以GPAW全盤探討金原子塊材的數值方法及其特性

Abstract

本論文係利用丹麥技術大學團隊開發之自由軟體GPAW，改善用來分析金原子塊材的科恩–沈呂九方程之計算方法。對於一個由金原子所組成的塊材來說，第一原理計算所採用的密度泛函理論包括了兩個計算量相當大的方程，其一為科恩–沈呂九方程的非線性特徵值問題，其二則是哈特里電位勢的帕松方程計算。對於密度泛函方程的離散，在本研究中考慮非週期邊界條件並採用實空間網格的有限差分法。 除了在實空間網格上執行科恩–沈呂九方程的有限差分法外，本文的目標是改進在大尺寸情形下的特徵值方程的計算，進而保存適合的疊代法。由於非週期性邊界條件及材料中原子的不規則分佈，造成所涉及的矩陣方程具不對稱的性質。在此，本文將共軛梯度法逐步發展成雙共軛梯度法與廣義最小殘量法，並比較這兩個新方法的正確性與計算效率。

To this thesis, the open source code GPAW, which had been developing by DTU Physics, is used to analyze the gold bulk system via different numeric methods. Simulations based on first-principle density functional theory for gold atom involves solving two most computationally expensive equations, namely, the nonlinear eigenvalue problem for Kohn-Sham equation and the inhomogeneous calculation of Poisson equation for the indispensable Hartree potential. For the discretization of the density-functional equation, in this study, adopting finite difference mode could implement it in motion real-space grids for the considerations of a reliable treatment of non-periodic boundary conditions and for an efficient parallelization of the eigenvalue and Poisson equations that require most of the computing time. Besides running the finite difference Kohn-Sham equation on real-space grids, aiming to improve the computation of large-sized eigenvalue equation saving the iterative solvers that are suitable implemented in parallel. Due to the asymmetric nature of the involved matrix equations owing to the non-periodic boundary conditions and the indefinite nature resulting from irregular distribution of atoms in the materials, this thesis develop the conjugate gradient method into biconjugate gradient method and generalized minimal residual method, and investigate for their performance between input materials.