Predicting electronic structure properties of transition metal complexes with neural networks

On the stability constants of metal-nitrate complexes in aqueous solutions

Stability constants of simple reactions involving addition of the NO3- ion to hydrated metal complexes, [M(H2O)x]n+ are calculated with a computational workflow developed using cloud computing resources. The computational workflow performs conformational searches for metal complexes at both low and high levels of theories in conjunction with a continuum solvation model (CSM). The low-level theory is mainly used for the initial conformational searches, which are complemented with high-level density functional theory conformational searches in the CSM framework to determine the coordination chemistry relevant for stability constant calculations. In this regard, the lowest energy conformations are found to obtain the reaction free energies for the addition of one NO3- to [M(H2O)x]n+ complexes, where M represents Fe(II), Fe(III), Sr(II), Ce(III), Ce(IV), and U(VI), respectively. Structural analysis of hundreds of optimized geometries at high-level theory reveals that NO3- coordinates with Fe(II) and Fe(III) in either a monodentate or bidentate manner. Interestingly, the lowest-energy conformations of Fe(II) metal-nitrate complexes exhibit monodentate or bidentate coordination with a coordination number of 6 while the bidentate seven-coordinated Fe(II) metal-nitrate complexes are approximately 2 kcal mol-1 higher in energy. Notably, for Fe(III) metal-nitrate complexes, the bidentate seven-coordinated configuration is more stable than the six-coordinated Fe(II) complexes (monodentate or bidentate) by a few thermal energy units. In contrast, Sr(II), Ce(III), Ce(IV), and U(VI) metal ions predominantly coordinate with NO3- in a bidentate manner, exhibiting typical coordination numbers of 7, 9, 9, and 5, respectively. Stability constants are accordingly calculated using linear free energy approaches to account for the systematic errors and good agreements are obtained between the calculated stability constants and the available experimental data.

Discovery of Spin-Crossover Materials with Equivariant Graph Neural Networks and Relevance-Based Classification

Swift discovery of spin-crossover materials for their potential application in electronic and quantum devices requires techniques that enable efficient identification of suitable candidates. To this end, we screened the Cambridge Structural Database to develop a specialized database of 1439 materials and computed spin-switching energies from density functional theory for each material. The database was used to train an equivariant graph convolution neural network to predict the magnitude of the spin-conversion energy. A test mean absolute error was 360 meV. For candidate identification, we equipped the system with a relevance-based classifier. This approach leads to a nearly 4-fold improvement in identifying potential spin-crossover systems of interest as compared to conventional high-throughput screening.

High-Throughput Prediction of Metal-Embedded Complex Properties with a New GNN-Based Metal Attention Framework

Metal-embedded complexes (MECs), including transition metal complexes (TMCs) and metal-organic frameworks (MOFs), are important in catalysis, materials science, and molecular devices due to their unique metal atom centrality and complex coordination environments. However, modeling and predicting their properties accurately is challenging. A new metal attention (MA) framework for graph neural networks (GNNs) was proposed to address the limitations of traditional methods, which fail to differentiate core coordination structures from ordinary covalent bonds. This MA framework converts heterogeneous graphs of complexes into homogeneous ones with distinct metal features by highlighting key metal-feature coordination through hierarchical pooling and a metal cross-attention. To assess its performance, 11 widely used GNN algorithms, three of which are heterogeneous, were compared. Experimental results indicate significant improvements in accuracy: an average of 32.07% for predicting TMC properties and up to 23.01% for MOF CO2 absorption. Moreover, tests on the framework's robustness regarding data set size variation and comparison with a larger non-MA model show that the enhanced performance stems from the architecture, not merely increasing model capacity. The MA framework's potential in predicting metal complex properties offers a potent statistical tool for optimizing and designing new materials like catalysts and gas storage systems.

Data Generation for Machine Learning Interatomic Potentials and Beyond

The field of data-driven chemistry is undergoing an evolution, driven by innovations in machine learning models for predicting molecular properties and behavior. Recent strides in ML-based interatomic potentials have paved the way for accurate modeling of diverse chemical and structural properties at the atomic level. The key determinant defining MLIP reliability remains the quality of the training data. A paramount challenge lies in constructing training sets that capture specific domains in the vast chemical and structural space. This Review navigates the intricate landscape of essential components and integrity of training data that ensure the extensibility and transferability of the resulting models. We delve into the details of active learning, discussing its various facets and implementations. We outline different types of uncertainty quantification applied to atomistic data acquisition and the correlations between estimated uncertainty and true error. The role of atomistic data samplers in generating diverse and informative structures is highlighted. Furthermore, we discuss data acquisition via modified and surrogate potential energy surfaces as an innovative approach to diversify training data. The Review also provides a list of publicly available data sets that cover essential domains of chemical space.

Exploring nonlinear correlations among transition metal nanocluster properties using deep learning: a comparative analysis with LOO-CV method and cosine similarity

A novel approach is introduced for the rapid and accurate correlation analysis of nonlinear properties in Transition Metal (TM) clusters utilizing the Deep Leave-One-Out Cross-Validation technique. This investigation demonstrates that the Deep Neural Network (DNN)-based approach offers a more efficient predictive method for various properties of fourth-row TM nanoclusters compared to conventional Density Functional Theory methods, which are computationally intensive and time-consuming. The feature space, also known as descriptors, is established based on a broad spectrum of electronic and physical characteristics. Leveraging the similarities among these clusters, the DNN-based model is employed to explore the correlations among TM cluster properties. The proposed method, in conjunction with cosine similarity, achieves remarkable accuracy up to 10-9 for predicting total energy, lowest vibrational mode, binding energy, and HOMO-LUMO energy gap of TM2, TM3, and TM4nanoclusters. By analyzing correlation errors, the most closely coupled TM clusters are identified. Notably, Mn and Ni clusters exhibit the highest and lowest levels of energy coupling with other TMs, respectively. Generally, energy prediction for TM2, TM3, and TM4clusters exhibit similar trends, while an alternating behavior is observed for vibrational modes and binding energies. Furthermore, Ti, V, and Co demonstrate the highest binding energy correlations with TM2, TM3, and TM4sets, respectively. Regarding energy gap predictions, Ni exhibits the strongest correlation in the smallest TM2clusters, while Cr shows the highest dependence in TM3and TM4sets. Lastly, Zn displays the largest error in HOMO-LUMO energy gap across all sets, indicating distinctive independent energy gap characteristics.

Augmenting genetic algorithms with machine learning for inverse molecular design

Evolutionary and machine learning methods have been successfully applied to the generation of molecules and materials exhibiting desired properties. The combination of these two paradigms in inverse design tasks can yield powerful methods that explore massive chemical spaces more efficiently, improving the quality of the generated compounds. However, such synergistic approaches are still an incipient area of research and appear underexplored in the literature. This perspective covers different ways of incorporating machine learning approaches into evolutionary learning frameworks, with the overall goal of increasing the optimization efficiency of genetic algorithms. In particular, machine learning surrogate models for faster fitness function evaluation, discriminator models to control population diversity on-the-fly, machine learning based crossover operations, and evolution in latent space are discussed. The further potential of these synergistic approaches in generative tasks is also assessed, outlining promising directions for future developments.

Automated prediction of ground state spin for transition metal complexes

Exploiting crystallographic data repositories for large-scale quantum chemical computations requires the rapid and accurate extraction of the molecular structure, charge and spin from the crystallographic information file. Here, we develop a general approach to assign the ground state spin of transition metal complexes, in complement to our previous efforts on determining metal oxidation states and bond order within the cell2mol software. Starting from a database of 31k transition metal complexes extracted from the Cambridge Structural Database with cell2mol, we construct the TM-GSspin dataset, which contains 2063 mononuclear first row transition metal complexes and their computed ground state spins. TM-GSspin is highly diverse in terms of metals, metal oxidation states, coordination geometries, and coordination sphere compositions. Based on TM-GSspin, we identify correlations between structural and electronic features of the complexes and their ground state spins to develop a rule-based spin state assignment model. Leveraging this knowledge, we construct interpretable descriptors and build a statistical model achieving 98% cross-validated accuracy in predicting the ground state spin across the board. Our approach provides a practical way to determine the ground state spin of transition metal complexes directly from crystal structures without additional computations, thus enabling the automated use of crystallographic data for large-scale computations involving transition metal complexes.

In Silico Chemical Experiments in the Age of AI: From Quantum Chemistry to Machine Learning and Back

Computational chemistry is an indispensable tool for understanding molecules and predicting chemical properties. However, traditional computational methods face significant challenges due to the difficulty of solving the Schrödinger equations and the increasing computational cost with the size of the molecular system. In response, there has been a surge of interest in leveraging artificial intelligence (AI) and machine learning (ML) techniques to in silico experiments. Integrating AI and ML into computational chemistry increases the scalability and speed of the exploration of chemical space. However, challenges remain, particularly regarding the reproducibility and transferability of ML models. This review highlights the evolution of ML in learning from, complementing, or replacing traditional computational chemistry for energy and property predictions. Starting from models trained entirely on numerical data, a journey set forth toward the ideal model incorporating or learning the physical laws of quantum mechanics. This paper also reviews existing computational methods and ML models and their intertwining, outlines a roadmap for future research, and identifies areas for improvement and innovation. Ultimately, the goal is to develop AI architectures capable of predicting accurate and transferable solutions to the Schrödinger equation, thereby revolutionizing in silico experiments within chemistry and materials science.

Paving the road towards automated homogeneous catalyst design

In the past decade, computational tools have become integral to catalyst design. They continue to offer significant support to experimental organic synthesis and catalysis researchers aiming for optimal reaction outcomes. More recently, data-driven approaches utilizing machine learning have garnered considerable attention for their expansive capabilities. This Perspective provides an overview of diverse initiatives in the realm of computational catalyst design and introduces our automated tools tailored for high-throughput in silico exploration of the chemical space. While valuable insights are gained through methods for high-throughput in silico exploration and analysis of chemical space, their degree of automation and modularity are key. We argue that the integration of data-driven, automated and modular workflows is key to enhancing homogeneous catalyst design on an unprecedented scale, contributing to the advancement of catalysis research.

Understanding the Photoprocesses in Biological Systems: Need for Accurate Multireference Treatment

Light-matter interaction is crucial to life itself and revolves around many of the central processes in biology. The need for understanding these photochemical and photophysical processes cannot be overemphasized. Interaction of light with biological systems starts with the absorption of light and subsequent phenomena that occur in the excited states of the system. However, excited states are typically difficult to understand within the mean field approximation of quantum chemical methods. Therefore, suitable multireference methods and methodologies have been developed to understand these phenomena. In this Perspective, we will describe a few methods and methodologies suitable for these descriptions and discuss some persisting difficulties.

Machine Learning-Enhanced Quantum Chemistry-Assisted Refinement of the Active Site Structure of Metalloproteins

Understanding the fine structural details of inhibitor binding at the active site of metalloenzymes can have a profound impact on the rational drug design targeted to this broad class of biomolecules. Structural techniques such as NMR, cryo-EM, and X-ray crystallography can provide bond lengths and angles, but the uncertainties in these measurements can be as large as the range of values that have been observed for these quantities in all the published structures. This uncertainty is far too large to allow for reliable calculations at the quantum chemical (QC) levels for developing precise structure-activity relationships or for improving the energetic considerations in protein-inhibitor studies. Therefore, the need arises to rely upon computational methods to refine the active site structures well beyond the resolution obtained with routine application of structural methods. In a recent paper, we have shown that it is possible to refine the active site of cobalt(II)-substituted MMP12, a metalloprotein that is a relevant drug target, by matching to the experimental pseudocontact shifts (PCS) those calculated using multireference ab initio QC methods. The computational cost of this methodology becomes a significant bottleneck when the starting structure is not sufficiently close to the final one, which is often the case with biomolecular structures. To tackle this problem, we have developed an approach based on a neural network (NN) and a support vector regression (SVR) and applied it to the refinement of the active site structure of oxalate-inhibited human carbonic anhydrase 2 (hCAII), another prototypical metalloprotein target. The refined structure gives a remarkably good agreement between the QC-calculated and the experimental PCS. This study not only contributes to the knowledge of CAII but also demonstrates the utility of combining machine learning (ML) algorithms with QC calculations, offering a promising avenue for investigating other drug targets and complex biological systems in general.

Modeling Zinc Complexes Using Neural Networks

Understanding the energetic landscapes of large molecules is necessary for the study of chemical and biological systems. Recently, deep learning has greatly accelerated the development of models based on quantum chemistry, making it possible to build potential energy surfaces and explore chemical space. However, most of this work has focused on organic molecules due to the simplicity of their electronic structures as well as the availability of data sets. In this work, we build a deep learning architecture to model the energetics of zinc organometallic complexes. To achieve this, we have compiled a configurationally and conformationally diverse data set of zinc complexes using metadynamics to overcome the limitations of traditional sampling methods. In terms of the neural network potentials, our results indicate that for zinc complexes, partial charges play an important role in modeling the long-range interactions with a neural network. Our developed model outperforms semiempirical methods in predicting the relative energy of zinc conformers, yielding a mean absolute error (MAE) of 1.32 kcal/mol with reference to the double-hybrid PWPB95 method.

Modeling Fe(II) Complexes Using Neural Networks

We report a Fe(II) data set of more than 23000 conformers in both low-spin (LS) and high-spin (HS) states. This data set was generated to develop a neural network model that is capable of predicting the energy and the energy splitting as a function of the conformation of a Fe(II) organometallic complex. In order to achieve this, we propose a type of scaled electronic embedding to cover the long-range interactions implicitly in our neural network describing the Fe(II) organometallic complexes. For the total energy prediction, the lowest MAE is 0.037 eV, while the lowest MAE of the splitting energy is 0.030 eV. Compared to baseline models, which only incorporate short-range interactions, our scaled electronic embeddings improve the accuracy by over 70% for the prediction of the total energy and the splitting energy. With regard to semiempirical methods, our proposed models reduce the MAE, with respect to these methods, by 2 orders of magnitude.

Using Machine Learning to Predict the Antibacterial Activity of Ruthenium Complexes

Rising antimicrobial resistance (AMR) and lack of innovation in the antibiotic pipeline necessitate novel approaches to discovering new drugs. Metal complexes have proven to be promising antimicrobial compounds, but the number of studied compounds is still low compared to the millions of organic molecules investigated so far. Lately, machine learning (ML) has emerged as a valuable tool for guiding the design of small organic molecules, potentially even in low-data scenarios. For the first time, we extend the application of ML to the discovery of metal-based medicines. Utilising 288 modularly synthesized ruthenium arene Schiff-base complexes and their antibacterial properties, a series of ML models were trained. The models perform well and are used to predict the activity of 54 new compounds. These displayed a 5.7x higher hit-rate (53.7 %) against methicillin-resistant Staphylococcus aureus (MRSA) compared to the original library (9.4 %), demonstrating that ML can be applied to improve the success-rates in the search of new metalloantibiotics. This work paves the way for more ambitious applications of ML in the field of metal-based drug discovery.

Machine Learning Prediction of the Experimental Transition Temperature of Fe(II) Spin-Crossover Complexes

Spin-crossover (SCO) complexes are materials that exhibit changes in the spin state in response to external stimuli, with potential applications in molecular electronics. It is challenging to know a priori how to design ligands to achieve the delicate balance of entropic and enthalpic contributions needed to tailor a transition temperature close to room temperature. We leverage the SCO complexes from the previously curated SCO-95 data set [Vennelakanti et al. J. Chem. Phys. 159, 024120 (2023)] to train three machine learning (ML) models for transition temperature (T1/2) prediction using graph-based revised autocorrelations as features. We perform feature selection using random forest-ranked recursive feature addition (RF-RFA) to identify the features essential to model transferability. Of the ML models considered, the full feature set RF and recursive feature addition RF models perform best, achieving moderate correlation to experimental T1/2 values. We then compare ML T1/2 predictions to those from three previously identified best-performing density functional approximations (DFAs) which accurately predict SCO behavior across SCO-95, finding that the ML models predict T1/2 more accurately than the best-performing DFAs. In addition, we study ML model predictions for a set of 18 SCO complexes for which only estimated T1/2 values are available. Upon excluding outliers from this set, the RF-RFA RF model shows a strong correlation to estimated T1/2 values with a Pearson's r of 0.82. In contrast, DFA-predicted T1/2 values have large errors and show no correlation to estimated T1/2 values over the same set of complexes. Overall, our study demonstrates slightly superior performance of ML models in comparison with some of the best-performing DFAs, and we expect ML models to improve further as larger data sets of SCO complexes are curated and become available for model training.

Zeolite encapsulated organometallic complexes as model catalysts

Heterogeneities in the structure of active centers in metal-containing porous materials are unavoidable and complicate the description of chemical events occurring along reaction coordinates at the atomic level. Metal containing zeolites include sites of varied local coordination and secondary confining environments, requiring careful titration protocols to quantify the predominant active sites. Hybrid organometallic-zeolite catalysts are useful well-defined platform materials for spectroscopic, kinetic, and computational studies of heterogeneous catalysis that avoid the complications of conventional metal-containing porous materials. Such materials have been synthesized and studied previously, but catalytic applications were mostly limited to liquid-phase oxidation and electrochemical reactions. The hydrothermal stability, time-on-stream stability, and utility of these materials in gas-phase oxidation reactions are under-studied. The potential applications for single-site heterogeneous catalysts in fundamental research are abundant and motivate future synthetic, spectroscopic, kinetic, and computational studies.

Comparison of the Performance of Density Functional Methods for the Description of Spin States and Binding Energies of Porphyrins

This work analyzes the performance of 250 electronic structure theory methods (including 240 density functional approximations) for the description of spin states and the binding properties of iron, manganese, and cobalt porphyrins. The assessment employs the Por21 database of high-level computational data (CASPT2 reference energies taken from the literature). Results show that current approximations fail to achieve the "chemical accuracy" target of 1.0 kcal/mol by a long margin. The best-performing methods achieve a mean unsigned error (MUE) <15.0 kcal/mol, but the errors are at least twice as large for most methods. Semilocal functionals and global hybrid functionals with a low percentage of exact exchange are found to be the least problematic for spin states and binding energies, in agreement with the general knowledge in transition metal computational chemistry. Approximations with high percentages of exact exchange (including range-separated and double-hybrid functionals) can lead to catastrophic failures. More modern approximations usually perform better than older functionals. An accurate statistical analysis of the results also casts doubts on some of the reference energies calculated using multireference methods. Suggestions and general guidelines for users are provided in the conclusions. These results hopefully stimulate advances for both the wave function and the density functional side of electronic structure calculations.

On the potentially transformative role of auxiliary-field quantum Monte Carlo in quantum chemistry: A highly accurate method for transition metals and beyond

Approximate solutions to the ab initio electronic structure problem have been a focus of theoretical and computational chemistry research for much of the past century, with the goal of predicting relevant energy differences to within "chemical accuracy" (1 kcal/mol). For small organic molecules, or in general, for weakly correlated main group chemistry, a hierarchy of single-reference wave function methods has been rigorously established, spanning perturbation theory and the coupled cluster (CC) formalism. For these systems, CC with singles, doubles, and perturbative triples is known to achieve chemical accuracy, albeit at O(N7) computational cost. In addition, a hierarchy of density functional approximations of increasing formal sophistication, known as Jacob's ladder, has been shown to systematically reduce average errors over large datasets representing weakly correlated chemistry. However, the accuracy of such computational models is less clear in the increasingly important frontiers of chemical space including transition metals and f-block compounds, in which strong correlation can play an important role in reactivity. A stochastic method, phaseless auxiliary-field quantum Monte Carlo (ph-AFQMC), has been shown to be capable of producing chemically accurate predictions even for challenging molecular systems beyond the main group, with relatively low O(N3 - N4) cost and near-perfect parallel efficiency. Herein, we present our perspectives on the past, present, and future of the ph-AFQMC method. We focus on its potential in transition metal quantum chemistry to be a highly accurate, systematically improvable method that can reliably probe strongly correlated systems in biology and chemical catalysis and provide reference thermochemical values (for future development of density functionals or interatomic potentials) when experiments are either noisy or absent. Finally, we discuss the present limitations of the method and where we expect near-term development to be most fruitful.

Insights into the deviation from piecewise linearity in transition metal complexes from supervised machine learning models

Virtual high-throughput screening (VHTS) and machine learning (ML) with density functional theory (DFT) suffer from inaccuracies from the underlying density functional approximation (DFA). Many of these inaccuracies can be traced to the lack of derivative discontinuity that leads to a curvature in the energy with electron addition or removal. Over a dataset of nearly one thousand transition metal complexes typical of VHTS applications, we computed and analyzed the average curvature (i.e., deviation from piecewise linearity) for 23 density functional approximations spanning multiple rungs of "Jacob's ladder". While we observe the expected dependence of the curvatures on Hartree-Fock exchange, we note limited correlation of curvature values between different rungs of "Jacob's ladder". We train ML models (i.e., artificial neural networks or ANNs) to predict the curvature and the associated frontier orbital energies for each of these 23 functionals and then interpret differences in curvature among the different DFAs through analysis of the ML models. Notably, we observe spin to play a much more important role in determining the curvature of range-separated and double hybrids in comparison to semi-local functionals, explaining why curvature values are weakly correlated between these and other families of functionals. Over a space of 187.2k hypothetical compounds, we use our ANNs to pinpoint DFAs for which representative transition metal complexes have near-zero curvature with low uncertainty, demonstrating an approach to accelerate screening of complexes with targeted optical gaps.

DFT-Based Multireference Diagnostics in the Solid State: Application to Metal-Organic Frameworks

When a many-body wave function of a system cannot be captured by a single determinant, high-level multireference (MR) methods are required to properly explain its electronic structure. MR diagnostics to estimate the magnitude of such static correlation have been primarily developed for molecular systems and range from low in computational cost to as costly as the full MR calculation itself. We report the first application of low-cost MR diagnostics based on the fractional occupation number calculated with finite-temperature DFT to solid-state systems. To compare the behavior of the diagnostics on solids and molecules, we select metal-organic frameworks (MOFs) as model materials because their reticular nature provides an intuitive way to identify molecular derivatives. On a series of closed-shell MOFs, we demonstrate that the DFT-based MR diagnostics are equally applicable to solids as to their molecular derivatives. The magnitude and spatial distribution of the MR character of a MOF are found to have a good correlation with those of its molecular derivatives, which can be calculated much more affordably in comparison to those of the full MOF. The additivity of MR character discussed here suggests the set of molecular derivatives to be a good representation of a MOF for both MR detection and ultimately for MR corrections, facilitating accurate and efficient high-throughput screening of MOFs and other porous solids.

Unraveling the Contributions to Spin-Lattice Relaxation in Kramers Single-Molecule Magnets

The study of how spin interacts with lattice vibrations and relaxes to equilibrium provides unique insights into its chemical environment and the relation between electronic structure and molecular composition. Despite its importance for several disciplines, ranging from magnetic resonance to quantum technologies, a convincing interpretation of spin dynamics in crystals of magnetic molecules is still lacking due to the challenging experimental determination of the correct spin relaxation mechanism. We apply ab initio spin dynamics to a series of 12 coordination complexes of Co2+ and Dy3+ ions selected among ∼240 compounds that largely cover the literature on single-molecule magnets and well represent different regimes of spin relaxation. Simulations reveal that the Orbach spin relaxation rate of known compounds mostly depends on the ions' zero-field splitting and little on the details of molecular vibrations. Raman relaxation is instead found to be also significantly affected by the features of low-energy phonons. These results provide a complete understanding of the factors limiting spin lifetime in single-molecule magnets and revisit years of experimental investigations by making it possible to transparently distinguish Orbach and Raman relaxation mechanisms.

Quantum Chemical Roots of Machine-Learning Molecular Similarity Descriptors

In this work, we explore the quantum chemical foundations of descriptors for molecular similarity. Such descriptors are key for traversing chemical compound space with machine learning. Our focus is on the Coulomb matrix and on the smooth overlap of atomic positions (SOAP). We adopt a basic framework that allows us to connect both descriptors to electronic structure theory. This framework enables us to then define two new descriptors that are more closely related to electronic structure theory, which we call Coulomb lists and smooth overlap of electron densities (SOED). By investigating their usefulness as molecular similarity descriptors, we gain new insights into how and why Coulomb matrix and SOAP work. Moreover, Coulomb lists avoid the somewhat mysterious diagonalization step of the Coulomb matrix and might provide a direct means to extract subsystem information that can be compared across Born-Oppenheimer surfaces of varying dimension. For the electron density, we derive the necessary formalism to create the SOED measure in close analogy to SOAP. Because this formalism is more involved than that of SOAP, we review the essential theory as well as introduce a set of approximations that eventually allow us to work with SOED in terms of the same implementation available for the evaluation of SOAP. We focus our analysis on elementary reaction steps, where transition state structures are more similar to either reactant or product structures than the latter two are with respect to one another. The prediction of electronic energies of transition state structures can, however, be more difficult than that of stable intermediates due to multi-configurational effects. The question arises to what extent molecular similarity descriptors rooted in electronic structure theory can resolve these intricate effects.

Computational design of magnetic molecules and their environment using quantum chemistry, machine learning and multiscale simulations

Having served as a playground for fundamental studies on the physics of d and f electrons for almost a century, magnetic molecules are now becoming increasingly important for technological applications, such as magnetic resonance, data storage, spintronics and quantum information. All of these applications require the preservation and control of spins in time, an ability hampered by the interaction with the environment, namely with other spins, conduction electrons, molecular vibrations and electromagnetic fields. Thus, the design of a novel magnetic molecule with tailored properties is a formidable task, which does not only concern its electronic structures but also calls for a deep understanding of the interaction among all the degrees of freedom at play. This Review describes how state-of-the-art ab initio computational methods, combined with data-driven approaches to materials modelling, can be integrated into a fully multiscale strategy capable of defining design rules for magnetic molecules.

Machine learning-based analysis of overall stability constants of metal-ligand complexes

The stability constants of metal(M)-ligand(L) complexes are industrially important because they affect the quality of the plating film and the efficiency of metal separation. Thus, it is desirable to develop an effective screening method for promising ligands. Although there have been several machine-learning approaches for predicting stability constants, most of them focus only on the first overall stability constant of M-L complexes, and the variety of cations is also limited to less than 20. In this study, two Gaussian process regression models are developed to predict the first overall stability constant and the n-th (n > 1) overall stability constants. Furthermore, the feature relevance is quantitatively evaluated via sensitivity analysis. As a result, the electronegativities of both metal and ligand are found to be the most important factor for predicting the first overall stability constant. Interestingly, the predicted value of the first overall stability constant shows the highest correlation with the n-th overall stability constant of the corresponding M-L pair. Finally, the number of features is optimized using validation data where the ligands are not included in the training data, which indicates high generalizability. This study provides valuable insights and may help accelerate molecular screening and design for various applications.

Machine learning of material properties: Predictive and interpretable multilinear models

Machine learning models can provide fast and accurate predictions of material properties but often lack transparency. Interpretability techniques can be used with black box solutions, or alternatively, models can be created that are directly interpretable. We revisit material datasets used in several works and demonstrate that simple linear combinations of nonlinear basis functions can be created, which have comparable accuracy to the kernel and neural network approaches originally used. Linear solutions can accurately predict the bandgap and formation energy of transparent conducting oxides, the spin states for transition metal complexes, and the formation energy for elpasolite structures. We demonstrate how linear solutions can provide interpretable predictive models and highlight the new insights that can be found when a model can be directly understood from its coefficients and functional form. Furthermore, we discuss how to recognize when intrinsically interpretable solutions may be the best route to interpretability.

Improving Results by Improving Densities: Density-Corrected Density Functional Theory

Density functional theory (DFT) calculations have become widespread in both chemistry and materials, because they usually provide useful accuracy at much lower computational cost than wavefunction-based methods. All practical DFT calculations require an approximation to the unknown exchange-correlation energy, which is then used self-consistently in the Kohn-Sham scheme to produce an approximate energy from an approximate density. Density-corrected DFT is simply the study of the relative contributions to the total energy error. In the vast majority of DFT calculations, the error due to the approximate density is negligible. But with certain classes of functionals applied to certain classes of problems, the density error is sufficiently large as to contribute to the energy noticeably, and its removal leads to much better results. These problems include reaction barriers, torsional barriers involving π-conjugation, halogen bonds, radicals and anions, most stretched bonds, etc. In all such cases, use of a more accurate density significantly improves performance, and often the simple expedient of using the Hartree-Fock density is enough. This Perspective explains what DC-DFT is, where it is likely to improve results, and how DC-DFT can produce more accurate functionals. We also outline challenges and prospects for the field.

Local Kernel Regression and Neural Network Approaches to the Conformational Landscapes of Oligopeptides

The application of machine learning to theoretical chemistry has made it possible to combine the accuracy of quantum chemical energetics with the thorough sampling of finite-temperature fluctuations. To reach this goal, a diverse set of methods has been proposed, ranging from simple linear models to kernel regression and highly nonlinear neural networks. Here we apply two widely different approaches to the same, challenging problem: the sampling of the conformational landscape of polypeptides at finite temperature. We develop a local kernel regression (LKR) coupled with a supervised sparsity method and compare it with a more established approach based on Behler-Parrinello type neural networks. In the context of the LKR, we discuss how the supervised selection of the reference pool of environments is crucial to achieve accurate potential energy surfaces at a competitive computational cost and leverage the locality of the model to infer which chemical environments are poorly described by the DFTB baseline. We then discuss the relative merits of the two frameworks and perform Hamiltonian-reservoir replica-exchange Monte Carlo sampling and metadynamics simulations, respectively, to demonstrate that both frameworks can achieve converged and transferable sampling of the conformational landscape of complex and flexible biomolecules with comparable accuracy and computational cost.

Machine Learning for the Discovery, Design, and Engineering of Materials

Machine learning (ML) has become a part of the fabric of high-throughput screening and computational discovery of materials. Despite its increasingly central role, challenges remain in fully realizing the promise of ML. This is especially true for the practical acceleration of the engineering of robust materials and the development of design strategies that surpass trial and error or high-throughput screening alone. Depending on the quantity being predicted and the experimental data available, ML can either outperform physics-based modes, be used to accelerate such models, or be integrated with them to improve their performance. We cover recent advances in algorithms and in their application that are starting to make inroads toward (a) the discovery of new materials through large-scale enumerative screening, (b) the design of materials through identification of rules and principles that govern materials properties, and (c) the engineering of practical materials by satisfying multiple objectives. We conclude with opportunities for further advancement to realize ML as a widespread tool for practical computational materials design. Expected final online publication date for the Annual Review of Chemical and Biomolecular Engineering, Volume 13 is October 2022. Please see http://www.annualreviews.org/page/journal/pubdates for revised estimates.

Ligand Optimization of Exchange Interaction in Co(II) Dimer Single Molecule Magnet by Machine Learning

Designing single-molecule magnets (SMMs) for potential applications in quantum computing and high-density data storage requires tuning their magnetic properties, especially the strength of the magnetic interaction. These properties can be characterized by first-principles calculations based on density functional theory (DFT). In this work, we study the experimentally synthesized Co(II) dimer (Co₂(C₅NH₅)₄(μ-PO₂(CH₂C₆H₅)₂)₃) SMM with the goal to control the exchange energy, ΔE_J, between the Co atoms through tuning of the capping ligands. The experimentally synthesized Co(II) dimer molecule has a very small ΔE_J < 1 meV. We assemble a DFT data set of 1081 ligand substitutions for the Co(II) dimer. The ligand exchange provides a broad range of exchange energies, ΔE_J, from +50 to -200 meV, with 80% of the ligands yielding a small ΔE_J < 10 meV. We identify descriptors for the classification and regression of ΔE_J using gradient boosting machine learning models. We compare one-hot encoded, structure-based, and chemical descriptors consisting of the HOMO/LUMO energies of the individual ligands and the maximum electronegativity difference and bond order for the ligand atom connecting to Co. We observe a similar overall performance with the chemical descriptors outperforming the other descriptors. We show that the exchange coupling, ΔE_J, is correlated to the difference in the average bridging angle between the ferromagnetic and antiferromagnetic states, similar to the Goodenough-Kanamori rules.

On the Investigation of Effective Factors on Electronic Structure Properties of Transition Metal Complexes: Robust Modeling Using GPR Approach

Materials discovery is usually done using high-throughput computational screening. The use of costly and complex direct density functional theory (DFT) simulation methods has been commonly used to determine subtle trends in spin-state ordering and inorganic bonding of inorganic materials and, in general, to predict the electronic structure properties of transition metal complexes. A Gaussian process regression (GPR) framework consisting of four kernel functions is introduced for spin-state splitting estimation through inorganic chemistry-appropriate empirical inputs. To this end, the present study reviewed an extensive range of data values from earlier works. According to statistical analysis, the GPR model showed very good performance. The coefficients of determination were calculated to be 0.986 for the exponential and Matern kernel functions, suggesting the highest predictive power of these methods. Moreover, the sensitivity of output to inputs was measured. Artificial intelligence (AI) helped accurately predict the target values through various input ranges.

Accurate calculation of spin-state energy gaps in Fe(III) spin-crossover systems using density functional methods

Fe(III) complexes are receiving ever-increasing attention as spin crossover (SCO) systems because they are usually air stable, as opposed to Fe(II) complexes, which are prone to oxidation. Here, we present the first systematic study exclusively devoted to assess the accuracy of several exchange-correlation functionals when it comes to predicting the energy gap between the high-spin (S = 5/2) and the low-spin (S = 1/2) states of Fe(III) complexes. Using a dataset of 24 different Fe(III) hexacoordinated complexes, it is demonstrated that the B3LYP* functional is an excellent choice not only for predicting spin-state energy gaps for Fe(III) complexes undergoing spin-transitions but also for discriminating Fe(III) complexes that are either low- or high-spin in the whole range of temperatures. Our benchmark study has led to the identification of a very versatile Fe(III) compound whose SCO properties can be engineered upon changing a single axial ligand. Overall, this work demonstrates that B3LYP* is a reliable functional for screening new spin-crossover systems with tailored properties.

Recent Progress and Prospects in Catalytic Water Treatment

Presently, conventional technologies in water treatment are not efficient enough to completely mineralize refractory water contaminants. In this context, the implementation of catalytic processes could be an alternative. Despite the advantages provided in terms of kinetics of transformation, selectivity, and energy saving, numerous attempts have not yet led to implementation at an industrial scale. This review examines investigations at different scales for which controversies and limitations must be solved to bridge the gap between fundamentals and practical developments. Particular attention has been paid to the development of solar-driven catalytic technologies and some other emerging processes, such as microwave assisted catalysis, plasma-catalytic processes, or biocatalytic remediation, taking into account their specific advantages and the drawbacks. Challenges for which a better understanding related to the complexity of the systems and the coexistence of various solid-liquid-gas interfaces have been identified.

Deciphering Cryptic Behavior in Bimetallic Transition-Metal Complexes with Machine Learning

We demonstrate an alternative, data-driven approach to uncovering structure-property relationships for the rational design of heterobimetallic transition-metal complexes that exhibit metal-metal bonding. We tailor graph-based representations of the metal-local environment for these complexes for use in multiple linear regression and kernel ridge regression (KRR) models. We curate a set of 28 experimentally characterized complexes to develop a multiple linear regression model for oxidation potentials. We achieve good accuracy (mean absolute error of 0.25 V) and preserve transferability to unseen experimental data with a new ligand structure. We also train a KRR model on a subset of 330 structurally characterized heterobimetallics to predict the degree of metal-metal bonding. This KRR model predicts relative metal-metal bond lengths in the test set to within 5%, and analysis of key features reveals the fundamental atomic contributions (e.g., the valence electron configuration) that most strongly influence the behavior of these complexes. Our work provides guidance for rational bimetallic design, suggesting that properties, including the formal shortness ratio, should be transferable from one period to another.

Machine learning dynamic correlation in chemical kinetics

Lattice models are a useful tool to simulate the kinetics of surface reactions. Since it is expensive to propagate the probabilities of the entire lattice configurations, it is practical to consider the occupation probabilities of a typical site or a cluster of sites instead. This amounts to a moment closure approximation of the chemical master equation. Unfortunately, simple closures, such as the mean-field and the pair approximation (PA), exhibit weaknesses in systems with significant long-range correlation. In this paper, we show that machine learning (ML) can be used to construct accurate moment closures in chemical kinetics using the lattice Lotka-Volterra model as a model system. We trained feedforward neural networks on kinetic Monte Carlo (KMC) results at select values of rate constants and initial conditions. Given the same level of input as PA, the ML moment closure (MLMC) gave accurate predictions of the instantaneous three-site occupation probabilities. Solving the kinetic equations in conjunction with MLMC gave drastic improvements in the simulated dynamics and descriptions of the dynamical regimes throughout the parameter space. In this way, MLMC is a promising tool to interpolate KMC simulations or construct pretrained closures that would enable researchers to extract useful insight at a fraction of the computational cost.

Machine learning to tame divergent density functional approximations: a new path to consensus materials design principles

Virtual high-throughput screening (VHTS) with density functional theory (DFT) and machine-learning (ML)-acceleration is essential in rapid materials discovery. By necessity, efficient DFT-based workflows are carried out with a single density functional approximation (DFA). Nevertheless, properties evaluated with different DFAs can be expected to disagree for cases with challenging electronic structure (e.g., open-shell transition-metal complexes, TMCs) for which rapid screening is most needed and accurate benchmarks are often unavailable. To quantify the effect of DFA bias, we introduce an approach to rapidly obtain property predictions from 23 representative DFAs spanning multiple families, “rungs” (e.g., semi-local to double hybrid) and basis sets on over 2000 TMCs. Although computed property values (e.g., spin state splitting and frontier orbital gap) differ by DFA, high linear correlations persist across all DFAs. We train independent ML models for each DFA and observe convergent trends in feature importance, providing DFA-invariant, universal design rules. We devise a strategy to train artificial neural network (ANN) models informed by all 23 DFAs and use them to predict properties (e.g., spin-splitting energy) of over 187k TMCs. By requiring consensus of the ANN-predicted DFA properties, we improve correspondence of computational lead compounds with literature-mined, experimental compounds over the typically employed single-DFA approach.

Machine learning (ML)-based feature analysis reveals universal design rules regardless of density functional choices. Using the consensus among multiple functionals, we identify robust lead complexes in ML-accelerated chemical discovery.

Combining artificial intelligence and physics-based modeling to directly assess atomic site stabilities: from sub-nanometer clusters to extended surfaces

The performance of functional materials is dictated by chemical and structural properties of individual atomic sites. In catalysts, for example, the thermodynamic stability of constituting atomic sites is a key descriptor from which more complex properties, such as molecular adsorption energies and reaction rates, can be derived. In this study, we present a widely applicable machine learning (ML) approach to instantaneously compute the stability of individual atomic sites in structurally and electronically complex nano-materials. Conventionally, we determine such site stabilities using computationally intensive first-principles calculations. With our approach, we predict the stability of atomic sites in sub-nanometer metal clusters of 3-55 atoms with mean absolute errors in the range of 0.11-0.14 eV. To extract physical insights from the ML model, we introduce a genetic algorithm (GA) for feature selection. This algorithm distills the key structural and chemical properties governing the stability of atomic sites in size-selected nanoparticles, allowing for physical interpretability of the models and revealing structure-property relationships. The results of the GA are generally model and materials specific. In the limit of large nanoparticles, the GA identifies features consistent with physics-based models for metal-metal interactions. By combining the ML model with the physics-based model, we predict atomic site stabilities in real time for structures ranging from sub-nanometer metal clusters (3-55 atom) to larger nanoparticles (147 to 309 atoms) to extended surfaces using a physically interpretable framework. Finally, we present a proof of principle showcasing how our approach can determine stable and active nanocatalysts across a generic materials space of structure and composition.

Spin-free formulation of the multireference driven similarity renormalization group: A benchmark study of first-row diatomic molecules and spin-crossover energetics

We report a spin-free formulation of the multireference (MR) driven similarity renormalization group (DSRG) based on the ensemble normal ordering of Mukherjee and Kutzelnigg [J. Chem. Phys. 107, 432 (1997)]. This ensemble averages over all microstates of a given total spin quantum number, and therefore, it is invariant with respect to SU(2) transformations. As such, all equations may be reformulated in terms of spin-free quantities and they closely resemble those of spin-adapted closed-shell coupled cluster (CC) theory. The current implementation is used to assess the accuracy of various truncated MR-DSRG methods (perturbation theory up to third order and iterative methods with single and double excitations) in computing the constants of 33 first-row diatomic molecules. The accuracy trends for these first-row diatomics are consistent with our previous benchmark on a small subset of closed-shell diatomic molecules. We then present the first MR-DSRG application on transition-metal complexes by computing the spin splittings of the [Fe(H₂O)₆]²⁺ and [Fe(NH₃)₆]²⁺ molecules. A focal point analysis (FPA) shows that third-order perturbative corrections are essential to achieve reasonably converged energetics. The FPA based on the linearized MR-DSRG theory with one- and two-body operators and up to a quintuple-ζ basis set predicts the spin splittings of [Fe(H₂O)₆]²⁺ and [Fe(NH₃)₆]²⁺ to be -35.7 and -17.1 kcal mol^-1, respectively, showing good agreement with the results of local CC theory with singles, doubles, and perturbative triples.

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.

Ab Initio Machine Learning in Chemical Compound Space

Chemical compound space (CCS), the set of all theoretically conceivable combinations of chemical elements and (meta-)stable geometries that make up matter, is colossal. The first-principles based virtual sampling of this space, for example, in search of novel molecules or materials which exhibit desirable properties, is therefore prohibitive for all but the smallest subsets and simplest properties. We review studies aimed at tackling this challenge using modern machine learning techniques based on (i) synthetic data, typically generated using quantum mechanics based methods, and (ii) model architectures inspired by quantum mechanics. Such Quantum mechanics based Machine Learning (QML) approaches combine the numerical efficiency of statistical surrogate models with an ab initio view on matter. They rigorously reflect the underlying physics in order to reach universality and transferability across CCS. While state-of-the-art approximations to quantum problems impose severe computational bottlenecks, recent QML based developments indicate the possibility of substantial acceleration without sacrificing the predictive power of quantum mechanics.

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.

Machine-Learned Energy Functionals for Multiconfigurational Wave Functions

We introduce multiconfiguration data-driven functional methods (MC-DDFMs), a group of methods which aim to correct the total or classical energy of a qualitatively accurate multiconfigurational wave function using a machine-learned functional of some featurization of the wave function such as its density, on-top density, or both. On a data set of carbene singlet-triplet energy splittings, we show that MC-DDFMs are able to achieve near-benchmark performance on systems not used for training with a robust degree of active-space independence. Beyond demonstrating that the density and on-top density hold the information necessary to correct the singlet-triplet energy splittings of multiconfigurational wave functions, this approach shows great promise for the development of functionals for MC-PDFT because corrections to the classical energy appear to be more transferable to types of molecules not included in the training data than corrections to total energies such as those yielded by CASSCF or NEVPT2.

Trends in computational molecular catalyst design

Computational methods have emerged as a powerful tool to augment traditional experimental molecular catalyst design by providing useful predictions of catalyst performance and decreasing the time needed for catalyst screening. In this perspective, we discuss three approaches for computational molecular catalyst design: (i) the reaction mechanism-based approach that calculates all relevant elementary steps, finds the rate and selectivity determining steps, and ultimately makes predictions on catalyst performance based on kinetic analysis, (ii) the descriptor-based approach where physical/chemical considerations are used to find molecular properties as predictors of catalyst performance, and (iii) the data-driven approach where statistical analysis as well as machine learning (ML) methods are used to obtain relationships between available data/features and catalyst performance. Following an introduction to these approaches, we cover their strengths and weaknesses and highlight some recent key applications. Furthermore, we present an outlook on how the currently applied approaches may evolve in the near future by addressing how recent developments in building automated computational workflows and implementing advanced ML models hold promise for reducing human workload, eliminating human bias, and speeding up computational catalyst design at the same time. Finally, we provide our viewpoint on how some of the challenges associated with the up-and-coming approaches driven by automation and ML may be resolved.

Using collective knowledge to assign oxidation states of metal cations in metal-organic frameworks

Knowledge of the oxidation state of metal centres in compounds and materials helps in the understanding of their chemical bonding and properties. Chemists have developed theories to predict oxidation states based on electron-counting rules, but these can fail to describe oxidation states in extended crystalline systems such as metal-organic frameworks. Here we propose the use of a machine-learning model, trained on assignments by chemists encoded in the chemical names in the Cambridge Structural Database, to automatically assign oxidation states to the metal ions in metal-organic frameworks. In our approach, only the immediate local environment around a metal centre is considered. We show that the strategy is robust to experimental uncertainties such as incorrect protonation, unbound solvents or changes in bond length. This method gives good accuracy and we show that it can be used to detect incorrect assignments in the Cambridge Structural Database, illustrating how collective knowledge can be captured by machine learning and converted into a useful tool.