As a result of recent successes in describing and predicting properties of materials, electronic structure calculations have become increasingly important in the fields of physics and chemistry over the past decade, especially with the advent of present-day, high-performance computers. Assuming a knowledge of the types of atoms comprising any given material, a computational approach enables us to answer two basic questions: What is the atomic structure of the material and what are its electronic properties ? The methods used to derive answers to these questions will be the subject of the course.
The student will be introduced to computational methods used in electronic structure calculations at various levels of sophistication: tight-binding approximation, density-functional theory, Hartree-Fock methods, etc. Lectures on concepts will be combined with practical exercises designed to enable the student not only to use the various standard codes, but also to develop new ones, if required. Special attention will be given to the electronic structure of nanosystems, such as carbon nanotubes and atomic clusters.
Programming and mathematical exercises are given during the course, but not every week. The programming exercises should be preferably solved in an Unix environment, but also solutions written under other environments in strict adherence to the Fortran90, Fortran(77) or ANSI C standards (so that they can be compiled anywhere) are acceptable.
Exercises (60 %)
Final exam (40 %)
Lecture notes (to appear in this page).Some parts of the material can also be found in:
Lecture 1. Introduction to the course. Review of the basics of quantum mechanics. The bras and kets formalism. One-particle systems. The hydrogen atom.Lecture notes in PS format.
Lecture 2. The Born-Oppenheimer approximation. Outline of empirical, semi-empirical and ab initio methods. Molecular dynamics and other methods of equilibrium/metastable configuration calculations. Variational methods in quantum mechanics.Lecture notes in PS format.
Lecture 3. Diatomic molecules. The nature of chemical bonding. Many-electron systems. The exclusion principle. Multi electron wave function. Non-interacting fermion systems. HOMO, LUMO.Lecture notes in PS format.
Lecture 4. Hartree and Hartree-Fock approximations. Self-consistent field method. Restricted and unrestricted Hartree-Fock. Gaussian- and Slater basis functions. Hartree-Fock-Roothaan method. Structure of a Hartree-Fock code.Lecture notes in PS format.
Lecture 5. More about orbitals. Fast integral evaluation. Semi-empirical methods. The Mulliken charges, orbital population. Vibration analysis.Lecture notes in PS format.
Lecture 6. Tight-binding approximation. Orthogonal, non-orthogonal tight-binding. The Slater-Koster method. Linear scaling algorithms.Lecture notes in PS format.
Lecture 7. From the finite to the infinite. Basic concept from the solid state physics. Reciprocal space. The Fermi surface. Band energy and bond energy. Tight binding for periodic solids. The density of states: total and local. The recursion method. The Peierls transition.Lecture notes in PS format.
Lecture 8. Muffin tin approximation and augmentod plane wave formalism. Pseudopotentials.Lecture notes in PS format.
Lecture 9. Density-functional theory. Local density approximation (LDA). Beyond LDA. Plane wave formalism. The fast Fourier transform.Lecture notes in PS format.
Lecture 10. Car-Parinello simulations. Quantities calculated in atomistic simulations and their relation to experimentally measurable characteristics.Lecture notes in PS format.
Lecture 11. Correlations. Many-body perturbation theory. Configuration interaction and coupled-cluster methods.Lecture notes in PS format.
Lecture 12. Electronic structure of solids near surfaces. Peculiarities of semi-infinite crystal calculations. The Tamm and the Shockley states. Experimental methods which measure the surface properties of solids. Scanning tunneling microscopy (STM), experimental setup and simulation of STM images. Electronic structure of nanosystems (1). Carbon nanotubes. Atomic and electronic structure of carbon nanotubes. The relation between the electronic structure of graphite and the nanotube electronic structure.Lecture notes in PS format. Quite large (34 Mb)! Try downloading pdf-file (5 Mb)
Lecture 13. Electronic structure of nanosystems (2). Atomic metal and semiconductor clusters. Magic numbers. The DFT-based jellium model. Quantum dots.Lecture notes in PS format. Quite large (24 Mb)! Try downloading pdf-file (2 Mb)
Lecture 14. Summary of the course. Review of commercial (and non-commercial) ab initio electronic structure calculation software.Lecture notes in PS format.
Examination questions in PS format.
Proof of the Hellmann-Fyenmann theorem for a non-orthogonal TB model.
Exercise 1. Variational problems. The direct iteration method.Exercise explanation in PS format.
Exercise 2. Hartree-Fock methods and Gaussian basis sets by the example of helium atom.Exercise explanation in PS format.
Exercise 3. Writing a tight-binding code.Exercise explanation in PS format.
Exercise 4. Tight-binding models and the eigen value problem.Exercise explanation in PS format.
Exercise 5. Car-Parrinello method.Exercise explanation in PS format.
Exercise 6. Hydrogen-like atoms and perturbation theory.Exercise explanation in PS format.
Exercise 7. Local density of states. Recursion methods.Exercise explanation in PS format.