Introduction to the Basic Terms of Band Structures©
Gion Calzaferri
Department of Chemistry and Biochemistry
University of Berne
Freiestr. 3
Bern CH-3012
mail to: gion.calzaferri@iac.unibe.ch

Stephan Glaus
Department of Chemistry and Biochemistry
University of Berne
Freiestr. 3
Bern CH-3012

Marc Meyer
Department of Chemistry and Biochemistry
University of Berne
Freiestr. 3
Bern CH-3012


In this document you will find an explanation of the concepts and application of band structures, translational symmetry, crystal orbitals, Bloch functions, wave vectors, the Peierls distortion, density of states (DOS), crystal orbital overlap population (COOP) and Brillouin zones by means of a Mathcad program.

Extended structures starting from molecules as building blocks have been well elucidated in earlier publications. [1 - 4] Nevertheless, the concept of energy bands with the related terms and ideas is often unfamiliar to chemistry students. Since the topic is abstract and based on Bloch's theorem, which is not descriptive at all, there is a strong need for exhaustive visualization and interactivity if the theory should be made easier accessible to learners. We have therefore created a Mathcad course acting as an introduction to the theory of band structures, which has been successfully used for several years. It intensively makes use of interactivity and visualization. Because of a better understanding all examples are made with the zero differential overlap (ZDO) approximation. The formulas are kept simple and the numerical effort small. Furthermore the model compounds selected have been chosen in order to keep things as simple as possible.

The QBand Mathcad course is best used as a computer class in which the students interactively learn and practice individually or in small groups until they master the topic. A teacher will be needed for the discussion of the problems included in the course. How deep the students should go into theory can be decided by themselves or by the teacher since the course is built up of a main document containing the indispensable parts of the theory, while more involved, more detailed and more mathematical contents are included in files that can be opened as popups using hyperlinks within the main document. The course can also well be self-studied, especially by advanced quantum chemistry students. The problems range from simple visualization exercises to quite challenging problems that relentlessly test the thorough understanding of theory.

We have worked hard to draw an easily comprehensible flow from elementary quantum mechanics to research level topics such as the quantum-chemical description of three dimensional crystalline systems. At the end of the course, the student is therefore capable and encouraged to use our research-level tight binding program package BICON-CEDiT, which includes oscillator strength calculations and many more options. It is available with examples free of charge. [5] At the end of the course the students should also be able to understand and to benefit from the research publications [6 - 10] that will still deepen and broaden their understanding. In [8], for example, the terms related to band structures are transferred to large cluster structures. The publications are cited in the course, where suitable.

Overview of the Mathcad document

  1. Preliminary Information and Introduction
    The goal of the course, prerequisites, performance objectives and a short introduction are given.
  2. Finite, one-dimensional systems - One basis function
    Most chemists are more familiar with the discrete energies of molecular orbitals than with band structures of crystal orbitals although the two approaches are essentially similar. Our didactical approach is therefore to start with the description of the p molecular orbitals of finite linear and cyclic unsaturated hydrocarbon chains.
  3. Finite, one-dimensional systems - Three basis functions
    The chains and rings of chapter 2 are no longer described using only pp AOs but also using ps and ss orbitals. This enhances the understanding of the effects resulting from different shape and orientation of the AOs.
  4. Infinite rings - Three basis functions
    For the treatment of infinite rings or chains, the Bloch theorem is introduced as well as Bloch orbitals, the wave vector and the first Brillouin zone. Infinite rings are then described using Bloch crystal orbital coefficients and energy bands.
  5. Infinite rings - One basis function - Alternating / Non alternating bond lengths
    The effect of bond length alternation on the shape of the energy bands is investigated. The concepts of back-folding and the Peierls distortion are introduced.
  6. Infinite rings - Three basis functions - Alternating bond lengths
    The bond length alternation is applied to the expanded basis set, resulting in pp, ps, and ss bands.
  7. Density of states (DOS)
    The DOS is introduced and calculated for the bands discussed in chapter 6.
  8. Crystal Orbital Overlap Population (COOP)
    After introduction of the term, different COOP contributions and the total COOP are calculated for the pp basis.
  9. Overview: Energy band, DOS and COOP
  10. Band structure of a two dimensional carbon lattice
    The idea of this last part is to expand the learned band structure concept to two-dimensional square structures. In the course of this, the understanding of reciprocal space is deepened.
  11. Summary

Technical details

The course was written for Mathcad 2001i. For users with an older version, a Mathcad 8 release of the course is provided. This document works with minor curtailments (e.g. the formatting is not perfect due to the lack of tabs, and hyperlinks to regions within the document will not work while links to external files should work). All files are optimized for a display resolution of 96 dpi. 120 dpi also work, but the layout is significantly better using 96 dpi. A pdf file of the main program is provided for potential user examination. Interested faculty should download the zip files and open them in an appropriate directory. The version 8 file would work with Mathcad version 8 up to Mathcad 2001. The Mathcad file 2001i would be better to use with Mathcad 2001i and higher versions. For versions of Mathcad below 2001i the internal links within Qband.mcd do not work while external links do work. The external links arel files placed in the same directory as Qband.mcd. These are supplements that enrich the material and increase the depth to which students can use the documents to learn band structure.


[1] R. Hoffmann, Solids and Surfaces, VCH, New York, 1988
[2] John P. Lowe, Quantum Chemistry, Academic Press, San Diego, 1993
[3] B. J. Duke, B. O'Leary, Journal of Chemical Education, 1988, 65, 319
B. J. Duke, B. O'Leary, Journal of Chemical Education, 1988, 65, 379
B. J. Duke, B. O'Leary, Journal of Chemical Education, 1988, 65, 513
[4] A. Pisanty, Journal of Chemical Education, 1991, 68, 804
[5] M. Brändle, R. Rytz, S. Glaus, M. Meyer, and G. Calzaferri, BICON-CEDiT (tight binding program package, including oscillator strength calculations), Available at http://dcbwww.unibe.ch/groups/calzaferri
[6] M. Brändle and G. Calzaferri, Helv. Chim. Acta, 1993, 76, 2350
[7] R. Hoffmann, C. Janiak and C. Kollmar, Macromolecules,1991, 24, 3725
[8] S. Glaus, G. Calzaferri, R. Hoffmann, Chemistry - A European Journal, 2002, 8, 1785
[9] G. Calzaferri and R. Rytz, J. Phys. Chem., 1996, 100, 11122
[10] S. Glaus and G. Calzaferri, Photochem. & Photobiol. Sci., 2003, 2, in press

Editor's Commentary
Audiences: Upper-Division Undergraduate, Graduate Education / Research
Pedagogies: Computer-Based Learning
Domains: Physical Chemistry
Topics: Computational Chemistry, Mathematics / Symbolic Mathematics, MO Theory, Quantum Chemistry, Solid State Chemistry
File NameDescriptionSoftware TypeSoftware Version
BandInstructorNotes.pdf Read-Only Document
BandStructure.pdf Read-Only Document
BandStructure2001i.zip JCE Archive of Mathcad 2001i Computational Documents
QBAND11.pdf Read-Only Document
QBAND11.zip JCE Mathcad 11 ZIP file
JCE JCE Subscribers only: name and password or institutional IP number access required.
Meyer, Marc; Glaus, Stephan; Calzaferri, Gion. J. Chem. Educ. 2003, 80, 1221.
Comments to: gion.calzaferri@iac.unibe.ch
©Copyright 2003 Journal of Chemical Education