7.2 Hybrid functionals and exact-exchange controls
7.2.1 From a PBE baseline to a different Hamiltonian
A hybrid calculation introduces nonlocal exchange; it is not a switch that automatically produces experimental band gaps. The basis, k sampling, geometry, pseudopotential construction and screened interaction still matter. This case asks a narrower question: how can you attribute a change in the silicon electronic structure to the functional rather than to an uncontrolled exchange approximation?
The conditional templates here follow ABACUS 3.9.0 LCAO documentation and have not been executed. The first LCAO silicon parent uses LDA assets: first replace its pseudopotential and numerical orbitals together with a verified, matched PBE family, explicitly set PBE, and reconverge the primary basis and k sampling. Preserve its fixed geometry. Only this independently converged PBE parent is the baseline below; do not merely change the functional while retaining the teaching LDA assets. Use an ABACUS build with the required Libxc and exact-exchange dependencies. A binary that supports ordinary PBE is not proof that the HSE path and all selected solver combinations were compiled. Read the matching build guide, the startup banner and the accepted functional in the log before trusting any result.
The course does not promise a PW hybrid workflow, hybrid forces, SOC hybrids, or every available accelerator combination. Such combinations need release-specific validation beyond this LCAO single-point exercise. Do not combine undocumented advanced switches merely because each appears independently in a manual.
7.2.2 Screened exchange and auxiliary representation
For a screened hybrid, the short-range exchange is mixed schematically as
The splitting uses \(1/r=\operatorname{erfc}(\omega r)/r+\operatorname{erf}(\omega r)/r\). Changing the exchange fraction or screening parameter changes the physical functional; tightening an integral threshold instead reduces a numerical approximation. Keep these two categories separate in your notebook.
In the LCAO implementation, products of numerical atomic orbitals can be represented with auxiliary functions. Reducing that representation saves work but introduces another accuracy control beyond the primary orbital basis. A larger exx_pca_threshold removes more auxiliary information. Agreement between two coarse k meshes cannot demonstrate convergence of that independent approximation.
The analytic curves illustrate the exchange range split. The error-decay panel depicts a controlled study, not measured timings or a silicon band structure.
7.2.3 Worked delta and a controlled threshold study
Make a new folder from the converged silicon LCAO parent. Keep STRU, KPT, species order, pseudopotential and orbital files identical. Replace these existing INPUT entries and add the exchange controls:
# ABACUS 3.9.0 conditional LCAO HSE single point
suffix Si_HSE_fixed
basis_type lcao
dft_functional hse
exx_hybrid_alpha 0.25
exx_pca_threshold 1e-4
exx_c_threshold 1e-4
exx_dm_threshold 1e-4
exx_v_threshold 0
The exchange fraction is dimensionless. These thresholds are documented numerical controls, not energy cutoffs in eV or Ry. Their starting values are illustrative; your observable determines how tightly to converge them. Leave exx_hse_omega at the documented release default for this initial named-functional comparison and record the resolved value from the manual/output. Do not silently borrow a screening number expressed in inverse ångströms from another code and insert it under a different unit convention.
First change only the PCA threshold, for example from 1e-3 to 1e-4 and 1e-5. Then test the coefficient and density-matrix truncations. Keep the Coulomb-matrix truncation at zero for this baseline. Record both accepted parameters and realized auxiliary dimensions where printed. If the executable does not recognize a requested control, stop and reconcile its version; do not remove warnings just to finish a run.
7.2.4 Workflow and outputs
Converge the PBE parent without exact exchange. Start each HSE branch from a deliberately documented initialization; distinguish a fresh atomic-density start from reuse of a compatible density. Check every self-consistency loop. A small density residual inside a nested procedure does not alone prove that the exchange update is converged. If using exx_separate_loop, follow the documented outer-loop limits and mixing semantics of the pinned release rather than assuming the PBE scf_nmax controls everything.
The main evidence is the complete OUT.<suffix>/running_scf.log, accepted method, termination reason, final energy and eigenvalues. Preserve all exchange-related files for reproducibility, but do not treat processor-specific exchange restart files as universally portable. Extract a gap only after verifying the sampling and band indexing. A sparse SCF mesh may miss the indirect conduction minimum; a gap read from those eigenvalues is not automatically the fundamental gap.
Keep the first comparison at the same geometry. If you later optimize with a version-supported hybrid force implementation, label that as a second experiment and perform finite-difference force checks. Separate vertical electronic changes from geometry-induced changes. No computed gap, speedup or force accuracy is supplied here.
7.2.5 Blank convergence record
| Primary orbital hash | k mesh | PCA threshold | Other EXX thresholds | All loops converged? | Gap definition / value (eV) | Energy (eV) | Wall time (s) |
|---|---|---|---|---|---|---|---|
| 1e-3 | recorded | ||||||
| 1e-4 | recorded | ||||||
| 1e-5 | recorded |
Inspect whether the gap and the energy difference of interest are stable under independent changes in orbital quality, k sampling and exchange truncation. An accidentally stable total energy can hide a moving conduction-band edge. Use a target appropriate to your task, and measure performance only after comparable numerical work has been established. Tightening several controls simultaneously may improve accuracy, but it does not identify which approximation caused the original discrepancy.
7.2.6 Pitfalls, exercises and answers
Pitfalls. Calling any exchange fraction “HSE06” without recording the range convention is ambiguous. Comparing a PBE optimized structure with a differently strained HSE structure obscures causality. Mixing orbital libraries while keeping the same cutoff changes the primary basis. Assuming a library name in the build configuration proves it was loaded at runtime ignores dynamic-library changes. Copying warnings from an unrecognized latest-documentation keyword into an older binary can conceal an unintended fallback.
Exercise 1. A tighter PCA threshold changes the gap but almost not the energy. Which quantity controls acceptance? Answer: the requested observable; convergence is not certified solely by total energy. Track both band edges and the gap definition under the same sampling.
Exercise 2. A larger exact-exchange fraction gives a more attractive gap. Is that numerical convergence? Answer: no. It is a functional sensitivity or model calibration study, requiring its own justification and validation set. Numerical convergence keeps the functional fixed while reducing representation and integration errors.
7.2.7 Sources and next case
- ABACUS 3.9.0 exact-exchange controls.
- ABACUS 3.9.0 build dependencies.
- Heyd, Scuseria and Ernzerhof, screened hybrid formulation.
- University-linked ABACUS+LibRI hands-on lesson: auxiliary-basis workflow context; its historical examples must be reconciled with the pinned release.