Kikuchi Akihito. Computer Algebra and Material Design

Tokyo: Canon Inc., 2016. — 192 p.This work is intended to an introductory lecture in material physics, in which the modern computational group theory and the electronic structure calculation are in collaboration. The effort 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 the computer algebra for the applications in the solid-state simulation, by means of one of the computer algebra package, the GAP system. The 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 of material design, which could be executed in mathematically rigorous and systematic way. Some technical details and some computations which require the knowledge of a little higher mathematics (but computable easily by the computer algebra) are also given. The selected topics will provide the reader with some insights toward the dominating role of the 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 to the system without periodicity). The principal object of the study is to illustrate the hierarchical change of the quantum-physical properties of the molecule, in accordance with the reduction of the symmetry (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 computer 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.Introduction
Computation of group theoretical properties using GAP
Some preliminaries
Projection operator
Space group
Crystallographic group
Theoretical set-up for Wyckoff positions
Units in the computation
Application 1: Identification of wavefunctions to irreducible representations
The simplest case: at Γ point
Character table computation in super-cell
Application 2: A systematic way of the material designing
How to manage quantum-dynamics in crystal?
Control of band gaps by means of the Wyckoff positions
A possible plan of material design in the super-cell
Technical details
Determination of crystal symmetry
Symmetry operations in the reciprocal space
The computation of the compatibility relation
The character table at the boundary of the Brillouin zone
The semidirect product and the induced representation
The computation of the character table in the cubic diamond by GAP
Crystal lattice and Cohomology
Remarks to this section
Symmetry in C60
The initial set-up
The analysis of the eigenstates
The analysis again, against the failure in the identification
The perturbation and the symmetry
The possible deformation in the reduced symmetry
The reduction of the symmetry, from the view point of the orbit in the permutation
The symmetry in the eigenstates (hidden one)
The super-symmetries
Remarks to this section
Analysis of vibrational mode in C60
The vibrational modes of C60
One technical problem
Remark to this section
Conclusion
Symmetry operations in the diamond structure (real space)
Symmetry operations in the hexagonal lattice (real space)
Symmetry operations in the diamond structure (wave-number space)
Symmetry operations in the hexagonal lattice (wave-number space)
The computation of Wyckoff positions by GAP
Subgroup lattice
The analytical representation of the energy spectra of C60.
References