I. Pre-Phd Courses in Physical Sciences |
I. Pre-Phd Courses in Physical Sciences
Selected courses are offered each semester from the following list:
- PS 427: Computational Physics (3 credits)
- PS 603: Topics in Classical and Quantum Mechanics (3 credits)
- PS 621: Advanced Statistical Physics (3 credits)
- PS 633: Stochastic Phenomena (3 credits)
- PS 651: Dynamical Systems and Chaos (3 credits)
- PS 662: Experimental Methods in Physics (3 credits)
- PS 664: Special Topics in Condensed Matter Physics (3 credits)
- PS 682: Advanced Laser Physics (3 credits)
- PS 721: Nonequilibrium Statistical Mechanics (3 credits)
- PS 722: Phase Transitions and Critical Phenomena (3 credits)
- PS 723: Introduction to String Theory (3 credits)
- PS 761: Many-Body Theory (3 credits)
- PS 762: Introduction to Computational Neuroscience (3 credits)
- & 15. PS 699 & 799: Research Course – I and II (3 credits each)
Note: The Pre-Ph.D. students may take any of the courses listed under M.Sc. Semester IV, as recommended by the Ph.D. Student Advisor.
Overview of computer organization, hardware, software, scientific programming in FORTRAN and/or C, C++.
Numerical Techniques
Sorting, interpolation, extrapolation, regression, numerical integration, quadrature, random number generation, linear algebra and matrix manipulations, inversion, diagonalization, eigenvectors and eigenvalues, integration of initial-value problems, Euler, Runge-Kutta, and Verlet schemes, root searching, optimization, fast Fourier transforms.
- Simulation Techniques
Monte Carlo methods, molecular dynamics, simulation methods for the Ising model and atomic fluids, simulation methods for quantum-mechanical problems, time-dependent Schrödinger equation, discussion of selected problems in percolation, cellular automata, nonlinear dynamics, traffic problems, diffusion-limited aggregation, celestial mechanics, etc. - Parallel Computation
Introduction to parallel computation
Suggested Texts:
- V. Rajaraman, Computer Programming in Fortran 77.
- W.H. Press, B.P. Flannery, S.A. Teukolsky and W.T. Vetterling, Numerical Recipes in FORTRAN 77: The Art of Scientific Computing. (Similar volumes in C, C++.)
- H.M. Antia, Numerical Methods for Scientists and Engineers.
- D.W. Heermann, Computer Simulation Methods in Theoretical Physics.
- H. Gould and J. Tobochnik, An Introduction to Computer Simulation Methods.
- J.M. Thijssen, Computational Physics.
PS 603: Topics in Classical and Quantum Mechanics
This course is intended as a refresher course for incoming Ph.D. students. It reviews material that has been taught in M.Sc. courses, but with greater emphasis on problem-solving. The actual content and format of the course will depend upon the instructor and the composition of the class. The topics to be covered will be selected from the list presented below.
Course Outline:
Classical Mechanics
- Particle motion in 1, 2 and 3 dimensions; conservation laws; non-inertial frames.
- Generalized coordinates; Lagrangian method and examples; two-body problem; bound states and scattering.
- Small oscillations; Hamiltonian formalism; canonical transformations, Hamilton-Jacobi theory.
- Rigid-body motion; motion of tops; rotation matrices.
- Special relativity; relativistic kinematics.
Quantum Mechanics
- One-dimensional Schrödinger equation; particle in a square-well potential; bound states; transmission and reflection from step potentials; W.K.B. method for bound states; tunneling; harmonic oscillator; operator method of solution.
- Two-level and other finite-dimensional Hilbert space problems.
- Three-dimensional Schrödinger equation, angular problem algebra of angular momentum; square well in 3 dimensions; hydrogen atom.
- Atoms in electric magnetic fields; spin-orbit coupling; examples of time-dependent perturbation theory; scattering theory.
PS 621: Advanced Statistical Physics
This course is also intended as a refresher course for incoming Ph.D. students and reviews material that may have been taught in M.Sc. courses. The topics to be covered will be selected from the list presented below:
- Review of thermodynamics; thermodynamic potentials; equation of state; phase transitions.
- Statistical mechanics; theory of ensembles; phase space and ergodicity; quantum statistics; density matrix; maximum entropy principle.
- Ising model; mean-field theory; exact solution in 1 dimension; Onsager solution in 2 dimensions; transfer matrix.
- Landau theory of second-order phase transitions; scaling hypothesis; critical exponents and universality classes; correlation length; importance of fluctuations near critical point; concept of renormalization group.
- Liquid-solid transitions; density-functional theory of freezing.
- Computer simulations; molecular dynamics/; Monte Carlo methods; Cellular Automata models.
- Bose-Einstein condensation; quantum fluids; superfluidity.
- Non-equilibrium statistical mechanics; linear response theory; Kubo formula; Onsager relations; Boltzmann equation; D.C. conductivity of metals.
- Hydrodynamics; conserved and broken-symmetry variables; Goldstone theorem; spin dynamics; Navier-Stokes equation and viscous hydrodynamics.
- Disordered systems; spin glasses; Sherrington-Kirkpatrick model; topological defects; dislocations; vortex unbinding and Kosterlitz-Thouless transition.
This course introduces students to the modeling and characterization of stochastic phenomena. Theoretical concepts are illustrated via many physical examples.
Course Outline:
- Stochastic variables; multivariate distributions; Gaussian distributions; central limit theorem.
- Random events; Poisson distribution; correlation functions; waiting time.
- Stochastic processes in physics; stationary processes; vibrating string and random field.
- Markov processes; Chapman-Kolmogorov equation; jump and telegraph processes; Kubo-Anderson process; Kangaroo process.
- Fokker-Planck and Langevin equation; Brownian motion; Rayleigh particle; Kramers’ equation; stationary solutions; decay of metastable states; first-passage time problem.
- Master equation; closed and isolated physical systems; principle of detailed balance.
- Quantum dissipation; quantum diffusion.
PS 651: Dynamical Systems and Chaos
This course is intended for students planning to pursue research in the areas of nonlinearity, classical and quantum chaos and related subjects. Depending on the interests of the students/instructor, the topics to be covered will be chosen from the list below:
Course Outline:
Dynamical Systems
One-dimensional dynamics; logistic map; Sharkovskii theorem; hyperbolic systems; symbolic dynamics; chaos and bifurcation theory; period-doubling and other routes to chaos; Hopf bifurcations; Ruelle-Takens scenario; strange attractors.
Control of chaotic systems; synchronization and applications; experiments on chaotic systems; electronic circuits; chemical chaos; turbulence.
Classical Chaos
Qualitative analysis of ordinary differential equations; dissipative and conservative systems; Lagrangian and Hamiltonian formulation; integrability and Hamilton-Jocobi equations; perturbation methods; K.A.M. theory; chaotic dynamics.
Attractors – simple and strange; Lyapunov exponents; Lorenz system.
Quantum Chaos
Semiclassical mechanics; W.K.B. and E.B.K. quantization; periodic orbit theory; random matrices and applications to the study of eigenvalue spectra; periodic quantum systems; Floquet theory and recurrence; quantum maps.
Suggested Texts:
- R.L. Devaney, Introduction to Dynamical System
- E. Ott, Chaos in Dynamical Systems, Cambridge University Press (1992).
- A.J. Lichtenberg and M.A. Lieberman, Regular and Chaotic Dynamics, Springer-Verlag (1992).
- V.I. Arnold, Mathematical Methods of Classical Mechanics, Springer-Verlag (1979)
PS 662: Experimental Methods in Physics
This course introduces the student to various important experimental techniques in physics.
Course Outline:
- Signal Processing: Signal transmission and impedance matching; noise sources; signal noise optimization; pre-amplifiers, amplifiers and pulse shaping.
- Vacuum Techniques and Sample Preparation: Vacuum chamber; types of pumps; gauges, controls and leak detection techniques; sample preparation methods.
- Detectors and Transducers: Interaction of particles and radiation with matter; energy, position and timing detectors; particle identification; gaseous, solid state and scintillation detectors; channel multipliers.
- Measurement Techniques and Data Acquisition: Measurement of voltage, current, charge, frequency, etc.; overview of digital and analog systems in measurement; data acquisition.
- Data Reduction and Error Analysis: Statistical characterization of data; systematic errors; propagation of errors; least squares method; goodness of fit; reliability.
- Transport Measurement: General expression for resistivity; band structure and electrical resistivity; models and pseudo-potentials in non-simple metals; local spin fluctuations and spin glasses; resistivity at critical points.
- Spectroscopy: Infrared and Raman spectroscopy applications; Mossbauer spectroscopy; electron spectroscopy; high-resolution nuclear magnetic resonance.
PS 664: Special Topics in Condensed Matter Physics
The topics to be covered will be chosen from the list below.
- Mesoscopic Systems: Low-dimensional systems; characteristic lengths; transverse mode or magneto-electric sub-bands; resistance of a ballistic conductor; Landauer formula; reformulation of Ohm’s law; Landauer-Buttiker formula; transmission function and S-conductance fluctuations.
- Quantum Hall Effect : Classical Hall effect; integral quantum Hall effect (IQHE); fractional quantum Hall effect (FQHE) and Laughlin’s theory.< >Nanoscale Science: Synthesis and Fabrication methods (Physical and chemical approaches), characterization methods (microscopy, diffraction, spectroscopy techniques), surface analysis and depth profiling, techniques for physical property measurement, processing and properties of inorganic nanomaterials, special nanomaterials, Thermodynamics and statistical mechanics of small systems, Nucleation and growth of nanocrystals; kinetics of phase transformations
Nanotechnology: Introduction and classification, effects of nanometer length scales, self assembling nanostructures molecular materials and devices, applications of nanomaterials: molecular electronics and nanoelectronics; nano-biotechnology; quantum devices; nanomagnetic materials and devices : magnetism, nanomagnetic materials, magnetoresistance; nanomechanics
Conventional Superconductors: Occurrence; Meissner effect; phase diagram of Type I and Type II; thermodynamics; energy gap; isotope effect; London’s equation; Landau-Ginzburg theory; flux quantization; Josephson effect; BCS-pairing theory.
Metal-Insulator Transition: Phenomenology of metal-insulator transition; transitional metal oxides; doped semiconductors; Hubbard model; mean-field solutions; metallic and insulating limits; magnetism; Hartree-Fock picture; itinerant vs. localized systems.
High-Tc Superconductivity: Materials structure; magnetic behaviour; isotope effect; normal state anomalies; evidence for d-wave pairing; correlations; CuO2-plane; anti-ferromagnetism; Mott insulator; low doping vs. high doping. Strange metals; conjectures about d-wave.PS 682: Advanced Laser Physics
Course Outline:
Quantization of free electromagnetic field; coherent states of electromagnetic field; blackbody radiation; density operator r; photon statistics; diagonal coherent-state representation of r.
Interaction of two-level atom with classical field; Bloch equations; Rabi problem.
Semi-classical theory of single-mode laser; Lamb equation; Langevin equation for field amplitude; Fokker-Planck equation; photon statistics.
Relationship between quantum and semi-classical theories; coherent state representation of laser field; steady-state solution of master equation.
Suggested Texts:
- L. Mandel and E. Wolf, Optical Coherence and Quantum Optics, Cambridge University Press (1995).
- M. Sargent, M.O. Scully and W.E. Lamb, Laser Physics, Addison-Wesley (1977).
- P. Meystre and M. Sargent, Elements of Quantum Optics, Springer-Verlag (1990).
- B.B. Laud, Lasers and Nonlinear Optics, Wiley-Eastern (1991).
PS 721: Non-equilibrium Statistical Mechanics
Course Outline:
- Kinetic Ising model; 1-dimensional solution; mean-field theory; time-dependent Ginsburg-Landau equation; dynamical scaling and exponents; renormalization group theory.
Liouville equation; response function and susceptibility; fluctuation-dissipation theorem; two-level systems; irreversibility and the master equation; Boltzmann equation; conductivity; Kubo formula.
Brownian motion; Langevin, Fokker-Planck and diffusion equations; hydrodynamic fluctuations and structure factor.
PS 722: Phase Transitions and Critical Phenomena
Course Outline:
- Scaling functions and scaling relations; renormalization group; Ginzburg-Landau free-energy functional; momentum-space renormalization group; Î-expansion; real-space renormalization group.
Criteria for thermodynamic stability; first-order phase transitions; Van der Waals’ theory; Gibbs phase rule.
Examples of phase transitions and phase diagrams; criticl points; second-order phase transitions; order parameter; critical exponents.
Universality; Landau theory for phase transitions.
Ising model; mean-field approximation; transfer matrix method; Onsager solution of 2-dimensional Ising model; Yang-Lee theorem.
PS 723: An Introduction to String Theory
Prerequisite: Lagrangian mechanics, Special theory of relativity, Electromagnetism, Quantum Mechanics.
Course outline:
- Motivation.
Motion of a non-relativistic string, normal modes.
Review of Special Theory of Relativity, light cone coordinates, Lorentz invariance in diverse dimensions, small and compact dimensions, square-well and quantum mechanics of extra dimensions, motion of a relativistic particle, world-line.
Review of electromagnetism, electric field in diverse dimensions, point particle with electric charge.
Newton’s law of gravitation in diverse dimensions, Planck units of mass, length and time, Newton’s constant, gravity and geometry.
Relativistic string, world-sheet, invariants on the worldsheet, area of embedded surfaces, Nambu-Goto action, equations of motion, boundary conditions and branes, static gauge, tension and energy of strings.
Classical motion of a string, dynamics on the world-sheet, conserved quantities.
Point particle in light-cone gauge, quantization.
String in light-cone gauge, string as oscillators, normal modes and particle spectrum (open and closed string), quantization.
String thermodynamics: counting of states, partition function.
Overview of results.
Suggested Texts:
- B. Zwiebach, A first course in string theory (Cambridge University Press)
- H. Goldstein, Classical mechanics (Addison-Wesley)
- J. Hartle, Gravity: An introduction to Einstein’s general relativity (Pearson education)
- J. Sakurai, Modern quantum Mechanics (Pearson education)
- L. Schiff, Quantum mechanics (McGraw Hill)
Course Outline:
Methods of second quantization; Green’s function; adiabatic switching and interaction picture; Wick’s theorem; diagrammatic analysis of perturbation theory; linear response theory.
Fermi systems; Hartree-Fock approximation; hard sphere Fermi gas; uniform electron gas polarization and screening; correlation energy.
Finite-temperature formalism; thermal Green’s functions; perturbation theory and diagrammatic analysis; electron gas at finite temperature; linear response at finite temperature.
Linear theory of phonons; phonon-phonon and electron-phonon interaction; field theory for coupled electron-phonon system.
PS 762: Introduction to Computational Neuroscience
A major effort is currently under way to understand the operation of the central nervous system, and more specifically, of neuronal networks in the brain. This is of great importance at not only the theoretical level, but also for the possibility of understanding the causes and cures for diseases such as Alzheimer’s and Parkinson’s.
The approach presently being taken includes both experimental studies and theoretical and computational modeling to jointly address questions that arise in this area of research. There is an increasing need for scientists trained at the interface of these disciplines who possess a strong analytic background together with a solid understanding of biological phenomena.
The present course will teach students the basic set of mathematical and computational techniques required for them to pursue higher level research in the field of neuroscience. It would also prepare them, in part, to be able to move on to various industry jobs that require quantitative and analytic skills. For example, several pharmaceutical companies are actively seeking employees with the background to model and simulate processes on the computer prior to production and testing. The set of lectures will cover necessary techniques to be able to understand various biological questions, to address them mathematically and computationally and then to translate results into a language that is accessible to experimentalists. It is envisioned that students who pass this class will be able to immediately utilize their course work in either an experimental lab or on the way towards a PhD. Further the mathematical content of the course is sufficiently general that it will also allow students to work in modeling of biological problems outside of neuroscience, in fields such as genomics, protein signaling networks and even ecology.
This Pre-Ph.D. course will be accessible to final year students of the M.Sc.(Physics) program as well as those of the M.Sc.(Life Sciences) and that it could be an optional course in the M Tech. (Systems Biology) programme as well.
Pre-requisites: Calculus of many variables, a basic understanding of differential equations, ability to use computer software such as MATLAB and the ability to code in Fortran, C, C++ etc.
Course Outline [Approximate number of lectures per topic]
Introduction to neuroscience with description of some specific neuronal systems. [2]
Mathematical background – Introduction to dynamical Systems, review of basics of differential equations, introduction to phase plane analysis, dimensional reduction techniques including timescale separation ideas. [5]
Computational techniques – Introduction to relevant computer software such as XPP and Matlab. Classes during this time to be held in a computer lab in a tutorial manner with demonstrations of software usage. [5]
Models of single neurons – Derivation of the Hodgkin-Huxely equations and various reductions such as the FitzHugh-Nagumo and Morris-Lecar models. Analysis of these and other basic models such as the Integrate and Fire model. [6]
Models of synaptic interactions – Description of synapses and neurotransmitter release. Mathematical models for excitatory and inhibitory synapses. Models for short-term synaptic plasticity. [6]
Small network dynamics – Focus on understanding and characterizing the dynamics of small networks of excitatory, inhibitory or mixed-type neurons. Detailed analysis of conditions leading to complete synchronization, phase locking or chaotic behavior in such networks. [8]
Case studies –The dynamics of several specific biological examples will be explored including problems from the following areas: place cells in the hippocampus, sleep rhythms and oscillations of the thalamus, irregular activity in the basal ganglia, working memory models of the cortex and phase lag models of central pattern generators. [8]
Textbooks:
The following are list of suggested textbooks, although the course will initially be taught from a set of lecture notes.
1. Theoretical Neuroscience: Computational and Mathematical Modeling of Neural Systems, by Peter Dayan and Larry F. Abbott. The MIT Press, 2001. ISBN 0-262-04199-5
2. Dynamical Systems in Neuroscience: The Geometry of Excitability and Bursting, by Eugene M. Izhikevich. The MIT Press, 2007. ISBN 0-262-09043-8
3. Simulating, Analyzing, and Animating Dynamical Systems: A Guide to XPPAUT for Researchers and Students, by Bard Ermentrout, SIAM 2002 ISBN 0-89871-506-7
II. Pre-Ph.D. Courses in Chemical Sciences
Core Courses:
PS 611C Concepts in Chemistry (3 credits)
Optional Courses:
PS 612C Analytical Methods in Chemistry (3 credits)
PS 613C Computational Chemistry and its Applications (3 credits)
PS 614C Advanced Spectroscopy and its Applications (3 credits)
PS 615C Supramolecular Chemistry (3 credits)
PS 616C Molecular Materials (3 credits)
PS 617C Research Course (3 credits)
Note: Students can take the necessary Physics courses from the Pre-Ph.D. courses in Physical Sciences according to their interests and needs.
1. PS 611C Concepts in Chemistry (3 credits)
Course outline:
General Introduction
Different types of forces- Covalent bonding, ionic bonding, H-bonding, Van der Waals forces; Theory of chemical bonding; Symmetry and Group theory.
Physical Chemistry
Thermodynamics- Conservation of energy; Review of Enthalpy, Entropy, Free energy with examples involving chemical systems.
Chemical kinetics and catalysis- Discussion of reaction rate theory; Collision and transition state theory; Potential energy surfaces; Catalysis; Enzyme catalysis.
Organic Chemistry
- Delocalized chemical bonding- Aromaticity, Hyperconjugation, Tautomerism.
- Acids and Bases.
- Stereochemistry- Optical activity; Chirality; Topicity and prostereoisomerism; Conformations of acyclic and cyclic molecules.
- Mechanism, reactivity and reactions of- Aliphatic and aromatic nucleophilic substitution; Aliphatic and aromatic electrophilic substitution; Addition to C-C and C-hetero multiple bonds; Elimination and Rearrangements; Oxidation and reductions.
- Named reactions; New reagents; Catalysts and their applications.
Inorganic Chemistry
Structure of molecules- VSEPR theory; Bents rule; Berry Pseudorotation; Molecular orbital treatment for homonuclear, heteronuclear and delocalized molecules.
Acids and Bases- Measures of acid-base strength; Hard and soft acids and bases.
Redox reactions- Standard electrode potentials; Electromotive forces; Electrochemical series; Use of reduction potentials.
Coordination chemistry- Bonding in coordination compounds; VBT, CFT and MOT; Electronic spectra of complexes; Magnetic properties of complexes.
Suggested Texts:
- Peter Atkins and de Paula, Physical Chemistry, 7th edition, Oxford University Press Inc., New York.
- J. E. Huheey, E. A. Keiter; R. L. Keiter, Inorganic Chemistry- Principles of Structure and Reactivity, 4th edition, Pearsons education.
- J. March, Advanced Organic Chemistry, 5th edition, John Wiley and Sons.
- F. Albert Cotton, Chemical Applications of Group Theory, 3rd edition, John Wiley and Sons.
2. PS 612C Analytical Methods in Chemistry (3 credits)
Analytical chemistry is one of the important divisions of chemistry that aids researchers in experimental (classical and applied) chemistry to characterize chemical compounds. This course has been designed to introduce students to the methods and concepts of analytical chemistry and would give an overview of the instruments involved for characterization.
In this course, students would be taught how to characterize molecules using various molecular spectroscopic and electro-analytical methods and bulk or aggregated structures using various surface and microscopic techniques.
Course outline:
Molecular Spectroscopy
Infra-red spectroscopy- Theory; Instrumentation; Sample Handling; Interpretation of Spectra; Characteristic group absorptions of molecules.
Mass spectroscopy- Instrumentation; Mass spectrum; Determination of molecular formula; Recognition of molecular ion peak; Ionization techniques; Fragmentation and rearrangements; Quantitative applications of mass spectrometry.
- Nuclear Magnetic Resonance spectroscopy- Continuous-Wave (CW) NMR spectrometry; Relaxation; Pulsed Fourier Transform Spectrometry; Rotating Frame of Reference; Instrumentation and sample handling; Chemical shift; Spin coupling; Chemical shift equivalence; Magnetic equivalence; Geminal and vicinal coupling; Nuclear Overhauser Effect; 13C NMR spectroscopy; Correlation NMR spectrometry; 1H-1H COSY; Double-Quantum filtered 1H-1H COSY; 1H-13H COSY.
Electroanalytical Chemistry
Potentiometry, Coulometry and Voltammetry.
Separation Methods
An introduction to chromatographic separations; Gas chromatography; Gas chromatographic columns and stationary phases; Principles and applications of gas-liquid chromatography; High-Performance Liquid Chromatography (HPLC); Thin Layer and column Chromatography; Ion-Exchange Chromatography; Size-Exclusion chromatography.
Surface Characterization by Spectroscopy and Microscopy
Reciprocal space map:
X-ray diffraction- Lattice; Lattice symmetry; Characterization of powder and thin films; Line shape analysis.Real space map: Near Field- AFM, STM
Far field: SEM, TEM.
Thermal Methods
Thermogravimetric methods; Differential thermal analysis; Differential Scanning Calorimetry.
Statistical Analysis- Evaluating Data
Confidence limits, statistical aids to hypothesis testing; Detecting gross errors; Least square methods.
Suggested Texts:
- Principles of Instrumental Analysis by Douglas A. Skoog, F. James Holler, Timothy A. Nieman; Saunders Golden Sunburst Series.
- Modern Analytical Chemistry by David T Harvey; McGraw-Hill Science.
- Spectrometric Identification of Organic Compounds by R. M. Silverstein, F. X. Webster; John Wiley and sons
- Analytical Chemistry by Gary D. Christian; John Wiley and sons
- 200 and More NMR Experiments: A Practical Course by Stefan Berger, Siegmar Braun; Wiley
3. PS 613C Computational Chemistry and its Applications (3 credits)
This course is intended for the incoming PhD students to provide the basic knowledge about computational chemistry methods and its use in connection to the experimental research. The aim of this course is to provide students with basic background on computational methods and molecular modeling, including some hands-on experiences to get started in modeling the physicochemical properties of molecules. The basic theoretical background will be provided in this course, and the emphasis will be given on hand-on application of the computational methods to model molecular properties. The topics to be covered will be selected from the list presented below depending on the availability of resources and time.
Course Outline:
Overview of Basic Quantum Chemistry:
Introduction; Operators; Eigenvalues; Time-dependent Schrödinger equation; Particle in a box; Variation and perturbation methods; Born-Oppenheimer approximation.
Molecular Mechanics:
Fundamentals; Potential energy functions; Force fields; Electrostatic interactions; van der Waals interactions; Geometry optimization; Energy minimization methods.
Simulation Methods:
An introduction; Overview of Molecular Dynamics and Monte Carlo simulations; Introduction to application of simulation methods to explore bio-macromolecules; Brief on conformational analysis; Basic hands-on experiences on simulating bio-molecules (like protein, DNA, lipids) using academic and/or commercially available simulation packages (depending on availability of resources).
Overview of Molecular Orbital (MO) Theory:
Trial wave functions; Hückel MO theory; Hartree-Fock (HF) theory; MO-LCAO formalism.
Semiempirical Implementation of MO Theory:
Extended Hückel theory; CNDO, INDO and NDDO formalisms; Ongoing developments; Computation of electronic structures; Hands-on experience in running semiempirical jobs and extracting various molecular properties using academic and/or commercially available quantum chemical packages (depending on availability of resources).
Ab Initio Hartree-Fock Theory:
Review of HF equation and variational principle; Basis sets; Practical issues; Electron correlation techniques; Configuration Integration; Geometry optimization; Electric dipole moment; Electric polarizability and Hyperpolarizability; Hands-on experience in running jobs to explore electronic structure of molecules using quantum chemical packages (depending on availability of resources).
Overview of Density Functional Theory (DFT):
Introduction; Hohenberg-Kohn theorem; Kohn-Sham theory; Exchange-correlation functionals; Advantages and disadvantages of DFT compared to MO theory; Computing electronic structure; Hands-on experience in running DFT jobs using quantum chemical packages (depending on availability of resources); Comparison of performances of various methods.
Brief Overview of Condensed-phase Calculations:
Condensed-phase effects in general; Continuum model and its implementation; Effect of continuum salvation on electronic spectra of molecules; Solvent models; Brief on Molecular Dynamics and Monte Carlo simulations; Introduction to hybrid QM/MM methods and its applications.
Suggested Texts:
Ira. N. Levine, Quantum Chemistry, 5th ed., Prentice Hall, NJ.
Donald A. McQuarrie, Quantum Chemistry, University Science Books, Mill Valley, CA.
Mark A. Ratner and George C. Schatz, Introduction to Quantum Mechanics in Chemistry, Prentice-Hall, NJ.
Christopher J Cramer, Essentials of Computational Chemistry, 2nd edition, John Wiley & Sons Ltd., England.
Frank Jensen, Introduction to Computational Chemistry, 2nd edition, John Wiley & Sons Ltd., England.
Andrew R. Leach, Molecular Modelling: Principles and Applications, 2nd edition, Longman Group, United Kingdom.
M. P. Allen and D. J. Tildesley, Computer Simulation of Liquids, Clarendon, Oxford.
Attila Szabo and Neil S. Ostlund, Modern Quantum Chemistry, Introduction to Advanced Electronic Structure Theory, 1st ed., revised, Dover.
4. PS 614C Advanced Spectroscopy and its Applications(3 credits)
This course is designed as a refresher course for incoming Ph.D. students with Physics or Chemistry background. It reviews some materials that may have been taught in the M.Sc. courses, and also includes material designed to familiarize students with some of the advanced spectroscopic techniques and their applications. This course will be beneficial also for students of Biophysics or Biochemistry who want to gain further knowledge about the application of some advanced spectroscopic techniques to study biological macromolecules. The topics to be covered will be selected from the list presented below.
Course Outline:
- Fundamentals of Photochemistry:
Overview
Laws of photochemistry; Interaction of radiation with matter; Transition between states; Fermi-Golden rule; Electronic transitions, dipole approximation and two-photon transitions.
Radiative Transitions – Absorption and Emission of Light
Types of photophysical processes; Absorption of light; Frank-Condon principle; Emission spectra; Fluorescence and phosphorescence; Excited state dipole moments.
Non-radiative Transitions
Internal conversion; Intersystem crossing; Energy gap law; Isotope effect; Temperature effect.
Various Photophysical Processes
Fluorescence Resonance Energy Transfer (FRET); Förster theory; Overlap integral; Solvent effect on absorption and emission; Solvation energy; Lippert equation; Solvent relaxation dynamics; Fluorescence anisotropy; Fluorescence anisotropy decay; Excited state proton and electron transfer.
- Spectroscopic Techniques and Applications
Laser Fundamentals
Introduction to Lasers; Coherence; Population inversion; Laser cavity modes; Pulsed laser operations; Q-switching and mode-locking; Pulsed laser sources (Diode lasers, Ti:Sapphire lasers); Frequency multiplication of lasers and nonlinear optical effects.
Some Spectroscopic Techniques
Uv-vis spectrophotometer; Fluorescence spectrometer; Absorption, emission and excitation spectra; Time-resolved techniques; Time-correlated Single Photon Counting (TCSPC); Brief introduction to time-resolved fluorescence up-conversion technique; Microscopic techniques and imaging; Single molecule spectroscopy; Fluorescence correlation spectroscopy (FCS); Raman Spectroscopy.
Applications
Application of steady-state and time-resolved fluorescence techniques to study FRET, Solvation relaxation dynamics, Fluorescence anisotropy, Excited state charge transfer in molecular assemblies and biomacromolecules (e.g. DNA, protein, lipid etc.); Introduction to the application of fluorescence and Raman microscopic techniques to study microscopic and single molecular systems; Summery of applications of FCS; Brief introduction to the application of Coherent anti-Stokes Raman Spectroscopy (CARS) and microscopy; One- and multi-photon fluorescence imaging of biological systems.
Suggested Texts:
- Nicholas J. Turro, Modern Molecular Photochemistry, The Benjamin/Cummings Publishing Co., Inc.
- K. K. Rohatgi-Mukherjee, Fundamentals of Photochemistry, New Age International (P) Limited.
- J. R. Lakowicz, Principles of Fluorescence Spectroscopy, 3rd edition, Kluwer Academic, New York.
- Colin N. Banwell, Elaine M. McCash, Fundamentals of Molecular Spectroscopy, Tata McGraw-Hill.
- William T. Silvast, Laser Fundamentals, Cambridge University Press.
- D. V. O’Connor and D. Phillips, Time Correlated Single Photon Counting, Academic Press, New York.
- Robert M. Silverstein, Francis X. Websterand, David J. Kiemle, Spectrometric Identification of Organic Compounds.
5. PS 615C Supramolecular Chemistry(3 credits)
This course is designed to introduce students to the interdisciplinary science of supramolecular chemistry. Thermodynamic and kinetic parameters involved in designing supramolecular systems would be taught in detailed in this course. The course would also give insight into the role of supramolecular chemistry of life and designing artificial mimics pertaining to nature.
Course Outline:
Principles of molecular recognition
Quantification of non-covalent forces and medium effects; Host design; Preorganization; Enthalpy and entropic contributions; Cooperativity and allosteric effects; Induced fit; Complexation selectivity.
Supramolecular Chemistry of Life
Alkali metal cations in biochemistry; Porphyrins and tetrapyrrole macrocycles; Plant photosynthesis; Uptake and transport of oxygen in Haemoglobin; Coenzyme B12; Neurotransmitters and Hormones, DNA; Biochemical self-assembly
Cation Binding Hosts
Lariat ethers and podands, crown ethers, cryptands, calyx[n]arenes, cucurbit[n]urils, spherands; Selectivity of cation complexation; Macrocyclic, macrobicyclic and template effects
Anion Binding Hosts
Concepts in anion host design; Guanidinium-based receptors; Organometallic receptors; Neutral receptors; Hydride sponge; Anticrowns; Biological Anion receptors
Binding of neutral molecules
Binding by cavitands, cyclodextrins, cucurbit[n]urils, dendrimers, molecular clefts and tweezers, cyclophane Hosts
Supramolecular reactivity and catalysis
Catalysis by cation, anion and neutral receptors; Supramolecular metallocatalysis; Cocatalysis; Biomolecular and abiotic catalysis
Transport processes and carrier design
Anion and cation carriers; Coupled transport processes; Electron coupled transport; Proton coupled transport; Light driven transport; Transport via transmembrane channels
Self Processes
Programmed supramolecular systems; Self-assembly; Self-organization; Self-recognition; Self-replication systems
Suggested Texts:
- Supramolecular Chemistry by J. W. Steed and J. L. Atwood, John Wiley and Sons, Ltd.
- Supramolecular Chemistry-Concepts and Perspectives by Jean –Marie Lehn, VCH.
- Principles and Methods in Supramolecular Chemistry by Hans-Jorg Schnider and Anatoly K. Yatsimirsky, John Wiley and Sons, Ltd.
- Supramolecular Chemistry of anions by Antonio Bianchi, Kristin Bowman James and Enrique Garcia-Espana, Wiley-VCH.
6. PS 616C Molecular Materials(3 credits)
This course is designed to introduce students to the basic concepts of chemical interactions and the principles and theories involved in the design of new-age applicative materials e.g., molecular sensors and switches, organic light emitting diodes, electrochromic materials etc. This course will also emphasize on how the change in molecular design and molecular interactions can tune or modulate the properties of these materials. Principles underlying the organic/inorganic synthesis and purification of materials through well-known named-reactions will also be covered in a lucid manner.
This course will be inter-disciplinary in nature and would help understand the basic theoretical as well as practical aspects (design, synthesis and properties) of new materials.
[Introductory classes would be planned for physics students]
Course Outline:
Nature of Chemical Interactions:
Covalent bond Vs ion-ion, ion-dipole, dipole-dipole, H-bonding (weak, moderate and strong), cation-p, anion-p, p-p interactions, van der Waals forces, hydrophobic effects. Examples and comparison of energy parameters.
How can we design giant structures using weak interactions? Self-assembling systems:
Carboxylic acid dimers, alcohol-amine, amides etc. Designing molecular squares and boxes, giant self-assembling capsules, molecular tennis-balls, rosettes, self-assembly of metal arrays etc.
Soft Materials
Micelles, Vesicles:
Amphiphiles, thermodynamic principles of self-association, DLVO (Derjaguin, Landau, Verwey, Overbeek) Theory. Factors affecting changes from one structure to another.
Liquid crystals
Designing principles and classifications. Thermotropic and lyotropic liquid crystals. Nematic, smectic, cholesteric and discotic phases. Liquid crystalline polymers, Metallo-organic liquid crystals. Application towards liquid crystal displays.
Organogels, Hydrogels
Designing principles and effects of H-bonding, p-p stacking, van der Waals interactions. Molecular visualization and characterization by SEM, TEM, AFM and XRD techniques. Application towards drug delivery and controlled release.
Glasses
- Design, properties and characterization.
Molecular devices
Molecular sensors and switches:
Design and principles of photochemical sensors, PET (Photo induced electron transfer) systems, ON-OFF molecular switches, Molecular logic gates (AND, NOT, OR, NAND etc.); Electrochemical sensors.
Organic Light Emitting Diodes (OLED’s):
Definition of electroluminescence and electroluminescence quantum effieciency, power efficiency. Design and characterization, examples of OLEDS: conjugated oligomers and polymers, low molecular weight materials. Application, commercialization and optimization of OLED’s.
Electrochromic materials:
Definition and designing principles, conducting polymers, metallo polymers, metallophthalocyanines, visible and infrared electrochromism. Applications to practical materials.
Nonlnear Optical (NLO) materials:
Definition of First-order and Second-order hyperpolarizabilities, Experimental techniques (Electric field induced second harmonic generation, Hyper Rayleigh Scattering) to determine b and c. Molecular designing principles and characterization of dipolar, multipolar and octupolar molecules. Imparting tunability in NLO molecules. Designing SHG active molecules (Influence of chirality, H-bonding, steric effects).
How to synthesize molecules?
A brief idea on the principles involved in organic synthesis, purification and characterization. Importance of reactions like Sonogashira, Suzuki and Heck coupling and their applications to synthesize materials.
Suggested Texts:
- The design of organic solids by G. R. Desiraju, Elsevier: Amsterdam.
- The weak hydrogen bond in structural chemistry and biology by Gautam Desiraju and Thomas Steiner, Oxford University Press.
- Supramolecular Chemistry by J. W. Steed and J. L. Atwood, John Wiley and Sons Ltd.
- Molecular Fluorescence: Principles and Applications by B. Valeur, Wiley-VCH.
- Organic electroluminescent materials and devices, by S. Miyata and H. S. Nalwa, Gordon and Breach Publishers, Amsterdam.
- Non Linear Optical properties of organic molecules and crystals, by D. S. Chemla and J. Zyss, Academic Press, Inc.
III. Pre-Ph.D. Courses in Mathematical Sciences
Core Courses:
Optional Courses:
PS 641M Algebra (3 credits)
PS 642M Analysis (3 credits)
PS 643M Topology (3 credits)
PS 644M Topological Groups and Lie Groups (2 credits)
PS 645M Functional Analysis and Operator Theory (2 credits)
- & 7. PS 646M, 647M Research Courses I & II (3 credits each)
- PS 712M: Algebraic Number Theory (2 credits)
Note: In addition, students may take the courses on Computational Physics: PS-427 (3 credit).
The core courses are essential for students with M.Sc. degree in mathematics to expand their knowledge of basic mathematics for doing research. Apart from the three core courses (3 credits each), students may either choose one Research Course (3 credits) and the above mentioned course PS 427 (3 credits) or take two Research Courses (3 credits each).
Core Courses
PS – 641M: Algebra (3 credits)
Groups: Nilpotent and solvable groups, Sylow’s theorems, free groups.
Representation theory of finite groups, Peter-Weyl theorem.
Rings and Modules: Commutative rings, Noetherian and Artinian rings and modules, principal ideal domains (PID), unique factorization domain, modules over PID, tensor products.
Field Theory: Algebraic and transcendental extensions, introduction to Galois theory.
Suggested Texts:
- M. Artin. Algebra.
- I.N. Herstein. Topics in Algebra.
- S. Lang. Algebra.
Measure Theory: General introduction, product measures, Fubini’s theorem, Lebesgue measure, complex measue, Radon-Nikodym theorem.
Harmonic Analysis: On R and Rn . convolution, Fourier transform, Fourier Inversion formula, Plancherel theorem.’
Lp-spaces: Introduction, Holder’s inequality and Monkowski inequality, Completeness, duality.
Differential Manifolds: tangent spaces, vector fields, implicit function theorem and inverse function theorem.
Suggested Texts:
- A. Deitmar. A First Course in Harmonic Analysis.
- W. Rudin. Real and Complex Analysis.
- H.L. Royden, Real Analysis.
- F.W. Warner. Foundations of Differentiable Manifolds and Lie Groups.
General Topology: Introduction, metric topology, separation axioms, compactness, Connectedness, product topology, introduction to manifolds, submanifolds.
Homotopy Theory. Covering spaces, homotopy maps, homotopy equivalence,Contractible spaces, deformation retraction.
Fundamental Groups: Universal cover and lifting problem for covering maps, Fundamental groups of S1 and Sn.
Introduction to Homology Theory.
Suggested texts:
- C.O. Christenson and W.L. Voxman. Aspects of Topology.
- J.R. Munkres. General Topology.
- I.M. Singer and J.A. Thorpe. Lecture Notes in Elementary Topology and Geometry.
Optional Courses
PS 644M: Topological Groups and Lie Groups (3 credits)
Topological Groups: Introduction, integration on locally compact spaces, Haar Measure, Character groups, group action.
Lie groups and lie algebras: Basic theory, linear groups.
Suggested texts:
- K. Chandrasekharan. A Course on Topological Groups.
- W. Fulton and J. Harris. Representation Theory.
- F.W. Warner. Foundations of Differentiable Manifolds and Lie Groups.
PS 645M: Functional Analysis and Operator Theory (2 credits)
Functional Analysis: Topological vector spaces, normed linear spaces, Banach spaces, Hilbert spaces, Hahn-Banach theorem, Open mapping theorem.
Operator Theory: Spectral theorem for compact normal operators, commutative Banach algebra and spectral theory, Gelfand-Naimark theorem.
Suggested texts:
- B.V. Limaye. Functional Analysis.
- W. Rudin. Functional Analysis.
- G.F. Simmons. Introduction to Topology and Modern Analysis.
PS 646M, 647M: Research Courses I & II (3 credits each)
The research courses are advanced courses to prepare students to work in a specific area. The details of these courses may be decided by the instructors.
PS 712M: Algebraic Number Theory (2 credits)
Number fields, number rings and their structure as Dedekind domains.
Factorization of prime ideals, quadratic and cyclotomic extensions.
Decomposition group, inertia group.
Group of units, ideal class group, theorems of Dedekind and Minkowski.
Introduction to zeta function, Dirichlet character.
Suggested texts:
- D. Marcus. Number Fields
- Borevich and Shafarevich. Number Theory
- Esmonde and Murty. Problems in Algebraic Number theory
- Frohlich and Taylor. Algebraic Number Theory
- Hasse. Number Theory