A new topology-based method predicts atomic charges in metal-organic frameworks from bond connectivity alone, making large-scale computational screening practical.
(Nanowerk Spotlight) Capturing carbon dioxide from industrial exhaust, storing hydrogen fuel at room temperature, delivering drugs to targeted tissues, separating water isotopes at the molecular level: these are problems that demand materials with precisely controlled pores at the nanoscale. Metal-organic frameworks (MOFs) are among the most promising candidates.
Built from metal nodes linked by organic molecules into porous crystalline lattices, MOFs can be engineered with specific pore sizes, surface chemistries, and mechanical properties simply by swapping out their component parts. Since the early 2000s, chemists have synthesized tens of thousands of distinct MOFs, and no laboratory can test them all.
Computational screening has therefore become essential: scientists use force fields, sets of mathematical functions that approximate interatomic forces through classical physics, to simulate MOF behavior at a fraction of the cost of full quantum-mechanical calculations.
Force fields, however, need accurate inputs. One of the most consequential is partial atomic charges: the small positive or negative electric charges assigned to each atom in a structure. These charges govern electrostatic interactions, which determine how guest molecules like carbon dioxide or methane behave inside MOF pores.
The standard way to obtain them is through quantum-mechanical calculations performed individually for each MOF, a workflow that does not scale when thousands of candidates require evaluation.
A faster alternative appeared in 1991, when Rappé and Goddard introduced the charge equilibration method, known as QEq. It distributes charge across a molecule by balancing electronegativity differences between atoms. Several groups later adapted QEq specifically for MOFs with mixed success, and a 2018 comparison by Ongari and colleagues found that none of the derived methods significantly outperformed the original.
A persistent limitation also remained: because QEq depends on precise three-dimensional atomic coordinates, charges must be recalculated every time the geometry changes. For rigid MOFs this is tolerable. For flexible frameworks, where no single reference geometry exists, the dependence introduces ambiguity.
A study published in Advanced Functional Materials (“Predicting Atomic Charges in MOFs by Topological Charge Equilibration”) by a team at Ruhr-Universität Bochum tackles this limitation. Their method, called Topological Charge Equilibration (TopoQEq), replaces real spatial distances with purely topological ones, distances measured not through three-dimensional space but along the shortest paths of the molecular bond graph.
Because bond connectivity does not change when a structure flexes or vibrates, the resulting charges become formally independent of geometry.
The core idea is straightforward. TopoQEq constructs a graph in which atoms are vertices and bonds are edges. Each edge receives a weight equal to the sum of the experimentally known covalent radii of the two connected atoms. An efficient shortest-path algorithm then computes the distance between every pair of atoms through the graph. These topological distances replace spatial distances in the electrostatic equations of standard QEq, producing atomic charges that satisfy overall charge neutrality by construction, without any manual correction.
To make the model transferable across different MOFs, the researchers defined three levels of atom classification with increasing chemical resolution. The simplest, “etypes,” distinguishes atoms only by element. The intermediate level, “ctypes,” adds information about how many neighbors each atom has. The most detailed, “atypes,” further encodes the elemental identity of those neighbors. A carbon atom bonded to two other carbons and one hydrogen, for instance, receives a different atype label than one bonded to three carbons.
Each atom type carries three adjustable parameters: a charge distribution width, an electronegativity value, and a hardness correction. The team optimized these against reference data from the QMOF database, a publicly available repository of quantum-mechanically computed properties for over 20,000 MOFs.
They curated a subset of about 11,915 structures containing commonly studied metals, including zinc, copper, zirconium, cobalt, and nickel, filtering for balanced representation across atom types. Reference charges came from the DDEC6 method, a widely used charge partitioning scheme for periodic materials, applied to electron densities calculated with the PBE-D3(BJ) density functional.
The curated data was split into a training set of 10,603 MOFs and a testing set of 1,312, with matching atom-type distributions maintained across both sets.
Parameter fitting employed the covariance matrix adaptation evolutionary strategy (CMA-ES), a global optimization algorithm that evolves a population of candidate parameter sets over many iterations, selecting the best performers and refining the search based on a loss function that measures deviation from reference charges. Because CMA-ES requires no gradient calculations, it simplifies implementation and keeps computational costs manageable even as the number of parameters grows.
The authors note, however, that CMA-ES offers no formal guarantee of reaching the global minimum, a risk they mitigated by running four independent optimizations per atom-typing level with different random seeds.
Results showed systematic improvement with increasing atom-type detail. The best atype model achieved a mean absolute error of 0.019 elementary charges (e) on the test set, with a maximum absolute error of 0.35 e. Predicted charges satisfied overall charge neutrality without any post-hoc corrections, a notable advantage since many existing methods produce charges that do not sum to zero, requiring arbitrary manual adjustments that can compromise physical consistency.
At the coarser etype and ctype levels, some parameters converged toward their boundary values even when those boundaries were extended, suggesting that these simpler classification schemes lack sufficient flexibility to fully capture the underlying chemistry.
In a focused case study, the model correctly captured how substituting a single hydrogen atom with chlorine on a copper-paddlewheel MOF polarized neighboring carbon atoms. This demonstrated sensitivity to chemical environment changes at all three classification levels, and the systematic quality gain from etypes through ctypes to atypes was clearly reflected.
Computational benchmarks revealed that prediction time scales with system size at a fitted exponent of 2.19, indicating roughly quadratic behavior. For structures larger than about 10,000 atoms, the team implemented a sparse variant that truncates long-range interactions beyond a cutoff distance. Corrections borrowed from the Wolf summation and damped shifted force methods ensured that truncation errors decreased smoothly with increasing cutoff.
Tests on a DUT-49 nanocrystallite containing 3,636 atoms confirmed the approach: the sparse implementation consumed orders of magnitude less memory while delivering faster predictions, especially when paired with conjugate gradient solvers.
What sets TopoQEq apart from recent machine-learning approaches, such as graph neural networks and decision-tree ensembles that also predict MOF charges with high accuracy, is its foundation in a closed-form linear system grounded in classical electrostatics. This keeps computational costs low and scaling favorable, making the method well suited to high-throughput screening and to modeling large nanocrystallites where surface effects matter.
By producing geometry-independent charges that naturally adapt to different chemical environments, TopoQEq removes a concrete bottleneck in automated force-field generation, a capability the authors describe as paving the way for high-throughput investigations using system-specific force fields.
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