Ti–N Machine-Learned Potential

• The paper develops a machine-learned interatomic potential that unifies predictions across diverse Ti–N states including ordered compounds, vacancies, and strained configurations.
• It employs moment tensor descriptors up to rank 22 and 421 basis functions to capture both short-range chemistry and angular interactions with high fidelity.
• Extensive training on DFT data from relaxed, strained, and AIMD configurations yields formation energy errors below 7 meV/atom and elastic constant predictions with less than 5% median error.

Searching arXiv for the specified Ti–N MTP paper and closely related MTP background. The Ti–N system comprises elemental titanium, titanium nitride, several ordered subnitrides, and nitrogen-in-titanium solid solutions, and it is characterized by substantial structural diversity across composition. A recent study develops a machine-learned interatomic potential for this system within the Moment Tensor Potential (MTP) formalism, with the stated objective of reliably predicting both mechanical properties and thermodynamic stability across the full Ti–N composition range (Rana et al., 25 Jul 2025). The resulting potential is trained against density functional theory (DFT) data spanning stoichiometric compounds, strained configurations, and ab initio molecular dynamics snapshots, and is reported to reproduce formation energies, elastic constants, and convex-hull trends with quantitatively small errors (Rana et al., 25 Jul 2025).

1. Ti–N chemical space and modeling objective

The Ti–N material system includes compounds with different stoichiometries, specifically Ti, TiN, Ti$_2$N, Ti$_3$N$_2$, Ti$_4$N$_3$, Ti$_6$N$_5$, and solid solutions of N in Ti, with a maximum of 23% solubility stated for nitrogen in titanium (Rana et al., 25 Jul 2025). The ordered compounds cited in the training and test protocol are structurally heterogeneous: Ti is hcp, Ti$_2$N is $P4_2/mnm$, Ti$_3$N$_3$0 is $_3$1, Ti$_3$2N$_3$3 is $_3$4, TiN is $_3$5, and the held-out Ti$_3$6N$_3$7 phase is also $_3$8 (Rana et al., 25 Jul 2025).

The central methodological claim is that transferability across this chemically and structurally diverse space depends critically on training-set selection that accounts for both structural similarity and structural dissimilarity among Ti–N phases (Rana et al., 25 Jul 2025). In that sense, the work is not limited to fitting a potential for a single stoichiometric crystal, but instead targets a unified atomistic model covering ordered compounds, vacancies, strains, and dilute-to-near-stoichiometric nitrogen content (Rana et al., 25 Jul 2025).

This suggests that the main technical challenge is not merely interpolation within one crystal family, but consistent representation of local environments across multiple coordination motifs and compositional regimes. The study frames the resulting MTP as a potential suitable for large-scale atomistic simulations of Ti–N materials because of this breadth of coverage (Rana et al., 25 Jul 2025).

2. Moment Tensor Potential formulation

The paper adopts the MTP framework of Shapeev, in which the total energy of an $_3$9-atom configuration $_2$0 is written as

$_2$1

where each $_2$2 is a local site potential defined on a finite collection of moment-tensor descriptors $_2$3 (Rana et al., 25 Jul 2025). The descriptors are

$_2$4

with $_2$5 tensor products, $_2$6, $_2$7 the atomic type, and $_2$8 a radial basis function (Rana et al., 25 Jul 2025).

In practical form, each local site energy is expanded linearly in a finite basis $_2$9,

$_4$0

so that the total energy is linear in the parameter vector $_4$1 (Rana et al., 25 Jul 2025). For the Ti–N model, the implementation uses 2621 moment-tensor components up to rank $_4$2, together with 421 site-basis functions and radial cutoffs $_4$3 and $_4$4 (Rana et al., 25 Jul 2025).

These numbers indicate a relatively high-capacity local descriptor expansion. A plausible implication is that the model is designed to capture both short-range chemistry and a broad range of local angular environments within one finite-cutoff potential, rather than relying on separate parameterizations for distinct stoichiometries.

3. Training-set construction and fitting protocol

The training set is explicitly constructed to ensure transferability across ordered compounds and solid solutions (Rana et al., 25 Jul 2025). It contains three principal data sources.

First, it includes 0 K relaxed stoichiometric bulk phases: Ti, Ti$_4$5N, Ti$_4$6N$_4$7, Ti$_4$8N$_4$9, TiN, and a representative hcp solid solution Ti–0.14 N, described as 17 at % N in hcp Ti (Rana et al., 25 Jul 2025).

Second, each equilibrium structure is deformed by uniaxial and shear strain modes appropriate to its crystal symmetry, using seven strain values $_3$0, which produces approximately 2,000 strained snapshots (Rana et al., 25 Jul 2025). This component is central for elastic-constant recovery, since it exposes the model to harmonic and near-harmonic distortions.

Third, thermalized supercells are generated by ab initio molecular dynamics in the $_3$1 ensemble over 10 K–400 K with a 1 fs time step, yielding approximately 8,000 configurations; from these, about 150 per phase are selected via the MaxVol algorithm to preserve diversity (Rana et al., 25 Jul 2025). After data selection, the final training pool contains 2,398 structures (Rana et al., 25 Jul 2025).

The test set contains two deliberately held-out chemistries: Ti$_3$2N$_3$3 ($_3$4) and a Ti–0.20 N solid solution (Rana et al., 25 Jul 2025). This held-out design is notable because it evaluates extrapolative behavior across both an ordered vacancy-rich phase and a more nitrogen-rich solid solution.

Model parameters are fitted by minimizing a weighted least-squares objective with quadratic $_3$5 regularization,

$_3$6

with three tested weight triplets: $_3$7, $_3$8, and $_3$9 (Rana et al., 25 Jul 2025). A small Tikhonov regularization $_6$0 is used to control overfitting, and the selected final model is $_6$1, chosen for balancing energy, force, and stress fidelity (Rana et al., 25 Jul 2025).

4. Energetic accuracy and error statistics

The principal quantitative benchmark is the root-mean-square error in formation energy per atom relative to DFT. For the training set, the reported RMSE is $_6$2; for the testing set, it is $_6$3 (Rana et al., 25 Jul 2025). These values are paired with absolute-error distributions obtained via kernel density estimation: the peak absolute error is $_6$4 on training structures and $_6$5 on held-out phases (Rana et al., 25 Jul 2025).

The study defines the formation energy for a phase $_6$6 as

$_6$7

and uses these values to construct convex-hull diagrams as the lower convex envelope in composition space (Rana et al., 25 Jul 2025). A phase on the hull is thermodynamically stable at 0 K, while $_6$8 indicates metastability (Rana et al., 25 Jul 2025).

For the selected $_6$9, the paper states that the DFT convex hull is reproduced within $_5$0 for training compositions (Rana et al., 25 Jul 2025). It further reports that newly generated intermediate structures, obtained by iterative N-atom insertion and removal between Ti, Ti$_5$1N, Ti$_5$2N$_5$3, Ti$_5$4N$_5$5, Ti$_5$6N$_5$7, and TiN, remain within $_5$8 above the hull in some cases, indicating potential metastable phases (Rana et al., 25 Jul 2025). The abstract similarly states that a maximum deviation of $_5$9 from the 0 K convex hull was observed for a few systems, while structures with N/Ti ratios ranging from 0 to 1 can be thermodynamically stable (Rana et al., 25 Jul 2025).

These results place the model in a regime where phase-ordering predictions are sufficiently accurate to resolve near-hull competition across compositions. This suggests that the potential is not confined to reproducing absolute energies, but also preserves the fine energy differences governing phase stability.

5. Elastic constants and mechanical-property prediction

Mechanical validation is performed through symmetry-adapted stress–strain calculations. In the harmonic regime, the strain energy under infinitesimal strain $_2$0 is

$_2$1

and, for a cubic biaxial $_2$2 strain along $_2$3, $_2$4 gives

$_2$5

(Rana et al., 25 Jul 2025). The authors apply symmetry-adapted strain patterns to extract all independent $_2$6 for each phase, using both DFT and the MTP (Rana et al., 25 Jul 2025).

For each elastic constant, the percentage error is defined by

$_2$7

Across all elastic constants, the statistical summaries are the median error, interquartile range, and mean error (Rana et al., 25 Jul 2025). For the selected $_2$8, the reported values are

$_2$9

ranking first among the three tested weight choices (Rana et al., 25 Jul 2025).

The abstract states that the distribution and variability of elastic constants across compositions were systematically evaluated and that the observed trends are consistent with DFT benchmarks (Rana et al., 25 Jul 2025). The conclusion summarizes this by stating that the selected potential predicts elastic constants to $P4_2/mnm$0 median error (Rana et al., 25 Jul 2025).

This mechanical benchmark is significant because it probes derivatives of the potential-energy surface rather than energies alone. A plausible implication is that inclusion of forces, stresses, and strained structures in the objective function is essential to achieving simultaneous thermodynamic and elastic fidelity.

6. Thermodynamic stability, finite-temperature behavior, and scope

The thermodynamic-stability analysis combines 0 K convex-hull reconstruction with finite-temperature molecular dynamics (Rana et al., 25 Jul 2025). At 0 K, the model reproduces the DFT hull accurately for training compositions and identifies additional intermediate structures that lie within $P4_2/mnm$1 above the hull, which the paper interprets as potential metastable phases (Rana et al., 25 Jul 2025). This phase search is performed by iterative N-atom insertion and removal along the sequence Ti $P4_2/mnm$2 Ti$P4_2/mnm$3N $P4_2/mnm$4 Ti$P4_2/mnm$5N$P4_2/mnm$6 $P4_2/mnm$7 Ti$P4_2/mnm$8N$P4_2/mnm$9 $_3$0 Ti$_3$1N$_3$2 $_3$3 TiN (Rana et al., 25 Jul 2025).

At finite temperature, $_3$4 simulations from 10 K to 300 K are carried out on supercells containing 36–96 atoms (Rana et al., 25 Jul 2025). The reported outcome is that no structural collapse is observed, and total-energy fluctuations remain within 1–10%, which is taken as confirmation of mechanical stability across Ti–N chemistries (Rana et al., 25 Jul 2025).

The study’s conclusion states that the $_3$5 potential accurately interpolates DFT energies, forces, and stresses of diverse Ti–N phases, achieves sub-meV/atom energy errors on-hull, predicts elastic constants to $_3$6 median error, and reproduces the convex hull within $_3$7 for new stoichiometries (Rana et al., 25 Jul 2025). It further describes the model as robust across ordering, vacancies, strains, and temperatures, thereby motivating its use for large-scale atomistic simulations of Ti–N materials (Rana et al., 25 Jul 2025).

An important limitation is implicit in the dataset design: the model is trained and tested within the Ti–N composition range and structural families explicitly represented by ordered phases, solid solutions, strained states, and low-temperature AIMD configurations up to 400 K in the data-generation stage (Rana et al., 25 Jul 2025). The reported performance therefore directly supports predictions within that envelope; broader transfer beyond it would require additional evidence.

7. Significance within atomistic modeling of nitrides

Within the study’s own framing, the main contribution is the construction of a single MTP capable of treating elemental Ti, ordered Ti–N compounds, substoichiometric vacancy-rich nitrides, and nitrogen-bearing Ti solid solutions in one coherent representation (Rana et al., 25 Jul 2025). The combination of low formation-energy RMSE, elastic-constant agreement, and convex-hull fidelity indicates that the potential is intended for problems where both phase selection and mechanical response matter.

This is particularly relevant for computational workflows that are difficult to execute entirely at the DFT level, such as large-supercell defect studies, finite-temperature sampling, or systematic searches over intermediate stoichiometries. The paper does not enumerate such applications in detail, but its conclusion that the potential is suitable for large-scale atomistic simulations of Ti–N materials suggests precisely that role (Rana et al., 25 Jul 2025).

More broadly, the work exemplifies a characteristic strategy in machine-learned interatomic potentials: transferability is pursued not only through expressive descriptors and regularized fitting, but through deliberate coverage of chemically distinct environments in the training corpus. In the Ti–N case, the paper identifies this dataset-design issue as crucial, and the reported results indicate that such coverage is sufficient to span the range from pure Ti to TiN while preserving both thermodynamic and mechanical accuracy (Rana et al., 25 Jul 2025).