If you have ever run a standard DFT calculation on silicon and found a band gap of roughly 0.6 eV against the experimental 1.17 eV, you have met the band gap problem head on. It is one of the most consistent and misunderstood failures of practical density functional theory, and it trips up newcomers who assume the calculation is simply wrong. It is not: the underestimate is systematic, well understood, and fixable if you know what you are correcting.
This post explains why local and semilocal functionals underestimate band gaps, distinguishes the Kohn-Sham gap from the true fundamental gap, and walks through the practical fixes: hybrid functionals such as HSE06, DFT+U, the GW approximation, and the pragmatic scissor operator.
What a Band Gap Actually Is
Before diagnosing the error, be precise about the quantity. The fundamental gap of a material is a difference of total energies for the N-electron system:
$$ E_{\mathrm{gap}} = \bigl[E(N+1) - E(N)\bigr] - \bigl[E(N) - E(N-1)\bigr] = I - A $$where \(I\) is the ionization energy (energy to remove an electron) and \(A\) is the electron affinity (energy released adding one). This is a ground-state, total-energy quantity in principle accessible to DFT.
The optical gap is different again: it is the lowest energy to create a bound electron-hole pair (an exciton), so it is smaller than the fundamental gap by the exciton binding energy. Confusing these three quantities — fundamental, optical, and the Kohn-Sham gap below — is the first source of apparent disagreement with experiment.
The Kohn-Sham Gap Is Not the Fundamental Gap
In a Kohn-Sham DFT calculation you obtain a set of eigenvalues, and the natural temptation is to read the gap straight off them:
$$ E_{\mathrm{gap}}^{\mathrm{KS}} = \varepsilon_{\mathrm{LUMO}} - \varepsilon_{\mathrm{HOMO}} \quad (\varepsilon_{\mathrm{CBM}} - \varepsilon_{\mathrm{VBM}}\ \text{in a solid}) $$The problem is that the Kohn-Sham eigenvalues are, strictly, Lagrange multipliers that enforce orthonormality. They are not electron addition and removal energies. Even with the exact exchange-correlation functional, the Kohn-Sham gap would still differ from the fundamental gap by a term called the derivative discontinuity:
$$ E_{\mathrm{gap}}^{\mathrm{fundamental}} = E_{\mathrm{gap}}^{\mathrm{KS}} + \Delta_{\mathrm{xc}} $$\(\Delta_{\mathrm{xc}}\) is the jump in the exchange-correlation potential \(v_{\mathrm{xc}}(\mathbf{r})\) as the electron number passes through an integer. For the exact functional this is a real, finite, positive quantity. LDA and GGA functionals are smooth analytic functions of the density, so their derivative discontinuity is essentially zero. That missing \(\Delta_{\mathrm{xc}}\) is a large part of the underestimate, and it exists even before you consider approximation errors in the functional itself.
flowchart TD A[True fundamental gap I - A] --> B[Kohn-Sham gap eps_CBM - eps_VBM] A --> C[Derivative discontinuity Delta_xc] B --> D[What LDA/GGA report] C --> E[Missing in semilocal functionals] D --> F[Underestimated gap] E --> F
Why LDA and GGA Underestimate Gaps
Two intertwined effects drive the systematic error.
Self-interaction error (SIE). In the exact theory, the spurious Coulomb repulsion of an electron with itself in the Hartree term is exactly cancelled by exchange. In LDA and GGA the cancellation is incomplete, so every electron feels a residual repulsion from its own charge density. This delocalizes states, raises occupied-orbital energies relative to unoccupied ones, and narrows the gap. SIE is worst for localized states — precisely the d and f electrons in transition-metal oxides.
Missing derivative discontinuity. As above, the smooth semilocal potential cannot reproduce the integer-particle-number jump \(\Delta_{\mathrm{xc}}\), so a chunk of the true gap is simply absent from the eigenvalue spectrum.
The result is remarkably consistent: LDA and PBE underestimate gaps of sp semiconductors by 30-50%, and can predict small-gap or narrow-gap materials to be outright metallic. Germanium and several transition-metal oxides are classic cases where GGA closes a gap that experiment clearly shows to be open.
Hybrid Functionals: HSE06 and PBE0
The most popular practical fix mixes a fraction of exact (Hartree-Fock) exchange into the functional. Exact exchange is self-interaction free and reintroduces a discontinuity-like behaviour, opening the gap.
PBE0 uses 25% exact exchange globally. HSE06 (Heyd-Scuseria-Ernzerhof) is a range-separated hybrid: it applies 25% exact exchange only at short range and screens the long-range part, which makes it far cheaper for periodic solids and, empirically, more accurate for many semiconductors.
$$ \begin{aligned} E_{\mathrm{xc}}^{\mathrm{HSE}} &= a\, E_x^{\mathrm{HF,SR}}(\omega) + (1-a)\, E_x^{\mathrm{PBE,SR}}(\omega) \\ &\quad + E_x^{\mathrm{PBE,LR}}(\omega) + E_c^{\mathrm{PBE}} \\ &\quad a = 0.25,\ \omega = 0.2\ \text{Å}^{-1} \end{aligned} $$HSE06 typically brings sp-semiconductor gaps to within 0.1-0.3 eV of experiment. The cost is steep: evaluating exact exchange in a plane-wave code scales poorly and can be 10-100x more expensive than PBE, especially with dense k-meshes. This is exactly the kind of workload where GPU acceleration pays off.
DFT+U for Localized d and f States
For strongly correlated systems — NiO, MnO, many oxides of Fe, Co, Ce — the dominant error is SIE on localized d/f orbitals. DFT+U adds an on-site Hubbard-like penalty that pushes occupied localized states down and empty ones up, opening the gap and correcting over-delocalization at almost no extra cost over plain GGA. It is not a universal gap fix (it only touches the chosen orbitals) but for transition-metal oxides it is often the right, cheap tool. See our dedicated discussion of the method in the DFT+U article.
The GW Approximation
The most rigorous mainstream approach abandons Kohn-Sham eigenvalues as quasiparticle energies and instead solves for the true single-particle excitations using many-body perturbation theory. The electron self-energy is approximated as:
$$ \Sigma = i\, G\, W $$the product of the Green’s function \(G\) and the screened Coulomb interaction \(W\). G0W0 (one-shot, on top of a DFT starting point) is the standard flavour and delivers gaps typically within 0.1-0.2 eV of experiment across a wide range of materials, without empirical parameters. The downsides: results depend somewhat on the DFT starting point, convergence with respect to empty states and k-points is demanding, and the computational cost scales steeply with system size. GW on a supercell is a serious calculation.
Meta-GGA and the TB-mBJ Potential
Between the cheap failures of GGA and the expense of hybrids and GW sits a class of meta-GGA functionals, which add the kinetic energy density \(\tau(\mathbf{r})\) to the ingredients. The most gap-relevant is the Tran-Blaha modified Becke-Johnson (TB-mBJ) exchange potential. It is a potential-only model — there is no corresponding total energy functional — but it reproduces sp-semiconductor and insulator gaps with accuracy approaching HSE06 at almost the cost of a plain GGA calculation.
$$ v_x^{\mathrm{mBJ}}(\mathbf{r}) = c\, v_x^{\mathrm{BR}}(\mathbf{r}) + \frac{3c-2}{\pi}\sqrt{\frac{5}{12}\cdot\frac{2\tau(\mathbf{r})}{n(\mathbf{r})}} $$The catch is that TB-mBJ is not variational: you cannot compute reliable forces or relax a structure with it. The standard workflow is to relax with PBE, then compute the band structure with mBJ on the fixed geometry. For rapid, cheap gap estimates across many materials it is an excellent screening tool, and it is why high-throughput databases often report mBJ gaps alongside PBE ones.
The Scissor Operator: A Pragmatic Shortcut
Sometimes you already know the experimental gap and simply need correct optical or transport properties. The scissor operator rigidly shifts all conduction bands upward by a constant to match the known gap:
$$ \varepsilon_{\mathrm{CB}} \rightarrow \varepsilon_{\mathrm{CB}} + \Delta_{\mathrm{scissor}} $$It is cheap and often adequate for optical spectra, but it is empirical, it does not correct the shape of the bands or the effective masses, and it requires knowing the answer in advance. Treat it as an engineering convenience, not a predictive method.
Comparison of Band Gap Methods
| Method | Typical gap error | Relative cost vs PBE | Best use case | Parameters needed |
|---|---|---|---|---|
| LDA / PBE (GGA) | -30% to -50% | 1x | Geometry, energetics, screening | None |
| DFT+U | Material dependent | ~1-2x | TM oxides, localized d/f | Hubbard U |
| HSE06 (hybrid) | 0.1-0.3 eV | 10-100x | sp semiconductors, insulators | Mixing, screening |
| G0W0 | 0.1-0.2 eV | 100-1000x | Predictive gaps, spectroscopy | Starting point |
| Scissor operator | Exact by construction | ~1x | Optical/transport post-hoc | Known gap |
The pattern is clear: accuracy and cost move together. There is no free lunch, and the right choice depends on the material class and the property you actually need.
A Practical Workflow
A sensible strategy avoids paying for accuracy you do not need:
flowchart TD
A[Relax geometry with PBE] --> B{Material class?}
B -->|sp semiconductor| C[HSE06]
B -->|Correlated oxide| D[DFT+U]
B -->|Need predictive gap| E[G0W0]
C --> F[Converge k-mesh aggressively]
D --> F
E --> F
F --> G[Compare to photoemission not optical onset]- Relax the geometry with PBE. Structures are usually well described by GGA even when gaps are not.
- Classify the material. sp semiconductor, correlated oxide, or molecular solid? This selects the correction.
- Apply the matched correction. DFT+U for localized d/f; HSE06 for main-group semiconductors; GW when you need a parameter-free predictive gap.
- Converge aggressively. Hybrid and GW gaps are sensitive to k-point density and, for GW, the number of empty bands. Under-convergence masquerades as functional error.
- Compare to the right experiment. Match fundamental gaps to photoemission/inverse-photoemission, not to optical absorption onsets, which include exciton binding.
Standard, well-tested pseudopotentials and functionals ship with Quantum ESPRESSO, so the main practical barrier to HSE06 and GW is not availability but compute time.
Run it on Simatra
The band gap problem is, in the end, a compute problem: the accurate methods — HSE06 hybrids and GW — are exactly the ones that overwhelm a workstation. Simatra runs DFT on GPU-accelerated clusters (instance GPU-Opt-V2) with up to 5x faster convergence, handling supercells up to ~2,000 atoms across two computational engines: the open-source Quantum ESPRESSO and Simatra’s native KRONOS. Spin up a hybrid-functional or GW job without babysitting a queue, and start with a free trial and $100 in credits at app.simatra.io.