Exact Generalized Kohn-Sham Theory for Hybrid Functionals

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.

Popular Summary

Density-functional theory is the most successful theory for computing properties of molecules and materials, based on only the constituent atoms and the laws of quantum physics. It is exact in principle but approximate in practice because of an unknown mapping between the electron density and the many-electron wave function. Therefore, much effort has gone into improving approximations for this mapping. To that end, we rigorously analyze the exact definitions and relations in a mathematical theory that is widely used in making such approximations, thus filling in a crucial gap in the quest to achieve better predictive accuracy with reduced computational cost.

The theory in question is called generalized Kohn-Sham theory. This theory allows one to mathematically recast the original system of interacting electrons as one that is partially interacting. In our work, we provide mathematical justification for smoothly connecting the many-electron system with one-electron orbitals for the very important case known as a hybrid functional. We use this to derive different definitions for exchange and correlation potentials, which are key components of the theory. We then present a new algorithm for obtaining exact potentials from electron densities and use it to gauge a popular approximate hybrid functional.

Our work hints at new strategies for addressing difficult many-electron problems via tailored hybrid density-functional-theory models based on sound physical principles.

Figure 1

A schematic representation of the generalized Kohn-Sham inversion algorithm used in this work. See text for details.

Figure 2

Exact remainder exchange-correlation potentials $V_{\mathrm{x}\mathrm{c}}^{\alpha }$, found by inversion for GKS maps with various fractions $\alpha$ of Fock exchange for (a) ${\mathrm{Li}}^{+}$ (b) ${\mathrm{Li}}^{-}$, (c) Be, and (d) ${\mathrm{F}}^{-}$.

Figure 3

Scaled remainder exchange potentials ${\tilde{v}}_{R,\mathrm{x}}^{\alpha }$ approximated using Eq. (47), found to coincide for various fractions $\alpha$ of Fock exchange for (a) ${\mathrm{Li}}^{+}$, (b) ${\mathrm{Li}}^{-}$, (c) Be, and (d) ${\mathrm{F}}^{-}$.

Figure 4

(a)–(e) Exact (inversion-based) and approximate ($\mathrm{PBE}{0}_{\alpha }$) remainder exchange-correlation potentials $V_{R,\mathrm{x}\mathrm{c}}^{\alpha }$ obtained for the Be atom from various choices of the fraction of Fock exchange $\alpha$. Also shown is the error in the $\mathrm{PBE}{0}_{\alpha }$ potential.

Figure 5

Exact (inversion-based) and approximate ($\mathrm{PBE}{0}_{\alpha }$) step potentials $V_{R,\mathrm{x}}^{\alpha }-(1-\alpha )V_{\mathrm{Sl}}$ obtained for the Be atom from various choices of the fraction of Fock exchange $\alpha$.

Figure 6

Difference of orbital energy from that of the highest occupied orbital, from exact and $\mathrm{PBE}{0}_{\alpha }$ GKS maps, as a function of $\alpha$. (a) ${\mathrm{Li}}^{-}$ $1s$, (b) Be $1s$, (c) ${\mathrm{F}}^{-}$ $1s$, (d) ${\mathrm{F}}^{-}$ $2s$.