NUMERICAL METHODS AND COMPUTATIONAL PHYSICS

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).

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.

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.

a) 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.

b)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 plotting 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. 

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. Plotting of diagrams with gnuplot. Calculation and plotting 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.