quantum chemistry

1. QUANTUM CHEMISTRY
Presented By:-
Saurav K. Rawat
Department of Chemistry,
St. John’s College, Agra

2. Introductory Quantum Mechanics


18 1 1
Historical   
       
 E = m c2
i f n n
 2 2
Bohr's Atom
1 h
p x x    
Heisenberg
Wave vs. Particle De Brogile's Hypothesis
Quantum Tools
^
Operator Algebra Postulates

n x
  
 
Applications
Translational
Vibrational
Rotational
Spectroscopy (NMR)





2.178 10 J
f i
hc
E h

H  E
( , , )
ˆ 2
    
2
2
V x y z
m



  
a
a n

sin
2
2
a
2   
2 2
n 
h
En  
8 m a

1
0
dx

 
2 2
E


h c
 

3. Genealogy of Quantum Mechanics
Classical Mechanics
(Newton)
Wave Theory of Light
(Huygens)
Maxwell’s
EM Theory
Electricity and Magnetism
(Faraday, Ampere, et al.)
Relativity Quantum Theory
Quantum Electrodyamics
High
Velocity
Low
Mass

4. Energy and Matter
Size of Matter Particle Property Wave Property
Large – macroscopic Mainly Unobservable
Intermediate – electron Some Some
Small – photon Few Mainly
E = m c2

5. The Wave Nature of Light
c   E  h
The speed of light is constant!

6.  Classical Mechanics is based on the Newton’s Law of
Motion – describes the dynamic proportion of the
macroscopic world .
 It failed to describe the behavior of particles at atomic
scale .
 The concept of quantum
• The experiments of Young and Fresnal on light showed the
latter behaved as waves.
• But with Planck’s Quantum , Einstein's Photon and Bohr’s
atom it confirmed by 1920 that despite of the wave like
properties of light (interference and diffraction), when it
came to transfer of energy and momentum light behaved like
a particle . This led to the concept of Quantum which means a
bundle or unit of any form of Physical Energy such as Photon
which represents a discrete amount of electromagnetic
radiant energy
•In 1924 de Broglie made a formulation that particle behaves
like waves

7. λ=h/p, where p is the momentum of the particle and Λ is the
wave length.
•All particles have a wave characteristics where they are
moving with a moving momentum
•The macroscopic objects which have a large mass have a
wave with very small wave length
•CONCLUSION:-
I. The particle and wave aspects of electromagnetic
radiations .
II. The wave aspect of the particle allows the calculation of
the probability of locating the particle
III. The prediction of the locations of Photons and sub-atomic
particle like electron , neutron , etc, probabilistic
IV.The probability is given by |E(r,t)|2

8. THE NEED OF NEW MECHANICS FOR SUB-ATOMIC PARTICLES:-
The concept of continuous energy absorption ( classical
mechanics) and emission was in conflict with atomic and sub
atomic phenomena ( black body radiation, photo electric
effect, Compton effect ,diffraction of electron and atomic
spectra of hydrogen)
The explanation led to the new mechanics called quantum
mechanics
SCHRODINGER EQUATION (characteristics of Ψ )
Ψ should be single valued
Ψ should be continuous
Ψ should finish for a bound state

9. APPLICATIONS OF SCHRODINGER EQUATION
•PARTICLE IN A BOX
•Hydrogen atom
•Rigid rotator
•Simple Harmonic
Oscillator

10. Particle in a Box (1D) - Interpretations
● Plots of Wavefunctions
● Plots of Squares of Wavefunctions
● Check Normalizations
n x
  
a
2   
● How fast is the particle moving? Comparison of macroscopic versus
microscopic particles.
Calculate v(min) of an electron in a 20-Angstrom box.
Calculate v(min) of a 1 g mass in a 1 cm-box





  
a
a n

sin
2
2
2 2
n 
h
En  
8 m a

1
0
dx

11. Particle in a Box
Region -I Region-II Region -III
V=α V=0 V=α
x = 0 x = a
Free particle – P.E. is same everywhere, i.e. V=0
Potential box – P.E. is 0 within the closed region and infinite (i.e. V=α) everywhere else

12. For one dimensional box-
Region-II, V=0
(1)
(2)
(3)
(4)
(5)
Schrodinger Equation-

13. Solution of Equation-
Ψ= A cos kx + B sin kx
• Region I + II
• Ψ=0, V=α
• At, x=0 Ψ=0 from -
• 0= A cos 0 + B sin 0
• A=0
• in
• Ψ= B sin kx (Ψ=0, x=0, x=a)
• B sin kx=0, B sin ka=0
Sin ka=0, ka=nπ, k= nπ/a
• n=0,1,1,3…….. allowed solution.
• n=1,2,3……….. acceptable solution.
(3)
(6)
(7)
(8)
(9)
(8) (6)
(6)

14. • Ψ= Ψn= B sin nπx/a ; n=1,2,3,…
• Wave Function for particle in a box-
• From (5) and (9)
• E= n2h2/ 8ma2
(10)
• E depends on quantum no. which can have
integral value, the energy levels of the particle
in a box are quantized.

15. Normalisation of ψ-
Normalisation Constant

16. • The solution of Schrödinger equation for a particle in a one
dimensional box-
• Ψ= √2/a sin(nπx)/a
• En= n2h2/8ma2 n=1,2,3
• The particle will have certain discrete values of
energy, so discrete energy levels. Hence energy of
the particle is quantized. These values, E depend
upon n which are independent of x. These are called
Eigen values. So a free particle can have all values of
energy but when it is confined within a certain range
of space, the energy values become quantized.

17. • n=1, E1=h2/8ma2
• n=2, E2=4h2/8ma2
Emin= h2/ma2
• Zero point energy (ZPE)- When the particle is
present in the potential box, the energy of the
lowest level n=1 is called zero potential energy.
• Eigen Function
• n=1 Ψ1=√2/a sin[ πx/a]
• n=2 Ψ2=√2/a sin[2 πx/a]
• n=3 Ψ3 ==√2/a sin[3 πx/a]

18. Nodes- The points were the probability of finding the
particle is zero in the particle wave.(n -1) nodes
• Greater the number of
nodes, more the curvature
in the particle wave. For a
potential box of fixed size,
as the curvature in the
wave function increases
the number of nodes
increases, the wavelength
decreases and the total
energy in the box, P.E.(V)
has been assumed to be
zero.

19. Ψ-Wave Function Ψ2 – Probability Function

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