MathMods :: Joint MSc

1. Things to be simulated
a. Structure of crystals and glasses. Quasiperiodic structures. Crystallization and glass formation.
b. Elements of crystallography. Bravais lattices, crystal symmetry, reciprocal lattice, diffraction.
c. Bonding in solids. Binding energies.
d. Lattice vibrations. Dispersion relations, normal modes, phonons. Specific heat of crystals, classical, Einstein and Debye models.
e. Lattice defects: point defects, dislocations, plane defects.
f. Free electron gas. Drude model. Electric and thermal conductivities of solids.
g. Optical properties of metals and dielectrics.
h. Magnetism of solids. Superconductivity.

2. Simulations with classical particles – molecular dynamics methods
a. Simulation box and periodic boundary conditions
b. Basic Molecular Dynamics algorithm
c. Numerical integration methods for equations of motion: basic Verlet algorithm and
velocity Verlet algorithm
d. Methods for describing interactions: interatomic potentials
e. Identification of nearest neighbors: Verlet list and linked-list cell algorithm
f. Starting simulation: choosing initial positions and initial velocities, system equilibration and velocity
rescaling
g. Calculating thermodynamic parameters: temperature and pressure
h. Simulations in various thermodynamic ensembles: thermostatic and barostating

1. Axioms of quantum mechanics

2. Exact solutions of Schrodinger equation:
a. free particle (in one dimension)
b. particle in box (one- and two-dimensional)
c. harmonic oscillator (one-dimension)
d. rigid rotor (two- and three-dimensional)
e. hydrogen atom (separation of motion of the center of mass, solutions - orbitals (s, p(x,y,z), d(5
orbitals), energies, quantum numbers, radial density of probability, average distance of
electron from nuclei, average kinetic energy)
f. variational method: optimization of parameter c in trial function f(r,c) = exp(-cr) (expected
values of hamiltonian and variational calculation).

3. Periodic quantum systems (Bloch states, SF, HF, band theory, APW, KKR, DFT, etc.)
a. The molecule and crystal Hamiltonians
b. The Born-Oppenheimer approximation
c. The variational principle
d. Spin. The Slater determinant
e. The Hartree-Fock equations
f. The Roothan equations
g. The Hartree-Fock equations
h. The density functional theory
i. The Bloch theorem. Common principles of solving the problems of zones. Dispersion equation
j. The Kronig-Penney model of a crystal. Conductors, insulators, semiconductors.
k. Electrons in empty lattice. The Fermi energy.
l. Computer practice in computer solving of equations of quantum mechanics
m. The model of weakly bond electrons
n. Computer practice in solving of equations of quantum mechanics
o. The model of tightly bond electrons

1. Applications of classical simulations with particles
a. Implementation of basic Molecular Dynamics algorithm
b. Visualization of molecular systems
c. Introduction to the LAMMPS program
d. Dispersion curve of argon dimmer
e. Specific heat of argon single crystal
f. Bulk modulus of argon single crystal
g. Melting temperature of argon single crystal
h. Analysis of structure of molten metals (Cu, Mn, Al, ...)

2. Applications of quantum simulations with particles
a. Choice of molecular structures for quantum calculations, proposition/prediction of possible reactions mechanisms, electronic-structure change during the processes.
b. Internal and cartesian coordinate systems, constrains.
c. SCF-based algorithms in quantum calculations, choice of the best methods for open-shell and closed-shell electronic configurations,
d. Introduction to GAMESS package of programs, operating with GAMESS, its documentation,
e. Optimization of C2H5OH molecular structure,
f. Optimization of stationary point's structures of substituted p-xylylene-based diradical dimmers by means of Hartree-Fock methods,
g. Calculation of energy profiles for reactions of substituted acethylene molecule with diradical p-xylylene dimmers.
h. Looking for saddle points of all reactions studied, calculations of hessians, vibrational normal modes analysis.
i. Calculations of relative energies.
j. Comparison of all results: for each substituent in acethylene structure and for each possible kinetic barriers with appropriate reactions of vinyls with diradical p-xylylenes.
k. Comparison with possible initiation of polymerization reactions of two acethylene molecules and two p-xylylene molecules.
l. Graphical represenation of the results: diagrams (gnuplot), electron orbitals (molden), vibrational modes (molden).

1. Fundamentals of Nanotechnology
a. Synthesis of nanomaterials: examples of processes top-down and bottom-up;
b. Imaging nanomaterials;
c. Examples of nanomaterials;
d. Selected mechanical and thermal properties of nanomaterials and nanostructured materials;
e. Selected electronic, optical and magnetic properties of nanomaterials;
f. Examples of applications of nanomaterials .

2. Computer modeling of structure and mechanical and thermodynamic properties of nanostructures
a. Molecular Dynamics simulation of graphehe: setting up a model
b. Molecular Dynamics calculations of Young's modulus of graphene
c. Molecular Dynamics calculations of Poisson's ratio of graphene
d. Molecular Dynamics studies of influence of defects on mechanical properties of graphene

Emil Roduner , Nanoscopic Materials. Size-dependent Phenomena, Stuttgart, Germany The Royal Society of Chemistry 2006

1. Basic continuum theory
a. Tensors and tensor fields
b. Analysis of Stress-Strain
c. Deformation and Strain
d. Shear Strain
e. General Elasticity
f. Waves
g. Introduction to thermoelasticity
h. Multi-dimensional theories of thermoelasticity

2. Plasticity, discrete-continuous models, granular matter, composites, non-newtonian liquids, etc.
a. Classical continual models of plasticity
b. Discrete models as generalized solutions of continuum equations
c. Review of modern discrete-continuous models: three kinds of models
d. Hybrid discrete-continuous models
e. Some mathematical results for the theory of elasticity equations
f. Numerical methods of solving of equations of continual models
g. Numerical methods of solving of equations of continual models
h. Newtonian and non-Newtonian fluids
i. Computer practice in computer solving of equations of discrete-continuum models
j. Newtonian and non-Newtonian fluids
k. Granular and multiphase materials, composite materials
l. Granular and multiphase materials. The case of high-concentrated materials.