Uncertainty quantification for quantum chemical models of complex reaction networks

Chemical Reaction Networks from Scratch with Reaction Prediction and Kinetics-Guided Exploration

Algorithmic reaction explorations based on transition state searches can now routinely predict relatively short reaction sequences involving small molecules. However, applying these algorithms to deeper chemical reaction network (CRN) exploration still requires the development of more efficient and accurate exploration policies. Here, an exploration algorithm, which we name yet another kinetic strategy (YAKS), is demonstrated that uses microkinetic simulations of the nascent network to achieve cost-effective, deep network exploration. Key features of the algorithm are the automatic incorporation of bimolecular reactions between network intermediates, compatibility with short-lived but kinetically important species, and incorporation of rate uncertainty into the exploration policy. In validation case studies of glucose pyrolysis, the algorithm rediscovers reaction pathways previously discovered by heuristic exploration policies and elucidates new reaction pathways for experimentally obtained products. The resulting CRN is the first to connect all major experimental pyrolysis products to glucose. Additional case studies are presented that investigate the role of reaction rules, rate uncertainty, and bimolecular reactions. These case studies show that naïve exponential growth estimates can vastly overestimate the actual number of kinetically relevant pathways in the physical reaction networks. In light of this, further improvements in exploration policies and reaction prediction algorithms make it feasible that CRNs might soon be routinely predictable in some contexts.

The second wave of formose research

This review article highlights key developments in the second wave of formose research (from approximately 2000), summarizing approximately 100 relevant studies. Section 1 introduces the basics of formose reaction and its historical context. Section 2 provides a brief overview of the pioneering works from the first wave of formose research (from 1970 to 1990). Section 3 then overviews the second wave of formose research, in which formose reactions under various conditions, mechanistic studies of the formose reaction, formose reactions and the origin of life, and applications of formose reactions are described. Finally, Section 4 offers summary and outlook.

Uncertainty-Aware First-Principles Exploration of Chemical Reaction Networks

Exploring large chemical reaction networks with automated exploration approaches and accurate quantum chemical methods can require prohibitively large computational resources. Here, we present an automated exploration approach that focuses on the kinetically relevant part of the reaction network by interweaving (i) large-scale exploration of chemical reactions, (ii) identification of kinetically relevant parts of the reaction network through microkinetic modeling, (iii) quantification and propagation of uncertainties, and (iv) reaction network refinement. Such an uncertainty-aware exploration of kinetically relevant parts of a reaction network with automated accuracy improvement has not been demonstrated before in a fully quantum mechanical approach. Uncertainties are identified by local or global sensitivity analysis. The network is refined in a rolling fashion during the exploration. Moreover, the uncertainties are considered during kinetically steering of a rolling reaction network exploration. We demonstrate our approach for Eschenmoser-Claisen rearrangement reactions. The sensitivity analysis identifies that only a small number of reactions and compounds are essential for describing the kinetics reliably, resulting in efficient explorations without sacrificing accuracy and without requiring prior knowledge about the chemistry unfolding.

A human-machine interface for automatic exploration of chemical reaction networks

Autonomous reaction network exploration algorithms offer a systematic approach to explore mechanisms of complex chemical processes. However, the resulting reaction networks are so vast that an exploration of all potentially accessible intermediates is computationally too demanding. This renders brute-force explorations unfeasible, while explorations with completely pre-defined intermediates or hard-wired chemical constraints, such as element-specific coordination numbers, are not flexible enough for complex chemical systems. Here, we introduce a STEERING WHEEL to guide an otherwise unbiased automated exploration. The STEERING WHEEL algorithm is intuitive, generally applicable, and enables one to focus on specific regions of an emerging network. It also allows for guiding automated data generation in the context of mechanism exploration, catalyst design, and other chemical optimization challenges. The algorithm is demonstrated for reaction mechanism elucidation of transition metal catalysts. We highlight how to explore catalytic cycles in a systematic and reproducible way. The exploration objectives are fully adjustable, allowing one to harness the STEERING WHEEL for both structure-specific (accurate) calculations as well as for broad high-throughput screening of possible reaction intermediates.

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.

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.

Pathfinder─Navigating and Analyzing Chemical Reaction Networks with an Efficient Graph-Based Approach

While the field of first-principles explorations into chemical reaction space has been continuously growing, the development of strategies for analyzing resulting chemical reaction networks (CRNs) is lagging behind. A CRN consists of compounds linked by reactions. Analyzing how these compounds are transformed into one another based on kinetic modeling is a nontrivial task. Here, we present the graph-optimization-driven algorithm and program Pathfinder to allow for such an analysis of a CRN. The CRN for this work has been obtained with our open-source Chemoton reaction network exploration software. Chemoton probes reactive combinations of compounds for elementary steps and sorts them into reactions. By encoding these reactions of the CRN as a graph consisting of compound and reaction vertices and adding information about activation barriers as well as required reagents to the edges of the graph yields a complete graph-theoretical representation of the CRN. Since the probabilities of the formation of compounds depend on the starting conditions, the consumption of any compound during a reaction must be accounted for to reflect the availability of reagents. To account for this, we introduce compound costs to reflect compound availability. Simultaneously, the determined compound costs rank the compounds in the CRN in terms of their probability to be formed. This ranking then allows us to probe easily accessible compounds in the CRN first for further explorations into yet unexplored terrain. We first illustrate the working principle on an abstract small CRN. Afterward, Pathfinder is demonstrated in the example of the disproportionation of iodine with water and the comproportionation of iodic acid and hydrogen iodide. Both processes are analyzed within the same CRN, which we construct with our autonomous first-principles CRN exploration software Chemoton [Unsleber, J. P.; J. Chem. Theory Comput. 2022, 18, 5393-5409] guided by Pathfinder.

Deep Learning-Based Increment Theory for Formation Enthalpy Predictions

Machine learning predictions of molecular thermochemistry, such as formation enthalpy, have been limited for large and complicated species because of the lack of available training data. Such predictions would be important in the prediction of reaction thermodynamics and the construction of kinetic models. Herein, we introduce a graph-based deep learning approach that can separately learn the enthalpy contribution of each atom in its local environment with the effect of the overall molecular structure taken into account. Because this approach follows the additivity scheme of increment theory, it can be generalized to larger and more complicated species not present in the training data. By training the model on molecules with up to 11 heavy atoms, it can predict the formation enthalpy of testing molecules with up to 42 heavy atoms with a mean absolute error of 2 kcal/mol, which is less than half of the error of the conventional increment theory. We expect that this approach will also enable rapid prediction of other extensive properties of large molecules that are difficult to derive from experiments or ab initio calculation.

Chemoton 2.0: Autonomous Exploration of Chemical Reaction Networks

Fueled by advances in hardware and algorithm design, large-scale automated explorations of chemical reaction space have become possible. Here, we present our approach to an open-source, extensible framework for explorations of chemical reaction mechanisms based on the first-principles of quantum mechanics. It is intended to facilitate reaction network explorations for diverse chemical problems with a wide range of goals such as mechanism elucidation, reaction path optimization, retrosynthetic path validation, reagent design, and microkinetic modeling. The stringent first-principles basis of all algorithms in our framework is key for the general applicability that avoids any restrictions to specific chemical systems. Such an agile framework requires multiple specialized software components of which we present three modules in this work. The key module, Chemoton, drives the exploration of reaction networks. For the exploration itself, we introduce two new algorithms for elementary-step searches that are based on Newton trajectories. The performance of these algorithms is assessed for a variety of reactions characterized by a broad chemical diversity in terms of bonding patterns and chemical elements. Chemoton successfully recovers the vast majority of these. We provide the resulting data, including large numbers of reactions that were not included in our reference set, to be used as a starting point for further explorations and for future reference.

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.

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.

New Tools for Taming Complex Reaction Networks: The Unimolecular Decomposition of Indole Revisited

The level of detail attained in the computational description of reaction mechanisms can be vastly improved through tools for automated chemical space exploration, particularly for systems of small to medium size. Under this approach, the unimolecular decomposition landscape for indole was explored through the automated reaction mechanism discovery program AutoMeKin. Nevertheless, the sheer complexity of the obtained mechanisms might be a hindrance regarding their chemical interpretation. In this spirit, the new Python library amk-tools has been designed to read and manipulate complex reaction networks, greatly simplifying their overall analysis. The package provides interactive dashboards featuring visualizations of the network, the three-dimensional (3D) molecular structures and vibrational normal modes of all chemical species, and the corresponding energy profiles for selected pathways. The combination of the joined mechanism generation and postprocessing workflow with the rich chemistry of indole decomposition enabled us to find new details of the reaction (obtained at the CCSD(T)/aug-cc-pVTZ//M06-2X/MG3S level of theory) that were not reported before: (i) 16 pathways leading to the formation of HCN and NH₃ (via amino radical); (ii) a barrierless reaction between methylene radical and phenyl isocyanide, which might be an operative mechanism under the conditions of the interstellar medium; and (iii) reaction channels leading to both hydrogen cyanide and hydrogen isocyanide, of potential astrochemical interest as the computed HNC/HCN ratios greatly exceed the calculated equilibrium value at very low temperatures. The reported reaction networks can be very valuable to supplement databases of kinetic data, which is of remarkable interest for pyrolysis and astrochemical studies.

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.

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SUPPLEMENTARY INFORMATION

The online version contains supplementary material available at 10.1007/s11244-021-01543-9.

AutoMeKin2021: An open-source program for automated reaction discovery

AutoMeKin2021 is an updated version of tsscds2018, a program for the automated discovery of reaction mechanisms (J. Comput. Chem. 2018, 39, 1922). This release features a number of new capabilities: rare-event molecular dynamics simulations to enhance reaction discovery, extension of the original search algorithm to study van der Waals complexes, use of chemical knowledge, a new search algorithm based on bond-order time series analysis, statistics of the chemical reaction networks, a web application to submit jobs, and other features. The source code, manual, installation instructions and the website link are available at: https://rxnkin.usc.es/index.php/AutoMeKin.

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 %.

Perspective on integrating machine learning into computational chemistry and materials science

Machine learning (ML) methods are being used in almost every conceivable area of electronic structure theory and molecular simulation. In particular, ML has become firmly established in the construction of high-dimensional interatomic potentials. Not a day goes by without another proof of principle being published on how ML methods can represent and predict quantum mechanical properties-be they observable, such as molecular polarizabilities, or not, such as atomic charges. As ML is becoming pervasive in electronic structure theory and molecular simulation, we provide an overview of how atomistic computational modeling is being transformed by the incorporation of ML approaches. From the perspective of the practitioner in the field, we assess how common workflows to predict structure, dynamics, and spectroscopy are affected by ML. Finally, we discuss how a tighter and lasting integration of ML methods with computational chemistry and materials science can be achieved and what it will mean for research practice, software development, and postgraduate training.

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'.

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.

The electrostatic potential as a descriptor for the protonation propensity in automated exploration of reaction mechanisms

We discuss the possibility of exploiting local minima of the molecular electrostatic potential for locating protonation sites in molecules in a fully automated manner. We implement and apply this concept to exploring the mechanism of proton reduction catalyzed by a hydrogenase model complex [Orthaber et al., Dalton Trans., 2014, 43, 4537]. A large number of distinct structures arising already in the early stages of the hydrogen evolution mechanism demonstrates the need for reliable, automated algorithms for the thorough analysis of catalytic processes.

Automated exploration of the low-energy chemical space with fast quantum chemical methods

We propose and discuss an efficient scheme for the in silico sampling for parts of the molecular chemical space by semiempirical tight-binding methods combined with a meta-dynamics driven search algorithm. The focus of this work is set on the generation of proper thermodynamic ensembles at a quantum chemical level for conformers, but similar procedures for protonation states, tautomerism and non-covalent complex geometries are also discussed. The conformational ensembles consisting of all significantly populated minimum energy structures normally form the basis of further, mostly DFT computational work, such as the calculation of spectra or macroscopic properties. By using basic quantum chemical methods, electronic effects or possible bond breaking/formation are accounted for and a very reasonable initial energetic ranking of the candidate structures is obtained. Due to the huge computational speedup gained by the fast low-cost quantum chemical methods, overall short computation times even for systems with hundreds of atoms (typically drug-sized molecules) are achieved. Furthermore, specialized applications, such as sampling with implicit solvation models or constrained conformational sampling for transition-states, metal-, surface-, or noncovalently bound complexes are discussed, opening many possible applications in modern computational chemistry and drug discovery. The procedures have been implemented in a freely available computer code called CREST, that makes use of the fast and reliable GFNn-xTB methods.

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.

A Trajectory-Based Method to Explore Reaction Mechanisms

The tsscds method, recently developed in our group, discovers chemical reaction mechanisms with minimal human intervention. It employs accelerated molecular dynamics, spectral graph theory, statistical rate theory and stochastic simulations to uncover chemical reaction paths and to solve the kinetics at the experimental conditions. In the present review, its application to solve mechanistic/kinetics problems in different research areas will be presented. Examples will be given of reactions involved in photodissociation dynamics, mass spectrometry, combustion chemistry and organometallic catalysis. Some planned improvements will also be described.

Automated Chemical Kinetic Modeling via Hybrid Reactive Molecular Dynamics and Quantum Chemistry Simulations

An automated scheme for obtaining chemical kinetic models from scratch using reactive molecular dynamics and quantum chemistry simulations is presented. This methodology combines the phase space sampling of reactive molecular dynamics with the thermochemistry and kinetics prediction capabilities of quantum mechanics. This scheme provides the NASA polynomial and modified Arrhenius equation parameters for all species and reactions that are observed during the simulation and supplies them in the ChemKin format. The ab initio level of theory for predictions is easily exchangeable, and the presently used G3MP2 level of theory is found to reliably reproduce hydrogen and methane oxidation thermochemistry and kinetics data. Chemical kinetic models obtained with this approach are ready to use for, e.g., ignition delay time simulations, as shown for hydrogen combustion. The presented extension of the ChemTraYzer approach can be used as a basis for methodological advancement of chemical kinetic modeling schemes and as a black-box approach to generate chemical kinetic models.

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.

Molecular Modelling for Reactor Design

Chemical reactor modelling based on insights and data on a molecular level has become reality over the last few years. Multiscale models describing elementary reaction steps and full microkinetic schemes, pore structures, multicomponent adsorption and diffusion inside pores, and entire reactors have been presented. Quantum mechanical (QM) approaches, molecular simulations (Monte Carlo and molecular dynamics), and continuum equations have been employed for this purpose. Some recent developments in these approaches are presented, in particular time-dependent QM methods, calculation of van der Waals forces, new approaches for force field generation, automatic setup of reaction schemes, and pore modelling. Multiscale simulations are discussed. Applications of these approaches to heterogeneous catalysis are demonstrated for examples that have found growing interest over the last few years, such as metal-support interactions, influence of pore geometry on reactions, noncovalent bonding, reaction dynamics, dynamic changes in catalyst nanoparticle structure, electrocatalysis, solvent effects in catalysis, and multiscale modelling.

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.

Calculation of Ligand Dissociation Energies in Large Transition-Metal Complexes

The accurate calculation of ligand dissociation (or equivalently, ligand binding) energies is crucial for computational coordination chemistry. Despite its importance, obtaining accurate ab initio reference data is difficult, and density-functional methods of uncertain reliability are chosen for feasibility reasons. Here, we consider advanced coupled-cluster and multiconfigurational approaches to reinvestigate our WCCR10 set of 10 gas-phase ligand dissociation energies [ J. Chem. Theory Comput. 2014, 10, 3092]. We assess the potential multiconfigurational character of all molecules involved in these reactions with a multireference diagnostic [ Mol. Phys. 2017, 115, 2110] in order to determine where single-reference coupled-cluster approaches can be applied. For some reactions of the WCCR10 set, large deviations of density-functional results including semiclassical dispersion corrections from experimental reference data had been observed. This puzzling observation deserves special attention here, and we tackle the issue (i) by comparing to ab initio data that comprise dispersion effects on a rigorous first-principles footing and (ii) by a comparison of density-functional approaches that model dispersion interactions in various ways. For two reactions, species exhibiting nonnegligible static electron correlation were identified. These two reactions represent hard problems for electronic structure methods and also for multireference perturbation theories. However, most of the ligand dissociation reactions in WCCR10 do not exhibit static electron correlation effects, and hence, we may choose standard single-reference coupled-cluster approaches to compare with density-functional methods. For WCCR10, the Minnesota M06-L functional yielded the smallest mean absolute deviation of 13.2 kJ mol^-1 out of all density functionals considered (PBE, BP86, BLYP, TPSS, M06-L, PBE0, B3LYP, TPSSh, and M06-2X) without additional dispersion corrections in comparison to the coupled-cluster results, and the PBE0-D3 functional produced the overall smallest mean absolute deviation of 4.3 kJ mol^-1. The agreement of density-functional results with coupled-cluster data increases significantly upon inclusion of any type of dispersion correction. It is important to emphasize that different density-functional schemes available for this purpose perform equally well. The coupled-cluster dissociation energies, however, deviate from experimental results on average by 30.3 kJ mol^-1. Possible reasons for these deviations are discussed.

Efficient prediction of reaction paths through molecular graph and reaction network analysis

Despite remarkable advances in computational chemistry, prediction of reaction mechanisms is still challenging, because investigating all possible reaction pathways is computationally prohibitive due to the high complexity of chemical space. A feasible strategy for efficient prediction is to utilize chemical heuristics. Here, we propose a novel approach to rapidly search reaction paths in a fully automated fashion by combining chemical theory and heuristics. A key idea of our method is to extract a minimal reaction network composed of only favorable reaction pathways from the complex chemical space through molecular graph and reaction network analysis. This can be done very efficiently by exploring the routes connecting reactants and products with minimum dissociation and formation of bonds. Finally, the resulting minimal network is subjected to quantum chemical calculations to determine kinetically the most favorable reaction path at the predictable accuracy. As example studies, our method was able to successfully find the accepted mechanisms of Claisen ester condensation and cobalt-catalyzed hydroformylation reactions.

Ab initio instanton rate theory made efficient using Gaussian process regression

Ab initio instanton rate theory is a computational method for rigorously including tunnelling effects into the calculations of chemical reaction rates based on a potential-energy surface computed on the fly from electronic-structure theory. This approach is necessary to extend conventional transition-state theory into the deep-tunnelling regime, but it is also more computationally expensive as it requires many more ab initio calculations. We propose an approach which uses Gaussian process regression to fit the potential-energy surface locally around the dominant tunnelling pathway. The method can be converged to give the same result as from an on-the-fly ab initio instanton calculation but it requires far fewer electronic-structure calculations. This makes it a practical approach for obtaining accurate rate constants based on high-level electronic-structure methods. We show fast convergence to reproduce benchmark H + CH4 results and evaluate new low-temperature rates of H + C2H6 in full dimensionality at a UCCSD(T)-F12b/cc-pVTZ-F12 level.

Kinetic isotope effects and how to describe them

We review several methods for computing kinetic isotope effects in chemical reactions including semiclassical and quantum instanton theory. These methods describe both the quantization of vibrational modes as well as tunneling and are applied to the ⋅H + H2 and ⋅H + CH4 reactions. The absolute rate constants computed with the semiclassical instanton method both using on-the-fly electronic structure calculations and fitted potential-energy surfaces are also compared directly with exact quantum dynamics results. The error inherent in the instanton approximation is found to be relatively small and similar in magnitude to that introduced by using fitted surfaces. The kinetic isotope effect computed by the quantum instanton is even more accurate, and although it is computationally more expensive, the efficiency can be improved by path-integral acceleration techniques. We also test a simple approach for designing potential-energy surfaces for the example of proton transfer in malonaldehyde. The tunneling splittings are computed, and although they are found to deviate from experimental results, the ratio of the splitting to that of an isotopically substituted form is in much better agreement. We discuss the strengths and limitations of the potential-energy surface and based on our findings suggest ways in which it can be improved.