摘要：
在凝態理論中，多電子與單電子的差別， 在於多電子中之電子與電子間的交互作用，使得系統變得異常複雜。Hohenberg and Kohn 提出， 用電子密度取代多電子波函數作為處理多電子問題。以電子密度作為架構一切的基礎， 這個想法把原本需要處理的三倍電子個數的維度大大的降低到只剩3 個維度。又被稱為密度泛函理論。
Kohn-Sham 將多電子中，之電子與電子間的交互作用簡化為，古典中的電子間的庫倫作用力和另一個比較小的部分稱為交換和相關(exchange-correlateion)作用。我們知道古典中的電子間的庫倫作用力的正確解，可是不瞭解交換和相關(exchange-correlateion)作用的解。最簡單的近似是假設電子雲是均勻分佈且
各項同性，稱為區域密度近似(Local-Density Approximation, LDA)。其後有考慮自旋極化(Spin-polarization)的(Local-Spin-Density Approximation,LSD)。
我們用Gunnarsson, Lundqvist, and Wilkins 所建議的關於交換和相關作用(exchange-correlateion)的方程式。並利用氫原子作為簡單的例子，瞭解LDA和LSD 如何實際影響氫原子中的電子與電子交互作用。
我們用Python 作為編譯程式。Python 對初學者而言是非常友善的，幾乎可以在所有應用系統中運行，且已有十多年的發展歷史，是一個成熟又穩定的語言。此外Python 具有非常強大的程式庫，通過它們可以快速完成絕大部分常用的任務。我們再利用Python 中專門設計來做數值計算功能的numpy 和scipy，或是具有繪圖功能的MATplotlib 來協助我們的工作。總的來說Python 被稱為一門易
讀性、易維護性好，並且被大量用戶所歡迎的、用途廣泛的語言。

Abstract
A major difference between the problems of single-electronic systems and that of the much more complicated many-electronic systems arises from the electron-electron interactions Uee. In the later case, we have to deal with
3N-dimensional wave functions , since each electron contains three spatial dimensions. The magnitude of the number of degrees of freedom for the wave functions was a major bottleneck for serious numerical calculations on these
systems.
Instead of dealing directory with 3N-dimensional wave functions, the density functional theory (DFT) recasts this problem in terms of the 3N dimensional electronic density distribution function n(r). The most important theoretic justification for DFT was provided by the work of Hohenberg
and Kohn (HK) . Hohenberg and Kohn proved that the external potential v(r) of a many-electronic-system can be readily deduced from the electronicdensity of the ground state n(r)
Further progress of the Density Functional Theory (DFT) was furnished by the Local-Density Approximation(LDA) proposed by Kohn and Sham. With the Local-Density Approximation(LDA), the Density Functional Theory(DFT) became apractical and an efficient tool for studying complicatedelectronic systems through ab inito calculations. The Local-Density Approximation(LDA) may be generalized by considering the contributions from distinctspin states , (σ =↑, ↓), as distinct contributions, this leads to the local spin density (LSD) approximation. In this thesis we follow the interpolation formulas by Gunnarsson, Lundqvist and Wilkin for the exchange and correlation functional of the LSD.
The hydrogen atom was used as a simple electronic system to check the applicability of the electron exchange and correlation energy approximation in the LDA and LSD approaches.
The goal of this thesis is to apply python to the atomic system calculation. As a first step for complicate atomic system we start with the hydrogen atom. The asymptotic behavior of the solution at r → 0 and r → ∞ were used for stable numerical solution. of the atomic Schrぴodinger equations.

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