Self-interaction & delocalization error

The single defect that almost every “beyond-DFT” correction is chasing. It wears two faces, and conflating them is the most common error in the literature, so I keep them strictly separate here: the one-electron self-interaction error (SIE) of Perdew–Zunger, and the many-electron delocalization error of the Yang group. They coincide for the hydrogen atom and diverge almost everywhere else.

One-electron SIE (Perdew–Zunger)

For a genuine one-electron density $n_i$ the Hartree and exact-exchange energies must cancel exactly,

$$E_{\mathrm{H}}[n_i]+E_{\mathrm{X}}^{\mathrm{exact}}[n_i]=0,$$

so that a single electron feels no Coulomb field from itself. Any approximate functional with a semilocal $E_{\mathrm{XC}}$ violates this: the self-Hartree term $E_{\mathrm{H}}[n_i]=\frac{1}{2}\iint \frac{n_i(\mathbf{r})n_i({\mathbf{r}}^{\prime })}{|\mathbf{r}-{\mathbf{r}}^{\prime }|}d\mathbf{r}d{\mathbf{r}}^{\prime }$ is not undone by the approximate exchange. The PZ self-interaction correction subtracts the spurious term orbital-by-orbital,

$$E_{\mathrm{XC}}^{\mathrm{PZ}}=E_{\mathrm{XC}}^{\mathrm{approx}}[n_{\uparrow },n_{\downarrow }]-\sum _{i\sigma }(E_{\mathrm{H}}[n_{i\sigma }]+E_{\mathrm{XC}}^{\mathrm{approx}}[n_{i\sigma },0]).$$

This is a one-electron statement: it is defined on individual orbital densities, it is not invariant under unitary mixing of the occupied orbitals (hence localized-orbital SIC), and it does not by itself fix the many-electron problem below. PZ-SIC over-corrects equilibrium properties about as often as it helps; its modern revival — Fermi–Löwdin orbital SIC (FLO-SIC), which restores unitary invariance by constructing the orbitals from real-space Fermi-orbital descriptors — is a separate track. (Perdew & Zunger, PRB 23, 5048 (1981); Pederson, Ruzsinszky & Perdew, J. Chem. Phys. 140, 121103 (2014).)

Many-electron delocalization error (the exact conditions)

The deeper, system-relevant statement is about fractional charge. Extend the energy to open systems with non-integer electron number $N=N_0+\delta$, $0\le \delta \le 1$, via the zero-temperature grand-canonical (ensemble) construction. The exact functional is then piecewise linear in $N$ between adjacent integers (Perdew–Parr–Levy–Balduz),

$$E(N_0+\delta )=(1-\delta )E(N_0)+\delta E(N_0+1),$$

with a derivative discontinuity in the chemical potential $\mu =\partial E/\partial N$ at integers equal to the fundamental gap $E_g=I-A$. Semilocal functionals are instead convex (bowing below the line): they spuriously lower the energy of fractional charge, which makes electrons want to delocalize — to spread fractional charge over fragments rather than localize an integer per fragment. This is the delocalization error. Hartree–Fock has the opposite sign (concave, localizing). The diagnostic is the deviation from the straight $E(N)$ line; flattening it to the exact piecewise-linear behavior is the convexity / linearity condition, and it is exactly the target a properly tuned $U$ or range-separation parameter restores.

Symptoms of delocalization error, all the same disease: too-small band gaps, over-delocalized $d$/$f$ states, spuriously metallic Mott insulators, barrier underestimation in transition states, charge-transfer excitations too low, and the fractional-charge dissociation of ${\mathrm{H}}_2^{+}$.

Fractional spin and the flat plane

The companion exact condition is constancy under fractional spin. For a degenerate ground state the energy must be flat as spin density is moved between degenerate components at fixed total $N$ — the prototype is the H atom with $(n_{\uparrow },n_{\downarrow })=(\frac{1}{2},\frac{1}{2})$, which must have the same energy as $(1,0)$. Semilocal functionals raise this energy (the static-correlation error / fractional-spin error), HF does too. Combining the fractional-charge line and the fractional-spin line gives the flat-plane condition in the $(N,M)$ plane: the exact energy is a set of planar facets meeting at a ridge with a derivative discontinuity along the integer-$N$ line. Reproducing both the flat line (delocalization) and the flat plane edge (static correlation) simultaneously is the real bar, and the reason a method can cure one error while worsening the other. (Cohen, Mori-Sánchez & Yang, Science 321, 792 (2008); Chem. Rev. 112, 289 (2012); flat plane: Mori-Sánchez, Cohen & Yang, PRL 102, 066403 (2009).)

Why this note is the hub

Every correction downstream is a piecewise-linearity (or flat-plane) restoration in disguise:

• DFT+U penalizes fractional occupation of a localized subspace $\mathrm{Tr}[n-n^2]$ — a local, cheap convexity patch (see DFT+U).
• Hybrid functionals admix exact exchange to oppose the convex semilocal curvature with HF’s concave curvature; tuning the fraction is enforcing linearity (see hybrid functionals).
• Density-driven error is distinct but adjacent — there the density itself is wrong; here the energy functional mis-scores even the right density (see density-driven error (DC-DFT)).

Prerequisites

Kohn–Sham DFT · Hartree–Fock · exchange–correlation functionals · the fractional-particle / ensemble extension

Builds toward: DFT+U · hybrid functionals · the beyond-DFT ladder · flat-band SIE

Key references

• Fractional charge, piecewise linearity & the gap — J. P. Perdew, R. G. Parr, M. Levy & J. L. Balduz, PRL 49, 1691 (1982); J. P. Perdew & M. Levy, PRL 51, 1884 (1983); L. J. Sham & M. Schlüter, PRL 51, 1888 (1983).
• One-electron SIC — J. P. Perdew & A. Zunger, PRB 23, 5048 (1981); FLO-SIC: M. R. Pederson, A. Ruzsinszky & J. P. Perdew, J. Chem. Phys. 140, 121103 (2014).
• Many-electron delocalization & static correlation (flat plane) — P. Mori-Sánchez, A. J. Cohen & W. Yang, J. Chem. Phys. 125, 201102 (2006) & PRL 102, 066403 (2009); A. J. Cohen, P. Mori-Sánchez & W. Yang, Science 321, 792 (2008) & Chem. Rev. 112, 289 (2012).
• Linearity restoration / tuning — T. Stein, H. Eisenberg, L. Kronik & R. Baer, PRL 105, 266802 (2010); I. Dabo, A. Ferretti, N. Poilvert, Y. Li, N. Marzari & M. Cococcioni, PRB 82, 115121 (2010).