530018 Introduction to electronic structure calculations, 3 sw

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).

• R.G. Parr and W. Yang, Density-Functional Theory of Atoms and Molecules, Oxford University Press, 1989.
  
• J.M.Thijssen, Computational Physics, Cambridge University Press, 2001.
  
• M.L. Cohen, J.R. Chelikowsky, Electronic Structure and Optical Properties of Semiconductors, Springer, 1988
  
• A.A. Katsnelson, V.S. Stepanyuk, A.I. Szasz, O.A. Farberovich, Computational Methods in Condensed Matter: Electronic structure.AIP 1992.
• A.P. Sutton, Electronic structure of materials, Clarendon Press, Oxford 1994.

List of contents

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.

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Lecture 6. Tight-binding approximation. Orthogonal, non-orthogonal tight-binding. The Slater-Koster method. Linear scaling algorithms.

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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.

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Lecture 8. Muffin tin approximation and augmentod plane wave formalism. Pseudopotentials.

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Lecture 9. Density-functional theory. Local density approximation (LDA). Beyond LDA. Plane wave formalism. The fast Fourier transform.

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Lecture 10. Car-Parinello simulations. Quantities calculated in atomistic simulations and their relation to experimentally measurable characteristics.

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Lecture 11. Correlations. Many-body perturbation theory. Configuration interaction and coupled-cluster methods.

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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)!

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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)!

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Lecture 14. Summary of the course. Review of commercial (and non-commercial) ab initio electronic structure calculation software.

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Example examination questions and some extra material

Examination questions in PS format.

Proof of the Hellmann-Fyenmann theorem for a non-orthogonal TB model.

Exercises

Exercise 1. Variational problems. The direct iteration method.

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Exercise 2. Hartree-Fock methods and Gaussian basis sets by the example of helium atom.

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Exercise 3. Writing a tight-binding code.

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Exercise 4. Tight-binding models and the eigen value problem.

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Exercise 5. Car-Parrinello method.

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Exercise 6. Hydrogen-like atoms and perturbation theory.

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Exercise 7. Local density of states. Recursion methods.

Exercise explanation in PS format.

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