A iterative finite method approach to Kohn-Sham equations of light atoms: Density Functional Theory tutorial for undergraduate students

TL;DR

This paper presents an iterative finite element method for solving Kohn-Sham equations in light atoms, providing a detailed tutorial suitable for undergraduates, including code and analysis of convergence and stability.

Contribution

It introduces a step-by-step FEM approach to density functional theory for helium, with comprehensive educational material and convergence analysis.

Findings

Successful implementation of FEM for helium's DFT calculations

Demonstration of convergence and stability of the iterative method

Provision of educational code for undergraduate learning

Abstract

In this article we are going to study the FEM solution to the Density Functional description of Helium. Solving self-consistently including electron-electron repulsion and exchange-correlation effects. This project will be split in four different consecutive task with different approaches: Numerical Hartree potential (Sect. I) and nuclear potential (Sect. II) for the hydrogen and resolution of Schrodinger equation for hydrogen (Sect. III) and helium (Sect. IV). The code is included in the format Sinajo with $n\in{0,1,2,3,4}$ corresponding with supporting material (code) to each task. The format of each section is standard in three cases. Beginning with a theoretical introduction, we mainly explain the code and show the principal results in form of values or graphics. Later on, we try to study the convergence and stability of the method proposed. Exception to this rule can be found in Section II, corresponding to the analytical deduction of a potential term.