Computer Algebra and Material Design

This article is intended to an introductory lecture in material physics, in which themodern computational group theory and the electronic structure calculation are in collab-oration. The effort of mathematicians in field of the group theory, have ripened as a newtrend, called “computer algebra”, outcomes of which now can be available as handy com-putational packages, and would also be useful to physicists with practical purposes. Thisarticle, in the former part, explains how to use the computer algebra for the applicationsin the solid-state simulation, by means of one of the computer algebra package, the GAPsystem. The computer algebra enables us to obtain various group theoretical propertieswith ease, such as the representations, the character tables, the subgroups, etc. Further-more it would grant us a new perspective of material design, which could be executed inmathematically rigorous and systematic way. Some technical details and some computa-tions which require the knowledge of a little higher mathematics (but computable easily bythe computer algebra) are also given. The selected topics will provide the reader with someinsights toward the dominating role of the symmetry in crystal, or, the “mathematical firstprinciples” in it. In the latter part of the article, we analyze the relation between the struc-tural symmetry and the electronic structure in C60 (as an example to the system withoutperiodicity). The principal object of the study is to illustrate the hierarchical change ofthe quantum-physical properties of the molecule, in accordance with the reduction of thesymmetry (as it descends down in the ladder of subgroups). As an application, this article also presents the computation of the vibrational modes of the C60 by means of the com-puter algebra. In order to serve the common interest of the researchers, the details of the computations (the required initial data and the small programs developed for the purpose) are explained as minutely as possible. ∗ akihito [email protected] (The corresponding author)