於量子力學中一個新密度函數理論之有限元素數值解初探

Abstract

在此研究中，我們利用有限元素法解一個由Hsu[Hsu, 2003]推導出的新密度函數理論公式，此公式不同於Kohn-Sham方程式，其沒有因exchange-correlation項而另外做特別的假設來逼近問題。以有限元素法線性形狀函數的Galerkin殘差加權法來獲得特徵值線性代數方程式，再使用Jacobi-Davison法求解特徵值方程式。發展一維和三維有限元素法的程式並和實驗或理論資料做比較。用來測試程式的問題包含一顆電子系統（如氫原子）沒有電子間的作用、兩顆電子系統（如類氦原子）和四顆電子系統（如類鈹原子）有電子間的作用。結果顯示一維和三維的程式所獲得的氫原子特徵狀態能量與實驗資料相當接近，對氦原子的穩態能量而言，其三維的程式尚在發展中，相關的結果希望能夠在論文口頭報告時呈現，另外由於三維有限元素法勁度矩陣的對角線優勢，三維程式的收斂速度遠快於一維程式。

In the current study, we have used the finite element method (FEM) to solve a new formulation in density functional theory by Hsu [Hsu, 2003], in which, unlike Kohn-Sham equation, there is no exchange-correlation term, often requiring ad hoc assumption to close the problem. In this finite element method, Galerkin weighted residual method with linear shape function is used to obtain the eigenvalued linear algebra equations. Resulting eigenvalued equations are then solved using Jacobi-Davison method. Both 1-D and 3-D FEM codes are developed and compared with experimental or theoretical data wherever available. Benchmark test problems include one-electron system (e.g., hydrogen atom) without electron-electron interaction, two-electron system (e.g., helium-like atoms) and four-electron system (e.g., beryllium-like atoms) with electron-electron interactions. Results show that the eigenstate energies of hydrogen atom obtained by both 1-D and 3-D codes approach the experimental data. The ground state energy of helium atom using 3-D FEM code is still in progress. Related results hopefully will be presented in the oral examination of my thesis. In addition, convergence rate in 3-D code is generally much faster than that in 1-D code due to the diagonal dominance in the stiffness matrix of 3-D FEM.