A first encounter with the Hartree-Fock self-consistent-field method

Abstract

This paper is intended to serve as a bridge between introductory textbooks on quantum mechanics, which typically do not cover the Hartree-Fock self-consistent-field method, and more advanced ones which treat this important computational method for fermionic many-body systems in an abstract and formal way. We derive the Hartree-Fock equation for the 1s orbital of a realistic two-electron atom. By employing a two-dimensional basis-set representation, we avoid the use of variational calculus and are able to visualize key aspects of the method. We explain the basic self-consistent-field algorithm and provide a python script to illustrate how the algorithm works in practice. Utilizing perturbation theory, we perform an analysis of the convergence behavior of the self-consistent-field algorithm, thereby facilitating a deeper understanding of the numerical examples presented. We expect that this work will be useful for teaching computational techniques to physics students.