XC functionals & Jacob's ladder

The exact exchange–correlation functional $E_{\mathrm{xc}}[n]$ of Kohn–Sham DFT is unknown; the approximations to it are organized as Jacob’s ladder — a hierarchy of rungs, each adding a new local ingredient and, ideally, climbing toward chemical accuracy. This note covers the three semilocal rungs, the ones built only from the density and its local features, and the exact constraints that anchor them. The framing to carry away is that the ladder measures one kind of accuracy and is orthogonal to a second, independent axis — localization / self-interaction — which it cannot fix.

The ladder, and what “semilocal” means

Perdew’s metaphor: each rung uses more local information about the density at point $\mathbf{r}$, and the angels ascend from the Hartree “earth” toward the “heaven” of chemical accuracy.

Rung	Ingredients at $\mathbf{r}$	Archetype
1 — LSDA	$n_{\sigma }(\mathbf{r})$	PW92
2 — GGA	$n_{\sigma },\nabla n_{\sigma }$	PBE
3 — meta-GGA	$n_{\sigma },\nabla n_{\sigma },{\tau }_{\sigma }$ (and/or ${\nabla }^2{n}_{\sigma }$)	TPSS, SCAN
4 — hyper-GGA / hybrid	+ exact (HF) exchange	PBE0, HSE
5 — RPA-type	+ virtual orbitals	RPA, double hybrids

Rungs 1–3 are semilocal: $E_{\mathrm{xc}}=\int {\epsilon }_{\mathrm{xc}}(n,\nabla n,\tau )nd\mathbf{r}$ with the energy density a pointwise function of local ingredients. Rungs 4–5 break this by reaching for the occupied/virtual orbitals (exact exchange, RPA correlation) and belong to a different cost class and a different physics — they are the beyond-DFT correctors, not re-derived here.

Rung 1 — LSDA. Take the exchange–correlation energy density of the uniform electron gas at the local density, point by point:

$$E_{\mathrm{xc}}^{\mathrm{LSDA}}[n_{\uparrow },n_{\downarrow }]=\int n(\mathbf{r}){\epsilon }_{\mathrm{xc}}^{\mathrm{unif}}(n_{\uparrow }(\mathbf{r}),n_{\downarrow }(\mathbf{r}))d\mathbf{r}.$$

Exchange is the analytic Dirac form ${\epsilon }_{\mathrm{x}}^{\mathrm{unif}}\propto n^{1/3}$; correlation is the PW92 parametrization of the Ceperley–Alder quantum-Monte-Carlo uniform-gas data. LSDA is the natural starting point because it is exact by construction in the uniform limit; it systematically overbinds and gives lattice constants a few percent too short, but its error is smooth and well understood.

Rung 2 — GGA. Add the dimensionless density gradient $s=|\nabla n|/(2k_{\mathrm{F}}n)$ as a second ingredient, $E_{\mathrm{xc}}^{\mathrm{GGA}}=\int n{\epsilon }_{\mathrm{xc}}(n,s)d\mathbf{r}$, through an enhancement factor $F_{\mathrm{xc}}(s)$ over LSDA exchange. PBE fixes this factor entirely from exact constraints (below) with no fitted parameters — the deliberately non-empirical design that made it the default for solids. GGAs correct LSDA’s overbinding (lattice constants and atomization energies improve markedly) but tend to overcorrect into mild underbinding.

Rung 3 — meta-GGA. Add the kinetic-energy density ${\tau }_{\sigma }(\mathbf{r})=\frac{1}{2}{\sum }_i^{\mathrm{occ}}|\nabla {\psi }_{i\sigma }{|}^2$ (an orbital-dependent but still semilocal ingredient). Its value relative to the von Weizsäcker and uniform-gas limits encodes the local bonding character — single-orbital, slowly-varying, or overlapping — letting the functional distinguish covalent, metallic, and weak regions. TPSS is the non-empirical meta-GGA built on PBE; SCAN (Strongly Constrained and Appropriately Normed) is constructed to satisfy all 17 known exact constraints a semilocal functional can, and to be exact or accurate for a set of “appropriate norms” (rare-gas dimers, the uniform gas). SCAN delivers near-chemical accuracy for many bonded systems but inherits the semilocal limitation below — and is numerically stiff, demanding dense integration grids.

Exact constraints — the anchors

The non-empirical functionals are derived from conditions the exact $E_{\mathrm{xc}}$ provably satisfies. The load-bearing ones:

• Uniform-electron-gas limit. As $\nabla n\to 0$, every functional must reduce to LSDA. This is the zeroth anchor and the reason the gas data matter; $F_{\mathrm{xc}}(s\to 0)\to 1$.
• Slowly-varying gradient expansion (GE). For a weakly inhomogeneous gas the exchange enhancement has a known small-$s$ expansion, $F_{\mathrm{x}}(s)=1+\mu s^2+\cdots$ with $\mu$ fixed by the second-order gradient coefficient. PBE for solids (PBEsol) restores the GE value of $\mu$ that PBE itself sacrifices for atomic energies — the canonical solid-vs-molecule tension.
• Lieb–Oxford bound. A rigorous lower bound on the exchange–correlation energy of any density, $$E_{\mathrm{xc}}[n]\ge -C_{\mathrm{LO}}\int n(\mathbf{r}{)}^{4/3}d\mathbf{r},C_{\mathrm{LO}}\approx 1.804,$$ which caps how negative $E_{\mathrm{xc}}$ can be. It is imposed by bounding the enhancement factor, $F_{\mathrm{xc}}(s)\le 1.804$ for all $s$, and is one of the constraints SCAN enforces exactly.
• Coordinate-scaling, spin-scaling, the high-density limit, and the asymptotic behaviour of the exchange hole supply the rest of the constraint set that pins down TPSS and SCAN.

These constraints are what make the climb systematic rather than fitted: a higher rung is built to satisfy strictly more exact conditions with strictly more ingredients.

The orthogonal axis: semilocal accuracy is not localization

Here is the framing the note exists to make. Climbing rungs 1→3 improves the description of slowly-to-moderately varying density — bond lengths, cohesive energies, vibrational properties of main-group solids — because the added ingredients ($\nabla n$, $\tau$) resolve local inhomogeneity better. But every semilocal rung shares the same structural disease: the exchange–correlation hole is local, so the spurious self-interaction is not cancelled and the energy is convex in fractional electron number $N$. The consequences — too-small gaps, over-delocalized $d/f$ states, spuriously metallic Mott insulators, transition-state barriers too low — are the same on every semilocal rung. SCAN’s gaps are a little better than PBE’s chiefly through its different exchange shape, not because it has solved the delocalization problem; it has not.

So the Jacob’s-ladder (semilocal-accuracy) axis is orthogonal to the localization / self-interaction axis. Moving up the ladder does not move you along the SIE axis. Curing self-interaction / delocalization requires non-local ingredients — exact exchange (rung 4 hybrids), a Hubbard penalty, or a many-body self-energy — which is precisely the content of the beyond-DFT ladder and is defined and diagnosed in self-interaction & delocalization error. This note deliberately does not re-derive the SIE statement, the fractional-charge / piecewise-linearity condition, or the $+U$ / hybrid / GW corrections — those are owned there. The two axes intersect only at rung 4, where exact exchange first enters and the ladder and the SIE axis stop being independent.

Prerequisites

Kohn–Sham DFT · self-interaction error

Builds toward: the beyond-DFT ladder · hybrid functionals (rung 4)

Key references

• Jacob’s ladder — J. P. Perdew & K. Schmidt, AIP Conf. Proc. 577, 1 (2001).
• LSDA correlation (PW92) — J. P. Perdew & Y. Wang, Phys. Rev. B 45, 13244 (1992).
• GGA (PBE) — J. P. Perdew, K. Burke & M. Ernzerhof, Phys. Rev. Lett. 77, 3865 (1996).
• meta-GGA — TPSS — J. Tao, J. P. Perdew, V. N. Staroverov & G. E. Scuseria, Phys. Rev. Lett. 91, 146401 (2003).
• meta-GGA — SCAN — J. Sun, A. Ruzsinszky & J. P. Perdew, Phys. Rev. Lett. 115, 036402 (2015).
• Lieb–Oxford bound — E. H. Lieb & S. Oxford, Int. J. Quantum Chem. 19, 427 (1981).