Myoglobin-Catalyzed Azide Reduction Proceeds via an Anionic Metal Amide Intermediate.

Nitrene transfer reactions catalyzed by heme proteins have broad potential for the stereoselective formation of carbon-nitrogen bonds. However, competition between productive nitrene transfer and the undesirable reduction of nitrene precursors limits the broad implementation of such biocatalytic methods. Here, we investigated the reduction of azides by the model heme protein myoglobin to gain mechanistic insights into the factors that control the fate of key reaction intermediates. In this system, the reaction proceeds via a proposed nitrene intermediate that is rapidly reduced and protonated to give a reactive ferrous amide species, which we characterized by UV/vis and Mössbauer spectroscopies, quantum mechanical calculations, and X-ray crystallography. Rate-limiting protonation of the ferrous amide to produce the corresponding amine is the final step in the catalytic cycle. These findings contribute to our understanding of the heme protein-catalyzed reduction of azides and provide a guide for future enzyme engineering campaigns to create more efficient nitrene transferases. Moreover, harnessing the reduction reaction in a chemoenzymatic cascade provided a potentially practical route to substituted pyrroles.

Quantitative Structure-Reactivity Relationships for Synthesis Planning: The Benzhydrylium Case.

Selective and feasible reactions are among the top targets in synthesis planning. Mayr's approach to quantifying chemical reactivity has greatly facilitated the planning process, but reactivity parameters for new compounds require time-consuming experiments. In the past decade, data-driven modeling has been gaining momentum in the field, as it shows promise in terms of efficient reactivity prediction. However, state-of-the-art models use quantum chemical data as input, which prevent access to real-time planning in organic synthesis. Here, we present a novel data-driven workflow for predicting reactivity parameters of molecules that takes only structural information as input, enabling de facto real-time reactivity predictions. We use the well-understood chemical space of benzhydrylium ions as an example to demonstrate the functionality of our approach and the performance of the resulting quantitative structure-reactivity relationships (QSRRs). Our results suggest that it is straightforward to build low-cost QSRR models that are accurate, interpretable, and transferable to unexplored systems within a given scope of application. Moreover, our QSRR approach suggests that Hammett σ parameters are only approximately additive.

The computational road to reactivity scales.

Reactivity scales are useful research tools for chemists, both experimental and computational. However, to determine the reactivity of a single molecule, multiple measurements need to be carried out, which is a time-consuming and resource-intensive task. In this Tutorial Review, we present alternative approaches for the efficient generation of quantitative structure-reactivity relationships that are based on quantum chemistry, supervised learning, and uncertainty quantification. First published in 2002, we observe a tendency for these relationships to become not only more predictive but also more interpretable over time.

Learning Conductance: Gaussian Process Regression for Molecular Electronics.

Experimental studies of charge transport through single molecules often rely on break junction setups, where molecular junctions are repeatedly formed and broken while measuring the conductance, leading to a statistical distribution of conductance values. Modeling this experimental situation and the resulting conductance histograms is challenging for theoretical methods, as computations need to capture structural changes in experiments, including the statistics of junction formation and rupture. This type of extensive structural sampling implies that even when evaluating conductance from computationally efficient electronic structure methods, which typically are of reduced accuracy, the evaluation of conductance histograms is too expensive to be a routine task. Highly accurate quantum transport computations are only computationally feasible for a few selected conformations and thus necessarily ignore the rich conformational space probed in experiments. To overcome these limitations, we investigate the potential of machine learning for modeling conductance histograms, in particular by Gaussian process regression. We show that by selecting specific structural parameters as features, Gaussian process regression can be used to efficiently predict the zero-bias conductance from molecular structures, reducing the computational cost of simulating conductance histograms by an order of magnitude. This enables the efficient calculation of conductance histograms even on the basis of computationally expensive first-principles approaches by effectively reducing the number of necessary charge transport calculations, paving the way toward their routine evaluation.

Uncertainty Quantification of Reactivity Scales.

According to Mayr, polar organic synthesis can be rationalized by a simple empirical relationship linking bimolecular rate constants to as few as three reactivity parameters. Here, we propose an extension to Mayr's reactivity method that is rooted in uncertainty quantification and transforms the reactivity parameters into probability distributions. Through uncertainty propagation, these distributions can be transformed into uncertainty estimates for bimolecular rate constants. Chemists can exploit these virtual error bars to enhance synthesis planning and to decrease the ambiguity of conclusions drawn from experimental data. We demonstrate the above at the example of the reference data set released by Mayr and co-workers [J. Am. Chem. Soc. 2001, 123, 9500; J. Am. Chem. Soc. 2012, 134, 13902]. As by-product of the new approach, we obtain revised reactivity parameters for 36 π-nucleophiles and 32 benzhydrylium ions.

Machine-learned potentials for next-generation matter simulations.

The choice of simulation methods in computational materials science is driven by a fundamental trade-off: bridging large time- and length-scales with highly accurate simulations at an affordable computational cost. Venturing the investigation of complex phenomena on large scales requires fast yet accurate computational methods. We review the emerging field of machine-learned potentials, which promises to reach the accuracy of quantum mechanical computations at a substantially reduced computational cost. This Review will summarize the basic principles of the underlying machine learning methods, the data acquisition process and active learning procedures. We highlight multiple recent applications of machine-learned potentials in various fields, ranging from organic chemistry and biomolecules to inorganic crystal structure predictions and surface science. We furthermore discuss the developments required to promote a broader use of ML potentials, and the possibility of using them to help solve open questions in materials science and facilitate fully computational materials design.

Theoretical Studies of the Acid-Base Equilibria in a Model Active Site of the Human 20S Proteasome.

The 20S proteasome is a macromolecule responsible for the chemical step in the ubiquitin-proteasome system of degrading unnecessary and unused proteins of the cell. It plays a central role both in the rapid growth of cancer cells and in viral infection cycles. Herein, we present a computational study of the acid-base equilibria in an active site of the human proteasome (caspase-like), an aspect which is often neglected despite the crucial role protons play in the catalysis. As example substrates, we take the inhibition by epoxy- and boronic acid-containing warheads. We have combined cluster quantum mechanical calculations, replica exchange molecular dynamics, and Bayesian optimization of nonbonded potential terms in the inhibitors. In relation to the latter, we propose an easily scalable approach for the reevaluation of nonbonded potentials making use of the hybrid quantum mechanics molecular mechanics dynamics information. Our results show that coupled acid-base equilibria need to be considered when modeling the inhibition mechanism. The coupling between a neighboring lysine and the reacting threonine is not affected by the presence of the studied inhibitors.

Exchange Spin Coupling from Gaussian Process Regression.

Heisenberg exchange spin coupling between metal centers is essential for describing and understanding the electronic structure of many molecular catalysts, metalloenzymes, and molecular magnets for potential application in information technology. We explore the machine-learnability of exchange spin coupling beyond linear regression, which has not been studied yet. We employ Gaussian process regression, since it can potentially deal with small training sets (as likely associated with the rather complex molecular structures required for exploring spin coupling) and since it provides uncertainty estimates ("error bars") along with predicted values. We compare a range of descriptors and kernels for 257 small dicopper complexes and find that a simple descriptor based on chemical intuition, consisting only of copper-bridge angles and copper-copper distances, clearly outperforms several more sophisticated descriptors when it comes to extrapolating toward larger experimentally relevant complexes. Exchange spin coupling is similarly easy to learn as the polarizability, while learning dipole moments is much harder. The strength of the sophisticated descriptors lies in their ability to linearize structure-property relationships, to the point that a simple linear ridge regression performs just as well as the kernel-based machine-learning model for our small dicopper data set. The superior extrapolation performance of the simple descriptor is unique to exchange spin coupling, reinforcing the crucial role of choosing a suitable descriptor and highlighting the interesting question of the role of chemical intuition vs systematic or automated selection of features for machine learning in chemistry and material science.

Gaussian Process-Based Refinement of Dispersion Corrections.

We employ Gaussian process (GP) regression to adjust for systematic errors in D3-type dispersion corrections. We refer to the associated, statistically improved model as D3-GP. It is trained on differences between interaction energies obtained from PBE-D3(BJ)/ma-def2-QZVPP and DLPNO-CCSD(T)/CBS calculations. We generated a data set containing interaction energies for 1248 molecular dimers, which resemble the dispersion-dominated systems contained in the S66 data set. Our systems represent not only equilibrium structures but also dimers with various relative orientations and conformations at both shorter and longer distances. A reparametrization of the D3(BJ) model based on 66 of these dimers suggests that two of its three empirical parameters, a1 and s8, are zero, whereas a2 = 5.6841 bohr. For the remaining 1182 dimers, we find that this new set of parameters is superior to all previously published D3(BJ) parameter sets. To train our D3-GP model, we engineered two different vectorial representations of (supra-)molecular systems, both derived from the matrix of atom-pairwise D3(BJ) interaction terms: (a) a distance-resolved interaction energy histogram, histD3(BJ), and (b) eigenvalues of the interaction matrix ordered according to their decreasing absolute value, eigD3(BJ). Hence, the GP learns a mapping from D3(BJ) information only, which renders D3-GP-type dispersion corrections comparable to those obtained with the original D3 approach. They improve systematically if the underlying training set is selected carefully. Here, we harness the prediction variance obtained from GP regression to select optimal training sets in an automated fashion. The larger the variance, the more information the corresponding data point may add to the training set. For a given set of molecular systems, variance-based sampling can approximately determine the smallest subset being subjected to reference calculations such that all dispersion corrections for the remaining systems fall below a predefined accuracy threshold. To render the entire D3-GP workflow as efficient as possible, we present an improvement over our variance-based, sequential active-learning scheme [ J. Chem. Theory Comput. 2018 , 14 , 5238 ]. Our refined learning algorithm selects multiple (instead of single) systems that can be subjected to reference calculations simultaneously. We refer to the underlying selection strategy as batchwise variance-based sampling (BVS). BVS-guided active learning is an essential component of our D3-GP workflow, which is implemented in a black-box fashion. Once provided with reference data for new molecular systems, the underlying GP model automatically learns to adapt to these and similar systems. This approach leads overall to a self-improving model (D3-GP) that predicts system-focused and GP-refined D3-type dispersion corrections for any given system of reference data.

Mechanism Deduction from Noisy Chemical Reaction Networks.

We introduce KiNetX, a fully automated meta-algorithm for the kinetic analysis of complex chemical reaction networks derived from semiaccurate but efficient electronic structure calculations. It is designed to (i) accelerate the automated exploration of such networks and (ii) cope with model-inherent errors in electronic structure calculations on elementary reaction steps. We developed and implemented KiNetX to possess three features. First, KiNetX evaluates the kinetic relevance of every species in a (yet incomplete) reaction network to confine the search for new elementary reaction steps only to those species that are considered possibly relevant. Second, KiNetX identifies and eliminates all kinetically irrelevant species and elementary reactions to reduce a complex network graph to a comprehensible mechanism. Third, KiNetX estimates the sensitivity of species concentrations toward changes in individual rate constants (derived from relative free energies), which allows us to systematically select the most efficient electronic structure model for each elementary reaction given a predefined accuracy. The novelty of KiNetX consists in the rigorous propagation of correlated free-energy uncertainty through all steps of our kinetic analyis. To examine the performance of KiNetX, we developed AutoNetGen. It semirandomly generates chemistry-mimicking reaction networks by encoding chemical logic into their underlying graph structure. AutoNetGen allows us to consider a vast number of distinct chemistry-like scenarios and, hence, to discuss the importance of rigorous uncertainty propagation in a statistical context. Our results reveal that KiNetX reliably supports the deduction of product ratios, dominant reaction pathways, and possibly other network properties from semiaccurate electronic structure data.

Statistical Analysis of Semiclassical Dispersion Corrections.

Semiclassical dispersion corrections developed by Grimme and co-workers have become indispensable in applications of Kohn-Sham density functional theory. A deeper understanding of the underlying parametrization might be crucial for well-founded further improvements of this successful approach. To this end, we present an in-depth assessment of the fit parameters present in semiclassical (D3-type) dispersion corrections by means of a statistically rigorous analysis. We find that the choice of the cost function generally has a small effect on the empirical parameters of D3-type dispersion corrections with respect to the reference set under consideration. Only in a few cases, the choice of cost function has a surprisingly large effect on the total dispersion energies. In particular, the weighting scheme in the cost function can significantly affect the reliability of predictions. In order to obtain unbiased (data-independent) uncertainty estimates for both the empirical fit parameters and the corresponding predictions, we carried out a nonparametric bootstrap analysis. This analysis reveals that the standard deviation of the mean of the empirical D3 parameters is small. Moreover, the mean prediction uncertainty obtained by bootstrapping is not much larger than previously reported error measures. On the basis of a jackknife analysis, we find that the original reference set is slightly skewed, but our results also suggest that this feature hardly affects the prediction of dispersion energies. Furthermore, we find that the introduction of small uncertainties to the reference data does not change the conclusions drawn in this work. However, a rigorous analysis of error accumulation arising from different parametrizations reveals that error cancellation does not necessarily occur, leading to a monotonically increasing deviation in the dispersion energy with increasing molecule size. We discuss this issue in detail at the prominent example of the C60 "buckycatcher". We find deviations between individual parametrizations of several tens of kilocalories per mole in some cases. Hence, in combination with any calculation of dispersion energies, we recommend to always determine the associated uncertainties for which we will provide a software tool.

Reliable Estimation of Prediction Uncertainty for Physicochemical Property Models.

One of the major challenges in computational science is to determine the uncertainty of a virtual measurement, that is the prediction of an observable based on calculations. As highly accurate first-principles calculations are in general unfeasible for most physical systems, one usually resorts to parameteric property models of observables, which require calibration by incorporating reference data. The resulting predictions and their uncertainties are sensitive to systematic errors such as inconsistent reference data, parametric model assumptions, or inadequate computational methods. Here, we discuss the calibration of property models in the light of bootstrapping, a sampling method that can be employed for identifying systematic errors and for reliable estimation of the prediction uncertainty. We apply bootstrapping to assess a linear property model linking the 57Fe Mössbauer isomer shift to the contact electron density at the iron nucleus for a diverse set of 44 molecular iron compounds. The contact electron density is calculated with 12 density functionals across Jacob's ladder (PWLDA, BP86, BLYP, PW91, PBE, M06-L, TPSS, B3LYP, B3PW91, PBE0, M06, TPSSh). We provide systematic-error diagnostics and reliable, locally resolved uncertainties for isomer-shift predictions. Pure and hybrid density functionals yield average prediction uncertainties of 0.06-0.08 mm s-1 and 0.04-0.05 mm s-1, respectively, the latter being close to the average experimental uncertainty of 0.02 mm s-1. Furthermore, we show that both model parameters and prediction uncertainty depend significantly on the composition and number of reference data points. Accordingly, we suggest that rankings of density functionals based on performance measures (e.g., the squared coefficient of correlation, r2, or the root-mean-square error, RMSE) should not be inferred from a single data set. This study presents the first statistically rigorous calibration analysis for theoretical Mössbauer spectroscopy, which is of general applicability for physicochemical property models and not restricted to isomer-shift predictions. We provide the statistically meaningful reference data set MIS39 and a new calibration of the isomer shift based on the PBE0 functional.

Error Assessment of Computational Models in Chemistry.

Computational models in chemistry rely on a number of approximations. The effect of such approximations on observables derived from them is often unpredictable. Therefore, it is challenging to quantify the uncertainty of a computational result, which, however, is necessary to assess the suitability of a computational model. Common performance statistics such as the mean absolute error are prone to failure as they do not distinguish the explainable (systematic) part of the errors from their unexplainable (random) part. In this paper, we discuss problems and solutions for performance assessment of computational models based on several examples from the quantum chemistry literature. For this purpose, we elucidate the different sources of uncertainty, the elimination of systematic errors, and the combination of individual uncertainty components to the uncertainty of a prediction.

Uncertainty quantification for quantum chemical models of complex reaction networks.

For the quantitative understanding of complex chemical reaction mechanisms, it is, in general, necessary to accurately determine the corresponding free energy surface and to solve the resulting continuous-time reaction rate equations for a continuous state space. For a general (complex) reaction network, it is computationally hard to fulfill these two requirements. However, it is possible to approximately address these challenges in a physically consistent way. On the one hand, it may be sufficient to consider approximate free energies if a reliable uncertainty measure can be provided. On the other hand, a highly resolved time evolution may not be necessary to still determine quantitative fluxes in a reaction network if one is interested in specific time scales. In this paper, we present discrete-time kinetic simulations in discrete state space taking free energy uncertainties into account. The method builds upon thermo-chemical data obtained from electronic structure calculations in a condensed-phase model. Our kinetic approach supports the analysis of general reaction networks spanning multiple time scales, which is here demonstrated for the example of the formose reaction. An important application of our approach is the detection of regions in a reaction network which require further investigation, given the uncertainties introduced by both approximate electronic structure methods and kinetic models. Such cases can then be studied in greater detail with more sophisticated first-principles calculations and kinetic simulations.

Heuristics-Guided Exploration of Reaction Mechanisms.

For the investigation of chemical reaction networks, the efficient and accurate determination of all relevant intermediates and elementary reactions is mandatory. The complexity of such a network may grow rapidly, in particular if reactive species are involved that might cause a myriad of side reactions. Without automation, a complete investigation of complex reaction mechanisms is tedious and possibly unfeasible. Therefore, only the expected dominant reaction paths of a chemical reaction network (e.g., a catalytic cycle or an enzymatic cascade) are usually explored in practice. Here, we present a computational protocol that constructs such networks in a parallelized and automated manner. Molecular structures of reactive complexes are generated based on heuristic rules derived from conceptual electronic-structure theory and subsequently optimized by quantum-chemical methods to produce stable intermediates of an emerging reaction network. Pairs of intermediates in this network that might be related by an elementary reaction according to some structural similarity measure are then automatically detected and subjected to an automated search for the connecting transition state. The results are visualized as an automatically generated network graph, from which a comprehensive picture of the mechanism of a complex chemical process can be obtained that greatly facilitates the analysis of the whole network. We apply our protocol to the Schrock dinitrogen-fixation catalyst to study alternative pathways of catalytic ammonia production.

Communication through molecular bridges: different bridge orbital trends result in common property trends.

Common trends in communication through molecular bridges are ubiquitous in chemistry, such as the frequently observed exponential decay of conductance/electron transport and of exchange spin coupling with increasing bridge length, or the increased communication through a bridge upon closing a diarylethene photoswitch. For antiferromagnetically coupled diradicals in which two equivalent spin centers are connected by a closed-shell bridge, the molecular orbitals (MOs) whose energy splitting dominates the coupling strength are similar in shape to the MOs of the dithiolated bridges, which in turn can be used to rationalize conductance. Therefore, it appears reasonable to expect the observed common property trends to result from common orbital trends. We illustrate based on a set of model compounds that this assumption is not true, and that common property trends result from either different pairs of orbitals being involved, or from orbital energies not being the dominant contribution to property trends. For substituent effects, an effective modification of the π system can make a comparison difficult.

Charge delocalization in an organic mixed valent bithiophene is greater than in a structurally analogous biselenophene.

A series of selenophenes with redox-active amine end-capping groups was synthesized and investigated. A combination of cyclic voltammetry, optical absorption, EPR spectroscopy, and quantum-chemical calculations based on Kohn-Sham density functional theory was used to explore charge delocalization in the monocationic mixed-valence forms of these selenophenes, and the results were compared to those obtained from analogous studies of structurally identical thiophenes. The striking finding is that the comproportionation constant (Kc) for the experimentally investigated biselenophene is more than 2 orders of magnitude lower than for its bithiophene counterpart (in CH3CN with 0.1 M TBAPF6), and the electronic coupling between the two amine end-capping groups in the mixed-valent biselenophene monocation is only roughly half as strong as in the corresponding bithiophene monocation. These are surprisingly large differences given the structural similarity between the respective biselenophene and bithiophene molecules. However, the computationally determined comproportionation constants for biselenophene and bithiophene are almost identical, and the electronic coupling in the monocationic biselenophene is only slightly smaller than that in the monocationic bithiophene. We assume that the external electric field may be responsible for the differences in monocation stabilities between experiment and computation. Our findings indicate that charge delocalization across individual selenophenes tends to be less pronounced than across individual thiophenes, and this may have important implications for long-range charge transfer across selenophene oligomers or polymers.