Exploring weight-dependent density-functional approximations for ensembles in the Hubbard dimer

• Published in: The European physical journal. B, Condensed matter physics Vol. 91; no. 7; pp. 1 - 18
• Main Authors: Deur, Killian, Mazouin, Laurent, Senjean, Bruno, Fromager, Emmanuel
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.07.2018; Springer; Springer Nature B.V; Springer-Verlag
• Subjects: Analysis; Chemical Sciences; Complex Systems; Condensed Matter; Condensed Matter Physics; Density; Density functional theory; Dependence; Dimers; Excitation; Fluid- and Aerodynamics; Interpolation; Mathematical analysis; Mathematical models; or physical chemistry; Physics; Physics and Astronomy; Regular Article; Solid State Physics; Strongly Correlated Electrons; Theoretical and; Topical issue: Special issue in honor of Hardy Gross
• ISSN: 1434-6028; 1434-6036
• DOI: 10.1140/epjb/e2018-90124-7

Gross–Oliveira–Kohn density-functional theory (GOK-DFT) is an extension of DFT to excited states where the basic variable is the ensemble density, i.e. the weighted sum of ground- and excited-state densities. The ensemble energy (i.e. the weighted sum of ground- and excited-state energies) can be obtained variationally as a functional of the ensemble density. Like in DFT, the key ingredient to model in GOK-DFT is the exchange-correlation functional. Developing density-functional approximations (DFAs) for ensembles is a complicated task as both density and weight dependencies should in principle be reproduced. In a recent paper [K. Deur et al., Phys. Rev. B 95 , 035120 (2017)], the authors applied exact GOK-DFT to the simple but nontrivial Hubbard dimer in order to investigate (numerically) the importance of weight dependence in the calculation of excitation energies. In this work, we derive analytical DFAs for various density and correlation regimes by means of a Legendre–Fenchel transform formalism. Both functional and density driven errors are evaluated for each DFA. Interestingly, when the ensemble exact-exchange-only functional is used, these errors can be large, in particular if the dimer is symmetric, but they cancel each other so that the excitation energies obtained by linear interpolation are always accurate, even in the strongly correlated regime.