The beyond-DFT ladder

A decision note, not a derivation: which deficiency of a semilocal Kohn–Sham calculation each “beyond-DFT” rung actually repairs, and when to reach for it. The single most common mistake is to treat these as interchangeable accuracy upgrades. They are not — they fix different errors, and a method that cures one routinely worsens another. The organizing question is always: which error am I hitting?

The four errors, separated

Error	What goes wrong	Symptom	Fix
Self-interaction / delocalization	convex $E(N)$; fractional charge spuriously stabilized	gaps too small, $d$/$f$ over-delocalized, Mott insulators turned metallic	$+U$, hybrids
Static (strong) correlation	single determinant inadequate; fractional-spin error	wrong multiplets, stretched bonds, some Mott/heavy-fermion physics	DMFT (sometimes $+U$)
Dynamic correlation	missing screened-exchange / correlation in the spectrum	quasiparticle gaps, satellites, band positions	GW
Density-driven error	the self-consistent density itself is wrong	abnormal anions / charge-transfer; corrupted forces	DC-DFT (HF-DFT)

The first three are functional errors — the energy or self-energy is mis-modelled even given a reasonable density. The fourth is orthogonal: the functional may be fine but the density it produces is not, so plugging in a better (e.g. HF) density beats the self-consistent answer. Conflating density-driven error with the functional errors is its own trap — see density-driven error (DC-DFT) and the hub note self-interaction & delocalization error.

The rungs, and what each is honestly for

DFT+U — a cheap SIE patch, not a correlation method. A local Hubbard penalty on a chosen $d$/$f$ subspace that restores convexity there by penalizing fractional occupation $\mathrm{Tr}[n-n^2]$. It is the right reach when the problem is delocalization error localized to a well-defined correlated shell and you want it for free. It is not a treatment of dynamic correlation, it gives no spectral function, and it is parameter- and subspace-dependent. See DFT+U. Two caveats that recur:

• Double counting (FLL vs AMF). The Hubbard energy added must have the part already present in the semilocal functional subtracted off — and the subtraction is not unique. Fully Localized Limit (FLL, a.k.a. the atomic limit) assumes integer occupations and suits genuine insulators; Around Mean Field (AMF) subtracts the uniform-occupation average and suits weaker correlation / more itinerant cases. They give different energies and different $U_{\mathrm{eff}}$, and the choice is physics, not a flag to leave at default.
• $U$ is not transferable across oxidation states or codes; the first-principles route (linear response / cRPA) is the honest one when the number matters.

Hybrid functionals — SIE via exact exchange. Admix a fraction of HF exchange to oppose the convex semilocal curvature with HF’s concave curvature; tuning the fraction (or the range-separation parameter) is enforcing piecewise linearity, globally rather than on one subspace. Better gaps and localization, but still functional-error prone, system-dependent in the fraction, and costly in plane-wave codes (screening, à la HSE, is what makes it tractable). See hybrid functionals. Overlap with $+U$ is real: both are SIE correctors, $+U$ local and cheap, hybrids global and dearer.

GW — dynamic correlation in the spectrum. A many-body self-energy $\Sigma =iGW$ that gives quasiparticle energies and the fundamental gap, restoring the derivative discontinuity that Kohn–Sham eigenvalues miss. Reach for it when you need band positions and gaps to spectroscopic accuracy, not total energies or structure. It does not by itself handle strong/static correlation. See GW & quasiparticles.

DMFT — static/strong, local correlation. Treats a local interacting impurity problem exactly (dynamically), capturing Mott physics, multiplets, and heavy-fermion behaviour that $+U$‘s static mean field cannot. The expensive, correct end of the local-correlation axis; kept here as the neighbour of GW (and developed only as context inside GW & quasiparticles and Green’s functions, per the topic map — no standalone DMFT note).

DC-DFT — the density axis. Orthogonal to all of the above: when density sensitivity is diagnosed, evaluate the functional on a better density (HF-DFT) rather than changing the functional. Cheap, and the only one of these that targets the density rather than the energy. See density-driven error (DC-DFT).

How to choose

1. Is the density suspect (anion, charge-transfer, abnormal system; bad forces)? → DC-DFT first; the functional fix may be unnecessary or even harmful on top of a corrected density.
2. Is it delocalization / gaps / over-itinerant $d$/$f$ with a single-reference character? → $+U$ if the correlation is shell-localized and you want it cheap; hybrids for a global SIE fix and better all-round gaps.
3. Do you need quasiparticle band positions / spectra? → GW (mind the $G_0{W}_0$ starting-point dependence).
4. Is the state genuinely multi-reference / Mott / heavy-fermion (fractional-spin / static correlation)? → DMFT; $+U$ is at best a crude stand-in here.

Prerequisites

self-interaction & delocalization error · DFT+U · hybrid functionals · density-driven error (DC-DFT)

Builds toward: GW & quasiparticles · exchange tensors from DFT · CrSBr

Key references

• DFT+U — V. I. Anisimov, J. Zaanen & O. K. Andersen, PRB 44, 943 (1991); S. L. Dudarev et al., PRB 57, 1505 (1998); M. Cococcioni & S. de Gironcoli, PRB 71, 035105 (2005).
• Hybrid functionals — C. Adamo & V. Barone, J. Chem. Phys. 110, 6158 (1999) (PBE0); J. Heyd, G. E. Scuseria & M. Ernzerhof, J. Chem. Phys. 118, 8207 (2003) (HSE).
• GW & DMFT — L. Hedin, Phys. Rev. 139, A796 (1965); A. Georges, G. Kotliar, W. Krauth & M. J. Rozenberg, Rev. Mod. Phys. 68, 13 (1996); G. Kotliar et al., Rev. Mod. Phys. 78, 865 (2006).
• DC-DFT & the error framework — M.-C. Kim, E. Sim & K. Burke, PRL 111, 073003 (2013); A. J. Cohen, P. Mori-Sánchez & W. Yang, Science 321, 792 (2008) & Chem. Rev. 112, 289 (2012).