Selectivity- and Activity-Aware Catalyst Descriptors for CO$_2$ Hydrogenation on Alloy Nanocatalysts using Machine-Learned Force Fields

Abstract: Adsorption energy distributions (AEDs) have emerged as a powerful and increasingly adopted descriptor for catalytic performance in high-entropy alloys and, more recently, in conventional metallic alloy nanocrystal catalysts. By accounting for diverse adsorption sites and crystallographic facets, AEDs more fully represent nanoparticle-based catalytic surfaces and show strong promise for accelerating rational design and discovery of heterogeneous catalysts, especially for CO$_2$ hydrogenation. However, previous approaches have not sufficiently resolved facet-specific contributions, despite the catalytic significance and prevalence of certain Miller planes in nanoscale catalysts, limiting their applicability in predicting activity and selectivity. Here, we introduce an updated facet-resolved framework for predicting catalytic activity, which also enables insight into selectivity toward C1 products. Universal machine-learned force fields trained on Open Catalyst Project data were employed to compute adsorption energetics across 226 experimentally observed metals, binary alloys, and ternary alloys, encompassing 1.4 million adsorption sites on 2,626 crystallographically distinct surfaces. Using statistical and unsupervised learning techniques, we analyzed facet-specific AEDs to identify highly active and methanol-selective facets. Our approach provides insight into the relationship between structure and catalytic performance metrics like activity and selectivity, and presents a set of alloy compositions and their respective surface orientations for experimental validation toward highly selective CO$_2$ hydrogenation.

• The paper introduces facet resolved AED descriptors to simulate nanocatalysts activity of alloy combinations in CO2 hydrogenation tentatively validate CuAu(ZnPd(lll).
• Adsorption energies for 5 intermediates were calculated in 2613 distinct facets across 226 alloy combinations across identified key catalysts CuAuZnPd226.
• The PCA analysis links AED moments (mean, median, etc.) to CO$_2$ hydrogenation products and identifies CuAu(lll) and ZnPd(ZnPd) as optimized alloys within the methanol window, inactive MW site homing one step closer to actionable.

Motivation and scope

Thermal CO$_2$ hydrogenation to methanol over alloy nanocatalysts requires simultaneous optimization of activity, stability, and C1 selectivity, a task for which single-site descriptors such as individual adsorption energies or the d-band center are inadequate because they ignore the site- and facet-heterogeneity of real nanoparticles. Building on their earlier material-averaged adsorption energy distribution (AED) framework (2605.07714), Pisal, Krejci, and Rinke introduce a facet-resolved extension: AEDs are computed per Miller plane rather than aggregated across all surfaces, allowing direct comparison of individual facets against an experimentally validated reference, Zn@Cu(211), which models the active site identified in industrial Cu/ZnO/Al$_2$O$_3$ catalysts. The central hypothesis is that distribution-level similarity to this reference serves as a proxy for apparent activity, while statistical moments of multi-adsorbate AEDs encode selectivity information toward C1 products.

Computational workflow

The material space comprises 226 experimentally observed pure metals, binary alloys, and ternary alloys drawn from Materials Project, restricted to 18 elements (K, V, Mn, Fe, Co, Ni, Cu, Zn, Ga, Y, Ru, Rh, Pd, Ag, In, Ir, Pt, Au). Bulk structures were relaxed with RPBE/VASP; all symmetrically distinct surfaces with $|hkl| \leq 2$ were generated with the fairchem/OCP tooling, retaining only the lowest-energy termination per Miller index based on GemNet-OC-relaxed 50 Å slabs. Wulff constructions from these surface energies provide facet abundance estimates.

Adsorption energetics were computed with EquiformerV2, trained on OC20 (reported accuracy 0.23 eV), for five intermediates—H, *CO, *OH, *OCHO, *OCH$_3$—explicitly without imposing scaling relations. The dataset spans approximately 1.4 million adsorption-site relaxations on 2,613 crystallographically distinct facets. MLFF predictions were spot-checked via DFT single-point calculations on min/median/max configurations per material–adsorbate pair; the resulting estimated mean absolute error (EMAE) was **0.11 eV*, below the pretrained model's reported accuracy, and materials exceeding an EMAE threshold of 0.25 eV were discarded.

A methodological caveat applies throughout: adsorption energies omit entropy, zero-point, and pressure corrections, so positive values do not strictly imply desorption. The facet-resolved AEDs also differ qualitatively from material-aggregated ones—they are discrete and discontinuous rather than broad continua—and *OCH$_3$ generally binds most strongly while *H and *OH frequently show positive energies whose ordering is strongly facet-dependent.

Activity screening via Wasserstein distance

Facets were ranked by the first Wasserstein distance ($l_1$) between their AEDs and that of Zn@Cu(211); the top 300 facets ($l_1 \leq 0.001$) were labeled "active." Binary alloys dominate this subset, followed by ternary alloys, with pure metals rare. The most common Miller indices among candidates are (110) and (100). Two findings stand out:

• Stability–activity tension: most of the 20 closest facets have very low (<1%) or zero Wulff abundance; even Ag(211) and Ru(211), among the closest matches, account for only 0.12% and 1.82% of the equilibrium morphology. The authors argue that vacuum Wulff constructions neglect kinetics, supports, synthesis conditions, and reaction environments that can stabilize otherwise transient terminations, implying that exploiting these candidates would require synthesis protocols that kinetically trap specific facets.
• Abundant exception: In$_4$Ag$_9$(110), at 64.8% Wulff abundance, is the most abundant facet in the top-300 set—though its PCA position maps it to CO/RWGS selectivity rather than methanol (see below).

Latent-space analysis and selectivity mapping

PCA over six statistical moments (mean, median, std, p5, min, max) of the facet–adsorbate AEDs yields a compact two-dimensional representation capturing 85.78% of the variance (PC1: 77.29%; PC2: 8.49%). The loading structure is physically interpretable: PC1 is dominated by minimum and fifth-percentile moments of oxygenated species (*OCH$_2$0, *OCHO, *OH), whereas *CO vectors dominate PC2 and are oriented orthogonally to the oxygenate directions, indicating that *CO binding trends are uncorrelated with oxygenate binding in this descriptor space. The map functions as a two-dimensional analogue of a volcano plot, with PC1 encoding oxygenate binding strength and PC2 encoding *CO binding strength.

Projecting literature-based selectivity trends onto this map partitions it into regions associated with distinct C1 products:

Region	Interpretation	Example facets
Intermediate *CO binding, reference-like oxygenates	CH$_2$1OH	CuAu(111), ZnPd(111), ZrZnCu$_2$2(100)
Weak *CO binding (positive PC2)	CO / RWGS	Ag(211), InAg$_2$3(210), In$_2$4Ag$_2$5(110), K(InAu$_2$6)$_2$7(001)
Strong *CO binding (negative PC2)	CH$_2$8	Ru(211), Ni(221), MnNi$_2$9(100), Co$_3$0(211)
Destabilized *OCHO (−1 to 0 eV)	HCOOH/formaldehyde	GaPd$_3$1(001), ZnPt(111), InPd$_3$2(001)

The retrospective consistency of this partitioning with known catalytic behavior—Ag- and In-based systems favoring RWGS, Ru/Ni/Co facets methanating, CuAu nanoalloys enhancing methanol production, Zn–Pd systems being methanol-selective—supports the claim that the AED–PCA map carries genuine selectivity information rather than merely clustering by composition. The authors emphasize that the *CO-dominated PC2 axis is the critical determinant separating methanol, methane, and CO-selective regions, while noting that the boundaries between product regimes are not perfectly sharp. Within the "methanol window," the requirement is moderate *CO binding—strong enough for activation but weak enough to avoid over-hydrogenation or poisoning—with CuAu(111) and ZnPd(111) identified as the closest matches to the reference in both $_3$3 distance and PCA projection.

Limitations and open questions

The paper is explicit about several constraints on its conclusions. First, the Wasserstein distance ranks overall distribution shape but is insensitive to uniform shifts in adsorption energy: facets with systematically stronger or weaker binding can appear close to the reference if the shift is shape-preserving, which is why the moment-based PCA is needed as a complementary view. Second, reference-based screening introduces compositional bias—the preponderance of binary alloys among "active" facets may partly reflect the metric's affinity for compositions stoichiometrically resembling CuZn, and the authors recommend ternary benchmarks in future experimental work. Third, the selectivity assignments rest on qualitative literature trends rather than microkinetic modeling, and the treatment of facet stability relies on vacuum Wulff constructions that may misrepresent surfaces under reaction conditions. Finally, whether the low-abundance facets dominating the candidate list can actually be synthesized and stabilized remains an open experimental question, as does the quantitative accuracy of the selectivity boundaries, which the authors themselves note are not sharply defined.

Conclusion

This work extends MLFF-driven AED screening from material-averaged to facet-resolved descriptors, demonstrating that distribution-level fingerprints contain information on both activity (via similarity to a validated reference facet) and C1 selectivity (via interpretable PCA structure of AED moments). The workflow produces a prioritized list of composition–facet combinations for experimental validation in CO$_3$4 hydrogenation, with CuAu(111) and ZnPd(111) as leading methanol candidates, and is transferable in principle to other reactions and material classes.