原子基態能量的計算

Abstract

基態能量在量子力學裡是代表最低能量。我們利用從Kohn-Sham 方程式以及local density approximation 中化簡出來之電子密度Eg 的基態能量泛函Eg(ρ) 來計算原子的基態能量。如果要計算Eg，我們必須先算出原子基態的波函數以及電子密度，而電子密度是跟波函數有關的函數。所以第一步，要利用解Kohn-Sham方程式HΨ =εΨ找出波函數。Kohn-Sham方程式是一個二階偏微分方程，Kohn-Sham 方程式中，ε 的最小值所對應的函數Ψ ，就是原子基態能量的波函數。為了在計算上的方便，我們將H 離散化，讓問題變成解特徵值的計算之後，利用自洽來解出波函數。此篇計算所得到的數據和實際值的數據誤差不超過15%。

The ground state is the lowest-energy state of the quantum mechanical system. We compute the ground state energy by using the ground state energy functional Eg[ρ] of electronic density ρ, which is deduced from the Kohn-Sham total-energy functional and local density approximation. To compute the ground state energy of atoms, we have to compute each electronic density of the ground state of atoms. The electronic density of atoms is a function about the wave function of atoms, so the first step is to determine the wave functions of ground state of atoms by solving the Kohn-Sham equation HΨ = εΨ. The Kohn-Sham equation is a problem of the second order partial differential equation. The wave function of the ground state of atoms is the function Ψ corresponding to the minimal ε of the Kohn-Sham equation. For the convenience of solving the Kohn-Sham equation, we discretize the Hamiltonian H of the Kohn-Sham equation and the problem become an eigenvalue problem. In this computation, we determine the wave function of the ground state of atoms by selfconsistency. The errors between the computation and the realistic values are less than 15%.