Reliable Estimation of Prediction Uncertainty for Physicochemical Property Models

Fluorescence sensing of water in various organic solvents based on a novel cyclic polymer

A sensor capable of sensing of water in various organic solvents ranging from water-soluble to water-miscible solvents is still a challenging task. In this research, a cyclic polymer fluorescence chemosensor (CPFC) has been developed for sensing of water by turn-on model in 9 organic solvents and turn-off model in DMA, where the broadest concentration range and the lowest detection limit was obtained for water in DMA (10 %-90 %) and dioxane (0.011 %), respectively. The sensing mechanism is explored by theory calculation and experimental investigation. The amphiphilic nature endows the polymer probe with great potential for measuring various contaminants from aqueous and nonaqueous mediums. Furthermore, the present search highlights the potential applications of cyclic polymer as fluorescence probes in the field of sensing.

Automated preparation of nanoscopic structures: Graph-based sequence analysis, mismatch detection, and pH-consistent protonation with uncertainty estimates

Structure and function in nanoscale atomistic assemblies are tightly coupled, and every atom with its specific position and even every electron will have a decisive effect on the electronic structure, and hence, on the molecular properties. Molecular simulations of nanoscopic atomistic structures therefore require accurately resolved three-dimensional input structures. If extracted from experiment, these structures often suffer from severe uncertainties, of which the lack of information on hydrogen atoms is a prominent example. Hence, experimental structures require careful review and curation, which is a time-consuming and error-prone process. Here, we present a fast and robust protocol for the automated structure analysis and pH-consistent protonation, in short, ASAP. For biomolecules as a target, the ASAP protocol integrates sequence analysis and error assessment of a given input structure. ASAP allows for pK a  prediction from reference data through Gaussian process regression including uncertainty estimation and connects to system-focused atomistic modeling described in Brunken and Reiher (J. Chem. Theory Comput. 16, 2020, 1646). Although focused on biomolecules, ASAP can be extended to other nanoscopic objects, because most of its design elements rely on a general graph-based foundation guaranteeing transferability. The modular character of the underlying pipeline supports different degrees of automation, which allows for (i) efficient feedback loops for human-machine interaction with a low entrance barrier and for (ii) integration into autonomous procedures such as automated force field parametrizations. This facilitates fast switching of the pH-state through on-the-fly system-focused reparametrization during a molecular simulation at virtually no extra computational cost.

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.

Graph Neural Tree: A novel and interpretable deep learning-based framework for accurate molecular property predictions

Determining various properties of molecules is a critical step in drug discovery. Recently, with the improvement of large heterogeneous datasets and the development of deep learning approaches, more and more scientists have turned their attention to neural network-based virtual preliminary screening to reduce the time and monetary cost of drug discovery. However, the poor interpretability of deep learning masks causality, so models' conclusions are often beyond the comprehension of human users, which reduces the credibility of the model and makes it difficult for chemists to further narrow the huge chemical space based on models' results. Thus, this study develops a novel framework consisting of Graph Neural Networks for feature extraction, Curriculum-Based Learning Strategies for optimization, and a Learning Binary Neural Tree (LBNT) for prediction, to improve the performance of neural networks and reveal their decision-making process to chemists. The framework encodes molecular graph data with graph neural networks (GNNs), then retrains the encoder with curriculum-based learning strategies to reduce uncertainty and improve accuracy, and finally uses LBNT as the predictor, which joint retrains with the encoder after independently training, for prediction and visualization. The framework is validated on the public datasets and compared to single GNNs with normal training strategies as well as GNN encoders with common machine learning predictors instead of the LBNT predictor. The result reveals that the proposed framework enhances the point prediction accuracy of the completely trained GNN and reduces its uncertainty through curriculum-based learning, and further improves the accuracy by combining LBNT. Besides, compared with common machine learning tools, the LBNT predictor generally has the best performance because of joint retraining with the GNN encoder. The decision-making process of LBNT is also better and easier to explain than that of other models.

DFT exchange: sharing perspectives on the workhorse of quantum chemistry and materials science

In this paper, the history, present status, and future of density-functional theory (DFT) is informally reviewed and discussed by 70 workers in the field, including molecular scientists, materials scientists, method developers and practitioners. The format of the paper is that of a roundtable discussion, in which the participants express and exchange views on DFT in the form of 302 individual contributions, formulated as responses to a preset list of 26 questions. Supported by a bibliography of 777 entries, the paper represents a broad snapshot of DFT, anno 2022.

The long road to calibrated prediction uncertainty in computational chemistry

Uncertainty quantification (UQ) in computational chemistry (CC) is still in its infancy. Very few CC methods are designed to provide a confidence level on their predictions, and most users still rely improperly on the mean absolute error as an accuracy metric. The development of reliable UQ methods is essential, notably for CC to be used confidently in industrial processes. A review of the CC-UQ literature shows that there is no common standard procedure to report or validate prediction uncertainty. I consider here analysis tools using concepts (calibration and sharpness) developed in meteorology and machine learning for the validation of probabilistic forecasters. These tools are adapted to CC-UQ and applied to datasets of prediction uncertainties provided by composite methods, Bayesian ensembles methods, and machine learning and a posteriori statistical methods.

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.

Autonomous Reaction Network Exploration in Homogeneous and Heterogeneous Catalysis

Autonomous computations that rely on automated reaction network elucidation algorithms may pave the way to make computational catalysis on a par with experimental research in the field. Several advantages of this approach are key to catalysis: (i) automation allows one to consider orders of magnitude more structures in a systematic and open-ended fashion than what would be accessible by manual inspection. Eventually, full resolution in terms of structural varieties and conformations as well as with respect to the type and number of potentially important elementary reaction steps (including decomposition reactions that determine turnover numbers) may be achieved. (ii) Fast electronic structure methods with uncertainty quantification warrant high efficiency and reliability in order to not only deliver results quickly, but also to allow for predictive work. (iii) A high degree of autonomy reduces the amount of manual human work, processing errors, and human bias. Although being inherently unbiased, it is still steerable with respect to specific regions of an emerging network and with respect to the addition of new reactant species. This allows for a high fidelity of the formalization of some catalytic process and for surprising in silico discoveries. In this work, we first review the state of the art in computational catalysis to embed autonomous explorations into the general field from which it draws its ingredients. We then elaborate on the specific conceptual issues that arise in the context of autonomous computational procedures, some of which we discuss at an example catalytic system.

The Impact of Computational Uncertainties on the Enantioselectivity Predictions: A Microkinetic Modeling of Ketone Transfer Hydrogenation with a Noyori-type Mn-diamine Catalyst

Selectivity control is one of the most important functions of a catalyst. In asymmetric catalysis the enantiomeric excess (e.e.) is a property of major interest, with a lot of effort dedicated to developing the most enantioselective catalyst, understanding the origin of selectivity, and predicting stereoselectivity. Herein, we investigate the relationship between predicted selectivity and the uncertainties in the computed energetics of the catalytic reaction mechanism obtained by DFT calculations in a case study of catalytic asymmetric transfer hydrogenation (ATH) of ketones with an Mn-diamine catalyst. Data obtained from our analysis of DFT data by microkinetic modeling is compared to results from experiment. We discuss the limitations of the conventional reductionist approach of e.e. estimation from assessing the enantiodetermining steps only. Our analysis shows that the energetics of other reaction steps in the reaction mechanism have a substantial impact on the predicted reaction selectivity. The uncertainty of DFT calculations within the commonly accepted energy ranges of chemical accuracy may reverse the predicted e.e. with the non-enantiodetermining steps contributing to e.e. deviations of up to 25 %.

Uncertainty quantification in classical molecular dynamics

Molecular dynamics simulation is now a widespread approach for understanding complex systems on the atomistic scale. It finds applications from physics and chemistry to engineering, life and medical science. In the last decade, the approach has begun to advance from being a computer-based means of rationalizing experimental observations to producing apparently credible predictions for a number of real-world applications within industrial sectors such as advanced materials and drug discovery. However, key aspects concerning the reproducibility of the method have not kept pace with the speed of its uptake in the scientific community. Here, we present a discussion of uncertainty quantification for molecular dynamics simulation designed to endow the method with better error estimates that will enable it to be used to report actionable results. The approach adopted is a standard one in the field of uncertainty quantification, namely using ensemble methods, in which a sufficiently large number of replicas are run concurrently, from which reliable statistics can be extracted. Indeed, because molecular dynamics is intrinsically chaotic, the need to use ensemble methods is fundamental and holds regardless of the duration of the simulations performed. We discuss the approach and illustrate it in a range of applications from materials science to ligand-protein binding free energy estimation. This article is part of the theme issue 'Reliability and reproducibility in computational science: implementing verification, validation and uncertainty quantification in silico'.

Influence of Electronic Structure Modeling and Junction Structure on First-Principles Chiral Induced Spin Selectivity

We have carried out a comprehensive study of the influence of electronic structure modeling and junction structure description on the first-principles calculation of the spin polarization in molecular junctions caused by the chiral induced spin selectivity (CISS) effect. We explore the limits and the sensitivity to modeling decisions of a Landauer/Green's function/two-component density functional theory approach to CISS. We find that although the CISS effect is entirely attributed in the literature to molecular spin filtering, spin-orbit coupling being partially inherited from the metal electrodes plays an important role in our calculations on ideal carbon helices, even though this effect cannot explain the experimental conductance results. Its magnitude depends considerably on the shape, size, and material of the metal clusters modeling the electrodes. Also, a pronounced dependence on the specific description of exchange interaction and spin-orbit coupling is manifest in our approach. This is important because the interplay between exchange effects and spin-orbit coupling may play an important role in the description of the junction magnetic response. Our calculations are relevant for the whole field of spin-polarized electron transport and electron transfer, because there is still an open discussion in the literature about the detailed underlying mechanism and the magnitude of physical parameters that need to be included to achieve a consistent description of the CISS effect: seemingly good quantitative agreement between simulation and the experiment can be caused by error compensation, because spin polarization as contained in a Landauer/Green's function/two-component density functional theory approach depends strongly on computational and structural parameters.

Probabilistic performance estimators for computational chemistry methods: Systematic improvement probability and ranking probability matrix. I. Theory

The comparison of benchmark error sets is an essential tool for the evaluation of theories in computational chemistry. The standard ranking of methods by their mean unsigned error is unsatisfactory for several reasons linked to the non-normality of the error distributions and the presence of underlying trends. Complementary statistics have recently been proposed to palliate such deficiencies, such as quantiles of the absolute error distribution or the mean prediction uncertainty. We introduce here a new score, the systematic improvement probability, based on the direct system-wise comparison of absolute errors. Independent of the chosen scoring rule, the uncertainty of the statistics due to the incompleteness of the benchmark datasets is also generally overlooked. However, this uncertainty is essential to appreciate the robustness of rankings. In the present article, we develop two indicators based on robust statistics to address this problem: P_inv, the inversion probability between two values of a statistic, and P_r, the ranking probability matrix. We demonstrate also the essential contribution of the correlations between error sets in these scores comparisons.

Evaluating Scalable Uncertainty Estimation Methods for Deep Learning-Based Molecular Property Prediction

Advances in deep neural network (DNN)-based molecular property prediction have recently led to the development of models of remarkable accuracy and generalization ability, with graph convolutional neural networks (GCNNs) reporting state-of-the-art performance for this task. However, some challenges remain, and one of the most important that needs to be fully addressed concerns uncertainty quantification. DNN performance is affected by the volume and the quality of the training samples. Therefore, establishing when and to what extent a prediction can be considered reliable is just as important as outputting accurate predictions, especially when out-of-domain molecules are targeted. Recently, several methods to account for uncertainty in DNNs have been proposed, most of which are based on approximate Bayesian inference. Among these, only a few scale to the large data sets required in applications. Evaluating and comparing these methods has recently attracted great interest, but results are generally fragmented and absent for molecular property prediction. In this paper, we quantitatively compare scalable techniques for uncertainty estimation in GCNNs. We introduce a set of quantitative criteria to capture different uncertainty aspects and then use these criteria to compare MC-dropout, Deep Ensembles, and bootstrapping, both theoretically in a unified framework that separates aleatoric/epistemic uncertainty and experimentally on public data sets. Our experiments quantify the performance of the different uncertainty estimation methods and their impact on uncertainty-related error reduction. Our findings indicate that Deep Ensembles and bootstrapping consistently outperform MC-dropout, with different context-specific pros and cons. Our analysis leads to a better understanding of the role of aleatoric/epistemic uncertainty, also in relation to the target data set features, and highlights the challenge posed by out-of-domain uncertainty.

The Exploration of Chemical Reaction Networks

Modern computational chemistry has reached a stage at which massive exploration into chemical reaction space with unprecedented resolution with respect to the number of potentially relevant molecular structures has become possible. Various algorithmic advances have shown that such structural screenings must and can be automated and routinely carried out. This will replace the standard approach of manually studying a selected and restricted number of molecular structures for a chemical mechanism. The complexity of the task has led to many different approaches. However, all of them address the same general target, namely to produce a complete atomistic picture of the kinetics of a chemical process. It is the purpose of this overview to categorize the problems that should be targeted and to identify the principal components and challenges of automated exploration machines so that the various existing approaches and future developments can be compared based on well-defined conceptual principles.

Towards theoretical spectroscopy with error bars: systematic quantification of the structural sensitivity of calculated spectra

Molecular spectra calculated with quantum-chemical methods are subject to a number of uncertainties (e.g., errors introduced by the computational methodology) that hamper the direct comparison of experiment and computation. Judging these uncertainties is crucial for drawing reliable conclusions from the interplay of experimental and theoretical spectroscopy, but largely relies on subjective judgment. Here, we explore the application of methods from uncertainty quantification to theoretical spectroscopy, with the ultimate goal of providing systematic error bars for calculated spectra. As a first target, we consider distortions of the underlying molecular structure as one important source of uncertainty. We show that by performing a principal component analysis, the most influential collective distortions can be identified, which allows for the construction of surrogate models that are amenable to a statistical analysis of the propagation of uncertainties in the molecular structure to uncertainties in the calculated spectrum. This is applied to the calculation of X-ray emission spectra of iron carbonyl complexes, of the electronic excitation spectrum of a coumarin dye, and of the infrared spectrum of alanine. We show that with our approach it becomes possible to obtain error bars for calculated spectra that account for uncertainties in the molecular structure. This is an important first step towards systematically quantifying other relevant sources of uncertainty in theoretical spectroscopy.

Uncertainty quantification is applied in theoretical spectroscopy to obtain error bars accounting for the structural sensitivity of calculated spectra.

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.

Designing in the Face of Uncertainty: Exploiting Electronic Structure and Machine Learning Models for Discovery in Inorganic Chemistry

Recent transformative advances in computing power and algorithms have made computational chemistry central to the discovery and design of new molecules and materials. First-principles simulations are increasingly accurate and applicable to large systems with the speed needed for high-throughput computational screening. Despite these strides, the combinatorial challenges associated with the vastness of chemical space mean that more than just fast and accurate computational tools are needed for accelerated chemical discovery. In transition-metal chemistry and catalysis, unique challenges arise. The variable spin, oxidation state, and coordination environments favored by elements with well-localized d or f electrons provide great opportunity for tailoring properties in catalytic or functional (e.g., magnetic) materials but also add layers of uncertainty to any design strategy. We outline five key mandates for realizing computationally driven accelerated discovery in inorganic chemistry: (i) fully automated simulation of new compounds, (ii) knowledge of prediction sensitivity or accuracy, (iii) faster-than-fast property prediction methods, (iv) maps for rapid chemical space traversal, and (v) a means to reveal design rules on the kilocompound scale. Through case studies in open-shell transition-metal chemistry, we describe how advances in methodology and software in each of these areas bring about new chemical insights. We conclude with our outlook on the next steps in this process toward realizing fully autonomous discovery in inorganic chemistry using computational chemistry.

Bayesian machine learning for quantum molecular dynamics

This article discusses applications of Bayesian machine learning for quantum molecular dynamics.

The parameter uncertainty inflation fallacy

Statistical estimation of the prediction uncertainty of physical models is typically hindered by the inadequacy of these models due to various approximations they are built upon. The prediction errors caused by model inadequacy can be handled either by correcting the model's results or by adapting the model's parameter uncertainty to generate prediction uncertainties representative, in a way to be defined, of model inadequacy errors. The main advantage of the latter approach (thereafter called PUI, for Parameter Uncertainty Inflation) is its transferability to the prediction of other quantities of interest based on the same parameters. A critical review of implementations of PUI in several areas of computational chemistry shows that it is biased, in the sense that it does not produce prediction uncertainty bands conforming to model inadequacy errors.

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.

The Matter Simulation (R)evolution

To date, the program for the development of methods and models for atomistic and continuum simulation directed toward chemicals and materials has reached an incredible degree of sophistication and maturity. Currently, one can witness an increasingly rapid emergence of advances in computing, artificial intelligence, and robotics. This drives us to consider the future of computer simulation of matter from the molecular to the human length and time scales in a radical way that deliberately dares to go beyond the foreseeable next steps in any given discipline. This perspective article presents a view on this future development that we believe is likely to become a reality during our lifetime.