Kikuchi Akihito. Computer Algebra and Material Design

Tokyo: Canon Inc., 2016. — 192 p.This work is intended to be an introductory lecture in material physics, in which the modern computational group theory and the electronic structure calculation are in collaboration. The efforts of mathematicians in field of the group theory, have ripened as a new trend, called “computer algebra”, outcomes of which now can be available as handy computational packages, and would also be useful to physicists with practical purposes. This article, in the former part, explains how to use computer algebra for applications in solid-state simulation, using one of the computer algebra packages, the GAP system. Computer algebra enables us to obtain various group theoretical properties with ease, such as the representations, the character tables, the subgroups, etc. Furthermore, it would grant us a new perspective on material design, which could be executed in a mathematically rigorous and systematic way. Some technical details and some computations that require the knowledge of a little higher mathematics (but computable easily by computer algebra) are also given. The selected topics will provide the reader with some insights into the dominating role of symmetry in crystal, or, the “mathematical first principles” in it. In the latter part of the article, we analyze the relation between the structural symmetry and the electronic structure in C60 (as an example of the system without periodicity). The principal object of the study is to illustrate the hierarchical change of the quantum-physical properties of the molecule, by the reduction of the symmetry (as it descends in thadder of subgroups). As an application, this article also presents the computation of the vibrational modes of the C60 using computer algebra. 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.