Overview of Chapters 12–17

There are three basic approaches to the calculation of independent-particle electronic states in materials. There are no fundamental disagreements: all agree when applied carefully and taken to convergence. Indeed, each of the approaches leads to instructive, complementary ways to understand electronic structure and each can be developed into a general framework for accurate calculations.

• Each method has its advantages: each is most appropriate for a range of problems and can provide particularly insightful information in its realm of application.

• Each method has its pitfalls: the user beware. It is all too easy to make glaring errors or over-interpret results if the user does not understand the basics of the methods.

The three types of methods and their characteristic pedagogical values are:

1. Plane wave and grid methods provide general approaches for solution of differential equations, including the Schrödinger and Poisson equations. At first sight, plane waves and grids are very different, but in fact each is an effective way of representing smooth functions. Furthermore, grids are involved in modern efficient plane wave calculations that use fast Fourier transforms.

Chapter 12 is devoted to the basic concepts and methods of electronic structure. Plane waves are presented first because of their simplicity and because Fourier transforms provide a simple derivation of the Bloch theorem.

Summary

Local spin density approximation (LSDA)

The local density approximation is based upon the exact expressions for the exchange energy, Eq. (5.15), and various approximations and fitting to numerical correlation energies for the homogeneous gas. Comparison of the forms is shown in Fig. 5.4. The first functions were the Wigner interpolation formula, Eq. (5.22), and the Hedin–Lundqvist [220] form; the latter is derived from many-body perturbation theory and is given below. As described in Ch. 5, the quantum Monte Carlo (QMC) calculations of Ceperley and Alder [297], and more recent work [298, 299, 303] provide essentially exact results for unpolarized and fully polarized cases. These results have been fitted to analytic forms for εc(rs), where rs is given by Eq. (5.1), leading to two widely used functionals due to Perdew and Zunger (PZ) [300] and Vosko, Wilkes, and Nusiar (VWN) [301], which are very similar quantitatively. Both functionals assume an interpolation form for fractional spin polarization, and Ortiz and Balone [298] report that their QMC calculations at intermediate polarization are somewhat better described by the VWN form.

The “TBPW” code is a modular code for Slater–Koster orthogonal tight-binding (TB, Sec. 14.4) and plane wave empirical pseudopotential (EPM, Sec. 12.6) calculations of electron energies in finite systems or bands in crystals. This appendix is a schematic description. A much more complete description, sample files, and the full source codes are available on-line at the site given in Ch. 24.

TBPW is meant to be an informative, instructional code that can bring out much of the physics of electronic structure. It has many of the main features of full density functional codes, but is much simpler. A main characteristic is that the codes are modular and are organized to separate the features common to all electronic structure calculations from the aspects that are specialized to a given method.

A sample input file is given below in Sec. N.4. The input file uses keywords that are recognized by the input routines, so that the same file can be used for either TB or PW calculations.

Summary

As emphasized in the previous chapter, the genius of the Kohn–Sham approach is two-fold: first, the construction of an auxiliary system leads to tractable independent-particle equations that hold the hope of solving interacting many-body problems. The famous Kohn–Sham equations are given in (7.11)–(7.13). Second, and perhaps more important, by explicitly separating out the independent-particle kinetic energy and the long-range Hartree terms, the remaining exchange–correlation functional Exc[n] can be reasonably approximated as a local or nearly local functional of the density. Even though the exact functional Exc[n] must be very complex, great progress has been made with remarkably simple approximations. This chapter is devoted to those approximations.

Summary

As emphasized in the overview, Sec. 2.10, two types of excitations are of primary importance for electronic structure: excitations in which an electron is added or subtracted from the system, and excitations in which the number of electrons remains fixed. The former are of great interest as the “one-electron excitations” in an interacting many-body system; however, in independent-particle theories, such as the Hartree–Fock or Kohn–Sham approaches, these excitations are just the eigenstates of the independent-particle hamiltonian. In a crystal, the eigenvalues form well-defined bands εik with none of the renormalization and broadening that can only be included in a many-body theory.

Summary

The theory of electrodynamics of matter [448, 790] (see App. E) is cast in terms of electric fields E(r, t) and currents j(r′, t′). (Here we ignore response to magnetic fields.) In metals, there are real currents and, in the static limit, electrons flow to screen all macroscopic electric fields.

Summary

Replacing one problem with another

The Kohn–Sham approach is to replace the difficult interacting many-body system obeying the hamiltonian (3.1) with a different auxiliary system that can be solved more easily. Since there is no unique prescription for choosing the simpler auxiliary system, this is an ansatz that rephrases the issues. The ansatz of Kohn and Sham assumes that the ground state density of the original interacting system is equal to that of some chosen non-interacting system.

It is not appropriate to summarize or conclude this volume on the basic theory and methods of electronic structure. The field is evolving rapidly with new advances in basic theory, algorithms, and computational methods. New developments and applications are opening unforeseen vistas. Volumes of information are now available on-line at thousands of sites.

It is more appropriate to provide a resource for information that will be updated in the future, on-line information available at a site maintained at the University of Illinois:

http://ElectronicStructure.org

A link is maintained at the Cambridge University Press site:

http://books.cambridge.org/0521782856.htm

Resources for materials computation are maintained by the Materials Computation Center at the University of Illinois, supported by the National Science Foundation:

http://www.mcc.uiuc.edu/

Additional sites include the Department of Physics, the electronic structure group at the University of Illinois, and the author's home page:

http://www.physics.uiuc.edu/

http://www.physics.uiuc.edu/research/ElectronicStructure/

http://w3.physics.uiuc.edu/~rmartin/homepage/

The on-line information includes:

• Additional material coordinated with descriptions in this book, as well as future updates, corrections, additions, and convenient feedback forms.

• Information related to many-body methods beyond the scope of this volume.

• Links to courses, tutorials, and codes maintained at the University of Illinois. Specific codes are meant for pedagogical use, teaching, or individual study, and are coordinated with descriptions in this book, for example, the general purpose empirical pseudopotential and tight-binding code (TBPW) in App. N.

• Links to codes for electronic structure calculations. This will include resources at the Materials Computation Center and many other sites.

• […]

Summary

Every textbook on solid state physics begins with the symmetry of crystals and the entire subject of electronic structure is cast in the framework of the eigenstates of the hamiltonian classified in k space by the Bloch theorem. So far this volume is no exception. However, the real goal is to understand the properties of materials from the fundamental theory of the electrons and this is not always the best approach, either for understanding or for calculations.