Exact Generalized Kohn-Sham Theory for Hybrid Functionals

Abstract

Hybrid functionals have proven to be of immense practical value in density-functional-theory calculations. While they are often thought to be a heuristic construct, it has been established that this is in fact not the case. Here, we present a rigorous and formally exact analysis of generalized Kohn-Sham (GKS) density-functional theory of hybrid functionals, in which exact remainder exchange-correlation potentials combine with a fraction of Fock exchange to produce the correct ground-state density. First, we extend formal GKS theory by proving a generalized adiabatic connection theorem. We then use this extension to derive two different definitions for a rigorous distinction between multiplicative exchange and correlation components-one new and one previously postulated. We examine their density-scaling behavior and discuss their similarities and differences. We then present a new algorithm for obtaining exact GKS potentials by inversion of accurate reference electron densities and employ this algorithm to obtain exact potentials for simple atoms and ions. We establish that an equivalent description of the many-electron problem is indeed obtained with any arbitrary global fraction of Fock exchange, and we rationalize the Fock-fraction dependence of the computed remainder exchange-correlation potentials in terms of the new formal theory. Finally, we use the exact theoretical framework and numerical results to shed light on the exchange-correlation potential used in approximate hybrid functional calculations and to assess the consequences of different choices of fractional exchange.

Exact Generalized Kohn-Sham Theory for Hybrid Functionals

“…First, we note that in the limit α ¼ 0, in which an exact KS map is retrieved, the obtained VðrÞ agree well with exact KS inversion results reported previously for the same systems using the same reference densities [67] (see Fig. S1 in the Supplemental Material [68]). We also note that as a second test, we successfully replicate the exact-exchange potential obtained in Ref.…”

“…4, are given in Fig. S7 of the Supplemental Material [68]. We find that for all systems studied in this work, the differences are overwhelmingly dominated by the functionaldriven component.…”

“…This is because although in general, jΦ S i ≡ jΦ 0 i and jΦ α i need not be the same (as they are only constrained to yield the same overall density), in practice they can often be close, which is indeed the case for the orbitals found by our inversion process (see Fig. S3 in the Supplemental Material [68] for an example).…”

“…We use Be as an example, with similar trends observed for all other systems; see Figs. S4-S6 of the Supplemental Material [68].…”

“…using orbitals from α ¼ 1 (varying α leads to differences of less than 1 mHa; see Fig. S3 in the Supplemental Material [68], which is indistinguishable from noise in the remainder potential). We adopt this approach because the Slater potential is known to serve as a rough approximation for the KS exchange potential corresponding to Hartree-Fock theory, and the step structure has been traced back to terms beyond the Slater potential [14,82,83].…”