Lies My Teacher Told Me About Density Functional Theory: Seeing Through Them with the Hubbard Dimer

TL;DR

This paper critically examines the fundamental aspects of density functional theory (DFT) using the Hubbard dimer model, clarifying misconceptions and visualizing key concepts like exchange-correlation functionals and excitation energies.

Contribution

It provides a clear, visual analysis of DFT's core principles and common misunderstandings through the simple Hubbard dimer model, enhancing conceptual understanding.

Findings

Visualization of the exact exchange-correlation functional

Insights into the behavior of XC potential under strong correlations

Clarification of the derivative discontinuity and charge gaps

Abstract

Most realistic calculations of moderately correlated materials begin with a ground-state density functional theory (DFT) calculation. While Kohn-Sham DFT is used in about 40,000 scientific papers each year, the fundamental underpinnings are not widely appreciated. In this chapter, we analyze the inherent characteristics of DFT in their simplest form, using the asymmetric Hubbard dimer as an illustrative model. We begin by working through the core tenets of DFT, explaining what the exact ground-state density functional yields and does not yield. Given the relative simplicity of the system, almost all properties of the exact exchange-correlation functional are readily visualized and plotted. Key concepts include the Kohn-Sham scheme, the behavior of the XC potential as correlations become very strong, the derivative discontinuity and the difference between KS gaps and true charge gaps, and how to extract optical excitations using time-dependent DFT. By the end of this text and accompanying exercises, the reader will improve their ability to both explain and visualize the concepts of DFT, as well as better understand where others may go wrong.