Comparing correlation components and approximations in Hartree-Fock and Kohn-Sham theories via an analytical test case study

TL;DR

This study analytically compares correlation energy components in Hartree-Fock and Kohn-Sham theories using the Hubbard dimer model, revealing differences in their contributions and evaluating the performance of specific correlation functionals.

Contribution

It provides a rigorous analytical comparison of correlation energies in HF and KS theories within the Hubbard dimer, highlighting the impact of reference choice on functional accuracy.

Findings

HF kinetic correlation energy can change sign.

Functionals perform better with HF reference in this model.

Errors arise when using strong-interaction limits for KS instead of HF.

Abstract

The asymmetric Hubbard dimer is a model that allows for explicit expressions of the Hartree-Fock (HF) and Kohn-Sham (KS) states as analytical functions of the external potential, $\Delta v$, and of the interaction strength, $U$. We use this unique circumstance to establish a rigorous comparison between the individual contributions to the correlation energies stemming from the two theories in the $\{U,\Delta v\}$ parameter space. Within this analysis of the Hubbard dimer, we observe a change in the sign of the HF kinetic correlation energy, compare the indirect repulsion energies, and derive an expression for the 'traditional' correlation energy, i.e. the one that corrects the HF estimate, in a pure site-occupation function theory spirit [Eq. (43)]. Next, we test the performances of the Liu-Burke and the Seidl-Perdew-Levy functionals, which model the correlation energy based on its weak- and strong-interaction limit expansions and can be used for both the traditional and the KS correlation energies. Our results show that, in the Hubbard dimer setting, they typically work better for the HF reference, despite having been originally devised for KS. These conclusions are somewhat in line with prior assessments of these functionals on various chemical data sets. However, the Hubbard dimer model allows us to show the extent of the error that may occur in using the strong-interaction ingredient for the KS reference in place of the one for the HF reference, as has been carried out in most of the prior assessments.