Lecture 1  Symmetry properties of crystals

Elements of points groups, rotations, reflection, inversion, rotary-reflection.The examples 
of point groups (Cn, Cnh, Dnh, T,  Td, Oh ). Translational symmetry. Basis vectors. Elemen- 
tary cell. Volume of elementary cell.  Simple and complex cells.  Niggly theorem.  Crystals 
classes. Bravais lattices. Examples of simple structures (cubic simple,  cubic  volume  and
centered lattices, NaCl, Diamond , Wurzite, CsCl, BaTiO3 ) lattices. Direct  and reciprocal
lattices. Base vectors for reciprocal lattice (examples of calculation).  Wigner-Zeits cell.
space groups.Symmetry property of physical tensors. Experimental definition of crystal stru-
cture.The Laue and Bragg equations for X-Ray diffraction in Crystals. Atomic and structural 
scattering factors (explanation). Miller indices.

Lecture 2   The Interaction of Atoms in Crystal

Ionic, covalent, metallic and Van-der-Waals chemical bonds. Basis characteristics.

Additional materials
• Calculation of electrostatic energy for ionic crystals (Ewald method)
  

Lecture 3   Vibrations of crystal lattices

Description of atomic vibration  for  simple 1-d linear (forces acting on each atom have a 
linear dependence on  deflection of atoms from equilibrium position) lattice. Equation  of 
motion for atoms. Solution  of the system of the equations  of motion  (harmonic traveling
wave). Dispersion relation. Density of vibrations.Total internal mechanical energy of lat-
tice. Transition  from classical to  quantum-mechanical  description  of the  total energy.  
Bose-Einstein distribution.Calculation of  heat capacity. Compare with experiment and with 
classical approach. One dimensional two atomic chain (two atoms in elementary cell). Equa-
tion  of  motion for atoms. Dispersion relations. Optical and acoustic branches. Nature of  
atomic motion for q=0. 3d crystals.Expanding the interatomic potential energy into a power 
series.Calculation of the interatomic force  constants. Equation of motion for atoms in 3d 
complex crystals. Solution of  the  equation of motion. Dynamical matrix.  Diagonalisation. 
Dispersion relations, eig-envectors (vectors of polarization) and eigenvalues (frequencies 
of vibrations). Density for CsCl, Nacl and BaTiO3.Long-wave(q=0) vibrations in ionic crys-
tals.Mechanism of vibrations. Example of  LT-splitting (Effective electrical field). Ewald 
method for calculation the electrostatic energy. Ewald method and Ewald constant.

Lecture 4  Adiabatic approximation

Schredinger  equation for crystal (Full Hamiltonian). Presentation of total wave-function. 
Use of the fact that the electron mass is much less than the nuclear masses (adiabatic ap-
proximation).  Schredinger  equations  for nucleus and electrons. Separation the motion of 
electrons and nucleuses.

Lecture 5 Simple models for describing the properties of electrons in solids

• 5.1.1 Free electrons approximation

  Schredinger equation for free electrons in 1d lattice. Solution of equation, periodical boun-
  dary conditions, Fermi energy. Density of states. Fermi-Dirac distribution. Heat capacity for 
  gas of free electrons. Schredinger equation for free electrons in 3d cubic lattice.  Solution 
  of Schredinger equation, periodical boundary conditions, Fermi energy. Density of states. Fer-
  mi-Dirac distribution.Heat capacity of gas of the free electrons.  Brillouine zone.Bloch func-
  tions.
  

• 5.1.2 Free electrons in empty periodical lattice

  Schredinger equation for free electrons in 1d lattice. Solution of equation, periodical boun-
  dary conditions, Fermi energy. Density of states. Fermi-Dirac distribution. Heat capacity for 
  gas of free electrons. Schredinger equation for free electrons in 3d cubic lattice.  Solution 
  of Schredinger equation, periodical boundary conditions, Fermi energy. Density of states. Fer-
  mi-Dirac distribution.Heat capacity of gas of the free electrons.  Brillouine zone.Bloch func-
  tions.
  

• 5.2 Weak potential approximation.

  Dependence of energy on the wave vector for electrons in 1d lattice. Property of periodicity 
  for this dependency.  Calculation of the electron band structure by perturbation theory (for 
  weak periodical potential). The energy spectrum of almost free electrons.Bands structure and 
  forbidden energy bands. The bandgap appearing. 
  

• 5.3 Tight binding approximation(TB)

  Schredinger equation for isolated atoms and for high localized electrons in  crystal. Formati-
  on of wave function.Calculation the electron energy. Band structure. Forbidden zone. Band gap. 
  

• 5.4 Effective mass, 5.5 Excitons of large-radius

  Motion of electrons and holes near the gamma point of IBZ,Schredinger equation. Physical mean-
  ing of the effective mass. Effective  mass  calculations. Free  electron and hole. Schrodinger 
  equation.Motion separation.Optical properties of excitons. Example for simple cubic lattice in 
  TB approximation.
  

• 5.6 LAPW and LCAO methods

  Classical and quantum theory of conductivity for metals and intrinsic semiconductors.
  

• 5.7 Introduction_to_optical_properties

  Descriptiopn of the optical properties of solids. Classical approache.  Dielectric function
  and it properties. Reflection, absorbtion and refraction coefficients and  relation to die-
  lectric function. Methods for dielectric function calculation. 
  

• 5.8 Magnetic properties of free electrons

  Classical and quantum theory of paramagnetism for gase of free electrons.
  

• 5.9 Transport phenomena

  Classical and quantum theory of conductivity for metals and intrinsic semiconductors.
  

Lecture 6   Hrtree-Fock method

The solution of the Schredinger equation for electrons. Total exact wave function for electrons. 
The  presentation  of this function as a multiplication of one-electron functions or by determi-
nant ( Pauli principle). Coulomb and exchange energies. Effective external  field. Algorithm of 
Hartree-Fock calculations. The contribution of correlations. Method of pseudopotentials.

Lecture 7   Density Functional Theory (DFT) approach

General idea of DFT. Hohenberg-Kohn theorems. Kohn-Sham equations. Pseudopotentials. Existing 
software packages for DFT calculations.

Lecture 8   Practical abinitio calculations by using VASP.

Practical calculation by VASP of basic physical properties of simple materials. Lattice structure 
optimization. Calculation of electron band structue, elastic modulus and frequency of atomic vib-
rations.