Nudged Elastic Band Calculations Accelerated with Gaussian Process Regression Based on Inverse Interatomic Distances

Flat-Bottom Elastic Network Model for Generating Improved Plausible Reaction Paths

Rapid generation of a plausible reaction path connecting a given reactant and product in advance is crucial for the efficient computation of precise reaction paths or transition states. We propose a computationally efficient potential energy based on the molecular structure to generate such paths. This potential energy has a flat bottom consisting of structures without atomic collisions while preserving nonreactive chemical bonds, bond angles, and partial planar structures. By combining this potential energy with the direct MaxFlux method, a recently developed reaction-path/transition-state search method, we can find the shortest plausible path passing within the bottom. Numerical results show that this combination yields lower energy paths compared to the paths obtained by the well-known image-dependent pair potential. We also theoretically investigate the differences between these two potential energies. The proposed potential energy and path generation routine are implemented in our Python version of the direct MaxFlux method, available on GitHub.

Physical Prior Mean Function-Driven Gaussian Processes Search for Minimum-Energy Reaction Paths with a Climbing-Image Nudged Elastic Band: A General Method for Gas-Phase, Interfacial, and Bulk-Phase Reactions

The climbing-image nudged elastic band (CI-NEB) method serves as an indispensable tool for computational chemists, offering insight into minimum-energy reaction paths (MEPs) by delineating both transition states (TSs) and intermediate nonstationary structures along reaction coordinates. However, executing CI-NEB calculations for reactions with extensive reaction coordinate spans necessitates a large number of images to ensure a reliable convergence of the MEPs and TS structures, presenting a computationally demanding optimization challenge, even with mildly costly electronic-structure methods. In this study, we advocate for the utilization of physically inspired prior mean function-based Gaussian processes (GPs) to expedite MEP exploration and TS optimization via the CI-NEB method. By incorporating reliable prior physical approximations into potential energy surface (PES) modeling, we demonstrate enhanced efficiency in multidimensional CI-NEB optimization with surrogate-based optimizers. Our physically informed GP approach not only outperforms traditional nonsurrogate-based optimizers in optimization efficiency but also on-the-fly learns the reaction path valley during optimization, culminating in significant advancements. The surrogate PES derived from our optimization exhibits high accuracy compared to true PES references, aligning with our emphasis on leveraging reliable physical priors for robust and efficient posterior mean learning in GPs. Through a systematic benchmark study encompassing various reaction pathways, including gas-phase, bulk-phase, and interfacial/surface reactions, our physical GPs consistently demonstrate superior efficiency and reliability. For instance, they outperform the popular fast inertial relaxation engine optimizer by approximately a factor of 10, showcasing their versatility and efficacy in exploring reaction mechanisms and surface reaction PESs.

Acceleration of hydrolytic ring opening of N7-alkylguanine by the terminal carbamoyl group of glycidamide

Hydrolytic ring opening of guanine N7-adducts with compounds containing an oxacyclopropane ring, namely glycidamide, glycidol and 1,2-epoxybutane, was analyzed, and the reaction of the glycidamide adduct was the fastest. The differences in the reaction rates were confirmed by theoretical calculations.

Shear-activation of mechanochemical reactions through molecular deformation

Mechanical stress can directly activate chemical reactions by reducing the reaction energy barrier. A possible mechanism of such mechanochemical activation is structural deformation of the reactant species. However, the effect of deformation on the reaction energetics is unclear, especially, for shear stress-driven reactions. Here, we investigated shear stress-driven oligomerization reactions of cyclohexene on silica using a combination of reactive molecular dynamics simulations and ball-on-flat tribometer experiments. Both simulations and experiments captured an exponential increase in reaction yield with shear stress. Elemental analysis of ball-on-flat reaction products revealed the presence of oxygen in the polymers, a trend corroborated by the simulations, highlighting the critical role of surface oxygen atoms in oligomerization reactions. Structural analysis of the reacting molecules in simulations indicated the reactants were deformed just before a reaction occurred. Quantitative evidence of shear-induced deformation was established by comparing bond lengths in cyclohexene molecules in equilibrium and prior to reactions. Nudged elastic band calculations showed that the deformation had a small effect on the transition state energy but notably increased the reactant state energy, ultimately leading to a reduction in the energy barrier. Finally, a quantitative relationship was developed between molecular deformation and energy barrier reduction by mechanical stress.

Curvature-weighted nudged elastic band method using the Riemann curvature

The nudged elastic band (NEB) method is utilized to find reaction paths (RPs) using discretized intermediate structures called "images" between a reactant and a product. In fact, NEB calculations do not always converge because of the bent of RPs. Mathematically, more images are needed for complex curves, and here, we focused on the curvature. In this study, we propose a new method for calculating the curvature of the RPs as well as a method for weighting the spring constant of the NEB with the curvature, which we named the curvature weighted NEB (CW-NEB) method. In addition, we will propose the CW-NEB method with the climbing image (CI) method (CW-CI-NEB). To show the efficiency of our method, calculations for the ene-reaction of the CW-CI-NEB method were performed and compared with those of the CI-NEB methods. The CW-CI-NEB methods converged in shorter iteration times than the CI-NEB calculation, which was found by gathering the bent of RPs.

De novo design of a polycarbonate hydrolase

Enzymatic degradation of plastics is currently limited to the use of engineered natural enzymes. As of yet, all engineering approaches applied to plastic degrading enzymes retain the natural $\alpha /\beta $-fold. While mutations can be used to increase thermostability, an inherent maximum likely exists for the $\alpha /\beta $-fold. It is thus of interest to introduce catalytic activity toward plastics in a different protein fold to escape the sequence space of plastic degrading enzymes. Here, a method for designing highly thermostable enzymes that can degrade plastics is described. With the help of Rosetta an active site catalysing the hydrolysis of polycarbonate is introduced into a set of thermostable scaffolds. Through computational evaluation, a potential PCase was selected and produced recombinantly in Escherichia coli. Thermal analysis suggests that the design has a melting temperature of >95$^{\circ }$C. Activity toward polycarbonate was confirmed using atomic force spectroscopy (AFM), proving the successful design of a PCase.

Guided discovery of chemical reaction pathways with imposed activation

Computational power and quantum chemical methods have improved immensely since computers were first applied to the study of reactivity, but the de novo prediction of chemical reactions has remained challenging. We show that complex reaction pathways can be efficiently predicted in a guided manner using chemical activation imposed by geometrical constraints of specific reactive modes, which we term imposed activation (IACTA). Our approach is demonstrated on realistic and challenging chemistry, such as a triple cyclization cascade involved in the total synthesis of a natural product, a water-mediated Michael addition, and several oxidative addition reactions of complex drug-like molecules. Notably and in contrast with traditional hand-guided computational chemistry calculations, our method requires minimal human involvement and no prior knowledge of the products or the associated mechanisms. We believe that IACTA will be a transformational tool to screen for chemical reactivity and to study both by-product formation and decomposition pathways in a guided way.

Perspective: Path Sampling Methods Applied to Enzymatic Catalysis

This Perspective reviews the use of Transition Path Sampling methods to study enzymatically catalyzed chemical reactions. First applied by our group to an enzymatic reaction over 15 years ago, the method has uncovered basic principles in enzymatic catalysis such as the protein promoting vibration, and it has also helped harmonize such ideas as electrostatic preorganization with dynamic views of enzyme function. It is now being used to help uncover principles of protein design necessary to artificial enzyme creation.

Facilitating ab initio QM/MM free energy simulations by Gaussian process regression with derivative observations

In combined quantum mechanical and molecular mechanical (QM/MM) free energy simulations, how to synthesize the accuracy of ab initio (AI) methods with the speed of semiempirical (SE) methods for a cost-effective QM treatment remains a long-standing challenge. In this work, we present a machine-learning-facilitated method for obtaining AI/MM-quality free energy profiles through efficient SE/MM simulations. In particular, we use Gaussian process regression (GPR) to learn the energy and force corrections needed for SE/MM to match with AI/MM results during molecular dynamics simulations. Force matching is enabled in our model by including energy derivatives into the observational targets through the extended-kernel formalism. We demonstrate the effectiveness of this method on the solution-phase SN2 Menshutkin reaction using AM1/MM and B3LYP/6-31+G(d,p)/MM as the base and target levels, respectively. Trained on only 80 configurations sampled along the minimum free energy path (MFEP), the resulting GPR model reduces the average energy error in AM1/MM from 18.2 to 5.8 kcal mol-1 for the 4000-sample testing set with the average force error on the QM atoms decreased from 14.6 to 3.7 kcal mol-1 Å-1. Free energy sampling with the GPR corrections applied (AM1-GPR/MM) produces a free energy barrier of 14.4 kcal mol-1 and a reaction free energy of -34.1 kcal mol-1, in closer agreement with the AI/MM benchmarks and experimental results.

Expanding the Material Search Space for Multivalent Cathodes

Multivalent batteries are an energy storage technology with the potential to surpass lithium-ion batteries; however, their performance have been limited by the low voltages and poor solid-state ionic mobility of available cathodes. A computational screening approach to identify high-performance multivalent intercalation cathodes among materials that do not contain the working ion of interest has been developed, which greatly expands the search space that can be considered for material discovery. This approach has been applied to magnesium cathodes as a proof of concept, and four resulting candidate materials [NASICON V₂(PO₄)₃, birnessite NaMn₄O₈, tavorite MnPO₄F, and spinel MnO₂] are discussed in further detail. In examining the ion migration environment and associated Mg²⁺ migration energy in these materials, local energy maxima are found to correspond with pathway positions where Mg²⁺ passes through a plane of anion atoms. While previous studies have established the influence of local coordination on multivalent ion mobility, these results suggest that considering both the type of the local bonding environment and available free volume for the mobile ion along its migration pathway can be significant for improving solid-state mobility.

Fast atomic structure optimization with on-the-fly sparse Gaussian process potentials. JOURNAL OF PHYSICS. CONDENSED MATTER : AN INSTITUTE OF PHYSICS JOURNAL 2022;34:344007. [PMID: 35675808 DOI: 10.1088/1361-648x/ac76ff]

We apply on-the-fly machine learning potentials (MLPs) using the sparse Gaussian process regression (SGPR) algorithm for fast optimization of atomic structures. Great acceleration is achieved even in the context of a single local optimization. Although for finding the exact local minimum, due to limited accuracy of MLPs, switching to another algorithm may be needed. For random gold clusters, the forces are reduced to ∼0.1 eV Å^-1within less than ten first-principles (FP) calculations. Because of highly transferable MLPs, this algorithm is specially suitable for global optimization methods such as random or evolutionary structure searching or basin hopping. This is demonstrated by sequential optimization of random gold clusters for which, after only a few optimizations, FP calculations were rarely needed.

Geometry meta-optimization

Recent work has demonstrated the promise of using machine-learned surrogates, in particular, Gaussian process (GP) surrogates, in reducing the number of electronic structure calculations (ESCs) needed to perform surrogate model based (SMB) geometry optimization. In this paper, we study geometry meta-optimization with GP surrogates where a SMB optimizer additionally learns from its past "experience" performing geometry optimization. To validate this idea, we start with the simplest setting where a geometry meta-optimizer learns from previous optimizations of the same molecule with different initial-guess geometries. We give empirical evidence that geometry meta-optimization with GP surrogates is effective and requires less tuning compared to SMB optimization with GP surrogates on the ANI-1 dataset of off-equilibrium initial structures of small organic molecules. Unlike SMB optimization where a surrogate should be immediately useful for optimizing a given geometry, a surrogate in geometry meta-optimization has more flexibility because it can distribute its ESC savings across a set of geometries. Indeed, we find that GP surrogates that preserve rotational invariance provide increased marginal ESC savings across geometries. As a more stringent test, we also apply geometry meta-optimization to conformational search on a hand-constructed dataset of hydrocarbons and alcohols. We observe that while SMB optimization and geometry meta-optimization do save on ESCs, they also tend to miss higher energy conformers compared to standard geometry optimization. We believe that further research into characterizing the divergence between GP surrogates and potential energy surfaces is critical not only for advancing geometry meta-optimization but also for exploring the potential of machine-learned surrogates in geometry optimization in general.

Progress towards machine learning reaction rate constants

Quantum and classical reaction rate constant calculations come at the cost of exploring potential energy surfaces. Due to the "curse of dimensionality", their evaluation quickly becomes unfeasible as the system size grows. Machine learning algorithms can accelerate the calculation of reaction rate constants by predicting them using low cost input features. In this perspective, we briefly introduce supervised machine learning algorithms in the context of reaction rate constant prediction. We discuss existing and recently created kinetic datasets and input feature representations as well as the use and design of machine learning algorithms to predict reaction rate constants or quantities required for their computation. Amongst these, we first describe the use of machine learning to predict activation, reaction, solvation and dissociation energies. We then look at the use of machine learning to predict reactive force field parameters, reaction rate constants as well as to help accelerate the search for minimum energy paths. Lastly, we provide an outlook on areas which have yet to be explored so as to improve and evaluate the use of machine learning algorithms for chemical reaction rate constants.

Structural Elucidation, Aggregation, and Dynamic Behaviour of N,N,N,N-Copper(I) Schiff Base Complexes in Solid and in Solution: A Combined NMR, X-ray Spectroscopic and Crystallographic Investigation

A series of Cu(I) complexes of bidentate or tetradentate Schiff base ligands bearing either 1-H-imidazole or pyridine moieties were synthesized. The complexes were studied by a combination of NMR and X-ray spectroscopic techniques. The differences between the imidazole- and pyridine-based ligands were examined by 1H, 13C and 15N NMR spectroscopy. The magnitude of the 15Nimine coordination shifts was found to be strongly affected by the nature of the heterocycle in the complexes. These trends showed good correlation with the obtained Cu-Nimine bond lengths from single-crystal X-ray diffraction measurements. Variable-temperature NMR experiments, in combination with diffusion ordered spectroscopy (DOSY) revealed that one of the complexes underwent a temperature-dependent interconversion between a monomer, a dimer and a higher aggregate. The complexes bearing tetradentate imidazole ligands were further studied using Cu K-edge XAS and VtC XES, where DFT-assisted assignment of spectral features suggested that these complexes may form polynuclear oligomers in solid state. Additionally, the Cu(II) analogue of one of the complexes was incorporated into a metal-organic framework (MOF) as a way to obtain discrete, mononuclear complexes in the solid state.

First-principles materials simulation and design for alkali and alkaline metal ion batteries accelerated by machine learning

The challenge of regeneration of batteries requires a performance improvement in the alkali/alkaline metal ion battery (AMIB) materials, whereas the traditional research paradigm fully based on experiments and theoretical simulations needs massive research and development investment. During the last decade, machine learning (ML) has made breakthroughs in many complex disciplines, which testifies to their high processing speed and ability to capture relationships. Inspired by these achievements, ML has also been introduced to bring a new paradigm for shortening the development of AMIB materials. In this Perspective, the focus will be on how this new ML technology solves the key problems of redox potentials, ionic conductivity and stability parameters in first-principles materials' simulation and design for AMIBs. It is found that ML not only accelerates the property prediction, but also gives physicochemical insights into AMIB materials' design. In addition, the final part of this paper summarizes current achievements and looks forward to the progress of a novel paradigm in direct/inverse design with the increasing number of databases, skills, and ML technologies for AMIBs.

Gaussian Process Regression for Materials and Molecules

We provide an introduction to Gaussian process regression (GPR) machine-learning methods in computational materials science and chemistry. The focus of the present review is on the regression of atomistic properties: in particular, on the construction of interatomic potentials, or force fields, in the Gaussian Approximation Potential (GAP) framework; beyond this, we also discuss the fitting of arbitrary scalar, vectorial, and tensorial quantities. Methodological aspects of reference data generation, representation, and regression, as well as the question of how a data-driven model may be validated, are reviewed and critically discussed. A survey of applications to a variety of research questions in chemistry and materials science illustrates the rapid growth in the field. A vision is outlined for the development of the methodology in the years to come.

Machine Learning for Electronically Excited States of Molecules

Electronically excited states of molecules are at the heart of photochemistry, photophysics, as well as photobiology and also play a role in material science. Their theoretical description requires highly accurate quantum chemical calculations, which are computationally expensive. In this review, we focus on not only how machine learning is employed to speed up such excited-state simulations but also how this branch of artificial intelligence can be used to advance this exciting research field in all its aspects. Discussed applications of machine learning for excited states include excited-state dynamics simulations, static calculations of absorption spectra, as well as many others. In order to put these studies into context, we discuss the promises and pitfalls of the involved machine learning techniques. Since the latter are mostly based on quantum chemistry calculations, we also provide a short introduction into excited-state electronic structure methods and approaches for nonadiabatic dynamics simulations and describe tricks and problems when using them in machine learning for excited states of molecules.

Combining Machine Learning and Computational Chemistry for Predictive Insights Into Chemical Systems

Machine learning models are poised to make a transformative impact on chemical sciences by dramatically accelerating computational algorithms and amplifying insights available from computational chemistry methods. However, achieving this requires a confluence and coaction of expertise in computer science and physical sciences. This Review is written for new and experienced researchers working at the intersection of both fields. We first provide concise tutorials of computational chemistry and machine learning methods, showing how insights involving both can be achieved. We follow with a critical review of noteworthy applications that demonstrate how computational chemistry and machine learning can be used together to provide insightful (and useful) predictions in molecular and materials modeling, retrosyntheses, catalysis, and drug design.

Computational Discovery of Transition-metal Complexes: From High-throughput Screening to Machine Learning

Transition-metal complexes are attractive targets for the design of catalysts and functional materials. The behavior of the metal-organic bond, while very tunable for achieving target properties, is challenging to predict and necessitates searching a wide and complex space to identify needles in haystacks for target applications. This review will focus on the techniques that make high-throughput search of transition-metal chemical space feasible for the discovery of complexes with desirable properties. The review will cover the development, promise, and limitations of "traditional" computational chemistry (i.e., force field, semiempirical, and density functional theory methods) as it pertains to data generation for inorganic molecular discovery. The review will also discuss the opportunities and limitations in leveraging experimental data sources. We will focus on how advances in statistical modeling, artificial intelligence, multiobjective optimization, and automation accelerate discovery of lead compounds and design rules. The overall objective of this review is to showcase how bringing together advances from diverse areas of computational chemistry and computer science have enabled the rapid uncovering of structure-property relationships in transition-metal chemistry. We aim to highlight how unique considerations in motifs of metal-organic bonding (e.g., variable spin and oxidation state, and bonding strength/nature) set them and their discovery apart from more commonly considered organic molecules. We will also highlight how uncertainty and relative data scarcity in transition-metal chemistry motivate specific developments in machine learning representations, model training, and in computational chemistry. Finally, we will conclude with an outlook of areas of opportunity for the accelerated discovery of transition-metal complexes.

Machine Learning for Electronically Excited States of Molecules

Electronically excited states of molecules are at the heart of photochemistry, photophysics, as well as photobiology and also play a role in material science. Their theoretical description requires highly accurate quantum chemical calculations, which are computationally expensive. In this review, we focus on not only how machine learning is employed to speed up such excited-state simulations but also how this branch of artificial intelligence can be used to advance this exciting research field in all its aspects. Discussed applications of machine learning for excited states include excited-state dynamics simulations, static calculations of absorption spectra, as well as many others. In order to put these studies into context, we discuss the promises and pitfalls of the involved machine learning techniques. Since the latter are mostly based on quantum chemistry calculations, we also provide a short introduction into excited-state electronic structure methods and approaches for nonadiabatic dynamics simulations and describe tricks and problems when using them in machine learning for excited states of molecules.

Geometry Optimization in Internal Coordinates Based on Gaussian Process Regression: Comparison of Two Approaches

Geometry optimization based on Gaussian process regression (GPR) was extended to internal coordinates. We used delocalized internal coordinates composed of distances and several types of angles and compared two methods of including them. In both cases, the GPR surrogate surface is trained on geometries in internal coordinates. In one case, it predicts the gradient in Cartesian coordinates and in the other, in internal coordinates. We tested both methods on a set of 30 small molecules and one larger Rh complex taken from the study of a catalytic mechanism. The former method is slightly more efficient, while the latter method is somewhat more robust. Both methods reduce the number of required optimization steps compared to GPR in Cartesian coordinates or the standard L-BFGS optimizer. We found it advantageous to use automatically adjusted hyperparameters to optimize them.

Perspective on computational reaction prediction using machine learning methods in heterogeneous catalysis

Heterogeneous catalysis plays a significant role in the modern chemical industry. Towards the rational design of novel catalysts, understanding reactions over surfaces is the most essential aspect. Typical industrial catalytic processes such as syngas conversion and methane utilisation can generate a large reaction network comprising thousands of intermediates and reaction pairs. This complexity not only arises from the permutation of transformations between species but also from the extra reaction channels offered by distinct surface sites. Despite the success in investigating surface reactions at the atomic scale, the huge computational expense of ab initio methods hinders the exploration of such complicated reaction networks. With the proliferation of catalysis studies, machine learning as an emerging tool can take advantage of the accumulated reaction data to emulate the output of ab initio methods towards swift reaction prediction. Here, we briefly summarise the conventional workflow of reaction prediction, including reaction network generation, ab initio thermodynamics and microkinetic modelling. An overview of the frequently used regression models in machine learning is presented. As a promising alternative to full ab initio calculations, machine learning interatomic potentials are highlighted. Furthermore, we survey applications assisted by these methods for accelerating reaction prediction, exploring reaction networks, and computational catalyst design. Finally, we envisage future directions in computationally investigating reactions and implementing machine learning algorithms in heterogeneous catalysis.

Baseline Comparisons of Complementary Sampling Methods for Assembly Driven by Short-Ranged Pair Potentials toward Fast and Flexible Hybridization

This work measures baseline sampling characteristics that highlight fundamental differences between sampling methods for assembly driven by short-ranged pair potentials. Such granular comparison is essential for fast, flexible, and accurate hybridization of complementary methods. Besides sampling speed, efficiency, and accuracy of uniform grid coverage, other sampling characteristics measured are (i) accuracy of covering narrow low energy regions that have low effective dimension (ii) ability to localize sampling to specific basins, and (iii) flexibility in sampling distributions. As a proof of concept, we compare a recently developed geometric methodology EASAL (Efficient Atlasing and Search of Assembly Landscapes) and the traditional Monte Carlo (MC) method for sampling the energy landscape of two assembling trans-membrane helices, driven by short-range pair potentials. By measuring the above-mentioned sampling characteristics, we demonstrate that EASAL provides localized and accurate coverage of crucial regions of the energy landscape of low effective dimension, under flexible sampling distributions, with much fewer samples and computational resources than MC sampling. EASAL's empirically validated theoretical guarantees permit credible extrapolation of these measurements and comparisons to arbitrary number and size of assembling units. Promising avenues for hybridizing the complementary advantages of the two methods are discussed.

Restricted-Variance Constrained, Reaction Path, and Transition State Molecular Optimizations Using Gradient-Enhanced Kriging

Gaussian process regression has recently been explored as an alternative to standard surrogate models in molecular equilibrium geometry optimization. In particular, the gradient-enhanced Kriging approach in association with internal coordinates, restricted-variance optimization, and an efficient and fast estimate of hyperparameters has demonstrated performance on par or better than standard methods. In this report, we extend the approach to constrained optimizations and transition states and benchmark it for a set of reactions. We compare the performance of the newly developed method with the standard techniques in the location of transition states and in constrained optimizations, both isolated and in the context of reaction path computation. The results show that the method outperforms the current standard in efficiency as well as in robustness.

Multi-fidelity machine-learning with uncertainty quantification and Bayesian optimization for materials design: Application to ternary random alloys

We present a scale-bridging approach based on a multi-fidelity (MF) machine-learning (ML) framework leveraging Gaussian processes (GP) to fuse atomistic computational model predictions across multiple levels of fidelity. Through the posterior variance of the MFGP, our framework naturally enables uncertainty quantification, providing estimates of confidence in the predictions. We used density functional theory as high-fidelity prediction, while a ML interatomic potential is used as low-fidelity prediction. Practical materials' design efficiency is demonstrated by reproducing the ternary composition dependence of a quantity of interest (bulk modulus) across the full aluminum-niobium-titanium ternary random alloy composition space. The MFGP is then coupled to a Bayesian optimization procedure, and the computational efficiency of this approach is demonstrated by performing an on-the-fly search for the global optimum of bulk modulus in the ternary composition space. The framework presented in this manuscript is the first application of MFGP to atomistic materials simulations fusing predictions between density functional theory and classical interatomic potential calculations.

Restricted-Variance Molecular Geometry Optimization Based on Gradient-Enhanced Kriging

Machine learning techniques, specifically gradient-enhanced Kriging (GEK), have been implemented for molecular geometry optimization. GEK-based optimization has many advantages compared to conventional—step-restricted second-order truncated expansion—molecular optimization methods. In particular, the surrogate model given by GEK can have multiple stationary points, will smoothly converge to the exact model as the number of sample points increases, and contains an explicit expression for the expected error of the model function at an arbitrary point. Machine learning is, however, associated with abundance of data, contrary to the situation desired for efficient geometry optimizations. In this paper, we demonstrate how the GEK procedure can be utilized in a fashion such that in the presence of few data points, the surrogate surface will in a robust way guide the optimization to a minimum of a potential energy surface. In this respect, the GEK procedure will be used to mimic the behavior of a conventional second-order scheme but retaining the flexibility of the superior machine learning approach. Moreover, the expected error will be used in the optimizations to facilitate restricted-variance optimizations. A procedure which relates the eigenvalues of the approximate guessed Hessian with the individual characteristic lengths, used in the GEK model, reduces the number of empirical parameters to optimize to two: the value of the trend function and the maximum allowed variance. These parameters are determined using the extended Baker (e-Baker) and part of the Baker transition-state (Baker-TS) test suites as a training set. The so-created optimization procedure is tested using the e-Baker, full Baker-TS, and S22 test suites, at the density functional theory and second-order Møller–Plesset levels of approximation. The results show that the new method is generally of similar or better performance than a state-of-the-art conventional method, even for cases where no significant improvement was expected.

Quantum Chemistry in the Age of Machine Learning

As the quantum chemistry (QC) community embraces machine learning (ML), the number of new methods and applications based on the combination of QC and ML is surging. In this Perspective, a view of the current state of affairs in this new and exciting research field is offered, challenges of using machine learning in quantum chemistry applications are described, and potential future developments are outlined. Specifically, examples of how machine learning is used to improve the accuracy and accelerate quantum chemical research are shown. Generalization and classification of existing techniques are provided to ease the navigation in the sea of literature and to guide researchers entering the field. The emphasis of this Perspective is on supervised machine learning.

Geometry optimization using Gaussian process regression in internal coordinate systems

Locating the minimum energy structure of molecules, typically referred to as geometry optimization, is one of the first steps of any computational chemistry calculation. Earlier research was mostly dedicated to finding convenient sets of molecule-specific coordinates for a suitable representation of the potential energy surface, where a faster convergence toward the minimum structure can be achieved. More recent approaches, on the other hand, are based on various machine learning techniques and seem to revert to Cartesian coordinates instead for practical reasons. We show that the combination of Gaussian process regression with those coordinate systems employed by state-of-the-art geometry optimizers can significantly improve the performance of this powerful machine learning technique. This is demonstrated on a benchmark set of 30 small covalently bonded molecules.