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The PseudoDojo: Training and grading a 85 element optimized norm-conserving pseudopotential table
M. J. van Setten, M. Giantomassi, E. Bousquet, M. J. Verstraete, D. R. Hamann, X. Gonze, G. -M. Rignanese
TL;DR
Norm-conserving pseudopotential tables often lack modern optimization, systematic validation, and high-accuracy design despite the formalism’s simplicity and broad support. The paper presents PseudoDojo, an open-source framework using ONCVPSP and automated tests to develop and assess pseudopotentials. The resulting PD-PBE set contains 141 pseudopotentials tested through around 70.000 calculations, while revealing effects of core-valence partitioning and non-linear core corrections on stability, convergence, and transferability.
Problem
Existing norm-conserving pseudopotential tables are often old, not systematically tested, and lack developments such as optimization, spin-orbit support, and flexible projector choices.
Method
PseudoDojo combines ONCVPSP generation with an open-source Python framework and automated ABINIT-based validation of pseudopotentials.
Results
141 pseudopotentials were organized into standard and stringent accuracy tables and tested through around 70.000 calculations.
Takeaways & Limitations
The development process provided insights into how core-valence partitioning and non-linear core corrections affect pseudopotential stability, convergence, and transferability.
Takeaways & Limitations
Lanthanide pseudopotentials with 4 f states frozen are limited when f-electron physics matters, and 4 f-in-valence testing is left for future work.
Abstract
from arXiv · showhide
First-principles calculations in crystalline structures are often performed with a planewave basis set. To make the number of basis functions tractable two approximations are usually introduced: core electrons are frozen and the diverging Coulomb potential near the nucleus is replaced by a smoother expression. The norm-conserving pseudopotential was the first successful method to apply these approximations in a fully ab initio way. Later on, more efficient and more exact approaches were developed based on the ultrasoft and the projector augmented wave formalisms. These formalisms are however more complex and developing new features in these frameworks is usually more difficult than in the norm-conserving framework. Most of the existing tables of norm- conserving pseudopotentials, generated long ago, do not include the latest developments, are not systematically tested or are not designed primarily for high accuracy. In this paper, we present our PseudoDojo framework for developing and testing full tables of pseudopotentials, and demonstrate it with a new table generated with the ONCVPSP approach. The PseudoDojo is an open source project, building on the AbiPy package, for developing and systematically testing pseudopotentials. At present it contains 7 different batteries of tests executed with ABINIT, which are performed as a function of the energy cutoff. The results of these tests are then used to provide hints for the energy cutoff for actual production calculations. Our final set contains 141 pseudopotentials split into a standard and a stringent accuracy table. In total around 70.000 calculations were performed to test the pseudopotentials. The process of developing the final table led to new insights into the effects of both the core-valence partitioning and the non-linear core corrections on the stability, convergence, and transferability of norm-conserving pseudopotentials. ...
1. Introduction
Pseudopotentials make planewave DFT tractable by replacing core electrons and the singular nuclear potential with a smoother effective description. The paper addresses shortcomings in legacy norm-conserving tables by introducing a systematically tested, optimized table and framework.
- Motivation: Pseudopotentials eliminate core states and describe valence electrons with smooth pseudo-wavefunctions, enabling efficient planewave calculations.Planewave completeness is controlled by the maximal kinetic energy, or energy cutoff Ec.
- Limitations of existing tables: Legacy norm-conserving tables often lack systematic validation, spin-orbit support, optimization, and flexible projector configurations for semicore states and high-energy scattering.Many were generated before optimization methods such as RRKJ and commonly use only one projector per angular channel.
- Motivation: Norm-conserving pseudopotentials remain useful because their formalism is simple, robust, widely supported, and often receives new first-principles-code developments first.Compared with ultrasoft and PAW methods, they are easier to implement, although they generally require higher kinetic-energy cutoffs.
- Contribution: The paper constructs the PD-PBE norm-conserving table with the ONCVPSP approach to provide softer and more accurate pseudopotentials.ONCVPSP supports multiple projectors for semicore states and can generate different exchange-correlation and spin-orbit variants.
- Contribution: PseudoDojo distributes ready-to-use pseudopotential files, inputs, notebooks, and an open-source Python toolbox for automated generation and validation.Files are provided in psp8, UPF2, and PSML 1.1 formats.
2. Formalism
ONCVPSP combines generalized norm conservation with convergence optimization to reproduce all-electron binding and scattering properties while reducing planewave convergence errors. Its construction uses multiple projectors, constrained basis expansions, and a kinetic-energy-error metric.
- Generalized norm conservation: ONCVPSP uses two projectors and generalized norm conservation to reproduce all-electron binding and scattering properties.Generalized norm conservation matches integrated charge inside the core radius for linear combinations of pseudo- and all-electron wavefunctions.
- Non-local operator: The non-local operator is built from projectors and a matrix whose symmetry makes the resulting operator Hermitian.The local potential agrees with the all-electron potential outside the core radius.
- Transferability: The construction preserves agreement of logarithmic derivatives and their energy derivatives with all-electron solutions outside the core radius.The matching applies at the selected bound-state or scattering energies.
- Convergence optimization: ONCVPSP minimizes a residual kinetic-energy error because this metric predicts planewave convergence errors in solid-state calculations.The optimization is based on the RRKJ convergence metric and is formulated independently of a particular solid.
- Convergence optimization: The pseudo-wavefunction is expanded in basis functions subject to continuity and norm constraints, then optimized through a residual basis formed by diagonalizing the residual-energy matrix.The resulting coefficients can be optimized for a prescribed cutoff and evaluated to obtain convergence profiles at other cutoffs.
- Convergence optimization: The constrained optimization reduces to solving for one coefficient by interval-halving because the relevant constraint sum is monotonic in its magnitude.The optimized coefficients are based on a prescribed qc, while the error profile can be evaluated for any cutoff q.
- Multiple projectors: The second projector is optimized after adding its overlap constraint with the first projector, while scattering states are handled through artificial bound states for optimization.The artificial barrier produces bound pseudo-wavefunctions with convergence properties comparable to valence functions.
3. The PseudoDojo
PseudoDojo is an open-source Python framework that combines reference data, pseudopotential-generation tools, automated crystalline tests, and web-based reports. It supports reproducible validation, visualization, ranking, and high-throughput workflows.
- Framework components: The framework contains a reference database, tools and graphical interfaces for generation and initial validation, and scripts for automated crystalline-environment tests.Reference data include all-electron results for the Δ-Gauge and GBRV benchmarks.
- Generation and validation: PseudoDojo provides a GUI for setting input parameters, comparing pseudopotentials with atomic all-electron references, and generating parameter series.Logarithmic derivatives are among the visualized comparison quantities.
- Reporting: Each pseudopotential receives a machine-generated DojoReport in JSON containing entries for its validation tests and results as functions of Ec.The reports are intended for record keeping, plotting, ranking, and setup of high-throughput calculations rather than direct use by ab-initio codes.
- Web interface: The web interface enables online visualization of internal and external validation results and download of pseudopotential and input files.HTML reports are generated automatically from Jupyter Notebooks without requiring installation of the Python package.
4. The PD-PBE tables
The PD-PBE tables balance accuracy, transferability, and convergence by offering tailored pseudopotentials across elements, functionals, and core-valence choices. Their design combines multiple projectors, continuity constraints, RRKJ-related considerations, and nonlinear core corrections.
- Scope and table variants: Localized d- and f-electron elements remain difficult to pseudize, so the project does not require all norm-conserving pseudopotentials to converge below one common cutoff.This distinguishes the table’s strategy from projects focused on uniformly low cutoff energies.
- Scope and table variants: The first PseudoDojo version focuses on GGA-PBE, while LDA-PW and PBEsol sets are also available and other functionals can be generated.Each functional is associated with standard and stringent accuracy versions.
- Project infrastructure: JSON stores project data because it is broadly supported for parsing, generation, and transmission between servers and web applications.
- Core-valence partitioning: Semicore variants are supplied where core-valence separation is ambiguous because retaining semicore states generally improves accuracy and transferability and may matter for GW calculations.The stated rationale is reduced error from the frozen-core approximation and better treatment of valence–semicore overlap.
- Scope and table variants: Normal and high versions are provided for slowly converging elements: the high-accuracy option uses smaller core radii, requires larger cutoffs, and is more transferable.Low-accuracy versions are additionally supplied when the standard version converges only above 40 Ha.
- Projectors and smoothness: Most pseudopotentials use two projectors per angular channel, giving close logarithmic-derivative agreement with all-electron results up to at least 3–5 Ha and often 10 Ha.A small number of listed exceptions use different projector counts.
- Projectors and smoothness: Fourth-order derivative continuity is generally enforced at the core radius, but using third-order continuity can reduce the required cutoff by approximately 5 Ha.The additional fourth-order requirement usually makes pseudopotentials slightly harder.
- Nonlinear core corrections: PD-PBE includes nonlinear core corrections for core-frozen pseudopotentials except third-row semicore Na-sp–Cl-sp and Ne, using Teter model core charges to improve smoothness and convergence.The construction did not show a clear correlation between pseudopotential quality and reproduction of all-electron chemical hardness, while Teter’s approach worked well for localized core charges.
5. Convergence and energy cutoff hints
PseudoDojo selects energy-cutoff hints from convergence studies across multiple tests and precision levels. The framework combines automated sampling and criteria with manual inspection where convergence behavior can be oscillatory.
- Tables and precision levels: Two tables, standard and stringent accuracy, each contain one pseudopotential per element; for about half the elements, the stringent table uses a different and more accurate potential.
- Tables and precision levels: High, normal, and low cutoff hints are provided because different physical properties converge differently with the plane-wave cutoff.Phonons and bulk moduli are cited as more cutoff-sensitive than total energy.
- Convergence workflow: The framework evaluates the ∆-Gauge, GBRV parameters, and Γ-point phonons over an energy-cutoff mesh initialized from ONCVPSP electronic-eigenvalue hints.The mesh uses a dense 2 Ha step around the initial value and a coarser 10 Ha step for absolute convergence.
- Convergence workflow: Each precision hint is defined by bounds on convergence observables, with additional cutoff points added until the curve has an approved converged tail.The largest grid point supplies the converged observable value, and all curves are manually inspected.
- Practical use: The three precision levels support workflows that screen many systems cheaply at low precision before refining promising candidates at normal or high precision.
- Convergence workflow: Automatic convergence assessment is limited because quantities such as the ∆-Gauge can oscillate and therefore require human inspection.
- Convergence criteria: Table 3 defines low, normal, and high hints using maximal electronic-energy deviations, the revised ∆-Gauge, and total electronic energy per atom.
- Reported results: Table 4 reports statistics for the low, normal, and high hints of the standard table, while Figure 1 compares hint distributions for the standard and high tables.
6. Discussion of the validation per table
Validation across ∆-Gauge, ∆’-Gauge, GBRV, convergence, and phonon tests shows generally converged structural results but distinct sensitivities across properties and elements.
- ∆-Gauge: 0.70 and 0.64 are the mean ∆-Gauge values for the standard and high tables when Cr, Mn, and Fe are excluded.These magnetic elements require harder pseudopotentials in the ∆-Gauge test.
- ∆’-Gauge: The ∆’-Gauge identifies Hg, Sr, and Ba as problematic elements in addition to the Cr, Mn, and Fe outliers seen in the standard table.A ∆’-Gauge value below 2 generally indicates an accurate potential for ground-state structural properties.
- GBRV: Low and normal energy-cutoff hints already give rather converged GBRV lattice parameters, while the high table does not significantly improve GBRV results.The ∆-Gauge shows stronger high-versus-low differences for magnetic systems, which are absent from GBRV tests.
- GBRV: The GBRV tests tend to underestimate all-electron lattice parameters, with a relatively symmetric error distribution peaked near the mean.Elements prominent in FCC and BCC tests, including F, S, and K, can perform better in compound tests.
- GBRV: F, S, Cs, Rb, and K show larger deviations in both NCPP and PAW tables, suggesting that frozen-core treatment contributes to the error.Adding additional valence states for these elements proved difficult.
- Phonons: Phonon modes are not converged at the low energy-cutoff hint, whereas acoustic-sum-rule errors remain within 2 cm−1 except for a few outliers.The acoustic phonon-mode error is decoupled from the other validation tests.
7. Discussion of individual pseudopotentials
Individual pseudopotentials use element-specific core partitions, projectors, and cutoff choices to balance transferability, accuracy, and efficiency.
- H and He: H and He use two s projectors, with 1s pseudization radii of 1.0 a.u. and 1.25 a.u., respectively.Their localized 1s wavefunctions require careful radius choices to balance accuracy and efficiency.
- Li and Be: Li and Be include 1s semicore electrons to improve transferability and accuracy, at the cost of higher energy cutoffs.The standard versions use indicative cutoffs of 35 Ha for Li and 42 Ha for Be.
- B to Ne: For B to Ne, the cutoff is governed by localized 2p states, and two projectors per angular channel are used to improve transferability.O and F additionally receive a single d projector.
- B to Ne: F shows large GBRV errors and also performs poorly for atomization energies and molecular systems.The same large GBRV error is observed in tested PAW tables.
- B to Ne: Ne is difficult to model with nonlinear core corrections because its all-electron core is strongly localized, leading to inaccurate equation-of-state curves and unstable self-consistency for many model cores.Adding nonlinear core corrections does not improve transferability for Ne.
7.4. Na, Mg
For Na and Mg, including 2s and 2p semicore states improves validation results, while alternative deeper-valence versions are mainly useful for convergence studies.
- Na, Mg: Including 2s and 2p in the valence significantly improves both ∆-Gauge and GBRV results for Na and Mg.For Mg, this choice also removes a ghost state near 80 eV.
- Na, Mg: The Na-sp and Mg-sp pseudopotentials apply no nonlinear core corrections when 2s and 2p are included in the valence.This reflects the difficulty of constructing transferable model core charges for strongly localized core states.
- Na, Mg: For the third-row series, 2s and 2p semicore versions are supplied for convergence studies and all-electron comparisons.These versions require high cutoffs and show nonsystematic accuracy improvements, so they are excluded from the high table.
- Na, Mg: The default versions retain 3s and 3p in the valence and use two d projectors to improve transferability.Reasonable cutoff hints and good test results place these versions in both the standard and high tables.
7.7. 3d transition metals
The 3d transition-metal pseudopotentials are challenging, especially for magnetic Δ-Gauge performance. Including semicore states and choosing between standard and high-accuracy versions balances transferability against cutoff requirements.
- 3d transition metals: For the 3d transition metals, 3s and 3p states are included in the valence partition.For Fe, lowering continuity at the pseudization radius to the third derivative avoids prohibitively large Ec requirements.
- 3d transition metals: Cr, Mn, and Fe are the most complicated elements in this series, particularly for magnetic Δ-Gauge tests.Standard versions with reasonable Ec hints can remain beyond acceptable Δ-Gauge limits.
- 3d transition metals: The high-accuracy versions fix the Δ-Gauge problem for Cr, Mn, and Fe but require considerably larger Ec values.Both standard and high-accuracy versions perform equally well in the non-magnetic GBRV tests.
- 3d transition metals: For Ga, Ge, As, and Se, 3d electrons are placed in valence, while they are frozen in the core for Br and Kr.The localized 3d states make this partitioning choice relevant to transferability.
- 3d transition metals: Br and I also have versions including their entire relevant shells in valence, mainly for accurate GW calculations.For ground-state calculations, the 3d or 4d valence choice is considered sufficiently accurate.
7.9. Rb, Sr
Rb and Sr achieve reasonable cutoff hints, while GBRV performance declines for heavier alkaline elements. Across the 4d series, semicore choices create explicit trade-offs between Δ-Gauge, GBRV, and Ec.
- Rb, Sr: Rb and Sr achieve very reasonable Ec hints, but alkaline elements from Rb downward show decreasing GBRV performance.Making the pseudopotentials harder and more accurate did not improve the GBRV results.
- 4d transition metals: The 4d transition-metal PSPs include 4s and 4p states in valence, yielding reasonable Ec energies and test results.Only Ru and Rh have Δ-Gauge results that are barely acceptable, while their Δ’-Gauge values remain acceptable.
- 4d transition metals: Cd-sp improves GBRV results over Cd but increases Ec and worsens Δ-Gauge performance.Cd-sp-high restores acceptable GBRV and Δ-Gauge results at a still larger Ec cost.
- In to I: For In through I, all three core-valence partitionings keep Δ-Gauge, Δ’-Gauge, and GBRV within acceptable ranges.Their main differences concern the description of unoccupied states, including logarithmic-derivative deviations above the Fermi level.
- Cs, Ba: For Cs, adding explicit f projectors improves transferability; for Ba, an excited-reference PSP improves Δ-Gauge but worsens GBRV similarly.Both Cs and Ba include 5s and 5p states in valence.
7.13. 5d transition metals
The 5d transition-metal PSPs require careful semicore and projector choices to avoid ghost states and balance Δ-Gauge against GBRV performance. Frozen 4f states remain a scope limitation for optical and GW properties.
- 5d transition metals: Including only 5d states in valence can produce ghost states a few eV above the Fermi level, so 5s and 5p are always included.This choice is made even when tests with only 5d states show good Δ-Gauge and GBRV results.
- 5d transition metals: For Hf and Ta, adding 4f states to the 5s and 5p valence partition significantly improves Δ-Gauge results.For Ta, GBRV results worsen; for W, the changes are small.
- 5d transition metals: Explicit f-projectors benefit all 5d-transition-metal PSPs even when 4f electrons remain frozen in the core.This projector choice is distinct from including 4f electrons in the valence partition.
- 5d transition metals: For W–Hg, freezing 4f electrons can describe ground-state properties well enough but may not suffice for optical properties and GW.The limitation also applies to elements such as Au, where 4f states lie about 3 Ha below the Fermi level.
- Lanthanides: La-sp underestimates the BCC lattice parameter by 0.1%, while Lu-fsp has a Δ-Gauge of 1.0.The reported low, normal, and high Ec hints are 50, 55, 65 Ha for La and 46, 50, 58 Ha for Lu.
- Lanthanides: For lanthanide nitrides, PD-PBE lattice parameters generally agree well with comparable VASP PAW results.Comparison with experiment supports freezing 4f states for Sm–Lu when assessing rocksalt-nitride structural properties.
7.15. Tl, Pb, Po, At, Rn
For Tl through Rn, pseudizing 6p valence electrons is not demanding. Tables with different core-valence choices show good and quickly converging Δ-Gauge and GBRV results.
- Tl, Pb, Po, At, Rn: For Tl–Rn, pseudizing the 6p valence electrons is not very demanding.The standard table includes versions with 5d in valence, while the high table also includes 5s and 5p.
- Tl, Pb, Po, At, Rn: Both the Δ-Gauge and GBRV results are good and converge quickly for the Tl–Rn pseudopotentials.The results are shown in Figure 25 for the standard and high-table choices.
8. Conclusions
The PseudoDojo combines open-source pseudopotential generation with systematic, complementary validation and produces a 141-PSP PD-PBE set in standard and stringent accuracy tables. Its development also yielded findings on testing coverage, core-valence partitioning, non-linear core corrections, and extra projectors.
- Conclusions: The PseudoDojo framework couples ONCVPSP generation, AbiPy-driven ABINIT testing, a graphical toolkit, and distribution in multiple pseudopotential formats.The project provides generation and validation tools, inputs for parameter tuning, and files in psp8, UPF2, and PSML 1.1 formats.
- Conclusions: Seven crystalline-environment tests provide complementary validation, with GBRV-FCC and GBRV-BCC as the only strongly correlated pair.The tests include the Δ-Gauge, Δ′-Gauge, GBRV tests, ghost-state detection, and Γ-point phonons, all executed with ABINIT.
- Conclusions: 141 PSPs form the PD-PBE set, divided into standard and stringent accuracy tables and tested through around 70,000 calculations.The calculations used PseudoDojo tools built on AbiPy's high-throughput framework.
- Conclusions: Non-linear core corrections dramatically improve magnetic 3d transition-metal results, but localized corrections can make pseudopotentials difficult to converge, especially in DFPT calculations.For some systems, adding non-linear core corrections instead provides no improvement and can cause instability.
- Conclusions: Core-valence partitioning improves the unoccupied-space description, while insufficient partitioning can leave ghost states and poor transferability despite good Δ-Gauge results.For fifth-row main-group elements, including the full n = 4 shell removes ghost-state signs up to several hundreds of eV; analogous issues occur in some 5d transition metals.
- Conclusions: Additional d and f projectors improve pseudopotentials for second-row B–F elements and fifth-row transition metals with frozen 4f states.