ ∂ 
+ + + ∂ ∂ 
 ∂ ∂Θ ∂ Φ 
= − + =  Θ ∂ ∂ Φ ∂  
2 2
2
2 2
2
2 2
1 2 2
1 1 1 1
sin
sin sin
e
R rQZe r
r E
R r r
µ µ
θ β
θ θ θ θ φ
LHS = f(r)=f(θ ,φ) =RHS
⇒ f(r)=f(θ ,φ) =constant=β

∂ ∂ 
+ + + = ∂ ∂ 
∂ ∂Θ ∂ Φ 
+ = − Θ ∂ ∂ Φ ∂ 
2 2
2
2 2
2
2 2
1 2 2
1 1 1 1
sin
sin sin
e
R rQZe r
r E
R r r
µ µ
β
θ β
θ θ θ θ φ
θ β
θ θ θ θ φ
∂ ∂Θ ∂ Φ 
+ = − Θ ∂ ∂ Φ ∂ 
2
2 2
1 1 1 1
sin
sin sin
Let us consider
θ2
Multiply with sin and rearrange
θ
θ β θ
θ θ φ
∂ ∂Θ ∂ Φ 
+ = − Θ ∂ ∂ Φ ∂ 
2
2
2
sin 1
sin sin

θ
θ β
θ θ
∂ ∂Θ 
+ = Θ ∂ ∂ 
2sin
sin m
φ
∂ Φ
= −
Φ ∂
2
2
2
1
m
θ
θ β θ
θ θ φ
∂ ∂Θ ∂ Φ 
+ = − Θ ∂ ∂ Φ ∂ 
2
2
2
sin 1
sin sin
LHS = f(θ)=f(φ) =RHS
⇒ f(θ)=f(φ) =constant=m2

∂ ∂ 
+ + + = ∂ ∂ 
2 2
2
2 2
1 2 2
e
R rQZe r
r E
R r r
µ µ
β
θ
θ β θ
θ θ
∂ ∂Θ 
+ = Θ ∂ ∂ 
2 2sin
sin sin m
φ
∂ Φ
= −
Φ ∂
2
2
2
1
m
We have separated out all the three variables r, θ and φ

Solution to ΦΦΦΦ part
φ
φ φ
φ
φ
φ
∂ Φ
+ =
Φ ∂
∂ Φ
= − Φ
∂
2
2
2
2
2
2
1 ( )
0
( )
( )
( )
m
m
Let is assume
as trial solution
φ
φ ±
Φ =( ) im
Ae
φ
φ
∂Φ
= ± Φ
∂
∂ Φ
= − Φ
∂
2
2
2
0im
m
Wavefunction has to be continuous
φ π φ⇒ Φ + = Φ( 2 ) ( )
‘φ’ ranges from 0 to 2π

Solution to ΦΦΦΦ part
φ π φ φ π φ
π π
+ − + −
− −
−
= =
= =
( 2 ) ( ) ( 2 ) ( )
(2 ) (2 )
and
1 and 1
im im im im
m m m m
im im
A e A e A e A e
e e
True only if m=0, ±1, ±2, ±3, ±4,….
m is the “magnetic quantum” number
m is restricted by another quantum number (orbital
Angular momentum), l, such that |m|<l
φ π φ⇒ Φ + = Φ( 2 ) ( )

The ΘΘΘΘ and the R part
∂ ∂ 
+ + + = ∂ ∂ 
2 2
2
2 2
1 2 2
e
R rQZe r
r E
R r r
µ µ
β
θ
θ β θ
θ θ
∂ ∂Θ 
+ = Θ ∂ ∂ 
2 2sin
sin sin m
 ∂ ∂ 
+ + − =  ∂ ∂   
2 2
2
2
( ) 2
( ) ( ) 0e
R r r QZe
r E R r R r
r r r
µ
β
θ
θ θ β θ
θ θ θ θ
∂ ∂Θ 
− Θ + Θ = ∂ ∂ 
2
2
1 ( )
sin ( ) ( ) 0
sin sin
m
Rearrange

Solve to get Θ(θ)
Need serious mathematical skill to solve these two equations.
We only look at solutions
The ΘΘΘΘ and the R part
 ∂ ∂ 
+ + − =  ∂ ∂   
2 2
2
2
( ) 2
( ) ( ) 0e
R r r QZe
r E R r R r
r r r
µ
β
θ
θ θ β θ
θ θ θ θ
∂ ∂Θ 
− Θ + Θ = ∂ ∂ 
2
2
1 ( )
sin ( ) ( ) 0
sin sin
m
Solve to get R(r)
Restriction on m are
due this this equation

The ΘΘΘΘ part
are known as Associated Legendre Polynomials
The new quantum number is ‘l’ called orbital / Azimuthal
quantum number
Restriction on m≤l
is due to this equation
θ
θ θ β θ
θ θ θ θ
∂ ∂Θ 
− Θ + Θ = ∂ ∂ 
2
2
1 ( )
sin ( ) ( ) 0
sin sin
m
θ θ θ
θ θ β
+
+
−
−
= − −
−
= − = +
+
2 22
( 1)
(cos ) (1 cos ) (cos 1)
2 !
( )!
(cos ) ( 1) (cos ) with ( 1)
( )!
m l m
m
m l
l l l m
m m m
l l
d
P
l dx
l m
P P l l
l m
Solution to Θ(θ) are
θ(cos )m
lP
l=0,1,2,3…

The angular (ΘΘΘΘ ΦΦΦΦ) part
The angular part of the solution
are called spherical harmonicsθ φ θ φ⇒ Θ ⋅Φ( , ) ( ) ( )m
lY
φ
θ φ θ
π
+ −
=
+
(2 1) ( )!
( , ) (cos )
4 ( )!
m m im
l l
l l m
Y P e
l m
l=0,1,2,3…
m=0, ±1, ±2, ±3… and |m|≤l

The R part
 ∂ ∂ 
+ + − =  ∂ ∂   
2 2
2
2
( ) 2
( ) ( ) 0e
R r r QZe
r E R r R r
r r r
µ
β
( )
( )
+
−
+
+
 − −     = −          +  
0
1
2 3
2
2 1
3
1 ! 2 2
( )
2 !
l
Zr
nal l
nl n l
n l Z Zr
R r r e L
na nan n l
Solution to R(r) are
Where are called associated Laguerre functions
The new quantum number is ‘n’ called principal quantum
number
+
+
 
 
 
2 1
0
2l
n l
Zr
L
na
= =
22
0
2 2
4
a
Q e e
πε
µ µ
Restriction on l<n

Energy of the Hydrogen Atom
( )= − = − = − ≈
−
=
2 2 4 2 4 2 4
2 2 2 2 2 2
0 0 0
2
2
8 8
13.6
n e
n
Q Z e Z e Z e
E m
n h n a n
eV
E
n
µ µ
µ
ε πε
Energy is dependent only on ‘n’
Energy obtained by full quantum mechanical treatment is
equal to Bohr energy
Potential energy term is only dependent on the Radial
part and has no contribution from the Angular parts

Quantum Numbers of Hydrogen Atom
n Principal Quantum number
Specifies the energy of the electron
l Orbital Angular Momentum Quantum number
Specifies the magnitude of the electron's orbital angular
momentum
m Z-component of Angular Momentum Quantum number
Specifies the orientation of the electron's orbital
angular momentum
s Orbital Angular Momentum Quantum number
Specifies the orientation of the electron's spin angular
momentum

Orbital Angular Momentum Quantum Number
l=0 ⇒⇒⇒⇒ s-Orbital
l=1 ⇒⇒⇒⇒ p-Orbital
l=2 ⇒⇒⇒⇒ d-Orbital
l=3 ⇒⇒⇒⇒ f-Orbital

Normalization
( )
( ) ( )
( )
∞
∗
∞ ∞
= ⋅
⇒
= ⋅ = ⋅
=
 = = 
=
∫ ∫ ∫
∫ ∫ ∫ ∫
∫ ∫
, , ,
2 22 2
, , , ,
2 22
, ,0 0 0
2 22
0 0 0 0
22
, ,0 0
( , , ) ( , )
( , , ) ( , ) ( , )
sin ( , , ) 1
sin ( , ) sin ( , ) ( , ) 1
m
n l m n l l
m m
n l m n l l n l l
n l m
m m m
l l l
n l n l
r R r Y
r R r Y R r Y
r dr d d r
d d Y d d Y Y
r dr R r dr R
π π
π π π π
ψ θ φ θ φ
ψ θ φ θ φ θ φ
θ θ φ ψ θ φ
θ θ φ θ φ θ θ φ θ φ θ φ
( ) ( )
∗
  =  , 1n lr R r
Normalize the Radial and Angular parts separately

Spherical Harmonics Yl
m
( )
φ
φ
φ
π
θ
π
θ
π
θ
π
θ θ
π
θ
π
±
±
±
 
= =  
 
 
= =  
 
 
= = ±  
 
 
= = − 
 
 
= = ±  
 
 
= = ±  
 
1 2
1 2
1 2
1 2
2
1 2
1 2
2 2
1
0; 0
4
3
1; 0 cos
4
3
1; 1 sin
8
3
2; 0 3cos 1
8
15
2; 1 cos sin
8
15
2; 2 sin
32
i
i
i
l m
l m
l m e
l m
l m e
l m e
( )
( ) φ
φ
φ
θ θ
π
θ θ
π
θ θ
π
θ
π
±
±
±
 
= = − 
 
 
= = ± − 
 
 
= = ±  
 
 
= = ±  
 
1 2
3
1 2
2
1 2
2 2
1 2
3 3
7
3; 0 5cos 3cos
16
21
3; 1 5cos 1 sin
64
105
3; 2 sin cos
32
35
3; 3 sin
64
i
i
i
l m
l m e
l m e
l m e
φ
θ φ θ
π
+ −
=
+
(2 1) ( )!
( , ) (cos )
4 ( )!
m m im
l l
l l m
Y P e
l m

Radial Functions
( )
( )
( )
ρ
ρ
ρ
ρ
ρ
ρ
ρ
ρ
ρ ρ
ρ
ρ
−
−
−
−
−
−
 
= =  
 
 
= = − 
 
 
= =  
 
 
= = − − 
 
 
= = − 
 
 
= =  
 
3 2
2
3 2
2
1
2
3 2
2
1
2
3 2
2 2
1
2
3 2
2
1
2
3 2
2 2
1
2
1; 0 2
1
2; 0 2
8
1
2; 1
24
1
3; 0 6 6
243
1
3; 1 4
486
1
3; 2
2430
Z
n l e
a
Z
n l e
a
Z
n l e
a
Z
n l e
a
Z
n l e
a
Z
n l e
a
ρ
πε
µ
µ
=
=
= =
2
0
2
0
2
4
(for )e
Zr
na
a
e
a a m
( )
( )
+
−
+
+
 − −     = −          +  
0
1
2 3
2
2 1
3
1 ! 2 2
( )
2 !
l
Zr
nal l
nl n l
n l Z Zr
R r r e L
na nan n l

Radial Functions of Hydrogen Atom
−
−
−
−
 
= =  
 
   
= = −   
   
   
= =    
   
      
 = = − −            

= =
3 2
3 2
2
1
2 0
3 2
2
1
2 0
3 2 2
3
0 0
1
2
1
1; 0 2
1 1
2; 0 2
8
1 1
2; 1
24
1 2 2
3; 0 2 1
3 3 27
1 1
3; 1
486
o
o
o
o
r
a
o
r
a
o
r
a
o
r
a
o
o
n l e
a
r
n l e
a a
r
n l e
a a
r r
n l e
a a a
n l
a
−
−
  
−   
   
   
= =    
   
3 2
3
0
3 2 2
3
1
2 0
2
4
3
1 1 2
3; 2
32430
o
o
r
a
r
a
o
r
e
a
r
n l e
a a
( )
( )
+
−
+
+
 − −    
 = −    
  +      
0
1
32
2
2 1
3
0 0
1 ! 2 2
( )
2 !
l
r
nal l
nl n l
n l r
R r r e L
na nan n l
ρ
πε
µ
µ
=
=
= =
2
0
2
0
2
4
(for )e
Zr
na
a
e
a a m

Wavefunctions of Hydrogen Atom
φ
ψ ψ
π
ψ ψ
π
ψ ψ θ
π
ψ ψ θ
π
ψ ψ
π
+
−
−
−
−
−
+
−
 
= =  
 
   
= = −   
   
   
= =    
   
   
= =    
   
 
= =  
 
1
1
3 2
1,0,0 1
3 2
2
2,0,0 2
0
3 2
2
2,1,0 2
0
3 2
2
2,1, 1 2
0
3 2
2,1, 1 2
0
1 1
1 1
2
4 2
1 1
cos
4 2
1 1
sin
8
1 1
8
o
o
o
z
o
r
a
s
o
r
a
s
o
r
a
p
o
r
a i
p
o
p
o
e
a
r
e
a a
r
e
a a
r
e e
a a
r
a a
φ
θ
−
− 
 
 
2
sino
r
a i
e e
( )ψ θ φ θ φ= ⋅, , ,( , , ) ( , )m
n l m n l lr R r Y
f(r)
f(r)
f(r,θ)
f(r,θ,φ)
f(r,θ,φ)

1s and 2s Orbitals
ψ ψ
π
ψ ψ
π
−
−
 
= =  
 
   
= = −   
   
3 2
1,0,0 1
3 2
2
2,0,0 2
0
1 1
1 1
2
4 2
o
o
r
a
s
o
r
a
s
o
e
a
r
e
a a
Functions of only ‘r’

φ
φ
ψ ψ θ
π
ψ ψ θ
π
ψ ψ θ
π
+
−
−
−
+
−
−
−
   
= =    
   
   
= =    
   
   
= =    
   
1
1
3 2
2
2,1,0 2
0
3 2
2
2,1, 1 2
0
3 2
2
2,1, 1 2
0
1 1
cos
4 2
1 1
sin
8
1 1
sin
8
o
z
o
o
r
a
p
o
r
a i
p
o
r
a i
p
o
r
e
a a
r
e e
a a
r
e e
a a
2p Orbitals
Functions of ‘r’, ‘θ’ and ‘φ’

φ
φ
ψ ψ θ
π
ψ ψ θ
π
ψ ψ θ
π
+
−
−
−
+
−
−
−
   
= =    
   
   
= =    
   
   
= =    
   
1
1
3 2
2
2,1,0 2
0
3 2
2
2,1, 1 2
0
3 2
2
2,1, 1 2
0
1 1
cos
4 2
1 1
sin
8
1 1
sin
8
o
z
o
o
r
a
p
o
r
a i
p
o
r
a i
p
o
r
e
a a
r
e e
a a
r
e e
a a
2p Orbitals
( )
( )
ψ θ φ ψ ψ
π
ψ θ φ ψ ψ
π
−
+ −
−
+ −
   
= +   
   
   
= −   
   
3 2
2
2 2,1, 1 2,1, 1
0
3 2
2
2 2,1, 1 2,1, 1
0
1 1 1
sin cos =
32 2
1 1 1
sin sin =
32 2
o
x
o
y
r
a
p
o
r
a
p
o
r
e
a a
r
e
a a i
Linear
combination

Radial functions
ρ
ψ −
′=100
1s N e ( )
ρ
ψ ρ
−
′′= −200 2
2 2s N e
ρ =
0
r
a
ρ
ψ ρ θ
−
′′′=210 2
2 coszp N e
For s-Orbitals the maximum probability
denisty of finding the electron is on the nucleus
For s-Orbitals the probability of finding the
electron on the nucleus zero

Surface plots
Surface plot of the ΨΨΨΨ2s ; 2s wavefunction (orbital) of the hydrogen atom. The
height of any point on the surface above the xy plane (the nuclear plane)
represents the magnitude of the ΨΨΨΨ2s function at the at point (x,y) in the
nuclear plane. Note that there is a negative region (depression) about the
nucleus; the negative region begins at r=2a0 an goes asymptotically to zero at
r=∞∞∞∞.
Surface plot of the |ΨΨΨΨ2s|
2
; the probability density associated with
the 1s wavefunction of the hydrogen atom. Note that the negative
region of the 2s plot on the left now appears as positive region.
Surface plot of the 1s wavefunction (orbital) of the hydrogen atom. The height
of any point on the surface above the xy plane (the nuclear plane) represents
the magnitude of the ΨΨΨΨ1s function at the at point (x,y) in the nuclear plane.
The nucleus is located in the xy place immediately below the ‘peak’
Surface plot of the |ΨΨΨΨ1s|
2
; the probability density associated with
the 1s wavefunction of the hydrogen atom.
1s
2s
(1s)
2
(2s)
2

Surface plots
R(2pz)
(2pz)
2
Surface plot of radial portion of a 2p wavefunction of the hydrogen
atom. The gird lines have been left transparent so that the inner
‘hollow’ portion is visible.
Profile of the radial portion of a 2p wavefunction of the hydrogen atom.
Profile of the 2pz orbital along the z-azis.
Surface plot of the 2pz wavefunction (orbital)
in the xz (or yz) plane for the hydrogen atom.
The ‘pit’ represents the negative lobe and the
‘hill’ the positive lobe of a 2p orbital.
Surface plot of the (2pz)
2
; the probability density
associated with the 2pz wavefunction of the
hydrogen atom. Each of the hills represents and
area in the xz (or yz) plane where the probability
density is the highest, The probability density
along the x (or y) axis passing through the nucleus
(0,0) is everywhere zero.
2pz2pz
R(2pz)

Surface plots
Surface plot of the 3dz2 wavefunction (orbital) in the xz (or yz) plane for the
hydrogen atom. The large hills correspond to the positive lobes and the
small pits correspond to the negative lobes.
Surface plot of the (3dz2 )2 the probability density associated with the 3dz2
orbital of the hydrogen atom. This figure is rotated with respect to the
figure on the left so that the small hill will be clearly visible. Another
smaller hill is hidden behind the large hill.
Surface plot of the 3dxy wavefunction (orbital) in the xz plane for the
hydrogen atom. The hills and the pits have same amplitude. Surface plot of the (3dxy )2 the probability density associated with the 3dxy
orbital of the hydrogen atom. Pits in the figure to the left appear has hills.

Radial and Radial Distribution Functions
π
π π
→
→ →
2 2
2
2 2 2
Probability of finding the electron
anywhere in a shell of thickness
at radius is 4 ( ) (for )
increasing function
4 ( ) 0 as 4 0
nl
nl
dr r r R r dr s
r
r R r dr r dr

Radial Distribution Functions
π 2 2
4 ( )nlr R r
3s: n=3, l=0
Nodes=2
3p: n=3, l=1
Nodes=1
3d: n=3, l=2
Nodes=2
= Ψ Ψns nsr r
Number of radial nodes = n-l-1

Shapes and Symmetries of the Orbitals
s-Orbitals
ψ ψ
π π
− −     
= = −     
     
3 2 3 2
2
1 2
0
1 1 1 1
2
4 2
o o
r r
a a
s s
o o
r
e e
a a a
Function of only r; No angular dependence
⇒⇒⇒⇒Spherical symmetric
n-l-1=0
l=0
n-l=0
radial nodes
angular nodes
Total nodes
n-l-1=1
l=0
n-l=1

Shapes and Symmetries of the Orbitals
p-Orbitals
Function of only r , θθθθ (and φφφφ)
⇒⇒⇒⇒Not Spherical symmetric
2pz Orbital: No φφφφ dependence
⇒⇒⇒⇒Symmetric around z-axis
radial nodes
angular nodes
Total nodes
n-l-1=0
l=1
n-l=1
ψ ψ θ
π
−   
= =    
   
3 2
2
210 2
0
1 1
cos
4 2
o
z
r
a
p
o
r
e
a a
xy nodal plane
Zero amplitude at nucleus

Angular Distribution Functions
p-Orbitals
ψ ψ θ
π
−   
= = =   
   
3 2
2
210 2
0
1 1
cos 0 case
4 2
o
z
r
a
p
o
r
e m
a a
+
–
θθθθ cosθθθθ
0 1.000
30 0.866
60 0.500
90 0.000
120 -0.500
150 -0.866
180 -1.000
210 -0.866
240 -0.500
270 0.000
300 0.500
330 0.866
360 1.000
ρ
ψ ψ ρ θ
−
= = 2
210 2 coszp N e
Angular part: Polar plot of 2pz --- cosθ
x
z

p-Orbitals
ρ
ψ ψ ρ θ
−
= = 2
210 2 coszp N e
ρ
ρ
ρ
ψ ρ θ
ψ ρ θ φ
ψ ρ θ φ
−
−
−
=
=
=
2
2
2
2
2
2
cos
sin cos
sin sin
z
x
x
p
p
p
N e
N e
N e
Color/shading are related to
sign of the wavefunction

d-Orbitals
ρ
ρ
ρ
ρ
ρ
ψ ρ θ
ψ ρ θ θ φ
ψ ρ θ θ φ
ψ ρ θ φ
ψ ρ θ φ
−
−
−
−
−
−
= −
=
=
=
=
2
2 2
2 2 3
13
2 3
3 2
2 3
3 3
2 2 3
3 4
2 2 3
3 5
(3cos 1)
(sin cos cos )
(sin cos sin )
(sin cos2 )
(sin sin2 )
z
xz
yz
x y
xy
d
d
d
d
d
N e
N e
N e
N e
N e
Angular part
Blue: -ve
Yellow: +ve
Angular + Radial
n=3; l=2; m=0,±1, ±2

f-Orbitals
n=4; l=3; m=0,±1, ±2, ±3
Green: -ve
Red: +ve

Hydrogen atom & Orbitals
Hydrogen atom has only one electron, so why bother
about all these orbitals?
1. Excited states
2. Spectra
3. Many electron atoms

Many Electron Atoms
Helium is the simplest many electron atom
+
-
-
r1
r2
r12= r1- r2
πε
 
= − ∇ − ∇ − ∇ − + − 
 
2 22 2 2 2
2 2 2
1 2
0 1 2 12
1
2 2 2 4
N N
N
N e e
Z e Z e e
H
m m m r r r
KE of
Nucleus
KE of
Electron1
KE of
Electron2
Attraction between
nucleus and Electron1
Attraction between
nucleus and Electron1
Repulsion between
Electron1 and Electron2

Helium Atom
 
= − ∇ − ∇ − ∇ − + − 
 
= − ∇ − ∇ − − ∇ − + =
= − ∇ = − ∇ − − ∇ − +
=
2 22 2 2 2
2 2 2
1 2
0 1 2 12
2 22 2 2 2
2 2 2
1 2
1 2 12 0
2 22 2 2 2
2 2 2
1 2
1 2 12
1
2 2 2 4
1
;
2 2 2 4
2 2 2
N N
N
N e e
N N
N
N e e
N N
N eN
N e e
N eN n N
Z e Z e e
H
m m m r r r
QZ e QZ e Qe
H Q
m m r m r r
QZ e QZ e Qe
H H
m m r m r r
H E H
πε
πε
χ χ ψ =e e eE ψ

Helium Atom
= − ∇ − − ∇ − +
= + +
= − ∇ − = − ∇ −
2 22 2 2
2 2
1 2
1 2 12
2
1 2
12
2 22 2
2 2
1 21 2
1 2
2 2
and
2 2
N N
e
e e
e
N N
e e
QZ e QZ e Qe
H
m r m r r
Qe
H H H
r
QZ e QZ e
H H
m r m r
The Hamiltonians Ĥ1 and Ĥ1 are one electron
Hamiltonians similar to that of hydrogen atom
= +
+
1 21 1 1 2 2 2 1 1 1 2 2 2 1 1 1 2 2 2
2
1 1 1 2 2 2
12
( , , , , , ) ( , , , , , ) ( , , , , , )
( , , , , , )
e e e e
e
H r r H r r H r r
Qe
r r
r
ψ θ φ θ φ ψ θ φ θ φ ψ θ φ θ φ
ψ θ φ θ φ

Orbital Approximation
ψ θ φ θ φ ψ θ φ ψ θ φ=1 1 1 2 2 2 1 1 1 1 2 2 2 2( , , , , , ) ( , , ) ( , , )e e er r r r
ψ φ φ φ φ≈ ⋅ ⋅ ⋅⋅⋅⋅⋅(1,2,3,... ) (1) (2) (3) ( )e n n
Orbital is a one electron wavefunction
The total electronic wavefunction of n number of
electrons can be written as a product of n one electron
wavefunctions

= +
+
1 21 1 1 2 2 2 1 1 1 2 2 2 1 1 1 2 2 2
2
1 1 1 2 2 2
12
( , , , , , ) ( , , , , , ) ( , , , , , )
( , , , , , )
e e e e
e
H r r H r r H r r
Qe
r r
r
ψ θ φ θ φ ψ θ φ θ φ ψ θ φ θ φ
ψ θ φ θ φ
ψ θ φ θ φ ψ θ φ ψ θ φ=1 1 1 2 2 2 1 1 1 1 2 2 2 2( , , , , , ) ( , , ) ( , , )e e er r r r
= +
+
1 21 1 1 1 2 2 2 2 1 1 1 1 2 2 2 2
2
1 1 1 1 2 2 2 2
12
( , , ) ( , , ) ( , , ) ( , , )
( , , ) ( , , )
e e e e e e
e e
H H r r H r r
Qe
r r
r
ψ ψ θ φ ψ θ φ ψ θ φ ψ θ φ
ψ θ φ ψ θ φ
Helium Atom: Orbital Approximation

ψ ψ θ φ ψ θ φ ψ θ φ ψ θ φ
ψ θ φ ψ θ φ
= +
+
1 21 1 1 1 2 2 2 2 1 1 1 1 2 2 2 2
2
1 1 1 1 2 2 2 2
12
( , , ) ( , , ) ( , , ) ( , , )
( , , ) ( , , )
e e e e e e
e e
H H r r H r r
Qe
r r
r
ψ ε ψ θ φ ψ θ φ ε ψ θ φ ψ θ φ
ψ θ φ ψ θ φ
= +
+
1 1 1 1 1 2 2 2 2 2 1 1 1 1 2 2 2 2
2
1 1 1 1 2 2 2 2
12
( , , ) ( , , ) ( , , ) ( , , )
( , , ) ( , , )
e e e e e e
e e
H r r r r
Qe
r r
r
[ ]ψ ε ε ψ θ φ ψ θ φ
 
= + + 
 
2
1 2 1 1 1 1 2 2 2 2
12
( , , ) ( , , )e e e e
Qe
H r r
r
µ
ε ε
ε
−
= = − =
2 4 2
1 2 2 2 2 2
0
13.6
8
Z e Z
eV
h n n

[ ]ψ ε ε ψ θ φ ψ θ φ
 
= + + 
 
2
1 2 1 1 1 1 2 2 2 2
12
( , , ) ( , , )e e e e
Qe
H r r
r
If we ignore the term
2
12
Qe
r
( )[ ]ψ ε ε ψ θ φ ψ θ φ= +1 2 1 1 1 1 2 2 2 2( , , ) ( , , )e e e eH r r
ε ε+ = −He 1 2E = 108.8eV
ψ ψ ψ
π π
− −      
   = ⋅ = ⋅   
         
3 2 3 2
1 1
1 1
(1) (2)o o
Zr Zr
a a
e s s
o o
Z Z
e e
a a

ε ε+ = − + = −
= − + = −
He 1 2
He
E = (54.4 54.4) 108.8
E (24.59 54.4) 78.99 (Experimental)
eV eV
eV eV
Ignoring is not justified! Need better approximation
2
12
Qe
r

Many Electron Atoms
α = = = >
= − ∇ − ∇ − +∑ ∑ ∑∑
2 2
2 2 2 2
1 1 1
1 1
2 2
n n n n
N i N
i i i j ie i ij
H QZ e Qe
m m r r
Nuclei are static
= = = >
= = >
= − ∇ − +
= +
∑ ∑ ∑∑
∑ ∑∑
2
2 2 2
1 1 1
2
1 1
1 1
2
1
n n n n
e i N
i i i j ie i ij
n n n
e i
i i j i ij
H QZ e Qe
m r r
H H Qe
r
Inter-electron repulsion term
leads to deviation from the
hydrogen atom. Unfortunately
CANNOT be ignored
    
≠ +       
    
1 1 1
ij i j
f g h
r r r
term in the Hamiltonian is not separable
1
ijr

Many Electron Atoms
= = = >
= = >
= − ∇ − +
= +
∑ ∑ ∑∑
∑ ∑∑
2
2 2 2
1 1 1
2
1 1
1 1
2
1
n n n n
e i N
i i i j ie i ij
n n n
e i
i i j i ij
H QZ e Qe
m r r
H H Qe
r
Hamiltonian is no longer spherically symmetric and
the Time-Independent Schrodinger Equation
(TISE) cannot be solved using analytical techniques
Numerical methods must be used solve the TISE

Many Electron Atoms: Orbital Approximation
He atom result indicate that neglecting the inter-electron
interaction is not a good idea
Improvement
The term in the Hamiltonian represents the interaction
between the electrons. Which mean the electron move in the
potential provided by the nucleus and rest of the electron.
Since the electron and nucleus have opposite charges, it can
be thought that the rest of the electrons reduce the charge felt
by a particular electron ⇒ Shielding
1
ijr
ψ φ φ φ φ≈ ⋅ ⋅ ⋅⋅⋅⋅⋅(1,2,3,... ) (1) (2) (3) ( )e n n

Effective Nuclear Charge
= = = >
= =
= − ∇ − +
= − ∇ −
∑ ∑ ∑∑
∑ ∑
2
2 2 2
1 1 1
2
2 2
1 1
1 1
2
1
2
n n n n
e i N
i i i j ie i ij
n n
eff
e i N
i ie i
H QZ e Qe
m r r
H QZ e
m r
Effective Nuclear Charge Zeff
For Helium atom
ψ ψ ψ
π π
− −      
    ′ ′= ⋅ = ⋅   
         
3 2 3 2
1 1
1 1
(1) (2)
eff eff
o o
Z r Z r
eff effa a
e s s
o o
Z Z
e e
a a
= − ∇ − ∇ − −
2 22 2
2 2
1 2
1 22 2
eff eff
N N
e
e N
QZ e QZ e
H
m m r r

Due to Shielding, the electrons do not see the full nuclear
charge Z, but Zeff = Z–σσσσ (σ = Shielding Constant)
( )
σ
σ
=
= −
 −
= ⋅  
 
= ⋅ +
∑
2
1
2 2
eff
N
i
Hatom
i i
He Hatom eff eff
Z Z
Z
E E
N
E E Z Z
For Helium atom
σ= − =
=
1.69
1
effZ Z
n
=
 
= ⋅  
 
− = −
−
∑
2
2
1
13.6 5.712 77.68
Compare with 78.99
eff
He Hatom
i i
Z
E E
n
X eV
There are methods such as
Perturbation Theory and
Variational Method to
estimate Zeff

Due to Shielding, the electrons do not see the full nuclear
charge Z, but Zeff = Z–σσσσ (σ = Shielding Constant)
σ
σ
=
= −
 −
= ⋅  
 
∑
2
1
eff
N
i
Hatom
i i
Z Z
Z
E E
N
Effective nuclear charge is same for electrons in the same
orbital, but varies greatly for electrons of different orbitals
(s,p,d,f) and n.
Zeff determines chemical properties of many electron atoms

Building-up (Aufbau) Principle
Effective nuclear charge varies for electrons of different
orbitals. Different orbitals corresponding to same n. are
no longer degenerate
How do we get 2p energy higher than 2s?
How does Radial distributions change?
How does Zeff affect atomic properties?

Orbital Angular Momentum
e-x
y
z
L
=L r X p
Orbital Angular Momentum ‘L’
= +
=
≤ −
=
= ± ± ± ±
( 1)
orbital angular momentum quantum number
1
0, 1, 2, 3,....,
z
L l l
l
l n
L m
m l

Spin Angular Momentum
Stern-Gerlach Experiment A beam of silver atoms
(4d
10
5s
1
) thorough an inhomogeneous magnetic field
and observer that the beam split into two of quantized
components
Classical, "spinning" particles,
would have truly random
distribution of their spin
angular momentum vectors.
This would produce an even
distribution on screen.
But electrons are deflected
either up or down by a specific
amount.
Uhlenbeck-Goudsmit
Suggested intrinsic spin
angular momentum for
electrons

Spin Angular Momentum
Spin Angular Momentum ‘S’
’ = +
=
=
=
=
( 1)
orbital angular momentum quantum number
1 2
1 2
z s
s
S s s
s
s
L m
m
Electrons are spin-1⁄2 particles. Only two possible spin angular
momentum values. “spin-up” (or α) and “spin-down” (or β)
The exact value in the z direction is ms= +ħ/2 or −ħ/2
Not a result of the rotating particles, otherwise would be
spinning impossibly fast (GREATER THAN SPEED OF LIGHT)
Spin S(ω) where ω is an unknown coordinate

Hydrogen Atom Wavefunctions: Redefined
Incorporate “spin” component to each of the 1-electron
wavefunctions. Each level is now doubly degenerate
1-Electron wavefunctions are now called SPIN ORBITALS
Total wavefunctions is a product of spatial and spin parts
H-atom wavefunctions now can be written as
Which are orthogonal and normalized. Quantum
numbers are n,l,m,ms
θ φ ω ψ θ φ ω α ω ψ θ φ ω β ω
ψ α ψ β
π π
− −
−
Ψ = ⋅ ⋅
   
= =   
   
3 2 3 2
1 11,0,0, 1,0,0,
2 2
( , , , ) ( , , , ) ( ) or ( , , , ) ( )
1 1 1 1o o
r r
a a
o o
r r r
e e
a a

Spin Orbitals and Exclusion Principle
Spin should always be included for
systems with more than one electron
Two electron wavefunctions should include four spin functions
The last two wavefunctions are strictly not allowed because the
two electron can be distinguished.
α α β β α β β α(1) (2) (1) (2) (1) (2) (1) (2)
Indistinguishability
Exchange Operator
Ψ = ±Ψ(1,2) (2,1)
[ ]
[ ]
α α β β α β β α
α β β α
+
−
1
(1) (2) (1) (2) (1) (2) (1) (2)
2
1
(1) (2) (1) (2)
2
Symmetric
Anti-symmetric

He atom wavefunction
Spin Orbitals and Exclusion Principle
Ψ = −Ψ(1,2) (2,1)
[ ]ψ ψ ψ α β β α= ⋅ −1 1
1
(1) (2) (1) (2) (1) (2)
2
He s s
Pauli’s Exclusion Principle (by Dirac!)
The complete wavefunction (both spin and spatial
coordinates) of a system of identical fermions (i.e. electrons)
must be anti-symmetric with respect to interchange of all
their coordinates (spatial and spin) of any two particles
If the two electrons in 1s orbital had same spin then the
wavefunction would be symmetric and hence it is not allowed

Helium Atom: Excited States
[ ] [ ]
α α
α β β α
β β
 = =


⋅ − ⋅ + = =

 = = −
(1) (2) ( 1; 1)
1 1
1 (1) 2 (2) 1 (2) 2 (1) (1) (2) (1) (2) ( 1; 0)
2 2
(1) (2) ( 1; 1)
s
s
s
s m
s s s s s m
s m
[ ][ ]α β β α⋅ + ⋅ − = =
1 1
1 (1) 2 (2) 1 (2) 2 (1) (1) (2) (1) (2) ( 0; 0)
2 2
ss s s s s m
If the second electron is in the 2s orbital then it could have
the same spin or the opposite spin.
He excited state 1s
1
.2s
1
(triplet)
He excited state 1s
1
.2s
1
(singlet)

1s (1)1s (2) The spatial part is symmetric
1s (1)2s (2) or 1s (2)2s (1) symmetric nor anti-symmetric
1s (1)2s (2) + 1s (2)2s (1) Symmetric
1s (1)2s (2) - 1s (2)2s (1) Anti-symmetric
[ ] [ ]
α α
α β β α
β β
 = =


⋅ − ⋅ + = =

 = = −
(1) (2) ( 1; 1)
1 1
1 (1) 2 (2) 1 (2) 2 (1) (1) (2) (1) (2) ( 1; 0)
2 2
(1) (2) ( 1; 1)
s
s
s
s m
s s s s s m
s m
[ ][ ]α β β α⋅ + ⋅ − = =
1 1
1 (1) 2 (2) 1 (2) 2 (1) (1) (2) (1) (2) ( 0; 0)
2 2
ss s s s s m
Helium Atom: Excited States

Helium Atom
1s (1)1s (2) The spatial part is symmetric
1s (1)2s (2) or 1s (2)2s (1) symmetric nor anti-symmetric
1s (1)2s (2) + 1s (2)2s (1) Symmetric
1s (1)2s (2) - 1s (2)2s (1) Anti-symmetric
[ ] [ ]
α α
α β β α
β β
 = =


⋅ − ⋅ + = =

 = = −
(1) (2) ( 1; 1)
1 1
1 (1) 2 (2) 1 (2) 2 (1) (1) (2) (1) (2) ( 1; 0)
2 2
(1) (2) ( 1; 1)
s
s
s
s m
s s s s s m
s m
[ ][ ]α β β α⋅ + ⋅ − = =
1 1
1 (1) 2 (2) 1 (2) 2 (1) (1) (2) (1) (2) ( 0; 0)
2 2
ss s s s s m
Homework – Write the correct wavefunctions

Bonding: H2
+
and H2 molecules
+
-
+
R
HA HB
rA rB
r
e-
+
-
+
RHA HB
r1A r1B
r1
e-
-
e-
r2Br2A
r2
( )+
= − ∇ − ∇ − ∇
− − +
2 2 2
2 2 2
2
2 2 2
2 2 2
A B e
A B e
A B
H H
m m m
e e e
Q Q Q
r r R
( ) = − ∇ − ∇
− ∇ − ∇
− − −
+ +
2 2
2 2
2
2 2
2 2
1 2
2 2 2 2
1 1 2 2
2 2
12
2 2
2 2
A B
A B
e e
e e
A B A B
H H
m m
m m
e e e e
Q Q Q Q
r r r r
e e
Q Q
r R

Born – Oppenheimer Approximation
( )+
= − ∇ − ∇ − ∇ − − +
2 2 2 2 2 2
2 2 2
2
2 2 2
A B e
A B e A B
e e e
H H Q Q Q
m m m r r R
Nuclei are STATIONARY with respect to electrons
( )+
= − ∇ − − +
2 2 2 2
2
2
2
e
e A B
e e e
H H Q Q Q
m r r R
( )+
= − ∇ − ∇ − ∇ − − +
2 2 2 2 2 2
2 2 2
2
2 2 2
A B e
A B e A B
e e e
H H Q Q Q
m m m r r R
ignore

Born – Oppenheimer Approximation
( ) = − ∇ − ∇ − ∇ − ∇
− − − + +
2 2 2 2
2 2 2 2
2 1 2
2 2 2 2 2 2
1 1 2 2 12
2 2 2 2
A B e e
A B e e
A B A B
H H
m m m m
e e e e e e
Q Q Q Q Q Q
r r r r r R
( ) = − ∇ − ∇ − − − + +
2 2 2 2 2 2 2 2
2 2
2 1 2
1 1 2 2 122 2
e e
e e A B A B
e e e e e e
H H Q Q Q Q Q Q
m m r r r r r R
ignore

Bonding: H2
+
Molecule
( )+
= − ∇ − − +
2 2 2 2
2
2
2
e
e A B
e e e
H H Q Q Q
m r r R
( ) ψ ψ+
⋅ = ⋅2 ( , ) ( ) ( , )H H r R E R r R
Difficult; but can be solved using elliptical polar co-ordinates

Bonding: H2 molecule
( ) ψ ψ⋅ = ⋅2 ( , ) ( ) ( , )H H r R E R r R
CANNOT be Solved
( ) = − ∇ − ∇ − − − + +
2 2 2 2 2 2 2 2
2 2
2 1 2
1 1 2 2 122 2
e e
e e A B A B
e e e e e e
H H Q Q Q Q Q Q
m m r r r r r R
For all the molecules except the simplest
molecule H2+ the Schrodinger equation cannot be
solved.
We have approximate solutions

Bonding
For all the molecules except the simplest
molecule H2+ the Schrodinger equation cannot be
solved.
We have only approximate solutions
Valance-Bond Theory
&
Molecular Orbital Theory
are two different models that solve the
Schrodinger equation in different methods

Valance Bond Theory
ψ ψΨ = ⋅(1) (2)A B
ψ ψ ψ ψΨ = ⋅ + ⋅(1) (2) (2) (1)A B A B
ψ (1)A
ψ (2)B
R=∞ R= Re
( ) ( )ψ ψ ψ ψ λ ψ ψ λ ψ ψ
λ λ+ − − +
Ψ = ⋅ + ⋅ + ⋅ + ⋅
Ψ = Ψ + Ψ + Ψ
(1) (2) (2) (1) (1) (2) (1) (2)
cov
A B A B A A B B
H H H H
Resonance
H−−−−−−−−H ←→←→←→←→ H
+
−−−−−−−−H
−−−−
←→←→←→←→ H
−−−−
−−−−−−−−H
+
Inclusion of Ionic terms

Valance Bond Theory
R= Re
λ λ+ − − +Ψ = Ψ + Ψ + Ψcov H H H H

+
-
+
R
HA HB
rA rB
r
e-
A molecular orbital is analogous concept to atomic orbital but
spreads throughout the molecule
It’s a polycentric one-electron wavefunction (Orbital!)
It can be produced by Linear Combination of Atomic Orbitals
LCAO-MO
ψ ψ
 
− ∇ − − + = ⋅ 
 
2 2 2 2
2
2
e
e A B
e e e
Q Q Q E
m r r R
Molecular Orbital Theory of H2
+

+
-
+
R
HA HB
rA rB
r
e-
ψ ψ
 
− ∇ − − + = ⋅ 
 
2 2 2 2
2
2
e
e A B
e e e
Q Q Q E
m r r R
LCAO-MO
ψ φ φ= +1 1 2 1A BMO s sC C
( )ψ φ φ φ φ= + +
2 2 2 2 2
1 1 2 1 1 2 1 12A B A BMO s s s sC C C C
= ⇒ = ±2 2
1 2 1 2
Symmetry requirement
C C C C
+

+
-
+
R
HA HB
rA rB
r
e-
= ⇒ = ±2 2
1 2 1 2
Symmetry requirement
C C C C
( ) ( )ψ φ φ
= =
= + = +
1 2
1 1 1 1 1A B
a
a s s a A B
C C C
C C s s ( ) ( )ψ φ φ
= − =
= − = −
1 2
2 1 1 1 1A B
b
b s s b A B
C C C
C C s s
+ +
( )ψ = +1 1Bonding a A BC s s
+ -
( )ψ − = −1 1Anti bonding b A BC s s
+

( )ψ = +1 1Bonding a A BC s s ( )ψ − = −1 1Anti bonding b A BC s s
+

Bracket Notation
φ φ τ φ φ δ
φ φ τ φ φ
∗
∗
= =
= =
∫
∫
i
i
j i j ij
allspace
j i j ij
allspace
d
A d A A
= =
= ≠
1 (for )
0 (for )
i j
i j

Normalization
( )( )
[ ]
[ ]
[ ]
ψ ψ φ φ φ φ
φ φ φ φ φ φ φ φ
= = + +
 = + + +
 
= +
=
+
=
−
2
1 1 1 1 1 1
2
1 1 1 1 1 1 1 1
2
1
1
1 2 2
1
2 2
Similarly
1
2 2
A B A B
A A B B A B B A
a s s s s
a s s s s s s s s
a
a
b
C
C
C S
C
S
C
S
1
S is called Overlap-Integral
φ φ φ φ
φ φ φ φ
= =
= =
1 1 1 1
1 1 1 1
1A A B B
A B B A
s s s s
s s s sS

Overlap Integral
Overlap-Integral S can be positive or negative or zero

[ ]
( )
[ ]
( )
ψ φ φ
ψ φ φ
= +
+
= −
−
1 1 1
2 1 1
1
2 2
1
2 2
A B
A B
s s
s s
S
S
ψ ψ
ψ ψ
=
=
1 1 1
1 2 2
E H
E H
+

[ ]
( )
[ ]
( )
[ ]
( ) ( )
[ ]
ψ ψ
φ φ φ φ
φ φ φ φ
=
= + +
+ +
= + +
+
= + + +
+
1 1 1
1 1 1 1 1
1 1 1 1 1
1 1 1 1 1 1 1 1 1
1 1
2 2 2 2
1
2 2
1
2 2
A B A B
A B A B
A A B B A B B A
s s s s
s s s s
s s s s s s s s
E H
E H
S S
E H
S
E H H H H
S
+

[ ]
( )
[ ]
( )
[ ]
( ) ( )
[ ]
ψ ψ
φ φ φ φ
φ φ φ φ
=
= − −
− +
= − −
−
= + − −
−
2 2 2
2 1 1 1 1
2 1 1 1 1
2 1 1 1 1 1 1 1 1
1 1
2 2 2 2
1
2 2
1
2 2
A B A B
A B A B
A A B B A B B A
s s s s
s s s s
s s s s s s s s
E H
E H
S S
E H
S
E H H H H
S
+

[ ]
( )
[ ]
( )
ψ φ φ
ψ φ φ
= +
+
= −
−
1 1 1
2 1 1
1
2 2
1
2 2
A B
A B
s s
s s
S
S
ψ ψ ψ ψ= =1 1 1 1 2 2E H E H
+
[ ]
φ φ φ φ φ φ φ φ= + − −
−
2 1 1 1 1 1 1 1 1
1
2 2 A A B B A B B As s s s s s s sE H H H H
S
[ ]
φ φ φ φ φ φ φ φ= + + +
+
1 1 1 1 1 1 1 1 1
1
S

[ ]
φ φ φ φ φ φ φ φ= + + +
+
1 1 1 1 1 1 1 1 1
1
S
+
φ φ φ φ
φ φ φ φ
φ φ φ φ
= = =
= = =
= = =
1 1 1 1
1 1 1 1
1 1 1 1
i i j j
i j j i
i j j i
s s ii jj s s
s s ij ji s s
s s ij ji s s
H H H H
H H H H
S S
Ĥ is Hermitian
[ ]
φ φ φ φ φ φ φ φ= + − −
−
2 1 1 1 1 1 1 1 1
1
S
+ +
= =
   + +   
− −
= =
   − −   
1
2
2 2
2 2 1
2 2
2 2 1
ii ij ii ij
ij ij
ii ij ii ij
ij ij
H H H H
E
S S
H H H H
E
S S

+
+ +
= =
   + +   
− −
= =
   − −   
1
2
2 2
2 2 1
2 2
2 2 1
ii ij ii ij
ij ij
ii ij ii ij
ij ij
H H H H
E
S S
H H H H
E
S S

+
= − ∇ − − +
 
= − ∇ − − + 
 
= − +
2 2 2 2
2
2 2 2 2
2
2 2
1
2
2
e
e A B
e
e A B
e
B
e e e
H Q Q Q
m r r R
e e e
H Q Q Q
m r r R
e e
H H Q Q
r R
φ φ
φ φ φ φ φ φ
= =
= + −
1 1
2 2
11 1 1 1 1 1
(or )
1 1
i i
i i i i i i
ii AA BB s s
es s s s s s
B
H H H H
H Qe Qe
R r

+
φ φ
φ φ φ φ φ φ
φ φ φ φ φ φ
= =
= + −
= + −
= + − ⋅
1 1
2 2
11 1 1 1 1 1
2
2
11 1 1 1 1 1
2
2
1
(or )
1 1
1
i i
i i i i i i
i i i i i i
ii AA BB s s
eii s s s s s s
B
eii s s s s s s
B
ii s
H H H H
H H Qe Qe
R r
Qe
H H Qe
R r
Qe
H E Qe J
R
φ φ
φ φ
=
=
1 1
1 1
1
1
i i
i i
s s
s s
B
J
r
Constant
at Fixed
Nuclear
Distance
J ⇒⇒⇒⇒ Coulomb Integral

+
φ φ
φ φ φ φ φ φ
φ φ φ φ φ φ
= =
= + −
= + −
= + − ⋅
1 1
2 2
11 1 1 1 1 1
2
2
11 1 1 1 1 1
2
2
1
(or )
1 1
1
i j
i j i j i j
i j i j i j
ij AB BA s s
eij s s s s s s
B
eij s s s s s s
B
ij s
H H H H
H H Qe Qe
R r
Qe
H H Qe
R r
Qe
H E S S Qe K
R
φ φ
φ φ
=
=
1 1
1 1
1
i j
i j
s s
s s
B
S
K
r
K ⇒⇒⇒⇒ Exchange Integral
Resonance Integral
Constant
K is purely a quantum mechanical
concept. There is no classical
counterpart

+
[ ]
[ ]
[ ] [ ] [ ]
[ ]
[ ]
[ ]
[ ]
[ ]
+     
= = + − + + −    + +      
 
= + + + − + 
+  
+
= + −
+
−     
= = + − − − −    − −      
= −
−
2 2
1 1 1
2
2
1 1
22
1 1
2 2
2 1 1
2 1
1 1
11
1
1 1
1
1
1 1
11
1
1
1
ii ij
s s
ij
s
s
ii ij
s s
ij
s
H H S
E E Qe J E S Qe K
S R RS
Qe
E E S S Qe J K
S R
Qe J KQe
E E
R S
H H S
E E Qe J E S Qe K
S R RS
E E S
S
[ ] [ ]
[ ]
[ ]
 
+ − − − 
 
−
= + −
−
2
2
22
2 1
1
1
s
Qe
S Qe J K
R
Qe J KQe
E E
R S

+
[ ]
( )
[ ]
[ ]
[ ]
( )
[ ]
[ ]
ψ φ φ
ψ φ φ
= +
+
+
= + −
+
= −
−
−
= + −
−
1 1 1
22
1 1
2 1 1
22
2 1
1
2 2
1
1
2 2
1
A B
A B
s s
s
s s
s
S
Qe J KQe
E E
R S
S
Qe J KQe
E E
R S

+
[ ]
[ ]
[ ]
[ ]
+
= + −
+
−
= + −
−
≤ ≤ < <
22
1 1
22
2 1
1
1
0 1; 0& 0
s
s
Qe J KQe
E E
R S
Qe J KQe
E E
R S
S J K
Destabilization of Anti-bonding orbital is more than
Stabilization of Bonding orbital
J - Coulomb integral -
interaction of electron
in 1s orbital around A
with a nucleus at B
K - Exchange integral
– exchange (resonance)
of electron between the
two nuclei.

+
E1
1E2

Sigma Bonding with 1s Orbitals

Bonding with 2p Orbitals
Note the signs, symmetries and nodes

Sigma Bonding with 2p Orbitals

Pi Bonding with 2p Orbitals
Note the signs, symmetries and nodes

Symmetry of Orbitals
Hydrogen molecule ion:
Bonding: Symmetric
→ σg
Anti-bonding: Antisymmetric
→ σu
Gerade (g) → Symmetric
Ungarade (u) →
Antisymmetric

( ) = − ∇ − ∇ − ∇ − ∇
− − − + +
2 2 2 2
2 2 2 2
2 1 2
2 2 2 2 2 2
1 1 2 2 12
2 2 2 2
A B e e
A B e e
A B A B
H H
m m m m
e e e e e e
Q Q Q Q Q Q
r r r r r R
( )
( )
( ) ( ) ( )
= − ∇ − ∇ − − − + +
   
= − ∇ + − ∇ + − + +   
   
= + − − + +
2 2 2 2 2 2 2 2
2 2
2 1 2
1 1 2 2 12
2 2 2 2 2 2 2 2
2 2
2 1 2
1 2 1 2 12
2 2 2 2
2 1 2
1 2 12
2 2
2 2
e e
e e A B A B
e e
e A e B B A
e e
B A
e e e e e e
H H Q Q Q Q Q Q
m m r r r r r R
e e e e e e
H H Q Q Q Q Q Q
m r m r r r r R
e e e e
H H H H H H Q Q Q Q
r r r R
ignore
Cannot be Solved

For H2
+
[ ]
( )ψ ψ φ φ= = +
+
1 1 1
1
2 2
A Bbonding s s
S
Place the second electron in the bonding orbital to get H2
[ ]
( )
[ ]
( ) [ ]
ψ ψ ψ
φ φ φ φ α β β α
= ⋅
       = + ⋅ + −     + +   
2 1 2
1 1 2 2
1 1 1 1
( )
1 1 1
(1) (2) (1) (2)
22 2 2 2
A B A B
bonding
s s s s
H
S S

[ ]
( ) ( ) [ ]
ψ
    = + ⋅ + −    +  
2
1 1 2 2
1 1 1 1
( )
1 1
(1) (2) (1) (2)
2 1 2A B A B
bonding
s s s s
H
S
[ ]
[ ]
[ ]
ψ φ φ φ φ φ φ φ φ = + + + +
⋅ + ⋅ + ⋅ + ⋅
+
1 2 1 2 1 2 1 2
1 1 1 1 1 1 1 1
1
2 1
1
1 (1) 1 (2) 1 (1) 1 (2) 1 (1) 1 (2) 1 (1) 1 (2)
2 1
A A B B A B B Abonding s s s s s s s s
A A B B A B B A
S
s s s s s s s s
S
Spatial Part

[ ]
( ) ( ) [ ]
ψ
−
    = − ⋅ − −    −  
2
1 1 2 2
1 1 1 1
( )
1 1
(1) (2) (1) (2)
2 1 2A B A B
anti bonding
s s s s
H
S
[ ]
[ ]
[ ]
ψ φ φ φ φ φ φ φ φ−
 = + − − −
⋅ + ⋅ − ⋅ − ⋅
−
1 2 1 2 1 2 1 2
1 1 1 1 1 1 1 1
1
2 1
1
1 (1) 1 (2) 1 (1) 1 (2) 1 (1) 1 (2) 1 (1) 1 (2)
2 1
A A B B A B B Aanti bonding s s s s s s s s
A A B B A B B A
S
s s s s s s s s
S
Spatial Part

[ ]
[ ]
[ ]
ψ φ φ φ φ φ φ φ φ = + + + +
⋅ + ⋅ + ⋅ + ⋅
+
1 2 1 2 1 2 1 2
1 1 1 1 1 1 1 1
1
2 1
1
1 (1) 1 (2) 1 (1) 1 (2) 1 (1) 1 (2) 1 (1) 1 (2)
2 1
A A B B A B B Abonding s s s s s s s s
A A B B A B B A
S
s s s s s s s s
S
[ ]
[ ]
[ ]
ψ φ φ φ φ φ φ φ φ−
 = + − − −
⋅ + ⋅ − ⋅ − ⋅
−
1 2 1 2 1 2 1 2
1 1 1 1 1 1 1 1
1
2 1
1
1 (1) 1 (2) 1 (1) 1 (2) 1 (1) 1 (2) 1 (1) 1 (2)
2 1
A A B B A B B Aanti bonding s s s s s s s s
A A B B A B B A
S
s s s s s s s s
S

Effective nuclear charge changes the absolute energy
Levels and the size of orbitals!
Matching of energies of AOs important for LCAO-MO
If energies are not close to each other, they would
Not interact to form MOs.

Diatoms of First Row: H2
+
, H2 ,He2, He2
+

Effective nuclear charge changes the absolute energy
levels and the size of orbitals!
Matching of energies of AOs important for LCAO-MO, if
the energies of two Aos are not close they will not interact
to form MOs.

Matching of AO energies for MO
Due to large difference in energy of 1s(H) and 1s(F),
LCAO-MO for both 1S is not feasible in HF.
Rather, 2pz(F) and 1s(H) form a sigma bond.
Both symmetry and energy
Matching is required for MO.
Valence electrons are most important

Bonding in First-Row Homo-Diatomic Molecules
1s
2s
2p
1s
2s
2p
The orbital energies of the two approaching atoms are
identical before they start interacting to form a BOND

1s
2s
2p
1s
2s
2p
1σ
1σ*
2σ
2σ*
3σ
3σ*
1π
1π*
The interaction between the energy and symmetry
matched orbitals leads to various types of BONDs

MO Energies of Dinitrogen
Experiments tell us this picture is incorrect!

1s
2s
2p
1s
2s
2p
The 2s and 2p orbitals are degenerate in Hydrogen.
However in the many electron atoms these two sets of
orbitals are no longer degenerate.

1s
2s
2p
1s
2s
2p
The difference in the energies of the 2s and 2p orbitals
increases along the period. Its is minimum for Li and
maximum for Ne

MO Energies of Dinitrogen
Mixing of 2s and 2p orbital occur because of small energy gap between them
2s and 2p electrons feels not so different nuclear charge.
Note how the MO of 2s→σ have p-type looks, while π-levels are clean

s-p Mixing: Hybridization of MO
Mixing of 2s and 2p orbital occur because of small
energy gap between them 2s and 2p electrons feels not
so different nuclear charge

s-p Mixing: Hybridization of MO
B2 is paramagnetic. This can only happen if the two
electrons with parallel spin are placed in the degenerate
π-orbitals and if π orbitals are energetically lower than
the σ orbital
Incorrect!

MO diagram of F2: No s-p Mixing
No Mixing of s and p orbital because of higher energy
Gap between 2s and 2p levels in Oxygen and Fluorine!
2s and 2p electrons feels very different nuclear charge

MO Energy Level Diagram for Homo-Diatomics
Upto N2 Beyond N2

Bond-Order and Other Properties
N2 : (1σg)
2
(1σ*u)
2
(2σg)
2
(2σ*u)
2
(1πux)
2
(1πuy)
2
(3σg)
2
BO = 3
All spins paired: diamagnetic
O2 : (1σg)
2
(1σ*u)
2
(2σg)
2
(2σ*u)
2
(3σg)
2
(1πux)
2
(1πuy)
2
(1πux)
1
(1π*uy)
1
BO = 2
2 spins unpaired: paramagnetic

MO Contours and Electron Density

Hetero-Diatomics: HF
Due to higher electronegativity
of F than H, the electron
distribution is lopsided

Hybridization
Linear combination of atomic orbitals within
an atom leading to more effective bonding
2s
2pz 2px 2py 2px 2py
αααα 2s-ββββ 2pz
αααα 2s+ββββ 2pz
The coefficients αααα and ββββ depend on field strength
Hybridization is close to VBT approach. Use of experimental information
All hybridized orbitals are equivalent and are ortho-normal to each other

Contribution from s=0.5; contribution from p=0.5
Have to normalize each hybridized orbital
 = − 1
1
2
s pψ ψ ψ
 = + 2
1
2
s pψ ψ ψ
2 equivalent hybrid orbitals
of the same energy and
shape (directions different)
Linear geometry with
Hybridized atom at the center
s and p orbital of the Same atom!
Not same as S (overlap)
s+p (sp)Hybridization

The other p-orbitals are
available for π bonding
s+p (sp)Hybridization

1
2
3
1 2
0
33
1 1 1
3 2 6
1 1 1
3 2 6
s px py
s px py
s px py
ψ ψ ψ ψ
ψ ψ ψ ψ
ψ ψ ψ ψ
= + ⋅ +
= + −
= − −
The other p-orbital are
s+2p (sp2)Hybridization

1
2
3
4
1 1 1 1
2 2 2 2
1 1 1 1
2 2 2 2
1 1 1 1
2 2 2 2
1 1 1 1
2 2 2 2
s px py pz
s px py pz
s px py pz
s px py pz
ψ ψ ψ ψ ψ
ψ ψ ψ ψ ψ
ψ ψ ψ ψ ψ
ψ ψ ψ ψ ψ
= + + +
= − − +
= + − −
= − + −
How to calculate the coefficients?
Use orthogonality of hybrid orbitals
and normalization conditions
There is no unique solution
1
2
3
4
1 3
0 0
2 2
1 2 1
0
2 3 2 3
1 1 1 1
2 6 2 2 3
1 1 1 1
2 6 2 2 3
s px py pz
s px py pz
s px py pz
s px py pz
ψ ψ ψ ψ ψ
ψ ψ ψ ψ ψ
ψ ψ ψ ψ ψ
ψ ψ ψ ψ ψ
= + ⋅ + ⋅ +
= + + ⋅ −
= − + −
= − − −

No other p-orbital is

s-p3-d2 & s-p3-d Hybridization
Sp3d2
Octahedral
Sp3d
Trignoal bipyramidal

Do Orbitals Really Exist?
Tomographic image of HOMO of N2
Nature; Volume 342; Year 2004; 867-871

Collaboration between Chemists and Engineers
The tensile strength of
spider silk is greater than
the same weight of steel and
has much greater elasticity

Gnp ch103-lecture notes

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