Towards first-principles calculations using localised spherical-wave basis sets

• Main Author: Gan, C. K.
• Published: University of Cambridge 2001
• Subjects: 530.1
• Online Access: http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.599293

First-principles calculations based on density-functional theory are important for the study of a wide range of systems. However, conventional density-functional-theory calculations are very expensive since the computational effort and memory requirement scale as the cube and square of the system size, respectively. Recently there has been a surge of activity to investigate linear-scaling methods (where the computational effort and memory requirement scale linearly with the system size), all of which use localised basis sets in their implementations. One localised basis set proposed for linear-scaling methods, the spherical-wave basis set, is interesting because while sharing some of the properties (such as the concept of energy cutoff) with the popular extended plane-wave basis set, it possesses other advantages such as each basis function being fully localised within a sphere. Even though this basis set has been used to implement a linear-scaling method which has been tested on bulk crystalline silicon, the basis set itself has not yet been studied. This dissertation investigates the accuracy and properties of this localised basis set using an iterative matrix diagonalisation approach, which frees us from having to consider other sources of error introduced in linear-scaling methods. The matrix diagonalisation approach requires an efficient solution of the generalised eigenvalue problems <I>Hx = ^ΕSx</I>. I have proposed a new and efficient iterative conjugate-gradient method to obtain the lowest few eigenvalues and corresponding eigenvectors of the generalised eigenvalue problem. This method exhibits linear convergence. A preconditioning scheme which uses the kinetic energy matrix is introduced to improve the convergence of the solutions. The scheme is controlled by a single parameter whose optimal value may be chosen automatically.