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Ab initio computational studies have made tremendous progress in describing the behavior of molecular (homogeneous) catalysts and crystalline versions of heterogeneous catalysts, but not for amorphous heterogeneous catalysts. Even widely used industrial amorphous catalysts like atomically dispersed Cr on silica remain poorly understood and largely intractable to computational investigation. The central problems are that (i) the amorphous support presents an unknown quenched disordered structure, (ii) metal atoms attach to various surface grafting sites with different rates, and (iii) the resulting grafted sites have different activation and catalytic reaction kinetics. This study combines kernel regression and importance sampling techniques to efficiently model grafting of metal ions onto a non-uniform ensemble of support environments. Our analysis uses a simple model of the quenched disordered support environment, grafting chemistry, and catalytic activity of the resulting grafted sites. 

Ab initio calculations have greatly advanced our understanding of homogeneous catalysts and crystalline heterogeneous catalysts. In contrast, amorphous heterogeneous catalysts remain poorly understood. The principal difficulties include (i) the nature of the disorder is quenched and unknown; (ii) each active site has a different local environment and activity; (iii) active sites are rare, often less than ∼20% of potential sites, depending on the catalyst and its preparation method. Few (if any) studies of amorphous heterogeneous catalysts have ever attempted to compute site-averaged kinetics, because the exponential dependence on variable activation energy requires an intractable number of ab initio calculations to converge. We present a new algorithm using machine learning techniques (metric learning kernel regression) and importance sampling to efficiently learn the distribution of activation energies. We demonstrate the algorithm by computing the site-averaged activity for a model amorphous catalyst with quenched disorder.