(472f) Identification of Optimally Stable Nanoparticle Geometries Via Mathematical Optimization and Density-Functional Theory

One of the key research objectives in the
area of transition metal nanoparticles is to determine the most stable
structure for particles of a given size (mass) and composition. In this work,
we propose a mathematical optimization based modeling framework, coupled with
density functional theory (DFT) calculations, for determining the proven
minimum energy structures of three-dimensional, mono-metallic nanoclusters of a
given size. More specifically, we apply this framework to mono-metallic copper
(Cu) nanoclusters, as previous work has shown potential uses for mono- and
bi-metallic Cu nanoparticles in CO₂ activation and reduction [1],[2],[3].

In our work, we consider the global minimum
energy Cu nanocluster of a given size to be the one that attains the maximum average
dimensionless cohesive energy, which is the average cohesive energy normalized
to the bulk energy. The cohesive energy measures the strength of inter-atomic
bonding between atoms and the overall stability. Thus, maximizing the average
dimensionless cohesive energy leads to the most stable, ground-state
nanocluster geometry. Our proposed model uses a cohesive energy function, first
proposed by Tomanek et al., which depends only on the geometric descriptor of
coordination number (CN) [4]. Because CN is a local geometric descriptor of the
nanoparticle (NP), it can be encoded within a rigorous, “bottom-up” design optimization
model. In fact, previous work has successfully utilized CN as a local
descriptor to design transition metal surfaces via a mixed-integer linear
programming (MILP) [5], a class of mathematical optimization problems that can
be solved readily and reliably with appropriate numerical optimization
techniques. The current work serves as a first step in extending such tools for
targeted nanoparticle design in three dimensions. It should be highlighted that
the proposed method deviates from the existing literature, inasmuch as existing
methods for determining lowest energy small nanoclusters typically involves heuristic
search of the design space and evaluation of incumbent solutions via inter-atomic
potentials [6],[7],[8],[9]. However, global minimum energy nanoclusters
reported in this way are only considered “putative," as the search methods
utilized provide no mathematical proof of optimality. In contrast, the use of
MILP based optimization techniques proposed in this work guarantees the
optimality of the final designs.

However, the cohesive energy model
originally proposed by Tomanek el al. also includes a repulsive potential energy
term. Whereas previous work has shown that neglecting the repulsive
contribution for sufficiently large nanoclusters compares favorably to the DFT
calculated cohesive energy [10], doing so at small NP sizes is not appropriate due
to surface contraction and quantum effects. In order to create an optimization
model that is accurate for small NPs, we first assemble a large collection of
nanoparticles of given size that feature a wide variety of coordination number
distributions, and then use first-principles DFT calculations to evaluate their
cohesive energy. We then perform linear regression to elucidate a correction
term that can be represented in our MILP optimization framework. This way,
optimally cohesive NPs at small sizes can be correctly and quickly predicted.

Following the above methodology, a pool of
optimal and near-optimal Cu NPs of different small sizes (up to 50 atoms) was
collected, and single-point DFT calculations (using the PBE
exchange-correlation functional [11]) were performed to determine the expected
cohesive energies for these structures. Using this data, a linear regression
model was developed in order to identify corrections for the cohesive energy
contribution of a single atom in a given Cu NP. These corrective factors were then
incorporated into the mixed-integer optimization model, which determined highly
cohesive nanoclusters for each given size, establishing morphological trends
related to stability in NPs of any size. The results confirm various
“magic-number” designs that are well-known to be highly stable, as well as
reveal a number of unintuitive designs that were not previously reported in the
literature, yet are confirmed to be highly stable after DFT geometry
optimizations. These particles can serve as model particles for a multitude of
follow-up studies within the realm of computational chemistry to include, among
others, adsorption and catalytic activity studies. Finally, by collecting
sequences of most cohesive designs for given NP sizes, our results can also be
used to determine expected distributions of clusters that arise during
experimental synthesis.

References

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