Optimization Algorithm for the Generation of ONCV Pseudopotentials

Martin Schlipf; Francois Gygi

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Optimization Algorithm for the Generation of ONCV Pseudopotentials

Martin Schlipf, Francois Gygi

arXiv:1502.00995v1Feb 3, 2015cond-mat.mtrl-sciphysics.comp-ph

TL;DR

The paper addresses systematic construction of transferable pseudopotentials by optimizing their input parameters against accuracy and computational cost. It introduces a quality function and applies it to ONCV pseudopotentials using training materials and an independent test set. The resulting SG15 set has low lattice-constant deviations from FLAPW and performs competitively with established USPP and PAW libraries.

Problem
Systematic databases of transferable pseudopotentials require construction parameters that perform well across materials rather than relying only on experience from all-electron atomic calculations.
Method
The paper optimizes ONCV construction parameters with a quality function combining lattice-constant agreement with FLAPW and the plane-wave cutoff, using training and independent test materials.
Results
The SG15 ONCV set has the lowest lattice-constant root-mean-square deviation from FLAPW among SG15, GBRV, and PSLIB, while 60 Ry yields the fewest inaccurate lattice constants for ONCV.
Takeaways & Limitations
With a modest energy-cutoff increase, the SG15 norm-conserving library provides a competitive alternative to USPP and PAW libraries and simplifies pseudopotential-based algorithm development.

Abstract

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We present an optimization algorithm to construct pseudopotentials and use it to generate a set of Optimized Norm-Conserving Vanderbilt (ONCV) pseudopotentials for elements up to Z=83 (Bi) (excluding Lanthanides). We introduce a quality function that assesses the agreement of a pseudopotential calculation with all-electron FLAPW results, and the necessary plane-wave energy cutoff. This quality function allows us to use a Nelder-Mead optimization algorithm on a training set of materials to optimize the input parameters of the pseudopotential construction for most of the periodic table. We control the accuracy of the resulting pseudopotentials on a test set of materials independent of the training set. We find that the automatically constructed pseudopotentials provide a good agreement with the all-electron results obtained using the FLEUR code with a plane-wave energy cutoff of approximately 60 Ry.

I. INTRODUCTION

Pseudopotentials simplify plane-wave electronic-structure calculations by replacing core states with an effective description focused on valence electrons. This work builds on ONCV construction and develops an optimization framework for systematically selecting construction parameters.

Pseudopotentials avoid explicit core-electron states, allowing plane-wave calculations to focus on valence electrons relevant to chemical bonding.
ONCV potentials retain norm conservation while achieving accuracy comparable to ultrasoft pseudopotentials at a moderately increased plane-wave cutoff.
Existing pseudopotential databases commonly select construction parameters from all-electron atomic calculations before testing transferability across crystal structures.
The ONCV construction introduces a wave-vector cutoff qc and minimizes residual kinetic energy above that cutoff when optimizing pseudo wave functions.
Increasing qc can represent more features of the physical potential but makes the pseudopotential harder and more costly to evaluate.
Projector radius rc defines the pseudoization region, while the pseudo wave function is constrained to match the all-electron wave function at that radius.

A. Quality function

The quality function reduces pseudopotential performance to a scalar that balances structural accuracy against the plane-wave cutoff needed for convergence. It targets a specified accuracy while favoring smoother potentials within that target.

The quality function compares the relative lattice-constant deviation between pseudopotential and all-electron calculations with the plane-wave cutoff required for convergence.
The accuracy and cutoff criteria compete because smoother pseudoization lowers the required cutoff but can reduce accuracy near the nucleus.
The target accuracy is δ0 = 0.2%, motivated by approximately 0.1% variation arising from different all-electron codes or input parameters.
For pseudopotentials within the target accuracy, the quality function includes a term proportional to 1/Ecut, favoring smoother potentials over harder ones.
For deviations larger than 2δ0, optimization focuses on the relative deviation through an 1/(δPP_alat)^2 term, with a smooth continuation between regimes.

B. Sets of materials

The optimization uses structurally simple but physically varied materials, separating the structures used to construct pseudopotentials from an independent set used to assess their accuracy.

The material set is chosen to represent relevant atomic environments while limiting structures to high symmetry and at most two atoms per unit cell.
Metallic environments are represented by simple cubic, body-centered cubic, face-centered cubic, and diamond-cubic structures.
Ionic environments are represented by rock-salt and zinc-blende compounds formed by combining elements.
The training set contains bcc and fcc structures plus rock-salt compounds, and is used to optimize pseudopotential construction.
The independent test set contains sc and dc structures plus zinc-blende compounds, enabling evaluation beyond the structures directly optimized.

C. Computational setup

The calculations compare pseudopotential results with converged all-electron FLAPW results and evaluate cutoff-dependent deviations. A monotonic correction ensures deviations behave consistently as the plane-wave cutoff increases.

DFT calculations: PBE calculations use an 8 × 8 × 8 Monkhorst-Pack mesh for both all-electron and pseudopotential calculations.The shared mesh is intended to make residual k-point errors comparable between the two calculations.
All-electron reference: FLAPW calculations are converged with additional unoccupied local orbitals, and lattice constants are obtained from a Murnaghan fit.The fit uses 11 data points surrounding the total-energy minimum.
Cutoff tests: The pseudopotential test uses Quantum ESPRESSO, beginning at Emax_cut = 160 Ry and decreasing the cutoff in 10 Ry steps to 40 Ry.The 160 Ry calculation is treated as the converged solution.
Deviation correction: The corrected deviation is monotonically decreasing with increasing cutoff, unlike the potentially nonmonotonic raw deviation.This correction makes the deviation at a given cutoff an upper bound to the deviation at any larger cutoff.

D. Optimizing pseudopotentials

The optimization constructs perturbed pseudopotentials, evaluates them on training materials, and uses Nelder-Mead simplex updates to improve their quality. A second optimization generation removes dependence on GBRV potentials.

Initialization: New pseudopotentials are generated by randomly perturbing all input parameters of a reasonable starting guess.Starting guesses come from ONCVPSP examples where available or are generated otherwise.
Evaluation: The quality function is evaluated over the training materials using the geometric average of the involved material scores.For rock-salt compounds, one potential is tested alongside a GBRV potential for the other element during the first stage.
Nelder-Mead optimization: After constructing N + 1 potentials, Nelder-Mead replaces the simplex’s worst corner with a better potential and contracts toward an optimum.The input parameters form a simplex in an N-dimensional space.
Two-generation refinement: After 80 to 200 iterations, a second generation restarts optimization from the converged first-generation potentials and runs for another 100 iterations.The second generation uses the first-generation potentials in compounds, making the resulting potentials independent of GBRV.

E. Refining the training set

The study refines the training and test strategy to expose failures on independent materials and compares SG15 with GBRV and PSLIB potentials. The optimization framework can also target other material properties when a quality function is available.

Training-set refinement: A few second-generation potentials fail to reproduce all-electron results on the independent test set.For early transition metals Sc to Mn, the sc structure must be added to the training set.
General applicability: The algorithm can optimize other material properties whenever a quality function maps pseudopotential results to a single number.This extends the approach beyond the properties used in the present optimization.
Comparative evaluation: The comparison evaluates SG15 ONCV potentials against GBRV USPP and high-accuracy PSLIB PAW potentials.The study examines lattice constants and bulk moduli across training and test materials.
Evaluation metrics: Table I analyzes bcc materials using average and rms deviations for lattice constants and bulk moduli, together with the fraction inaccurate at different cutoffs.The average indicates systematic bias, while the rms average measures error size.

A. Training set

Across bcc, fcc, and rock-salt training materials, ONCV potentials approach all-electron results as the cutoff increases. Their typical lattice-constant errors are small, while convergence costs differ among PP types.

bcc structures: For bcc materials, converged lattice-constant rms errors are around 0.1% for all pseudopotential types and smallest for ONCV.Only carbon and calcium exceed a 0.2% lattice-constant deviation with converged ONCV calculations.
fcc structures: For fcc materials, USPP requires the smallest cutoff but cannot improve further beyond 40 Ry, whereas PAW and ONCV converge most materials at 60 Ry.ONCV and PAW are somewhat better overall than USPP, although all remain close to the all-electron results.
fcc structures: Only cadmium lies outside the 0.2% lattice-constant boundary for converged ONCV calculations in the fcc results.The USPP deviation is similar for cadmium, while PAW is close to the FLAPW result.
rock-salt structures: For rock-salt compounds, ONCV potentials essentially reproduce all-electron results at large cutoffs, with very good lattice-constant accuracy at 60 Ry.About 10% of materials require a larger cutoff for the bulk modulus.
Overall training-set trend: Across all examined potentials, lattice constants agree very well with all-electron results, while USPP shows slightly lower accuracy and only a few large-error outliers.The comparison uses converged calculations at 160 Ry.

B. Test set

The ONCV potentials agree well with all-electron results on the independent test set, generally converging near 60 Ry, with a few structure- and material-specific exceptions.

sc structure: 0.1% for lattice constants and 4% for bulk moduli summarize the overall deviations in simple-cubic structures.Most ONCV lattice constants converge at 60 Ry, while GBRV lattice constants change little above 40 Ry.
sc structure: The ONCV potentials reproduce simple-cubic lattice constants within 0.2% except for calcium and lanthanum.Lanthanum is underestimated by ONCV, whereas USPP overestimates it; PAW does not converge for this material.
diamond structure: Diamond structures show the largest overall deviations from all-electron results, and ONCV frequently requires 60 Ry for convergence.ONCV gives the best agreement for bulk moduli, while PAW exceeds the 5% average-error tolerance.
diamond structure: 0.2% is exceeded by ONCV lattice constants for boron, chlorine, scandium, nickel, rubidium, and yttrium in diamond structures.The larger pseudoization differences may reflect diamond’s low space filling.
zincblende structure: A third of zincblende materials require 60 Ry with ONCV, but their average lattice-constant error remains below 0.2%.Only BeO exceeds the 0.2% deviation threshold for ONCV in this structure.
Overall test-set assessment: The test-set lattice-constant deviation is very small for all pseudopotential types, with ONCV accuracy not significantly worse than training-set accuracy.USPP has slightly larger deviations, partly because some lattice constants are overestimated by more than 0.4%.

C. Dimers and ternary compounds

The authors test transferability beyond the mono- and diatomic crystal sets using diatomic molecules, perovskites, and half-Heusler compounds.

Scope: The training and test sets are limited to mono- and diatomic crystals, motivating tests on diatomic molecules and ternary compounds.The additional systems probe whether the constructed ONCV potentials work outside the optimization scope.
Computational setup: Molecular bond lengths were optimized in a 15 Å × 15 Å × 30 Å box, with the long axis parallel to the molecule.Compound calculations used the same computational setup as the training and test materials.
Diatomic molecules: ONCV gives the smallest deviations for the investigated diatomic molecules, with errors above 0.2% only for O2 at 0.25% and F2 at 0.35%.USPP is outside the 0.2% target for every dimer except Br2, while PAW meets it only for H2.
Perovskites: Perovskites are accurately described by all pseudoizations, frequently agreeing with FLAPW within 0.1% in lattice constants.The worst ONCV case is LaAlO3, with a deviation of −0.13%.
Half-Heusler compounds: All half-Heusler materials fall within the desired accuracy for every pseudoization.ONCV has slightly larger deviations for GeAlCu and NMgLi, while several compounds closely match FLAPW.

V. CONCLUSION

The paper presents an optimization algorithm for pseudopotential construction and applies it to the SG15 ONCV dataset. The resulting potentials achieve competitive accuracy with modestly increased cutoffs, especially around 60 Ry.

Conclusion: The algorithm optimizes pseudopotential input parameters by mapping performance to a single quality value.The quality function combines lattice-constant agreement with FLAPW and the energy cutoff required for convergence.
Conclusion: Training and test sets separate parameter optimization from independent performance assessment, and the training set can be extended when accuracy targets are unmet.This provides an iterative route for improving a pseudopotential’s quality.
Conclusion: The SG15 set has the lowest lattice-constant root-mean-square deviation from FLAPW among SG15, GBRV, and PSLIB.At 60 Ry, ONCV also yields the fewest materials with lattice-constant deviations above 0.2%.
Conclusion: ONCV has the smallest root-mean-square deviation for the tested materials, while USPP achieves similar accuracy at a moderately lower cutoff.For bulk moduli, all pseudoization methods require larger energy cutoffs.
Conclusion: With only a modestly increased energy cutoff, SG15 offers a competitive norm-conserving alternative to USPP and PAW libraries.The authors encourage applying the algorithm to other functionals and construction methods.