Hybrid functionals

Rung 4 of Jacob’s ladder: the first rung to reach past the density and its local features to the occupied orbitals, by admixing a fraction of exact (Hartree–Fock) exchange into a semilocal functional. The point of the admixture is not accuracy in the slowly-varying limit — semilocal functionals already do that well — but to oppose the convex curvature of semilocal exchange with the concave curvature of HF, and so partially restore the piecewise-linearity that cures delocalization error. The adiabatic connection tells us a fixed fraction is the right kind of correction; the rest of the note is which fraction, how to make it affordable in a solid, and where the construction stops being transferable.

The adiabatic connection — why mix exact exchange at all

The exact exchange–correlation energy can be written without approximation as an integral over a coupling constant $\lambda$ that scales the electron–electron interaction from the non-interacting Kohn–Sham system ($\lambda =0$) to the fully interacting one ($\lambda =1$), at fixed density:

$$E_{\mathrm{xc}}={\int }_0^1{U}_{\mathrm{xc}}^{\lambda }d\lambda ,U_{\mathrm{xc}}^{\lambda }=⟨{\Psi }^{\lambda }|{\hat{V}}_{ee}|{\Psi }^{\lambda }⟩-E_{\mathrm{H}}[n],$$

where $U_{\mathrm{xc}}^{\lambda }$ is the potential exchange–correlation energy of the $\lambda$-scaled system (the kinetic correlation is generated automatically by the $\lambda$-integration). The two endpoints are known in character. At $\lambda =0$ the wavefunction is the single KS determinant, so $U_{\mathrm{xc}}^0=E_{\mathrm{x}}^{\mathrm{HF}}[\{{\psi }_i^{\mathrm{KS}}\}]$ is exactly the exchange energy of the KS orbitals — pure, self-interaction-free exact exchange, with no correlation. At $\lambda =1$ the integrand is well approximated by a semilocal functional, because the fully-screened coupling-strength hole is reasonably local. A semilocal functional implicitly models the whole $\lambda$-average, which is why it carries no exact exchange and inherits the convexity disease. The hybrid idea is to honour the $\lambda =0$ endpoint explicitly — replace part of the integrand near $\lambda =0$ with the true $E_{\mathrm{x}}^{\mathrm{HF}}$ — and approximate the rest semilocally. The fraction of exact exchange is then a statement about the shape of the $U_{\mathrm{xc}}^{\lambda }$ curve between the endpoints.

Global hybrids: PBE0 and the $\frac{1}{4}$ argument

The simplest model takes a single, density-independent fraction $\alpha$ of exact exchange across the whole system. PBE0 is the non-empirical realization,

$$E_{\mathrm{xc}}^{PBE0}=\alpha E_{\mathrm{x}}^{\mathrm{HF}}+(1-\alpha )E_{\mathrm{x}}^{\mathrm{PBE}}+E_{\mathrm{c}}^{\mathrm{PBE}},\alpha =\frac{1}{4},$$

mixing a quarter of exact exchange with three-quarters PBE exchange and keeping PBE correlation untouched. The value $\alpha =\frac{1}{4}$ is derived, not fitted (this is the Perdew–Ernzerhof–Burke argument). Model the coupling-constant integrand as a $\lambda$-power series $U_{\mathrm{xc}}^{\lambda }\approx U_{\mathrm{xc}}^0+(\text{higher order in }\lambda )$ and ask how many terms a semilocal functional reproduces: PBE-type functionals are argued to be accurate to roughly fourth order in the $\lambda$ expansion of the hole, i.e. the lowest order at which exact exchange must be supplied explicitly is set by the perturbation order $n$ at which the semilocal model breaks down, giving an optimal mixing $\alpha \approx 1/n$ with $n=4$, hence $\alpha =\frac{1}{4}$. This is why PBE0 has no system-specific parameter — the quarter comes from the perturbation-order argument, in deliberate contrast to Becke’s original half-and-half and to the fitted three-parameter B3LYP. The practical payoff is the standard one: gaps, atomization energies, barrier heights, and $d/f$ localization all improve markedly over PBE, because a quarter of the convex semilocal exchange is traded for concave exact exchange.

Range-separated hybrids: HSE and the screening of long-range exchange

A global fraction of bare exact exchange is a disaster in a metal or small-gap solid. The exchange integrals decay as $1/r$, their Fock sum over a periodic system converges painfully slowly with $k$-mesh, and — fatally — in a metal the unscreened Fock exchange produces a vanishing density of states at the Fermi level (the well-known logarithmic divergence of the HF band velocity / zero DOS at $E_{\mathrm{F}}$ for the electron gas). The cure is to recognize that in a real solid the long-range exchange is screened anyway, and to build that in. HSE splits the Coulomb operator into short- and long-range parts with the error function,

$$\frac{1}{r}=\underset{\text{short range (SR)}}{\underbrace{\frac{\mathrm{erfc}(\omega r)}{r}}}+\underset{\text{long range (LR)}}{\underbrace{\frac{\mathrm{erf}(\omega r)}{r}}},$$

where $\omega$ is the screening (range-separation) parameter setting the crossover length $\sim 1/\omega$. Exact exchange is then admixed only in the short-range channel, while the long-range exchange is taken entirely from PBE:

$$E_{\mathrm{xc}}^{\mathrm{HSE}}=\alpha E_{\mathrm{x}}^{\mathrm{HF},\mathrm{SR}}(\omega )+(1-\alpha )E_{\mathrm{x}}^{\mathrm{PBE},\mathrm{SR}}(\omega )+E_{\mathrm{x}}^{\mathrm{PBE},\mathrm{LR}}(\omega )+E_{\mathrm{c}}^{\mathrm{PBE}},$$

again with $\alpha =\frac{1}{4}$. The two limits of $\omega$ recover the endpoints of the construction:

• $\omega \to 0$: $\mathrm{erfc}(\omega r)\to 1$, the SR channel becomes the full $1/r$, and HSE collapses to PBE0.
• $\omega \to \infty$: the SR channel shrinks to nothing, all exact exchange is screened away, and HSE collapses to PBE.

The standard HSE06 sits between, with $\omega \approx 0.11{\text{bohr}}^{-1}$ ($\approx 0.21{\text{A˚}}^{-1}$; the $0.2{\text{A˚}}^{-1}$ value often quoted is the VASP convention). Screening makes solids tractable for two reasons. First, removing the long-range tail of the exact exchange removes the slow $1/r$ Fock sum and the $k$-convergence problem along with it, since the short-range exchange hole is spatially compact. Second, killing the long-range Fock exchange removes the spurious zero-DOS / divergence at $E_{\mathrm{F}}$, so metals and narrow-gap semiconductors stay well-behaved. HSE is therefore the default screened hybrid for periodic systems, where bare PBE0 is often impractical.

(A bookkeeping caution worth getting right: the original 2003 HSE paper as implemented — HSE03 — used two different screening lengths in the HF and PBE channels, and the 2006 erratum corrected that specification. HSE06 is a separate, later definition: Krukau, Vydrov, Izmaylov & Scuseria fixed a single $\omega \approx 0.11{\text{bohr}}^{-1}$ in both channels. Cite Krukau et al. (2006) for the HSE06 screening parameter, not the erratum.)

Gaps as partial SIE cancellation — and the empirical ceiling

Why does a fraction of exact exchange open gaps? Read it through self-interaction error: semilocal exchange is convex in fractional occupation (it under-penalizes spreading charge), while HF exchange is concave (it over-localizes). Mixing a fraction $\alpha$ adds back $\alpha$ of the missing derivative discontinuity, flattening the convex $E(N)$ curve toward the exact piecewise-linear line. A hybrid gap is therefore partial SIE cancellation: the residual gap error is governed by how much of the curvature the chosen $\alpha$ (and $\omega$) removes. This immediately exposes the method’s ceiling. The optimal $\alpha$ that linearizes $E(N)$ is system-dependent — physically it tracks the inverse dielectric screening, so a fraction near $\frac{1}{4}$ that works for a typical semiconductor is too small for a wide-gap insulator (weak screening, large effective $\alpha$) and too large for a small-gap or metallic system (strong screening, small effective $\alpha$). Likewise the optimal $\omega$ depends on the dielectric environment. A fixed $(\alpha ,\omega )$ hybrid is thus not transferable: it cannot be right for materials with very different screening using one parameter set. The principled responses — system-specific tuning of $\alpha$ to enforce the ionization-potential / linearity condition (optimally-tuned range-separated hybrids), or making the screening dielectric-dependent so $\alpha =1/{\epsilon }_{\infty }$ — are exactly the admission that the constant comes from the same SIE physics, now made material-aware. Beyond that, a self-energy that is non-local and energy-dependent (GW) is the next rung; a hybrid is the static, fixed-fraction approximation to it.

The Fock-exchange cost bottleneck

The price of leaving the semilocal world is the exchange operator. Semilocal $E_{\mathrm{xc}}$ is a single real-space quadrature; exact exchange is a non-local, four-index object,

$$E_{\mathrm{x}}^{\mathrm{HF}}=-\frac{1}{2}\sum _{ij}\sigma \iint \frac{{\psi }_i^{\ast }(\mathbf{r}){\psi }_j(\mathbf{r}){\psi }_j^{\ast }({\mathbf{r}}^{\prime }){\psi }_i({\mathbf{r}}^{\prime })}{|\mathbf{r}-{\mathbf{r}}^{\prime }|}d\mathbf{r}d{\mathbf{r}}^{\prime },$$

i.e. an integral over every pair of occupied orbitals. In a periodic plane-wave code this becomes a double sum over the $k$-mesh — exact exchange couples ${\psi }_{\mathbf{k}}$ to ${\psi }_{{\mathbf{k}}^{\prime }}$ for all pairs $(\mathbf{k},{\mathbf{k}}^{\prime })$ — so the cost grows like $N_k^2$ in the number of $k$-points (times the band and basis factors), against the single-$k$ cost of a semilocal functional, and the integrable $\mathbf{q}=0$ Coulomb singularity needs special treatment (auxiliary-function or truncation schemes). This $k$-mesh scaling, not the per-point work, is what makes hybrids one to two orders of magnitude more expensive than GGA in plane waves, and it is precisely the cost HSE’s screening alleviates by making the exchange short-ranged. The basis matters too: in a Gaussian local-orbital code the four-index integrals are analytic and can be screened by orbital overlap, so global hybrids like PBE0 are comparatively routine even for solids — a different cost regime from plane waves, and the reason “which hybrid is affordable” is a question about the code as much as the functional.

Prerequisites

XC functionals & Jacob’s ladder · Hartree–Fock

Builds toward: GW quasiparticles (the static, fixed-fraction limit of) · delocalization error · the beyond-DFT ladder

Key references

• Adiabatic connection / hybrid concept, half-and-half — A. D. Becke, J. Chem. Phys. 98, 1372 (1993); B3 three-parameter form, J. Chem. Phys. 98, 5648 (1993).
• The $\frac{1}{4}$ mixing argument — J. P. Perdew, M. Ernzerhof & K. Burke, J. Chem. Phys. 105, 9982 (1996).
• PBE0 — C. Adamo & V. Barone, J. Chem. Phys. 110, 6158 (1999).
• HSE (screened hybrid) — J. Heyd, G. E. Scuseria & M. Ernzerhof, J. Chem. Phys. 118, 8207 (2003); erratum J. Chem. Phys. 124, 219906 (2006) (correcting the original HSE03 specification). HSE06 (single $\omega =0.11{\text{bohr}}^{-1}$) — A. V. Krukau, O. A. Vydrov, A. F. Izmaylov & G. E. Scuseria, J. Chem. Phys. 125, 224106 (2006).
• Dielectric-dependent / optimally-tuned hybrids — the IP-tuning line is T. Stein, L. Kronik & R. Baer, Phys. Rev. Lett. 105, 266802 (2010); the $\alpha =1/{\epsilon }_{\infty }$ scheme is J. H. Skone, M. Govoni & G. Galli, Phys. Rev. B 89, 195112 (2014).