QUANTUM CHEMISTRY Presented By:- Dr. Susan Verghese P. Associate Professor, Department of Chemistry, St. John’s College, Agra

Introductory Quantum Mechanics E = m c 2

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

Energy and Matter Size of MatterParticle PropertyWave Property Large – macroscopicMainlyUnobservable Intermediate – electronSome Small – photonFewMainly E = m c 2

The Wave Nature of Light The speed of light is constant The speed of light is constant!

 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

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

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

APPLICATIONS OF SCHRODINGER EQUATION. PARTICLES IN A BOX Hydrogen atom Rigid rotator Simple Harmonic Oscillator

Particle in a Box (1D) - Interpretations ●Plots of WavefunctionsPlots of Wavefunctions ●Plots of Squares of Wavefunctions ●Check Normalizations ●How fast is the particle moving? Comparison of macroscopic versus microscopic particles.Comparison Calculate v(min) of an electron in a 20-Angstrom box. Calculate v(min) of a 1 g mass in a 1 cm-box

V=0V=α Region -I Region-IIRegion -III x = 0x = 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 Particle in a Box

Schrodinger Equation- For one dimensional box- Region-II V=0

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

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

The solution of Schrödinger equation for a particle in a one dimensional box- Ψ= √2/a sin(nπx)/a E n = n 2 h 2/ 8ma 2 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.

n=1, E 1 =h 2 /8ma 2 n=2, E 2 =4h 2 /8ma 2 E min = h 2 /ma 2 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 Ψ 1 =√2/a sin[ πx/a] n=2 Ψ 2 =√2/a sin[2 πx/a] n=3 Ψ 3 ==√2/a sin[3 πx/a]

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.