Constructing tractable exchange–correlation functionals with an explicit derivative discontinuity for strongly correlated systems

Exact density functional theory requires the exchange–correlation functional to be piecewise linear in fractional electron number and flat in fractional spin — two intersecting planes with an explicit derivative discontinuity $D_{xc}$ at the integer. Mori-Sánchez, Cohen and Yang (Phys. Rev. Lett. 102, 066403, 2009) proved this flat-plane structure, showed that the gap of fractional-spin systems (and of Mott insulators, whose unit cells carry fractional spins) can come only from an explicit discontinuity of the functional itself, and demonstrated a prototype orbital/occupation-dependent functional with a controlled $D_{xc}$ — leaving open the construction of general, computationally tractable approximations, ideally as pure density functionals, with quantitative dynamic correlation and validation on realistic strongly correlated extended systems. Subsequent work has produced deep conceptual progress through partial schemes — among them ensemble DFT, in which fractional electrons are treated as ensembles of integer-electron states and the discontinuity emerges in closed form, and generalized-Kohn–Sham model potentials with a built-in discontinuity term — but no general, computationally tractable functional with an explicit controlled $D_{xc}$ is known to date: the problem remains open.

Background

Kohn–Sham density functional theory (DFT) is the workhorse method of electronic structure: the interacting electron system is replaced by an auxiliary non-interacting one reproducing the same density $\rho$, and all many-electron effects beyond classical electrostatics are collected in the exchange–correlation (xc) energy functional $E_{xc}[\rho]$. For an open system with fractional average electron number $N$, the exact ground-state energy $E(N)$ is piecewise linear in $N$ — straight segments between integers (Perdew, Parr, Levy and Balduz, 1982). The slope jumps at each integer, and the corresponding abrupt change of the functional derivative $v_{xc}(\mathbf{r})=\delta E_{xc}/\delta\rho(\mathbf{r})$ is the derivative discontinuity, of total size $D_{xc}$. The fundamental gap of a system — ionization potential $I$ minus electron affinity $A$ — equals the Kohn–Sham eigenvalue gap plus this discontinuity contribution (Perdew–Levy and Sham–Schlüter relations, 1983). The standard approximations — the local-density approximation (LDA), generalized-gradient approximations (GGA), most hybrid functionals (which mix a fraction of exact exchange) and meta-GGAs (which add the kinetic-energy density as an ingredient) — are smooth functions of the density and orbitals; they have $D_{xc}=0$, curve $E(N)$ away from linearity (delocalization error), and fail to keep the energy constant under fractional spin (static correlation error). Static correlation is the extra correlation energy of near-degenerate, effectively multi-reference states, such as a molecule stretched toward dissociation; dynamic correlation is the smooth, single-reference remainder. A Mott insulator is a crystal that band theory would predict to be metallic but which is insulating because electron–electron repulsion localizes the electrons.

Mori-Sánchez, Cohen and Yang (2009) unified the two exact conditions: over fractional charge $n$ and fractional spin, the exact energy lies on two intersecting planes — linear in charge, flat in spin — with a line of discontinuity at the integer (the flat-plane condition). The cleanest witness is the two-site model H[$\frac{1}{2}$,$\frac{1}{2}$]: one electron shared between two infinitely separated sites with opposite fractional spins. Its $\alpha$ and $\beta$ orbitals are exactly degenerate, so the entire gap must come from an explicit discontinuity of the xc functional itself, not from an orbital-level jump; ordinary functionals produce a gap there only by spuriously breaking the orbital degeneracy — a gap of the wrong nature, with wrong spin and magnetic observables. The same physics underlies the gaps of Mott insulators, whose nonmagnetic unit cells carry fractional spins drawn from different magnetic configurations. The authors built a prototype functional (their Eq. (7)) — a functional of orbitals and occupation numbers, i.e. within reduced-density-matrix-functional thinking — that carries an explicit $D_{xc}$, recovers the qualitative plane structure, and gives the correct equal gaps of H[$\frac{1}{2}$,$\frac{1}{2}$] and H[1,0], but retains slight curvature and a slightly underpredicted electron affinity for $1<n\le 2$, attributable to approximate dynamic correlation. They noted the construction 'can be generalized and placed much more clearly within DFT if the normalization of the exact-exchange hole is used instead of the occupation number': the exchange hole is the depletion of same-spin pair density around an electron, and enforcing its unit normalization is a density-level constraint that could replace occupation numbers as the switch for the discontinuity. Hubbard-model-based density functionals (the Hubbard model being the standard lattice model of correlated electrons) were known exceptions with a non-zero discontinuity, but no general, practical continuum-DFT approximation with an explicit $D_{xc}$ existed. The Letter closed by declaring such approximations 'essential for the advancement of DFT towards the calculation of strongly correlated systems'.

Problem Statement

Construct, implement, and validate approximate exchange–correlation functionals that incorporate an explicit, controlled derivative discontinuity $D_{xc}$ in a general and computationally tractable way — preferably expressed as functionals of the density alone, rather than only through orbitals, occupations, or reduced density matrices. The functional must jointly satisfy, within a single construction: (a) the flat-plane exact conditions over fractional charge and fractional spin — piecewise linearity in fractional charge, constancy in fractional spin, and the correct line of discontinuity at integers — so that fractional-spin systems such as H[$\frac{1}{2}$,$\frac{1}{2}$] obtain their gap from $D_{xc}$ itself (with orbitals remaining degenerate) rather than from spurious symmetry breaking; (b) quantitative treatment of dynamic correlation, removing the residual curvature and the slight underprediction of the electron affinity for $1<n\le2$ exhibited by the source's prototype Eq. (7); and (c) extension from the illustrative two-site models to realistic strongly correlated extended systems, such as Mott insulators: the construction must be implemented for at least one strongly correlated extended system beyond the two-site models — for example a 3d transition-metal-oxide Mott insulator or another correlated periodic solid — and must reproduce, to within the accuracy of a stated benchmark (experiment or a high-level many-body calculation), both the fundamental gap and at least one magnetic or spectroscopic observable, with the gap originating in the explicit discontinuity $D_{xc}$ rather than in spurious symmetry breaking. The conditions (a)–(c) are one coupled objective — a usable discontinuous DFT for strong correlation: the model-system exact conditions define the mechanism, the density-only formulation is the intended generality, and materials validation demonstrates the capability.

References

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