Machine learning for potential energy surfaces: An extensive database and assessment of methods

Accelerating wavepacket propagation with machine learning

In this work, we discuss the use of a recently introduced machine learning (ML) technique known as Fourier neural operators (FNO) as an efficient alternative to the traditional solution of the time-dependent Schrödinger equation (TDSE). FNOs are ML models which are employed in the approximated solution of partial differential equations. For a wavepacket propagating in an anharmonic potential and for a tunneling system, we show that the FNO approach can accurately and faithfully model wavepacket propagation via the density. Additionally, we demonstrate that FNOs can be a suitable replacement for traditional TDSE solvers in cases where the results of the quantum dynamical simulation are required repeatedly such as in the case of parameter optimization problems (e.g., control). The speed-up from the FNO method allows for its combination with the Markov-chain Monte Carlo approach in applications that involve solving inverse problems such as optimal and coherent laser control of the outcome of dynamical processes.

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.

From Organic Fragments to Photoswitchable Catalysts: The OFF-ON Structural Repository for Transferable Kernel-Based Potentials

Structurally and conformationally diverse databases are needed to train accurate neural networks or kernel-based potentials capable of exploring the complex free energy landscape of flexible functional organic molecules. Curating such databases for species beyond "simple" drug-like compounds or molecules composed of well-defined building blocks (e.g., peptides) is challenging as it requires thorough chemical space mapping and evaluation of both chemical and conformational diversities. Here, we introduce the OFF-ON (organic fragments from organocatalysts that are non-modular) database, a repository of 7869 equilibrium and 67,457 nonequilibrium geometries of organic compounds and dimers aimed at describing conformationally flexible functional organic molecules, with an emphasis on photoswitchable organocatalysts. The relevance of this database is then demonstrated by training a local kernel regression model on a low-cost semiempirical baseline and comparing it with a PBE0-D3 reference for several known catalysts, notably the free energy surfaces of exemplary photoswitchable organocatalysts. Our results demonstrate that the OFF-ON data set offers reliable predictions for simulating the conformational behavior of virtually any (photoswitchable) organocatalyst or organic compound composed of H, C, N, O, F, and S atoms, thereby opening a computationally feasible route to explore complex free energy surfaces in order to rationalize and predict catalytic behavior.

Artificial Neural Network-Derived Unified Six-Dimensional Potential Energy Surface for Tetra Atomic Isomers of the Biogenic [H, C, N, O] System

Recognition of different structural patterns in different potential energy surface regions, such as in isomerizing quasilinear tetra atomic molecules, is important for understanding the details of underlying physics and chemistry. In this respect, using three variants of artificial neural networks (ANNs), we investigated the six-dimensional (6-D) singlet potential energy surfaces (PES) of tetra atomic isomers of the biogenic [H, C, N, O] system. At first, we constructed a separate ANN potential for each of the studied isomers. In the next step, a comparative assessment of the separate ANN models led to the setting up of a unified 6-D singlet PES equally and accurately describing all studied isomers. The constructed unified model yields relative energies comparable to those obtained either from the gold standard CCSD(T) method or from separate ANNs for each of the studied isomers. The accuracy of the unified singlet PES is on the order of 10^-4 Hartrees (0.1 kcal/mol). The developed PES in this work captures the main features of nonlinear and quasilinear tetra atomic isomers of this biogenic system.

OntoPESScan: An Ontology for Potential Energy Surface Scans

In this work, a new OntoPESScan ontology is developed for the semantic representation of one-dimensional potential energy surface (PES) scans, a central concept in computational chemistry. This ontology is developed in line with knowledge graph principles and The World Avatar (TWA) project. OntoPESScan is linked to other ontologies for chemistry in TWA, including OntoSpecies, which helps uniquely identify species along the PES and access their properties, and OntoCompChem, which allows the association of potential energy surfaces with quantum chemical calculations and the concepts used to derive them. A force-field fitting agent is also developed that makes use of the information in the OntoPESScan ontology to fit force fields to reactive surfaces of interest on the fly by making use of the empirical valence bond methodology. This agent is demonstrated to successfully parametrize two cases, namely, a PES scan on ethanol and a PES scan on a localized π-radical PAH hypothesized to play a role in soot formation during combustion. OntoPESScan is an extension to the capabilities of TWA and, in conjunction with potential further ontological support for molecular dynamics and reactions, will further progress toward an open, continuous, and self-growing knowledge graph for chemistry.

Machine Learning Diffusion Monte Carlo Forces

Diffusion Monte Carlo (DMC) is one of the most accurate techniques available for calculating the electronic properties of molecules and materials, yet it often remains a challenge to economically compute forces using this technique. As a result, ab initio molecular dynamics simulations and geometry optimizations that employ Diffusion Monte Carlo forces are often out of reach. One potential approach for accelerating the computation of "DMC forces" is to machine learn these forces from DMC energy calculations. In this work, we employ Behler-Parrinello Neural Networks to learn DMC forces from DMC energy calculations for geometry optimization and molecular dynamics simulations of small molecules. We illustrate the unique challenges that stem from learning forces without explicit force data and from noisy energy data by making rigorous comparisons of potential energy surface, dynamics, and optimization predictions among ab initio density functional theory (DFT) simulations and machine-learning models trained on DFT energies with forces, DFT energies without forces, and DMC energies without forces. We show for three small molecules─C₂, H₂O, and CH₃Cl─that machine-learned DMC dynamics can reproduce average bond lengths and angles within a few percent of known experimental results at one hundredth of the typical cost. Our work describes a much-needed means of performing dynamics simulations on high-accuracy, DMC PESs and for generating DMC-quality molecular geometries given current algorithmic constraints.

Machine learning, artificial intelligence, and chemistry: how smart algorithms are reshaping simulation and the laboratory

Machine learning and artificial intelligence are increasingly gaining in prominence through image analysis, language processing, and automation, to name a few applications. Machine learning is also making profound changes in chemistry. From revisiting decades-old analytical techniques for the purpose of creating better calibration curves, to assisting and accelerating traditional in silico simulations, to automating entire scientific workflows, to being used as an approach to deduce underlying physics of unexplained chemical phenomena, machine learning and artificial intelligence are reshaping chemistry, accelerating scientific discovery, and yielding new insights. This review provides an overview of machine learning and artificial intelligence from a chemist’s perspective and focuses on a number of examples of the use of these approaches in computational chemistry and in the laboratory.

Benchmarking structural evolution methods for training of machine learned interatomic potentials

When creating training data for machine-learned interatomic potentials (MLIPs), it is common to create initial structures and evolve them using molecular dynamics (MD) to sample a larger configuration space. We benchmark two other modalities of evolving structures, contour exploration (CE) and dimer-method (DM) searches against MD for their ability to produce diverse and robust density functional theory training data sets for MLIPs. We also discuss the generation of initial structures which are either from known structures or from random structures in detail to further formalize the structure-sourcing processes in the future. The polymorph-rich zirconium-oxygen composition space is used as a rigorous benchmark system for comparing the performance of MLIPs trained on structures generated from these structural evolution methods. Using Behler-Parrinello neural networks as our MLIP models, we find that CE and the DM searches are generally superior to MD in terms of spatial descriptor diversity and statistical accuracy.

Molecular Conformer Search with Low-Energy Latent Space

Identifying low-energy conformers with quantum mechanical accuracy for molecules with many degrees of freedom is challenging. In this work, we use the molecular dihedral angles as features and explore the possibility of performing molecular conformer search in a latent space with a generative model named variational auto-encoder (VAE). We bias the VAE towards low-energy molecular configurations to generate more informative data. In this way, we can effectively build a reliable energy model for the low-energy potential energy surface. After the energy model has been built, we extract local-minimum conformations and refine them with structure optimization. We have tested and benchmarked our low-energy latent-space (LOLS) structure search method on organic molecules with 5-9 searching dimensions. Our results agree with previous studies.

Fast Near Ab Initio Potential Energy Surfaces Using Machine Learning

A machine-learning based approach for evaluating potential energies for quantum mechanical studies of properties of the ground and excited vibrational states of small molecules is developed. This approach uses the molecular-orbital-based machine learning (MOB-ML) method to generate electronic energies with the accuracy of CCSD(T) calculations at the same cost as a Hartree-Fock calculation. To further reduce the computational cost of the potential energy evaluations without sacrificing the CCSD(T) level accuracy, GPU-accelerated Neural Network Potential Energy Surfaces (NN-PES) are trained to geometries and energies that are collected from small-scale Diffusion Monte Carlo (DMC) simulations, which are run using energies evaluated using the MOB-ML model. The combined NN+(MOB-ML) approach is used in variational calculations of the ground and low-lying vibrational excited states of water and in DMC calculations of the ground states of water, CH₅⁺, and its deuterated analogues. For both of these molecules, comparisons are made to the results obtained using potentials that were fit to much larger sets of electronic energies than were required to train the MOB-ML models. The NN+(MOB-ML) approach is also used to obtain a potential surface for C₂H₅⁺, which is a carbocation with a nonclassical equilibrium structure for which there is currently no available potential surface. This potential is used to explore the CH stretching vibrations, focusing on those of the bridging hydrogen atom. For both CH₅⁺ and C₂H₅⁺ the MOB-ML model is trained using geometries that were sampled from an AIMD trajectory, which was run at 350 K. By comparison, the structures sampled in the ground state calculations can have energies that are as much as ten times larger than those used to train the MOB-ML model. For water a higher temperature AIMD trajectory is needed to obtain accurate results due to the smaller thermal energy. A second MOB-ML model for C₂H₅⁺ was developed with additional higher energy structures in the training set. The two models are found to provide nearly identical descriptions of the ground state of C₂H₅⁺.

Representing Globally Accurate Reactive Potential Energy Surfaces with Complex Topography by Combining Gaussian Process Regression and Neural Network

There has been increasing attention in using machine learning technologies, such as neural network (NN) and Gaussian process regression (GPR), to model multidimensional potential energy surfaces (PESs). NN PES features...

Theoretical studies on triplet-state driven dissociation of formaldehyde by quasi-classical molecular dynamics simulation on machine-learning potential energy surface

The H-atom dissociation of formaldehyde on the lowest triplet state (T1) is studied by quasi-classical molecular dynamic simulations on the high-dimensional machine-learning potential energy surface (PES) model. An atomic-energy based deep-learning neural network (NN) is used to represent the PES function, and the weighted atom-centered symmetry functions are employed as inputs of the NN model to satisfy the translational, rotational, and permutational symmetries, and to capture the geometry features of each atom and its individual chemical environment. Several standard technical tricks are used in the construction of NN-PES, which includes the application of clustering algorithm in the formation of the training dataset, the examination of the reliability of the NN-PES model by different fitted NN models, and the detection of the out-of-confidence region by the confidence interval of the training dataset. The accuracy of the full-dimensional NN-PES model is examined by two benchmark calculations with respect to ab initio data. Both the NN and electronic-structure calculations give a similar H-atom dissociation reaction pathway on the T1 state in the intrinsic reaction coordinate analysis. The small-scaled trial dynamics simulations based on NN-PES and ab initio PES give highly consistent results. After confirming the accuracy of the NN-PES, a large number of trajectories are calculated in the quasi-classical dynamics, which allows us to get a better understanding of the T1-driven H-atom dissociation dynamics efficiently. Particularly, the dynamics simulations from different initial conditions can be easily simulated with a rather low computational cost. The influence of the mode-specific vibrational excitations on the H-atom dissociation dynamics driven by the T1 state is explored. The results show that the vibrational excitations on symmetric C-H stretching, asymmetric C-H stretching, and C=O stretching motions always enhance the H-atom dissociation probability obviously.

Choosing the right molecular machine learning potential

Quantum-chemistry simulations based on potential energy surfaces of molecules provide invaluable insight into the physicochemical processes at the atomistic level and yield such important observables as reaction rates and spectra. Machine learning potentials promise to significantly reduce the computational cost and hence enable otherwise unfeasible simulations. However, the surging number of such potentials begs the question of which one to choose or whether we still need to develop yet another one. Here, we address this question by evaluating the performance of popular machine learning potentials in terms of accuracy and computational cost. In addition, we deliver structured information for non-specialists in machine learning to guide them through the maze of acronyms, recognize each potential's main features, and judge what they could expect from each one.

Solving the Schrödinger equation using program synthesis

We demonstrate that a program synthesis approach based on a linear code representation can be used to generate algorithms that approximate the ground-state solutions of one-dimensional time-independent Schrödinger equations constructed with bound polynomial potential energy surfaces (PESs). Here, an algorithm is constructed as a linear series of instructions operating on a set of input vectors, matrices, and constants that define the problem characteristics, such as the PES. Discrete optimization is performed using simulated annealing in order to identify sequences of code-lines, operating on the program inputs that can reproduce the expected ground-state wavefunctions ψ(x) for a set of target PESs. The outcome of this optimization is not simply a mathematical function approximating ψ(x) but is, instead, a complete algorithm that converts the input vectors describing the system into a ground-state solution of the Schrödinger equation. These initial results point the way toward an alternative route for developing novel algorithms for quantum chemistry applications.

"pySiRC": Machine Learning Combined with Molecular Fingerprints to Predict the Reaction Rate Constant of the Radical-Based Oxidation Processes of Aqueous Organic Contaminants

We developed a web application structured in a machine learning and molecular fingerprint algorithm for the automatic calculation of the reaction rate constant of the oxidative processes of organic pollutants by ^•OH and SO₄^•- radicals in the aqueous phase-the pySiRC platform. The model development followed the OECD principles: internal and external validation, applicability domain, and mechanistic interpretation. Three machine learning algorithms combined with molecular fingerprints were evaluated, and all the models resulted in high goodness-of-fit for the training set with R² > 0.931 for the ^•OH radical and R² > 0.916 for the SO₄^•- radical and good predictive capacity for the test set with R_ext² = Q_ext² values in the range of 0.639-0.823 and 0.767-0.824 for the ^•OH and SO₄^•- radicals. The model was interpreted using the SHAP (SHapley Additive exPlanations) method: the results showed that the model developed made the prediction based on a reasonable understanding of how electron-withdrawing and -donating groups interfere with the reactivity of the ^•OH and SO₄^•- radicals. We hope that our models and web interface can stimulate and expand the application and interpretation of kinetic research on contaminants in water treatment units based on advanced oxidative technologies.

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.

Dynamically Polarizable Force Fields for Surface Simulations via Multi-output Classification Neural Networks

We present a general procedure to introduce electronic polarization into classical Molecular Dynamics (MD) force fields using a Neural Network (NN) model. We apply this framework to the simulation of a solid-liquid interface where the polarization of the surface is essential to correctly capture the main features of the system. By introducing a multi-input, multi-output NN and treating the surface polarization as a discrete classification problem, we are able to obtain very good accuracy in terms of quality of predictions. Through the definition of a custom loss function we are able to impose a physically motivated constraint within the NN itself making this model extremely versatile, especially in the modeling of different surface charge states. The NN is validated considering the redistribution of electronic charge density within a graphene based electrode in contact with an aqueous electrolyte solution, a system highly relevant to the development of next generation low-cost supercapacitors. We compare the performances of our NN/MD model against Quantum Mechanics/Molecular Dynamics simulations where we obtain a most satisfactory agreement.

Efficient Amino Acid Conformer Search with Bayesian Optimization

Finding low-energy molecular conformers is challenging due to the high dimensionality of the search space and the computational cost of accurate quantum chemical methods for determining conformer structures and energies. Here, we combine active-learning Bayesian optimization (BO) algorithms with quantum chemistry methods to address this challenge. Using cysteine as an example, we show that our procedure is both efficient and accurate. After only 1000 single-point calculations and approximately 80 structure relaxations, which is less than 10% computational cost of the current fastest method, we have found the low-energy conformers in good agreement with experimental measurements and reference calculations. To test the transferability of our method, we also repeated the conformer search of serine, tryptophan, and aspartic acid. The results agree well with previous conformer search studies.

Modeling microsolvation clusters with electronic-structure calculations guided by analytical potentials and predictive machine learning techniques

We propose a new methodology to study, at the density functional theory (DFT) level, the clusters resulting from the microsolvation of alkali-metal ions with rare-gas atoms. The workflow begins with a global optimization search to generate a pool of low-energy minimum structures for different cluster sizes. This is achieved by employing an analytical potential energy surface (PES) and an evolutionary algorithm (EA). The next main stage of the methodology is devoted to establish an adequate DFT approach to treat the microsolvation system, through a systematic benchmark study involving several combinations of functionals and basis sets, in order to characterize the global minimum structures of the smaller clusters. In the next stage, we apply machine learning (ML) classification algorithms to predict how the low-energy minima of the analytical PES map to the DFT ones. An early and accurate detection of likely DFT local minima is extremely important to guide the choice of the most promising low-energy minima of large clusters to be re-optimized at the DFT level of theory. In this work, the methodology was applied to the Li+Krn (n = 2-14 and 16) microsolvation clusters for which the most competitive DFT approach was found to be the B3LYP-D3/aug-pcseg-1. Additionally, the ML classifier was able to accurately predict most of the solutions to be re-optimized at the DFT level of theory, thereby greatly enhancing the efficiency of the process and allowing its applicability to larger clusters.

Mean-field density matrix decompositions

We introduce new and robust decompositions of mean-field Hartree-Fock and Kohn-Sham density functional theory relying on the use of localized molecular orbitals and physically sound charge population protocols. The new lossless property decompositions, which allow for partitioning one-electron reduced density matrices into either bond-wise or atomic contributions, are compared to alternatives from the literature with regard to both molecular energies and dipole moments. Besides commenting on possible applications as an interpretative tool in the rationalization of certain electronic phenomena, we demonstrate how decomposed mean-field theory makes it possible to expose and amplify compositional features in the context of machine-learned quantum chemistry. This is made possible by improving upon the granularity of the underlying data. On the basis of our preliminary proof-of-concept results, we conjecture that many of the structure-property inferences in existence today may be further refined by efficiently leveraging an increase in dataset complexity and richness.

On the synergy of matrix-isolation infrared spectroscopy and vibrational configuration interaction computations

The key feature of matrix-isolation infrared (MI-IR) spectroscopy is the isolation of single guest molecules in a host system at cryogenic conditions. The matrix mostly hinders rotation of the guest molecule, providing access to pure vibrational features. Vibrational self-consistent field (VSCF) and configuration interaction computations (VCI) on ab initio multimode potential energy surfaces (PES) give rise to anharmonic vibrational spectra. In a single-sourced combination of these experimental and computational approaches, we have established an iterative spectroscopic characterization procedure. The present article reviews the scope of this procedure by highlighting the strengths and limitations based on the examples of water, carbon dioxide, methane, methanol, and fluoroethane. An assessment of setups for the construction of the multimode PES on the example of methanol demonstrates that CCSD(T)-F12 level of theory is preferable to compute (a) accurate vibrational frequencies and (b) equilibrium or vibrationally averaged structural parameters. Our procedure has allowed us to uniquely assign unknown or disputed bands and enabled us to clarify problematic spectral regions that are crowded with combination bands and overtones. Besides spectroscopic assignment, the excellent agreement between theory and experiment paves the way to tackle questions of rather fundamental nature as to whether or not matrix effects are systematic, and it shows the limits of conventional notations used by spectroscopists.

Structure prediction of surface reconstructions by deep reinforcement learning

We demonstrate how image recognition and reinforcement learning combined may be used to determine the atomistic structure of reconstructed crystalline surfaces. A deep neural network represents a reinforcement learning agent that obtains training rewards by interacting with an environment. The environment contains a quantum mechanical potential energy evaluator in the form of a density functional theory program. The agent handles the 3D atomistic structure as a series of stacked 2D images and outputs the next atom type to place and the atomic site to occupy. Agents are seen to require 1000-10 000 single point density functional theory evaluations, to learn by themselves how to build the optimal surface reconstructions of anatase TiO2(001)-(1 × 4) and rutile SnO2(110)-(4 × 1).

Efficient sampling of high-energy states by machine learning force fields

A method extending the range of applicability of machine-learning force fields is proposed. It relies on biased subsampling of the high-energy states described by the predefined coordinate(s).