1. Applications of Electronic Structure calculations to physical systems
1. There are too many possibilities to list.  Prof. Martin can help you decide on a project and use available codes such as empirical pseudopotential and tight-binding codes, the "ab initio" packages ABINIT and VASP; chemistry packages; local orbital programs like SIESTA; and other codes.
2. Many-body problems in physical systems.  Too many possibilities to list.  Among them could be systems near metal insulator transitions.  (review in 1998 Rev Mod Phys by Imada, et al.)
1. Summary of current ideas on electronic structure of Hi-Tc materials, e.g. photoemision and controversies of interpretation*
2. Evidence for "non-Fermi-liquid behavior" in materials*
3. One dimensional behavior in nanotubes*
4. Summaries of the current ideas on the "Colossal Magneto resistance" materials which are near a Mott metal-insulator transition.*
5. The new discoveries of metal-insulator transitions in 2 dimensions*
3. Density Functional Theory
1. Theory and actual recent examples of GGAs*
2. The "Optimized Effective Potential" method to find Kohn-Sam potential that minimizes energy for a functional of the orbitals such as the exchange energy*
3. Density functional theory of the superconducting state. Recent work has shown how to extend DFT to include the quantum nuclear vibrations and the pairing field in superconductivity. This has led to the first quantitative calculations with no adjustable parameters, with predictions in extremely good agrement with experiment. References are on the preprint archives: cond-mat/0408685; cond-mat/0408686; and cond-mat/0408688
4. Theory of Electronic Structure
1. Theory of Pseudopotentials - starting from class notes
2. Linear scaling "order N" methods.  A project could be a comparison of various methods proposed recently.*
3. Theory of alloys using averaging approximations for disorder (and beyond).*
4. Theory of Atomic Sphere methods in solids.  This is barely covered in class.  A project could go further to show the usefulness of atomic sphere approximations to have simple calculations for solids.
5. Theory of polarization in solids and Berry's phases.
6. Theory of Wannier Functions: history and recent advances*
5. Construction of programs
1. Construct a working empirical tight binding program which evaluates forces on the atoms by a Hellman-Feynman-like formula.  Use it to relax the atoms to minimum energy in a molecule or crystal.   This requires  terms in the energy in addition to the band structure terms.

Prof. Martin can give references for several papers for total energies.
You can build upon the class program.
2. Construct a self-consistent plane-wave pseudopotential program program which evaluates total energies.  This requires several steps part of which are done in class.  This project may be done using class codes as a model.   One could also calculate forces on the atoms by a Hellmann-Feynman formula.  This can be used straightforwardly to find phonon frequencies.  A next step would be to relax the atoms find the positions of minimum energy .  (Prof. Martin  can give examples for comparison.)
3. Construct a pseudopotential generation program.  This can be built upon an atomic program like that in Koonin.  In this project there should also be a description of the theory of pseudopotentials.

Prof. Martin  can give examples for comparison.
4. Add non-local potentials to the plane-wave pseudopotential program program.  For this project, the potential s can be used from the Martins code on the Web.
5. Construct a working Car-Parrinello program (either based upon pseudopotentials or tight-binding) that can solve for the eigenvalues and ground state energies of complex systems and carry out thermal simulations and quenches.  If the potentials are local or the tight-binding parameters have a simple form, the program can be rather straightforward.   This could be applied, for example, to simple metals and semiconductors.
6. Perturbation theory ("2n+1 theorem")  Describe the theory and write a program to evaluate simple cases.*
7. Write a code to calculate polarization and localization using Berry's phases. Apply to simple model such as a one-dimensional model.

Prof. Martin  can give list of references.
8. Construction of Wannier functions.  There have been recent developments and it is straightforward to construct localized Wannier functions.*
9. Linear scaling program for calculation of total energy (and forces?).  Could be based upon the tight-binding program in class.*
10. Construct a working many-body program to exactly solve Hubbard-type models in finite clusters.  The programs can be made simple by finding simple algorithms to generate all the states (the number of which grows factorially with  the number of electrons) and to diagonalize sparse matrices.  This involves Lanczos algorithms and some programs are available.
11. Construct a quantum Diffusion Monte Carlo program that can solve problems with more than two electrons.  The program constructed in class is a relatively straightforward program - the reason QMC programs for many fermions are much more elaborate  is the difficulty introduced by the antisymmetry of the many-body wave function.   This can be handled approximately by a fixed-node condition.
6. Theory of electrons correlations and physical consequences
1. Luttinger "Theorem" on the volume enclosed by the Fermi surface.  Luttinger gave the argument which is very often quoted that in a many-body interacting electron system the volume of the Fermi surface is the same as in a non-interacting system.  His proof was based upon equating two expressions for the total number of electrons.  Luttinger was careful to state that his demonstration relies upon the assumption that perturbation theory converges starting from some non-interacting state - the conditions for validity of Fermi Liquid Theory.   For a term paper analyze Luttinger's proof in terms of diagrammatic summations, clarify the consequences for crystals, and give interesting applications, e.g., to CuO planes in Hi-Tc materials.

 J. M. Luttinger, PR 119, 1153 (1960).;   Abrikosov, et. al. textbook.
 R. M. Martin, PRL 48, 362 (1982), and references therein.
2.  Solitons and Fractional Charge.

 Reference:  M. J. Rice, et. al., PRL 23, 2136 (1983).
 Linear chains can have excitations with quantum numbers fundamentally different from those of the bare particles.  An example is provided by one of the simplest possible linear chains formed of carbon, which apparently exists under special circumstances based upon experimental evidence.  For this paper you should describe the ideas  of solitons and "charge fractionalization" and analyze a model 1-d problem that exhibits solitons. [This could be readily studied with a tight-binding program as described in the computational projects.]
3. "GW" Approximation for quasiparticle energies.

 This would be an extension of the topics covered in class to give  a more thorough analysis Hedin's formulation of the “GW” approximation for quasiparticle self-energies and the Plasmon Pole Approximation for evaluation of this term in a nearly-free-electron-like systems.  Discuss the RPA and simple vertex corrections that have been proposed for improving the RPA and results from at least two more recent applications of the GW approximation to solids.
 L. Hedin & S. Lundquist, Solid State Phys. vol. 23, 1 (1968).
 J. E. Northrup, et.al., PR 39, 8198 (1989).
4. A famous paper by W. Kohn (PR A 171 (1964)) entitled "Theory of the Insulating State" showed that electronic states can rigorously be considered to be localized in an insulator whereas they cannot in a metal.  This is well known in a crystal: because there is a band gap separating empty from filled bands, the latter can be described by localized Wannier states.  Kohn showed that in a disordered insulator with no gap in the density of states the eigenfunctions are also strictly localized.  A term paper could restate his theoretical analysis and describe the relation to metallic vs. insulating behavior in disordered materials.  More recently papers by Resta and by Ortiz describe operators to evaluate localization of electrons in solids.
5. Hubbard Model.  A topic for a term paper is a general summary on what is known about the Hubbard model and which physical systems it is argued to describe. A recent review is in Int. J. Mod. Phys. B 5 No. 6/7 (89).
6. Large D limit.  Describe the theory of "dynamical mean field theory" which is valid in the limit of large dimensions.  A recent review in in Rev Mod Phys by Georges, et al.  Also links to programs are given there.
7. Large N limit.  Describe the theory of large degeneracy expansions in impurity problems.  In this limit one can solve the Anderson impurity problem and find Kondo-type effects.  The theory due to Gunnarsson and Schonhammer allows calculations for cases like Ce where the large N limit appears to be very good approach.*

*Prof. Martin  can provide references to help start the project.