Numerical Methods Field of study: Physics
Programme code: W4-S2FZA22.2022

1. History of classical and ab initio simulation methods. 2. Inter-atomic interaction potentials. Models of rigid and non-rigid molecules, intra- and inter-molecular interactions. Constructing an intermolecular potential. Isolated and bulk molecular systems (periodic boundary conditions, the nearest image convention, spherical truncation of interaction). 3. Typical shapes of computer simulation box. Deterministic methods of computer simulations: Newtonian equations of motion for atomic systems (centers of molecular masses), methods for solution of ordinary differential equations (the Verlet algorithm, the leap-frog method, the velocity form of the Verlet algorithm, predictor corrector method). 4. Molecular dynamics of rigid molecules, description of rotational motion (quaternions), methods for solution of the Euler’s equations (the leap-frog and predictor-corrector method), constraint dynamics – SHAKE method, molecular dynamics of hard spheres. 5. The initial configuration (positions, orientations and velocities in accordance with the required temperature), elimination of the total momentum, reduced (internal) units, control parameters in the equilibration run, forces, shifted and shifted-force potentials. 6. Long-range interactions (Coulomb and dipole interactions), Ewald summation method, errors of summation in the real and reciprocal space – selection of the convergence parameter and cut-off radii in the Ewald method, partial charges in polar molecules. 7. Average values and fluctuations, generalized equipartition, simple thermodynamics averages (energy, temperature, pressure), transforming averages between statistical ensembles, the specific heat. 8. Structural properties (pair distribution function, structure factor), long-range correction of energy and pressure. 9. Time correlation functions and transport coefficients (the diffusion coefficient – the Einstein relation and the velocity correlation function), the diffusion equation in restricted space. 10. Molecular dynamics for micro-canonical, canonical (constraint method, velocity scaling, extended system and Berendsen method), isobaric and isobaric-isothermic ensembles. 11. Stochastic methods of computer simulations: brownian dynamics, Monte Carlo methods (the Metropolis method, isothermic-isobaric and grand canonical Monte Carlo). 12. Basic techniques of ab initio molecular dynamics: Ehrenfest molecular dynamics (EMD), Born-Oppenheimer one (BOMD) and Car-Parinello molecular dynamics (CPMD) (lagrangian and equations of motion). Hellmann-Feynman forces. Comparison of the ab initio molecular dynamics methods. 13. Conjunction of CPMD with the density functional theory. Implementation of the CPMD with plane waves. Electrostatic energy, exchange and correlation energy. Optimizing the Kohn-Sham orbitals. Program organization and layout. 14. Atoms with plan waves – pseudo-potentials, thermostats and barostats, hybrid quantum/classical molecular dynamics. 15. Application of the ab initio molecular dynamics – from materials to biomolecules. Properties from ab initio simulations: electronic structure analyses, infrared spectroscopy, NMR and EPR spectroscopy.

Ability to program in any language that allows procedural programming (Fortran 90/95 or C / C ++ recommended). Knowledge of the basics of mathematical analysis (differentiation and integration) and linear algebra.

[1] Tao Pang, An Introduction to Computational Physics, Cabridge University Press, Cambridge 2006 (CINiBA). [2] Forman Acton, Numerical Methods that Work, Harper and Row, New York 1970 (CINiBA).