Mixed-Space Cluster Expansion

• Mixed-space cluster expansion is a method that combines real-space clusters with reciprocal-space terms to overcome convergence issues in modeling long-range electrostatic and elastic interactions.
• Hybrid formulations integrate distance-dependent pair energetics with cluster correlations to capture local structural relaxations and temperature-dependent phenomena.
• Tensor-contraction and axiomatic reformulations enable efficient computation on complex lattices, facilitating GPU acceleration and accurate modeling of tensorial observables.

Mixed-space cluster expansion is a family of cluster-expansion formalisms in which a configurational property is not represented solely by truncated real-space clusters. Typically, the term refers to a formalism where both real-space clusters and reciprocal-space (Fourier) terms are used to represent properties, particularly to describe long-range electrostatic or elastic interactions. Related work also uses the term for hybrid decompositions that combine explicit distance-dependent pair energetics with occupation-based cluster correlations, and for tensor-contraction formulations in which reciprocal-space or tensorial indices can be added directly to the interaction tensors (Chang et al., 2018, Geng et al., 2012, Jeffries et al., 4 Sep 2025).

1. Definition within the cluster-expansion framework

The standard cluster expansion expresses a configurational property on a fixed lattice as a sum over symmetry-distinct clusters and their effective interactions. In one common form,

$$\mathcal{H}_\text{eff}(\sigma) = \sum_\alpha m_\alpha J_\alpha \Pi_\alpha(\sigma),$$

where $\alpha$ denotes a unique cluster type, $m_\alpha$ its multiplicity, $J_\alpha$ the effective cluster interaction, and $\Pi_\alpha(\sigma)$ the corresponding correlation function (Jeffries et al., 4 Sep 2025). In the implementation-oriented notation used by CLEASE, the same idea appears as

$$q(\bm{\sigma}) = J_0 + \sum_\alpha m_\alpha J_\alpha \phi_\alpha,$$

or, per atom,

$$q(\bm{\sigma}) = \sum_\alpha \widetilde{J}_\alpha \phi_\alpha,$$

with $\phi_\alpha$ the correlation functions and $\widetilde{J}_\alpha$ the effective cluster interactions per atom (Chang et al., 2018).

This real-space formalism is computationally valuable because it maps first-principles results onto a Hamiltonian that is much faster to evaluate than direct DFT calculations, and it is widely used for substitutional disorder in multicomponent alloys and related materials (Chang et al., 2018). Its principal limitation, in the mixed-space context, is slow convergence when long-range electrostatic or elastic interactions are important. For heterovalent, ionic, or polar compounds, mixed-space formulations are described as critical because real-space clusters alone converge slowly; the mixed-space remedy is to include both real-space cluster terms and reciprocal-space or Ewald-type contributions in the energy expression (Chang et al., 2018).

The mixed-space perspective is therefore not a rejection of cluster expansion, but a change in representational basis and partitioning of interactions. Real-space clusters remain central for short-range configurational effects, while reciprocal-space terms are introduced when the physics is inherently long-ranged or when a purely local truncation becomes inefficient (Chang et al., 2018).

2. Hybrid expansions for local structural relaxations

One important precursor to mixed-space thinking is the hybrid treatment of substitutional and displacive degrees of freedom. In alloys with significant local structural relaxation, standard cluster expansion on an ideal lattice converges poorly, may require long-ranged many-body interactions, and may fail to capture relaxation-induced energetic effects (Geng et al., 2012). The hybrid expansion introduced for local structural relaxations addresses this by combining pair potentials with a cluster expansion.

Its energy functional is written as

$$E = \sum_{\alpha \neq ij} v_{\alpha}\xi_{\alpha} + \frac{1}{2N} \sum_{mn,ij} \iint_{\Omega} V_{mn}(\mathbf{R}_{ij}+\mathbf{u}_i-\mathbf{u}_j)\, \rho_{mn}(\mathbf{u}_i,\mathbf{u}_j)\, d\mathbf{u}_i d\mathbf{u}_j .$$

Here the first term contains the non-pair cluster contributions through effective cluster interactions $\alpha$0 and correlation functions $\alpha$1, while the second term explicitly captures interatomic-distance-dependent pair energetics through $\alpha$2 and displacement distributions $\alpha$3 (Geng et al., 2012). The lattice-gas description includes both an occupation variable and a displacement vector at each lattice site, so substitutional disorder and local relaxation are handled within one Hamiltonian (Geng et al., 2012).

The operational decomposition is explicit. Pair potentials are first obtained either by an ab initio Möbius inverse method or by empirical fitting; pair contributions are then subtracted from total cohesive energies of ordered, unrelaxed structures; the residual energies are finally fit by a cluster expansion over non-pair clusters (Geng et al., 2012). The resulting Hamiltonian is compatible with the Cluster Variation Method and Monte Carlo sampling, and because the relaxational degrees of freedom remain explicit rather than being absorbed into pre-relaxed energies, the formalism can treat temperature-dependent relaxations, atom-exchange barriers, and related local distortion effects more realistically (Geng et al., 2012).

Within the terminology used in that work, this hybrid decomposition can be viewed as a special case of mixed-space cluster expansion: the pair-potential term captures the real-space, distance-dependent relaxational energetics, while the cluster-expansion term captures configurational correlations (Geng et al., 2012). The “mixed” character lies in the additive combination of interaction channels with different natural variables rather than solely in an explicit Fourier-space sum.

3. Tensor-contraction reformulation and generalized mixed-space representations

A more recent generalization is the tensor cluster expansion (TCE), introduced as a tensor-contraction-based formalism for multicomponent solids (Jeffries et al., 4 Sep 2025). Its motivation is twofold. First, standard cluster expansion is difficult to extend to exotic and/or low-symmetry lattices because it is often implemented by iterating over explicitly enumerated cluster types specific to each lattice. Second, the conventional per-cluster loops have irregular, scattered memory access and are not efficient for vectorization or massively parallel accelerators (Jeffries et al., 4 Sep 2025).

TCE replaces explicit enumeration by mixed sparse-dense tensor contractions. For a one-hot configuration tensor $\alpha$4, with $\alpha$5 if site $\alpha$6 is occupied by species $\alpha$7, the Hamiltonian is written as

$\alpha$8

The topology tensors $\alpha$9 and $m_\alpha$0 encode the lattice connectivity, while the learnable interaction tensors $m_\alpha$1 and $m_\alpha$2 replace conventional enumerated ECIs (Jeffries et al., 4 Sep 2025). Pair and triplet counts become contraction objects,

$m_\alpha$3

and triplet topology can be constructed from pair topologies through

$m_\alpha$4

This eliminates the need for cluster-type iteration and makes correlation-function evaluation well suited to GPUs and related hardware (Jeffries et al., 4 Sep 2025).

The mixed-space relevance of TCE is explicit. The framework is described as naturally extending to tensorial or mixed-space properties by adding indices to the interaction tensors and performing the corresponding contractions. In that sense, reciprocal-space information, tensor components, and other extended descriptors can be incorporated without changing the underlying computational pattern of the Hamiltonian (Jeffries et al., 4 Sep 2025).

A second major consequence is local-update efficiency. For two configurations differing only on a small set of sites $m_\alpha$5,

$m_\alpha$6

so only local feature differences must be recomputed. For a single swap, the stated operation count is reduced from $m_\alpha$7 to $m_\alpha$8, with further gains if topology tensors are stored sparsely; the authors describe the resulting energy-difference calculations as nearly $m_\alpha$9 (Jeffries et al., 4 Sep 2025).

The paper demonstrates this formalism on TaW and CoNiCrFeMn. For bcc Ta$J_\alpha$0W$J_\alpha$1, a TCE model fit to 250 DFT-generated configurations, including up to third neighbors and 3-body terms $J_\alpha$2 and $J_\alpha$3, reproduced the enthalpy-of-mixing curve with cross-validation errors on the order of 5–10 meV/atom. For fcc CoNiCrFeMn, a TCE model using up to second-nearest neighbors and $J_\alpha$4 three-body clusters reproduced Cowley short-range order parameters at 600 K, with a slight overestimate in Cr–Cr self-correlation but good agreement otherwise. The abstract characterizes the TaW and CoNiCrFeMn results overall as showing excellent agreement with ground-truth data (Jeffries et al., 4 Sep 2025).

4. Intrinsic, axiomatic, and tensor-valued formulations

A distinct line of work reconstructs cluster expansion on an axiomatic basis, without conventional cluster functions, by defining cluster components intrinsically through conditional expectation and Möbius inversion (Lammert et al., 2022). In this formulation, for each site subset $J_\alpha$5 there is a projector $J_\alpha$6 onto the space of observables depending only on $J_\alpha$7, and the cluster component $J_\alpha$8 of a configuration function $J_\alpha$9 is defined by

$\Pi_\alpha(\sigma)$0

or equivalently,

$\Pi_\alpha(\sigma)$1

The paper identifies this as Möbius inversion on the lattice of subsets and interprets $\Pi_\alpha(\sigma)$2 as conditional expectation, so the cluster expansion becomes a geometric decomposition in a finite-dimensional Hilbert space (Lammert et al., 2022).

This reconstruction addresses two issues that are directly relevant to generalized and mixed-space cluster expansions. First, it avoids a dependence on preselected basis functions and arbitrary encodings of species into site variables. Second, model fitting is grounded in orthogonal projection in Hilbert space, and the model is constructed directly from the data, which is stated to avoid the underdetermination problem associated with conventional basis-based fitting (Lammert et al., 2022). The optimal model is determined by projection into the intersection of the shadow space spanned by the sampled observables and the chosen model space.

Tensor observables are treated on an equal footing with scalar observables. The formalism accommodates observables valued in a finite-dimensional Hilbert tensor space, with symmetry incorporated through the action of the group on site indices and tensor values (Lammert et al., 2022). This is significant for mixed-space cluster expansion because generalized observables often include tensor components, reciprocal-space descriptors, or both.

The same paper states that the axiomatic, projection-based formalism can be extended to mixed-space cluster expansions or other generalized frameworks, provided that suitable notions of dependency subsets and conditional expectation can be defined. In that extension, the cluster components remain defined as projections and via Möbius inversion, so the formal structure is preserved even when the observable is no longer a simple scalar function of occupation variables alone (Lammert et al., 2022).

5. Implementations, computational practice, and adjacent usages

The existing software and algorithmic landscape makes the distinction between real-space and mixed-space formulations particularly clear.

Framework or formulation	Mixed-space role	Stated support or limitation
CLEASE	Real-space cluster expansion	No explicit mixed-space cluster expansion; no reciprocal-space or Ewald terms
tce-lib / TCE	Tensor-contraction formalism with extra indices	Mixed-space and tensorial properties can be added through added indices and contractions
QCE	Mixed objective built from multiple property QUBOs	Combines independent QUBOs such as stability and efficiency into $\Pi_\alpha(\sigma)$3

CLEASE, integrated into ASE, supports arbitrary bulk lattice structures, multicomponent and multisublattice systems, user-defined crystal structures, several basis-function choices, and several fitting strategies including OLS, $\Pi_\alpha(\sigma)$4 regularization, genetic algorithms, and Bayesian compressive sensing. Its thermodynamic sampling capabilities include Metropolis Monte Carlo and simulated annealing on large supercells. However, the paper explicitly states that CLEASE does not support an explicit mixed-space cluster expansion: its implementation is strictly real-space, its cluster generation is based on real-space diameters, and its Hamiltonians contain no reciprocal-space sums, Ewald terms, or explicit long-range electrostatics (Chang et al., 2018).

The TCE implementation in tce-lib occupies a different position. Because all cluster correlations are expressed as tensor contractions against precomputed topology tensors, adding reciprocal-space information or tensorial output channels becomes an index-management problem rather than a problem of writing new cluster-enumeration logic. The same structure also makes local energy-difference calculations and massively parallel execution natural consequences of the formalism (Jeffries et al., 4 Sep 2025).

A further, broader usage appears in the quantum-inspired cluster expansion (QCE) literature. There, a conventional cluster expansion is mapped into a QUBO by encoding site occupations with one-hot binary variables, reducing higher-order terms to quadratic form, and solving the resulting optimization with quantum annealers or quantum-inspired digital annealers. In that work, “mixed-space” appears in the context of combining multiple property-specific QUBOs into a single objective,

$\Pi_\alpha(\sigma)$5

with an example

$\Pi_\alpha(\sigma)$6

The reported benchmark on a quaternary Cu-Ni-Pd-Ag fcc alloy gives a 10-fold cross-validation MAE of 8 meV/atom for the mixing-energy model, and the digital-annealer implementation is described as 10–50 times faster than genetic algorithms and Bayesian optimization in chemical-space search (Choubisa et al., 2022). This is a computationally important extension of cluster-expansion practice, although its use of “mixed-space” is not identical to the reciprocal-space usage of the canonical materials-science definition.

6. Conceptual boundaries and recurring misconceptions

A persistent misconception is that enlarging a real-space cutoff is equivalent to a mixed-space treatment. The CLEASE description is explicit that increasing maximum real-space cluster diameter can partially capture longer-ranged interactions, but this is not equivalent to mixed-space or Ewald approaches (Chang et al., 2018). The distinction matters most when convergence is governed by genuinely long-range Coulombic or elastic physics rather than by a finite but larger local neighborhood.

A second misconception is that arbitrary lattice support by itself implies mixed-space generality. Real-space cluster expansions can be highly flexible with respect to lattice type, as CLEASE demonstrates for structures supported by ASE and user-defined crystals, without implementing reciprocal-space terms at all (Chang et al., 2018). Conversely, TCE shows that lattice generality and mixed-space adaptability can coexist when the lattice is encoded in topology tensors and the observable is evaluated by contractions rather than by hand-coded cluster loops (Jeffries et al., 4 Sep 2025).

The literature summarized here also suggests that the term “mixed-space cluster expansion” is not used uniformly. In one usage it denotes the combination of real-space clusters and reciprocal-space terms for long-range interactions (Chang et al., 2018). In another it denotes a hybrid decomposition in which distance-dependent pair potentials describe relaxational energetics while cluster expansion captures higher-order configurational correlations (Geng et al., 2012). In a more recent tensor-contraction setting it refers to the ability to add reciprocal-space or tensorial indices directly to the interaction tensors (Jeffries et al., 4 Sep 2025). In the QCE literature, it can additionally denote a mixed objective formed from multiple property expansions in a shared optimization formalism (Choubisa et al., 2022).

Despite this terminological variation, the common motivation is consistent. Mixed-space and mixed-representation formulations are introduced when a purely real-space, basis-fixed, cluster-enumeration workflow becomes inefficient, conceptually awkward, or poorly convergent—whether because of long-range electrostatics, local structural relaxations, tensor observables, exotic lattices, or multi-property optimization.