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icet - A Python library for constructing and sampling alloy cluster expansions

Mattias Ångqvist, William A. Muñoz, J. Magnus Rahm, Erik Fransson, Céline Durniak, Piotr Rozyczko, Thomas Holm Rod, Paul Erhart

arXiv:1901.08790v1cond-mat.mtrl-sci

TL;DR

Alloy cluster expansions address the need to sample multicomponent configuration spaces efficiently despite the computational cost of first-principles calculations. ICET introduces a modular Python toolkit combining CE construction, sampling, regression, and workflow integration, with demonstrations on Ag–Pd and Ba8AlxSi46−x. The examples show that the most effective regression method depends on the material system and that target quantities require assessment beyond CV-RMSE alone.

  • Problem

    Alloy cluster expansions are needed because first-principles calculations are accurate but too costly for statistically adequate sampling of multicomponent configuration spaces.

  • Method

    ICET combines modular CE construction and sampling with linear and Bayesian regression, feature selection, cross-validation, model ensembles, and Python-based workflow integration.

  • Results

    ARDR performed best for Ag–Pd, whereas RFE with OLS achieved the best results for Ba8AlxSi46−x; individual Ag–Pd CEs predicted critical temperatures from 673 to 803 K.

  • Takeaways & Limitations

    ICET provides an extensible environment for CE method development, extended computational workflows, structure handling, and data analysis.

Abstract

from arXiv · show

Alloy cluster expansions (CEs) provide an accurate and computationally efficient mapping of the potential energy surface of multi-component systems that enables comprehensive sampling of the many-dimensional configuration space. Here, we introduce \textsc{icet}, a flexible, extensible, and computationally efficient software package for the construction and sampling of CEs. \textsc{icet} is largely written in Python for easy integration in comprehensive workflows, including first-principles calculations for the generation of reference data and machine learning libraries for training and validation. The package enables training using a variety of linear regression algorithms with and without regularization, Bayesian regression, feature selection, and cross-validation. It also provides complementary functionality for structure enumeration and mapping as well as data management and analysis. Potential applications are illustrated by two examples, including the computation of the phase diagram of a prototypical metallic alloy and the analysis of chemical ordering in an inorganic semiconductor.

I. INTRODUCTION

ICET is introduced as a modular Python toolkit for efficiently constructing and sampling alloy cluster expansions, integrating first-principles workflows and machine-learning methods. It combines regression, sampling, and supplementary structure and data-analysis functionality.

  • Alloy cluster expansions balance first-principles accuracy with computationally efficient sampling of configuration space using Monte Carlo simulations.
  • ICET is modular, extensible, and computationally efficient, with Python integration and C++ implementations for demanding operations.
  • The toolkit interfaces with ASE-accessible first-principles codes, analysis tools, and state-of-the-art regression techniques through Scikit-learn.
  • Compared with earlier monolithic codes, ICET offers comparable or broader functionality and easier extension through Python and open-source development.
  • ICET supports multiple species, coexisting sublattices, regularized linear regression, cross-validation, and ensemble optimization.
  • Its complementary functionality includes Monte Carlo simulations, structure enumeration, structure mapping, convex-hull extraction, and ground-state finding.

II. CLUSTER EXPANSION FORMALISM

The cluster-expansion formalism represents configuration-dependent properties with symmetry-aware clusters and orthogonal point-function bases. Practical constructions truncate the formally infinite cluster set according to interaction range and computational requirements.

  • A. Clusters and orbits: The BCC-lattice illustration orders clusters by radius, measured as the average site distance from the cluster center in units of a0.
  • A. Clusters and orbits: A cluster of order k is a list of k lattice sites, commonly called a singlet, pair, triplet, or quadruplet according to order.
  • A. Clusters and orbits: Clusters related by lattice symmetry belong to the same orbit, represented by one symmetry-inequivalent cluster.
  • A. Clusters and orbits: Although infinitely many clusters exist, short-ranged interactions motivate retaining only shorter-ranged clusters with limited size and order.
  • B. Point functions: The configuration vector σ records the species occupying each of N lattice sites, while point functions Θ_n(σ_p) form the basis construction.
  • B. Point functions: Square-lattice examples distinguish representative pair, triplet, and quadruplet clusters from other symmetry-equivalent orbit members.
  • B. Point functions: For M allowed species, the point-function index n ranges from 0 to M−1, and the resulting functions are orthogonal over occupation numbers.
  • B. Point functions: Multiple sublattices may use different species counts, such as M = 3 for cations and M = 2 for anions in a zincblende alloy.

C. CE expression

The CE expression builds configuration functions by multiplying point functions across sites and then exploits cluster symmetry to sum over representative orbits. Effective cluster interactions are the fitted free parameters of this representation.

  • Basis functions Πα(σ) are products of point functions assigned across the sites of a structure.
  • The index α specifies point-function indices, with zero entries excluding sites from the corresponding cluster and nonzero entries including them.
  • In binary systems each cluster has one α, whereas multicomponent systems can have multiple α values for the same cluster.
  • The functions Πα(σ) form a complete orthogonal set, allowing them to represent any function of the configuration.
  • The CE omits the configuration-invariant α = 0 basis component in favor of Q0 and sums over symmetry-equivalent orbits.
  • Effective cluster interactions Jα are the free parameters targeted during training, while mα denotes each representative cluster’s multiplicity.

III. CLUSTER EXPANSION CONSTRUCTION

ICET constructs cluster expansions by fitting effective cluster interactions to reference configurations and target properties. Its workflow addresses underdetermined, correlated data through feature selection, regularization, Bayesian methods, and cross-validation.

  • Matrix form: Reference configurations and target data are organized into a sensing-matrix system whose parameters are the effective cluster interactions.The target data are usually obtained from first-principles calculations.
  • Matrix form: The CE parameters are determined by minimizing the squared residual between predicted and target values, with conventional least-squares suited to overdetermined systems.The optimization seeks a parameter vector J minimizing ∥ΠJ − Q∥2.
  • Linear regression techniques: Underdetermined or correlated sensing matrices motivate sparse feature selection, which can reduce overfitting, improve transferability, and lower sampling cost.The physical short-ranged character of interactions supports sparse solutions.
  • Linear regression techniques: ICET supports regularized regression, recursive feature elimination, Bayesian regression, and other optimization techniques with algorithm-specific hyper-parameters.Examples include elastic net, LASSO, RFE, ARDR, and Bayesian compressive sampling.
  • Model assessment: Cross-validation estimates predictive power by fitting on training subsets and measuring RMSE on validation subsets across multiple data splits.Averaging over splits reduces the statistical error of the RMSE estimate.
  • Model assessment: ICET exposes regression and cross-validation functionality through Python interfaces and can generate model ensembles to assess prediction sensitivity and stability.The sensing matrix and target data can also be passed directly to external machine-learning libraries.

IV. CLUSTER EXPANSION SAMPLING

ICET samples cluster expansions with Monte Carlo methods through the MCHAMMER module, supporting canonical, semi-grand canonical, and variance-constrained ensembles. Ensemble choice determines whether compositions are fixed or varied and whether free-energy integration can cross multiphase regions.

  • Monte Carlo sampling: MCHAMMER supports multiple thermodynamic ensembles and supplemental tools for managing and analyzing Monte Carlo data.The module is used to sample cluster expansions and extract information from simulations.
  • Monte Carlo sampling: Metropolis sampling accepts trial moves with probability P = min{1, exp(−β∆ψ)}, where β = 1/kBT and ∆ψ is the ensemble-specific thermodynamic-potential change.The thermodynamic potential excludes the entropy term.
  • Ensembles: The canonical ensemble conserves concentrations through species swaps and is suited to fixed-composition studies, structural order parameters, and simulated annealing.In this ensemble, the thermodynamic potential equals the internal energy.
  • Ensembles: The semi-grand canonical ensemble varies composition continuously by exchanging one site's occupation under a chemical-potential-dependent thermodynamic potential.The chemical-potential differences are defined relative to the first species.
  • Ensembles: The SGC-to-concentration mapping is multivalued in two-phase regions, preventing SGC from sampling across miscibility gaps and hindering free-energy integration.Discontinuous jumps can nevertheless help track phase boundaries.
  • Ensembles: VCSGC adds concentration mean and fluctuation constraints, making the parameter-to-concentration mapping one-to-one for sufficiently large κ and enabling free-energy recovery across multiphase regions.The phase diagram can then be constructed through standard free-energy minimization.

V. WORKFLOW

ICET integrates cluster-expansion construction, regression, and Monte Carlo sampling into a Python-centered workflow. Starting from reference data, it builds a cluster space and sensing matrix, fits parameters, creates a CE, and optionally samples a chosen supercell.

  • Workflow: ICET combines the CE formalism, linear regression, and sampling techniques into one workflow beginning with reference data for enumerated structures.Reference data may come from DFT calculations.
  • Workflow: A cluster space is constructed from a prototype structure, cutoff radii, allowed species, and lattice symmetries identified with SPGLIB.The cutoff radii determine which clusters enter the expansion.
  • Workflow: Cluster vectors are computed for the reference structures and stored with the sensing matrix and target-value vector.These data provide the inputs for subsequent regression.
  • Workflow: Linear regression and cross-validation determine an optimized parameter vector, which is combined with the cluster space to form a CE predicting properties for arbitrary prototype supercells.External optimization algorithms from SCIKIT-LEARN and SCIPY can be used.
  • Workflow: A CE calculator can be configured for a specific supercell and used for Monte Carlo simulations through MCHAMMER.This step supports efficient sampling of the constructed model.
  • Additional functionality: Additional ICET tools support structure enumeration, convex-hull extraction, mapping relaxed structures to ideal lattices, and ground-state searches.The workflow is supplemented by utilities beyond CE fitting and sampling.

A. Phase diagram of the Ag–Pd system

For Ag–Pd, ICET constructs cluster expansions from enumerated and DFT-relaxed structures, comparing regression methods through cross-validation. ARDR achieves low validation error with a sparse model, supporting efficient sampling.

  • CE construction: 631 distinct Ag–Pd supercells with up to 8 atoms were generated using ICET’s structure-enumeration feature.The reference structures were evaluated using DFT calculations.
  • CE construction: The cluster space contained 81 parameters: one zerolet, one singlet, 24 pairs, 20 triplets, and 35 quadruplets.Clusters up to fourth order were included with specified pair, triplet, and quadruplet cutoffs.
  • Model comparison: Cross-validation compared singular value decomposition, LASSO, RFE with OLS, and ARDR using shuffle-split estimates averaged over 50 splits.The final cluster expansion was trained on the complete data set.
  • Model comparison: In the underdetermined region, LASSO and ARDR outperform singular value decomposition and RFE with OLS on CV-RMSE.All methods produce similar CV-RMSE scores in the overdetermined region.
  • Model comparison: 2 meV/atom CV-RMSE is achieved by ARDR and RFE with OLS using only 30 features.LASSO reaches the same CV-RMSE with about 50 features; ARDR therefore provides a smaller sparse model for sampling.

3. Phase diagram from MC simulations

Monte Carlo sampling with ICET-derived Ag–Pd cluster expansions recovers free-energy behavior and constructs a phase diagram. The predicted diagram shows a Pd-rich miscibility gap, while ensemble models reveal sensitivity beyond CV-RMSE alone.

  • Monte Carlo sampling: 6.9 × 10^8 trial steps or 1.4 × 10^6 MC cycles were used to sample Ag–Pd from 100 to 900 K.Sampling used a 500-site FCC supercell, with SGC and VCSGC ensembles evaluated over composition-control parameters.
  • Free-energy recovery: The SGC and VCSGC free-energy derivatives coincide except on the Pd-rich side at lower temperatures, where SGC shows a two-phase discontinuity.The discontinuity is identified as the hallmark of a two-phase region.
  • Free-energy recovery: VCSGC sampling recovers the full free energy across the two-phase region, where mixing free energy becomes concave.The concave region appears at low temperature between c = 0.5 and 1.0.
  • Phase diagram: The predicted phase diagram exhibits a pronounced Pd-rich miscibility gap with critical temperature Tc = 718 K.The free energy was represented using temperature- and composition-dependent Redlich–Kister polynomials, akin to CALPHAD.
  • Model sensitivity: Individual bagged cluster expansions predict critical temperatures ranging from 673 to 803 K, while their averaged model gives Tc = 742 K.This variation shows that CV-RMSE alone is insufficient for assessing phase-diagram quality.

2. CE construction

For Ba8AlxSi46−x, ICET builds cluster expansions from randomly generated and DFT-relaxed occupations, then samples ordering with simulated annealing. The models reproduce temperature-dependent site occupancy behavior and ordered ground-state configurations.

  • CE construction: Because the 46-site unit cell prevents enumeration, 240 primitive-cell occupations for x = 13 … 16 were generated by randomly distributing Al and Si.The structures were subsequently relaxed using DFT calculations.
  • CE construction: The cluster basis contained 13 symmetry-inequivalent pairs and 23 symmetry-inequivalent triplets, with CEs generated by ARDR, LASSO, and RFE with OLS.Model selection used shuffle-split cross-validation with 50 splits.
  • Model comparison: RFE with OLS converges rapidly with training-set size and produces sparse solutions, whereas LASSO and ARDR require almost twice as many structures for similar CV-RMSE.ARDR remains sparse, while LASSO produces substantially more features.
  • CE sampling: Simulated annealing predicted Al site occupancy factors at x = 12 using a 268-site supercell and nearly 48 million trial steps.The simulations used the canonical ensemble while cooling from 1200 to 0 K.
  • Ordering behavior: At x = 12, nearest-neighbor Al–Al pairs remain absent across the full temperature range, and the predicted SOFs strongly deviate from the approximately 35% random limit.Below approximately 600 K, the SOFs converge toward a well-ordered configuration.

VII. CONCLUSIONS

ICET provides a modular Python package for constructing and sampling alloy cluster expansions, with regression, ensemble, Monte Carlo analysis, and structure tools. Applications show that the preferred regression method depends on the material system, while ensembles expose prediction sensitivity.

  • VII. CONCLUSIONS: ICET combines cluster-expansion construction and sampling with structure enumeration, mapping, and data-analysis functionality.Its modular design supports extension with other Python packages and integration into high-throughput workflows.
  • VII. CONCLUSIONS: Regression performance depends on the alloy: ARDR performed best for Ag–Pd, whereas RFE based on OLS performed best for Ba8AlxSi46−x.LASSO produced less optimal solutions in both systems.
  • VII. CONCLUSIONS: CE ensembles quantify how statistical uncertainty in the underlying models propagates to complex observables such as phase diagrams and SOFs.Models with numerically similar cross-validation scores can still produce different predictions.
  • VII. CONCLUSIONS: MCHAMMER extracts structural order parameters from Monte Carlo trajectories, including SOFs, neighbor counts, and short-range order parameters.These quantities support analysis of chemical ordering during simulations.
  • VII. CONCLUSIONS: ICET is well suited for efficient construction and sampling of alloy cluster expansions within extensible computational workflows.
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