Compositional Phase Stability and Convex Hulls

Part D branches by the observable required to answer the research question. This topic is chosen only when that quantity is needed; it is not a required step in every DFT study.

Start from the phase set a researcher can inspect

Open a phase-diagram or database interface, select the chemical system, and inspect the records that define the competing phase set. Check composition, calculation method, correction or reference-energy scheme, database version, and entry identity before exporting a table. Open suspicious structures rather than trusting labels alone.

Rebuild or view the hull as an actual composition-energy plot, click or read the vertices and tie lines, and compare the decomposition products with the original records. The decision is conditional on the included phases and compatible energy scheme. Browse structure and thermochemical data sources, electronic-property and phase tools, and literature sources before treating a database hull as exhaustive.

Use a compositional convex hull when the question is whether a represented phase can lower its thermodynamic potential by decomposing into other represented phases. The output must include the candidate ledger, hull vertices or facets, balanced decomposition coefficients, energy above hull, tolerance, and every phase-set decision.

Begin with Rebuild a Li-P Convex Hull from an OQMD Snapshot to inspect the attributed records, vertices, and decompositions. Then use Stress-Test a Hull Against a Missing Competitor to see which conclusion changes when one represented phase is withheld.

Start from a candidate ledger

Every row needs an exact composition vector, structure and state identity, raw energy, corrections, compatible final energy, common normalization, method identity, numerical evidence, source or search identity, and inclusion or exclusion reason.

Define the thermodynamic system

A closed system fixes components and overall composition. An open system exchanges declared species with reservoirs and requires a transformed potential. These answer different scientific questions.

A common per-atom formation energy is

\Delta E_{\mathrm f} = \frac{E_{\mathrm{compound}}-\sum_i n_i\mu_i^{\mathrm{ref}}} {\sum_i n_i}.

A negative $\Delta E_{\mathrm f}$ means that the selected elemental decomposition is uphill within the model. It does not exclude decomposition into compounds.

Represent every phase by $\mathbf{x}_j$ in one component basis. Primitive cells, conventional cells, and formula units are interchangeable only after explicit conversion. Partial occupancy, vacancies, charge, and molecular reservoirs require balanced components rather than informal formula matching.

Build and inspect the lower envelope

Let $\mathcal S$ be the declared phase set, including candidate $k$ when its energy above hull is evaluated. With every $G_j$ in the same normalization, solve

G_{\mathrm{hull}}^{\mathcal S}(\mathbf{x}) = \min_{\{\lambda_j\}} \sum_{j\in\mathcal S}\lambda_jG_j

subject to

\lambda_j\ge0,\qquad \sum_{j\in\mathcal S}\lambda_j=1,\qquad \sum_{j\in\mathcal S}\lambda_j\mathbf{x}_j=\mathbf{x}.

Every point on it is a macroscopic mixture of its endpoints, not an interpolated homogeneous crystal structure. In higher dimensions, use the full composition vectors rather than trusting a plotting projection.

Store the nonzero $\lambda_j$ , phase identities, and balanced reaction. Reconstruct both target composition and mixture energy. The scalar hull distance alone hides the products and whether they change when a competitor is added.

For candidate $k$ in the same phase set,

E_{\mathrm{above\,hull},k}^{\mathcal S} = G_k-G_{\mathrm{hull}}^{\mathcal S}(\mathbf{x}_k) \ge0.

Classify zero only within an explicit tolerance. Do not clamp a materially negative value to zero; inspect normalization, phase-set membership, optimizer constraints, and tolerance instead.

Control the phase set and compatibility model

At one composition, compare genuine polymorphs, magnetic states, and orderings under one compatibility model before allowing the lowest record onto the envelope. Retain higher states for provenance and metastability.

A computed hull is monotonic with respect to adding candidates: a newly admitted lower phase can leave the envelope unchanged or lower it. “On hull” is always conditional on the documented search, filters, versions, duplicate policy, and exclusions.

All phases must share compatible exchange-correlation, core or basis, relativistic and magnetic treatment, numerical quality, energy definition, and correction scheme. Preserve raw energy, each correction, corrected energy, scheme version, and eligibility separately.

Converge and stress-test the hull

Rebuild under numerical refinements that can shift competitors unequally. SCF convergence is necessary but does not establish convergence of energy above hull or decomposition identity.

If vertices, facets, or decomposition products change within plausible variation, report the result as unresolved or near-degenerate.

Withhold represented vertices, add plausible candidates, or perturb near-hull values under a documented uncertainty model. Create a new derived result for every phase set. Never delete an inconvenient lower phase from the source ledger.

Extend the thermodynamic model consistently

Before rebuilding, apply the same type of free-energy model to every phase:

G_j(T,p)=E_{\mathrm{DFT},j}+F_{\mathrm{vib},j} +F_{\mathrm{el},j}+F_{\mathrm{conf},j}+pV_j+G_{\mathrm{other},j}.

Partial thermal treatment is not a finite-temperature phase diagram.

For reservoir species in $\mathcal R$ ,

\Phi=G-\sum_{i\in\mathcal R}\mu_iN_i.

Chemical potentials remain constrained by host equilibrium, elemental precipitation, competing phases, and stated conditions.

A stability polygon in chemical-potential space must not be read as a range of bulk compositions. A composition-space tie simplex does not specify an experimental pressure or activity without a reservoir model.

Interpret and diagnose the result

These distributions do not provide a universal energy-above-hull threshold that separates synthesizable from impossible materials. Kinetics, surfaces, defects, entropy, pressure history, precursors, and model error can change accessibility without changing the static hull definition.

Check missing endpoints, inconsistent reduction, duplicate phases, negative or non-normalized fractions, mixed correction schemes, incompatible magnetic states, and asymmetric thermal terms. Rebuild representative facets from the machine-readable table.

Inspect the public-data example before generalizing

This is a real public DFT-data case, not a claim that this project reran the underlying calculations. Reopen the OQMD identities behind the plotted vertices and the reported Li2P decomposition before citing the result. The frozen Li-P snapshot supports parsing, attribution, normalization, and convex geometry at source precision; it does not independently establish energy accuracy or phase-space completeness.

Preserve the result and claim boundary

Store candidate rows, vectors, raw and corrected energies, source versions, phase-set decisions, tolerance, vertices, facets, decompositions, residual checks, and hashes. A hull can support bounded ground-state and decomposition claims for that set and model. It does not establish exhaustive search, mechanical or phonon stability, kinetic persistence, finite-temperature equilibrium without required terms, synthesis, or experimental realization.