AiiDA-Hubbard: Scalable U/V Computation

• aiida-hubbard is an automated framework for self-consistent, high-throughput computation of Hubbard U and V parameters using DFPT within the AiiDA ecosystem.
• It integrates with AiiDA’s workflow engine to orchestrate iterative DFPT evaluations, parallel primitive cell calculations, and dynamic intersite V parameter determination.
• The framework enables reproducible DFT+U(+V) studies for screening correlated materials, supporting error recovery and efficient handling of geometry variations.

aiida-hubbard is an open-source, automated framework designed for the self-consistent, high-throughput computation of Hubbard $U$ (onsite) and $V$ (intersite) parameters from first-principles, fully integrated within the AiiDA workflow ecosystem. By leveraging density-functional perturbation theory (DFPT), the framework parallelizes tasks efficiently using concurrent primitive cell calculations and introduces dynamic, on-the-fly determination of intersite $V$ parameters to accommodate structural relaxations and variable coordination environments. aiida-hubbard targets reproducible and scalable evaluations required for accurate correction of electron correlation in DFT+$U$ and DFT+U+V studies, facilitating high-throughput screening of correlated and redox-active materials (Bastonero et al., 3 Mar 2025).

1. Architecture and Workflow Integration

aiida-hubbard is implemented as a standard AiiDA plugin (PyPI: aiida-hubbard), utilizing AiiDA’s provenance graph, data models, workflow engine, and error handling for automated submission and orchestration. A key data type, HubbardStructureData, extends AiiDA’s StructureData to encapsulate the atomic structure, choice of Hubbard functional flavor (e.g., Dudarev, Liechtenstein), projector type (atomic or ortho-atomic), and a structured list of interaction parameters ($U$, $V$), alongside atomic indices and translation vectors.

The hierarchical workflow structure is organized as follows:

• SelfConsistentHubbardWorkChain: Top-level workflow for the iterative, self-consistent calculation of $U$ and $V$, optionally invoking geometry optimization (via PwRelaxWorkChain), single-point DFT calculation (PwBaseWorkChain), and DFPT-based parameter evaluation (HpWorkChain), iterating until user-defined thresholds $|\Delta U|<\delta U$, $|\Delta V|<\delta V$ are achieved.
• HpWorkChain: Handles distributed DFPT evaluations, enabling parallelization over inequivalent Hubbard atom pairs (HpParallelAtomsWorkChain) and, within each, parallelization over irreducible $V$0-points (HpParallelQpointsWorkChain). Each $V$1-point task invokes hp.x for fully automated input generation, mixing control, and error recovery.
• DFPT in primitive cells: Employs monochromatic perturbations at discrete reciprocal lattice vectors, avoiding the computational cost of large supercells. The charge-density response is decomposed over $V$2-points, yielding independent calculations suitable for high parallel efficiency.

2. Formalism and Self-Consistent Computation of Hubbard Parameters

aiida-hubbard computes Hubbard corrections for ground-state DFT in the form:

$V$3

with

$V$4

where $V$5 represents the occupation matrices in the chosen Hubbard manifold and $V$6 indicates summation over unique $V$7 pairs.

The framework defines Hubbard parameters by linear response:

• $V$8
• $V$9

Rather than perturbing occupations directly, response matrices

• $V$0
• $V$1 are computed to yield
• $V$2
• $V$3

The self-consistent cycle involves:

1. Initialization: $V$4, $V$5 (default 0).
2. Optional geometry optimization with DFT+U+V.
3. Single-point DFT+U+V calculation; metallicity determined via smearing.
4. For insulators, occupations fixed to avoid $V$6 divergence.
5. DFPT calculation for $V$7.
6. Convergence check: $V$8, $V$9; otherwise, update ($U$0 mixing of $U$1) and repeat.

Intersite $U$2 manifolds are dynamically identified at every iteration via Voronoi tessellation using pymatgen’s CrystalNN, therefore reflecting changes due to atomic relaxation and coordination environment.

3. High-Throughput Protocols and Performance Benchmarks

The high-throughput capabilities of aiida-hubbard were demonstrated on a dataset of 115 Li-containing bulk solids (up to 32 atoms/cell, up to five elements). The overall workflow exhibited:

• Success rate: 91% (105/115 systems converged)
• Mean workflow cycles: ≈2.9 per system (combining geometry relaxations, DFT+U+V, DFPT)
• Failure recovery: 6% of DFT+U+V or DFPT tasks failed but were automatically recovered in all except two cases (closed-shell divergence)

Recommended computational protocols and parameters:

Protocol	k-point Δk (Å⁻¹)	q-point Δq (Å⁻¹)	δU (eV)	δV (eV)
"precise"	0.2	0.4	0.01	0.005
"moderate"	0.4	0.8	0.1	0.01
"fast"	0.6	1.2	0.2	0.1

Other standardized settings: Exchange-correlation PBEsol, SSSP v1.3 pseudopotentials (efficiency or precision), Lӧwdin-orthogonalized atomic orbitals, and relaxation tolerances of 10⁻⁶ Ry/atom (energy), 10⁻⁴ Ry/Bohr (forces), and 0.5 kbar (pressure).

4. Statistical Analysis and Case Study: Li-Containing Solids

The distribution and variation of computed $U$3 and $U$4 values highlight their dependence on oxidation state and coordination geometry. Key findings for transition-metal 3d orbitals include:

• Fe 3d:
  • Fe²⁺: $U$5 eV, mean ≈5.3 eV, σ ≈0.4 eV
  • Fe³⁺: $U$6 eV, mean ≈4.9 eV, σ ≈0.3 eV
• Mn 3d:
  • Mn²⁺: $U$7 eV, mean ≈4.5 eV
  • Mn³⁺: $U$8 eV, mean ≈6.0 eV
  • Mn⁴⁺: $U$9 up to ≈9.2 eV; shifts up to 1–2 eV compared to Mn²⁺

Correlations illustrate that $U$0 increases with oxidation state and varies with local geometry, e.g., Fe²⁺ in tetrahedral FeSi₂Li₄O₈ ($U$1=5.4 eV) vs square-planar FeP₂Li₂O₈ ($U$2=6.0 eV, +0.6 eV shift), or a 0.5 eV shift in Fe²⁺ vs Fe³⁺ in octahedral coordination.

For intersite $U$3 (M–O pairs):

• Fe–O: $U$4 eV, mean ≈0.8 eV, σ ≈0.2 eV; decreases from ≈1 eV at $U$5 Å to ≈0.4 eV at $U$6 Å
• Mn–O: $U$7 eV, mean ≈0.7 eV, σ ≈0.2 eV; similar distance-dependent decay

This systematic dataset establishes reference values and scaling for $U$8, $U$9 relevant to battery materials, redox chemistry, and electronic-structure modeling of correlated oxides.

5. Workflow Usage, Configuration, and Best Practices

aiida-hubbard can be installed via $U$0 (together with aiida_core and aiida_quantumespresso). A typical workflow instance involves script-based definition and launch in an AiiDA profile:

$U$1

Results are then accessed via the workflow’s AiiDA provenance: the final Hubbard parameters can be retrieved as onsite $V$0 and intersite $V$1 via methods on the output HubbardStructureData. Best practices include starting with the "moderate" protocol for screening, upgrading to "precise" for publication-quality results, always including geometry optimization when structure is variable, monitoring convergence thresholds ($V$2, $V$3), and ensuring nonzero occupation in the relevant Hubbard manifold to avoid $V$4 divergences.

6. Limitations and Prospective Developments

Current limitations include manual selection of Hubbard manifolds, with the risk of divergent $V$5 values if the relevant orbitals have little frontier-state character (notably in closed-shell or strong charge-transfer insulators). At present, only Quantum ESPRESSO’s DFT+U(+V) and DFPT (hp.x) are fully supported, although data models are code-agnostic. Hund’s $V$6, spin–orbit, and noncollinear spin corrections are not yet automated. Intersite $V$7 is limited to nearest-neighbor pairs identified via Voronoi tessellation; expansion to long-range interactions requires manual intervention.

Prospective directions include automated manifold selection using Wannier or occupation-based criteria, incorporation of machine learning for $V$8, $V$9 estimation or convergence acceleration, extension to other electronic structure codes (e.g., ABINIT, VASP) via transition layers, and integration with workflows for phonons, molecular dynamics, and defect formation energies through cooperation with packages such as aiida-vibroscopy and aiida-defects.

aiida-hubbard thus constitutes a robust, scalable infrastructure for first-principles, automated Hubbard parameter evaluation, enabling reproducible high-throughput materials screening critical for energy storage, redox-active materials, and the broader field of correlated electron systems (Bastonero et al., 3 Mar 2025).