Materials science often involves computing properties of numerous atomic configurations on a defined lattice. While first-principles methods like density-functional theory (DFT) are ideal, the combinatorial explosion of configurations with system size makes direct ab initio approaches impractical for many complex systems. The cluster expansion (CE) method offers a solution by using generalized Ising-like models parameterized with ab initio data to describe configuration-dependent properties. This significantly reduces computational cost while maintaining high accuracy, bridging length scales and enabling statistical-thermodynamics descriptions. However, technologically relevant materials frequently exhibit complexities that traditional CE methods struggle with, including multi-component settings, multiple sublattices, large parent lattices, and surfaces/interfaces. This necessitates sophisticated code capable of handling these intricacies, along with tasks like data-driven model training and evaluation. This work introduces the CELL Python package, providing a modular solution to address these challenges. CELL supports systems with varying substituent species, sublattices, parent lattice sizes, and dimensionality (1D, 2D, 3D), thus enabling the efficient study of complex materials, such as surface alloys.

The cluster expansion (CE) method has a long history in materials science, with early work focusing on binary alloys. The seminal paper by Connolly and Williams (1983) applied DFT to phase transformations in transition-metal alloys, laying the groundwork for subsequent developments. Sanchez, Ducastelle, and Gratias (1984) introduced a generalized cluster description for multicomponent systems, extending the applicability of CE. The development of the Alloy Theoretic Automated Toolkit (ATAT) by van de Walle and Ceder (2002) significantly advanced the field by automating first-principles phase diagram calculations. More recently, several Python-based CE packages have emerged, each with its own strengths and limitations. However, the ability of CELL to efficiently and robustly manage the complexities associated with multicomponent, multi-sublattice systems with large unit cells is a key differentiator. The focus on integrating machine learning techniques within the framework also makes CELL a powerful tool for addressing modern challenges in materials science.

CELL's core methodology centers around the cluster expansion formalism, specifically adapted for multi-component and multi-sublattice systems. The energy (or other property) of an arbitrary configuration is expressed as a linear combination of effective cluster interactions (ECIs) and cluster functions. The cluster functions are defined using orthonormal basis sets, such as discrete Chebyshev polynomials or trigonometric functions, providing flexibility in model construction. The choice of clusters significantly affects the accuracy and efficiency of the model. CELL employs strategies to address this, including special quasirandom structures (SQSs) and variance reduction schemes to optimize training datasets. The CE model construction is formulated as a machine learning problem, allowing the use of various estimators from scikit-learn (e.g., ridge regression, LASSO) and CELL's own native estimators. The process of selecting the optimal set of clusters is critical and involves minimizing a cost function that balances prediction accuracy (e.g., mean squared error) and model complexity (e.g., using L1 or L0 regularization). CELL offers different approaches for cluster selection, including combinatorial search, LASSO-based selection, and cross-validation, enabling users to choose the most suitable technique for their specific problem. Thermodynamic analysis is performed using Monte Carlo (MC) methods (Metropolis MC and Wang-Landau) which account for configurational entropy and facilitate the investigation of temperature-dependent properties. These techniques are implemented efficiently to allow for simulations on very large supercells and the computation of the configurational density of states. CELL's architecture uses a well-defined inheritance map of Python classes for structure generation and manipulation, built upon the Atomic Simulation Environment (ASE). This includes ParentLattice, SuperCell, and Structure classes facilitating seamless integration and manipulation of structures.

The paper demonstrates CELL's capabilities through several case studies. For the O-Pt/Cu(111) surface alloy, CELL successfully models the adsorption energy considering both alloying and adsorption phenomena simultaneously. The thermodynamic analysis revealed a temperature-driven order-disorder transition in the Pt-Cu surface alloy, confirming the previously known p(2x2) ordered phase. In the Si-Ge alloy study, CELL accurately predicted the energy of mixing and lattice parameters, confirming the system's tendency to phase separate. Analysis using Wang-Landau sampling provided insights into the demixing transition temperature (~200 K), which is consistent with literature values. The negative bowing of the lattice constant, deviating from Vegard's law, was also reproduced. For the complex Ba8Al16Si46 clathrate compound, CELL's iterative CE approach successfully identified the ground-state structures across a range of Al concentrations (x = 6-16), requiring only 40 ab initio calculations. This demonstrates CELL's effectiveness in handling materials with large unit cells where full enumeration is infeasible. The convergence of the iterative CE model, as assessed by cross-validation, showcases the efficiency of the workflow. In summary, the key findings highlight CELL's versatility in modeling complex materials and the effectiveness of its integrated workflow. The parallel capabilities of CELL allowed for simulations involving supercells with thousands of atoms, pushing the limits of computational materials science. The results of the Si-Ge study emphasize the utility of the Wang-Landau sampling in the canonical ensemble to access phase-separated states and better understand the thermodynamic behavior around a phase transition.

CELL addresses a crucial need in computational materials science by providing a flexible and powerful tool for cluster expansion and thermodynamic analysis of complex alloys. The modular design, integration with existing Python libraries (ASE and scikit-learn), and advanced sampling techniques enable researchers to efficiently study materials with complexities beyond the capabilities of traditional CE codes. The case studies presented demonstrate the effectiveness of CELL's approach in predicting ground state structures and thermodynamic properties in diverse systems, from simple binary alloys to complex clathrates with large unit cells. The ability to readily handle multi-component and multi-sublattice systems opens up possibilities for modeling realistic materials and gaining deeper insights into their behavior. The integration of machine learning techniques, such as LASSO regression, enhances the robustness and efficiency of model construction, particularly when data is scarce. The application of Wang-Landau sampling provides a significant advantage over traditional Metropolis Monte Carlo methods, especially for the calculation of quantities that require thermodynamic integration, like free energy. The success of CELL in accurately predicting the phase transition temperatures and lattice parameters, along with its handling of complex materials with large unit cells, positions it as a valuable contribution to the field of computational materials science.

CELL is a comprehensive Python package for cluster expansion, incorporating state-of-the-art methods for model construction, selection, and thermodynamic analysis. Its modular architecture, integration with machine learning techniques, and parallel capabilities enable efficient study of diverse complex materials. The presented case studies demonstrate CELL's versatility and accuracy in modeling systems ranging from surface alloys to complex intermetallics. Future work will focus on expanding CELL's functionality to include additional model types, sampling techniques, and property prediction capabilities. The integration of advanced machine learning algorithms and the development of more efficient algorithms for cluster selection are also important avenues for future development.

While CELL provides a powerful framework, certain limitations exist. The accuracy of CE models relies heavily on the quality and quantity of the ab initio training data. The computational cost of obtaining this data can be significant, especially for complex systems. The choice of clusters and the selection of appropriate regularization parameters influence the model's accuracy and generalizability; careful consideration and experimentation are necessary. The assumptions underlying the cluster expansion formalism (e.g., the linearity of the energy expression) might not hold perfectly for all systems and properties. Finally, the accuracy of thermodynamic calculations depends on the accuracy of the underlying CE model and the convergence of the sampling techniques.