Numerical Methods and Computational Physics

Paper Code: 
PHY 323
Credits: 
4
Contact Hours: 
60.00
Max. Marks: 
100.00
Objective: 

This course will enable the students to :

1.     This courseTo understand the basic Numerical methods and programming.

2.     To learn techniques to apply numerical methods into research areas

3.     To acquire working knowledge and practice for electronic structure studies of materials by using WIEN2K and Quantum Espresso softwares based on DFT.

Course outcomes (COs):

 

Course

Learning outcomes

(at course level)

Learning and teaching strategies

Assessment

Strategies

PAPER CODE

Paper Title

PHY 323

 

 

Numerical Methods  and Computational Physics

(Theory)

 

 

 

The students will be able to:

 

CO75: find numerical solutions of the system of linear equations with accuracy and obtain numerical solutions of algebraic transcendental equations.

CO76: get knowledge about various interpolating and extrapolating methods.

CO77: solve initial and boundary value problems in differential equations using numerical methods and apply various numerical methods in real-life problems.

CO78: gain the knowledge of WIEN2k software and plot DOS, electron structure and band structure.

CO79: obtain a brief idea about crystalline and amorphous substances, about lattice, unit cell, miller indices, reciprocal lattice, concept of Brillouin zones and diffraction of X-rays by crystalline materials by using the WIEN2k Software.

Approach in teaching:

Interactive Lectures, Discussion, Tutorials, , Demonstration, Problem Solving in tutorials.

 

 

Learning activities for the students:

Self learning assignments, Effective questions,  Seminar presentation, Solving numerical,

Additional learning through online videos and MOOC courses

Class test, Semester end examinations, Quiz, Solving problems, Assignments, Presentations

 

11.00
Unit I: 
Errors in Numerical Analysis:

Source of Errors, Round off error, Arithmetic error, error analysis, Condition and stability, method of undetermined coefficients, use of interpolation formula, iterated interpolation, inverse interpolation, Hermite interpolation and Spline interpolation (linear method).

 

12.00
Unit II: 
Solution of Linear and Nonlinear equations

Solution of Linear equations : Direct and Iterative methods, Jacobi and Gauss Seidal method

Solution of Nonlinear equation: Bisection method, Newton-Raphson method, Generalized Newton Raphson method, method of iteration, Newton Raphson method for the case of nearly equal roots, double root and multiple roots.

 

13.00
Unit III: 
III

Integration of a function: Trapezoidal and Simpson's rules. Gaussian quadrature formula, Singular integrals, Double integration.

Integration of Ordinary differential equation: Predictor-corrector methods, Runga-Kutta method. Simultaneous and Higher order equations.

Numerical differentiation of Data, Least-Squares Approximations, Fast Fourier Transformation. 

12.00
Unit IV: 
Theoretical methods for study of Electric Structure of Materials:
  1. Basic concepts only of: Periodic boundary conditions, Brillouin zone, Symmetry points in Brillouin zone of common types of lattices, Fermi Energy, Energy Band Diagram, effective mass of electron, Density of States, Fermi surface, Nature of Wave-function of the system, its plane wave expansion, pseudo potentials, Form factor, Structure factor, Dielectric screening, Exchange and Correlation, Various contributions to total energy of a system, Density Functional Theory, The LAPW Method, The APW+lo method.
  1. WIEN2k Software: its structure, W2web server, Brief description of using W2web for:

Creating a new session and directory, Creating the input and its setting of RMT values, sample structure file (Case. struct) file, Viewing the structure, Initialization of the calculation. Setting up of RMT*Kmax and k-points. The SCF calculation and convergence limits, flow of WIENk2k program, Saving the calculation, spin polarized calculation, spin- orbit interaction. Calculation and PSOtting of Electron Density distribution, Density of States, X-ray spectra, Band Structure, Fermi Surface. Volume Optimization, Super cell creation and addition of impurity atoms to the system. Serial and Parallel execution of WIEN2k, Working on a cluster, Working on a remote computing system.

 

12.00
Unit V: 
V

Quantum Espresso: Essential Linux Commands. Pseudopotentials in Quantum Espresso. Structure of a program in Quantum Espresso: Symbolizing lattice types, Cell parameters, Atomic Species, Atomic positions, Irreducible Brillouin zone sampling, k-point sampling and other parameters. A sample PW scf code. Self consistent solution of Schrödinger equation. Total energy, Various contributions to total energy, convergence tests for ecut-wfc, ecut-rho and k-points. Structure-stability considerations. Energy lattice constant Diagram and Equation of state.

 

Structure of files for calculation of electron density, density of states and band structure. PSOtting of diagrams with gnuPSOt. Calculation and PSOtting of Fermi surface. Super cell construction, introduction of impurity in the cell. Sample code for a system with impurity. Running of Quantum Espresso programs in serial and parallel mode and on a cluster of computers.

 

References: 
  • A Ralston and P. Rabinowitz, “A First Course in Numerical Analysis”, McGraw Hill (1985).

·         S.S. Sastry, “ Introductory Methods of Numerical Analysis”, Prentice-Hall of India (1979).

·         P. Blaha, K. Schwarz, G. Madsen, D. Kvasnicka, and J. Luitz, “WIEN2k- An augmented Plan Wave Plus Local Orbitals Program for Calculating Crystal Properties”, User’s Guide, Vienna University of Technology, Vienna (Austria)

·         D. Singh, Plane Waves, Pseudopotentials and the LAPW method, Kluwer Academic (1994).

·         Users’ Guide for Quantum Espresso (V.6.2).

·         Cottenier, “Density Functional Theory and the family of (L) APW-methods: a step by step introduction”, WIEN2k website (2013).

Academic Year: 
2021-2022
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