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arXiv:1405.0864 · Physical Review B

Kohn-Sham calculations with the exact functional

Lucas O. Wagner, Thomas E. Baker, Edwin Miles Stoudenmire, Kieron Burke, Steven R. White

Published Jul 9, 2014• 64 citations
arXivHTMLPDF

Abstract

The authors illustrate the convergence properties of the exact Kohn-Sham scheme for both weakly and strongly correlated systems, hence, they set up an important benchmark for first-principles DFT-based calculations.
DOI
10.1103/physrevb.90.045109

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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.
We explore the hypercomplex Kohn-Sham (HCKS) formalism and develop a third exact functional theory beyond the popular Kohn-Sham density functional theory (KS-DFT) and reduced density matrix functional theory (RDMFT), called 1-HCKS. The orbitals of hierarchical correlation introduced in 1-HCKS are interesting because their occupations are neither necessarily integers, as in KS-DFT, nor overly flexible, as in RDMFT; rather, they can capture strong correlation with fractional occupations, governed by stringent constraints. This feature allows 1-HCKS to combine the advantages of KS-DFT and RDMFT in dealing with dynamic correlation and strong correlation, while eliminating the issues of computational convergence and basis set dependence encountered in RDMFT. Thus, 1-HCKS is a promising alternative for implementing DFT beyond KS-DFT, especially for the treatment of strongly correlated systems.