Multiscale Quantum Mechanics/Molecular Mechanics Simulations with Neural Networks

Modeling Intermolecular Interactions with Exchange-Hole Dipole Moment Dispersion Corrections to Neural Network Potentials

Neural network potentials (NNPs) are an innovative approach for calculating the potential energy and forces of a chemical system. In principle, these methods are capable of modeling large systems with an accuracy approaching that of a high-level ab initio calculation, but with a much smaller computational cost. Due to their training to density-functional theory (DFT) data and neglect of long-range interactions, some classes of NNPs require an additional term to include London dispersion physics. In this Perspective, we discuss the requirements for a dispersion model for use with an NNP, focusing on the MLXDM (Machine Learned eXchange-Hole Dipole Moment) model developed by our groups. This model is based on the DFT-based XDM dispersion correction, which calculates interatomic dispersion coefficients in terms of atomic moments and polarizabilities, both of which can be approximated effectively using neural networks.

Modelling chemical processes in explicit solvents with machine learning potentials

Solvent effects influence all stages of the chemical processes, modulating the stability of intermediates and transition states, as well as altering reaction rates and product ratios. However, accurately modelling these effects remains challenging. Here, we present a general strategy for generating reactive machine learning potentials to model chemical processes in solution. Our approach combines active learning with descriptor-based selectors and automation, enabling the construction of data-efficient training sets that span the relevant chemical and conformational space. We apply this strategy to investigate a Diels-Alder reaction in water and methanol. The generated machine learning potentials enable us to obtain reaction rates that are in agreement with experimental data and analyse the influence of these solvents on the reaction mechanism. Our strategy offers an efficient approach to the routine modelling of chemical reactions in solution, opening up avenues for studying complex chemical processes in an efficient manner.

Nanoscale chemical reaction exploration with a quantum magnifying glass

Nanoscopic systems exhibit diverse molecular substructures by which they facilitate specific functions. Theoretical models of them, which aim at describing, understanding, and predicting these capabilities, are difficult to build. Viable quantum-classical hybrid models come with specific challenges regarding atomistic structure construction and quantum region selection. Moreover, if their dynamics are mapped onto a state-to-state mechanism such as a chemical reaction network, its exhaustive exploration will be impossible due to the combinatorial explosion of the reaction space. Here, we introduce a "quantum magnifying glass" that allows one to interactively manipulate nanoscale structures at the quantum level. The quantum magnifying glass seamlessly combines autonomous model parametrization, ultra-fast quantum mechanical calculations, and automated reaction exploration. It represents an approach to investigate complex reaction sequences in a physically consistent manner with unprecedented effortlessness in real time. We demonstrate these features for reactions in bio-macromolecules and metal-organic frameworks, diverse systems that highlight general applicability.

Amber free energy tools: Interoperable software for free energy simulations using generalized quantum mechanical/molecular mechanical and machine learning potentials

We report the development and testing of new integrated cyberinfrastructure for performing free energy simulations with generalized hybrid quantum mechanical/molecular mechanical (QM/MM) and machine learning potentials (MLPs) in Amber. The Sander molecular dynamics program has been extended to leverage fast, density-functional tight-binding models implemented in the DFTB+ and xTB packages, and an interface to the DeePMD-kit software enables the use of MLPs. The software is integrated through application program interfaces that circumvent the need to perform "system calls" and enable the incorporation of long-range Ewald electrostatics into the external software's self-consistent field procedure. The infrastructure provides access to QM/MM models that may serve as the foundation for QM/MM-ΔMLP potentials, which supplement the semiempirical QM/MM model with a MLP correction trained to reproduce ab initio QM/MM energies and forces. Efficient optimization of minimum free energy pathways is enabled through a new surface-accelerated finite-temperature string method implemented in the FE-ToolKit package. Furthermore, we interfaced Sander with the i-PI software by implementing the socket communication protocol used in the i-PI client-server model. The new interface with i-PI allows for the treatment of nuclear quantum effects with semiempirical QM/MM-ΔMLP models. The modular interoperable software is demonstrated on proton transfer reactions in guanine-thymine mispairs in a B-form deoxyribonucleic acid helix. The current work represents a considerable advance in the development of modular software for performing free energy simulations of chemical reactions that are important in a wide range of applications.

Electronic and Nuclear Quantum Effects on Proton Transfer Reactions of Guanine-Thymine (G-T) Mispairs Using Combined Quantum Mechanical/Molecular Mechanical and Machine Learning Potentials

Rare tautomeric forms of nucleobases can lead to Watson-Crick-like (WC-like) mispairs in DNA, but the process of proton transfer is fast and difficult to detect experimentally. NMR studies show evidence for the existence of short-time WC-like guanine-thymine (G-T) mispairs; however, the mechanism of proton transfer and the degree to which nuclear quantum effects play a role are unclear. We use a B-DNA helix exhibiting a wGT mispair as a model system to study tautomerization reactions. We perform ab initio (PBE0/6-31G*) quantum mechanical/molecular mechanical (QM/MM) simulations to examine the free energy surface for tautomerization. We demonstrate that while the ab initio QM/MM simulations are accurate, considerable sampling is required to achieve high precision in the free energy barriers. To address this problem, we develop a QM/MM machine learning potential correction (QM/MM-ΔMLP) that is able to improve the computational efficiency, greatly extend the accessible time scales of the simulations, and enable practical application of path integral molecular dynamics to examine nuclear quantum effects. We find that the inclusion of nuclear quantum effects has only a modest effect on the mechanistic pathway but leads to a considerable lowering of the free energy barrier for the GT*⇌G*T equilibrium. Our results enable a rationalization of observed experimental data and the prediction of populations of rare tautomeric forms of nucleobases and rates of their interconversion in B-DNA.

Machine Learning of Reactive Potentials

In the past two decades, machine learning potentials (MLPs) have driven significant developments in chemical, biological, and material sciences. The construction and training of MLPs enable fast and accurate simulations and analysis of thermodynamic and kinetic properties. This review focuses on the application of MLPs to reaction systems with consideration of bond breaking and formation. We review the development of MLP models, primarily with neural network and kernel-based algorithms, and recent applications of reactive MLPs (RMLPs) to systems at different scales. We show how RMLPs are constructed, how they speed up the calculation of reactive dynamics, and how they facilitate the study of reaction trajectories, reaction rates, free energy calculations, and many other calculations. Different data sampling strategies applied in building RMLPs are also discussed with a focus on how to collect structures for rare events and how to further improve their performance with active learning.

Accelerating reliable multiscale quantum refinement of protein-drug systems enabled by machine learning

Biomacromolecule structures are essential for drug development and biocatalysis. Quantum refinement (QR) methods, which employ reliable quantum mechanics (QM) methods in crystallographic refinement, showed promise in improving the structural quality or even correcting the structure of biomacromolecules. However, vast computational costs and complex quantum mechanics/molecular mechanics (QM/MM) setups limit QR applications. Here we incorporate robust machine learning potentials (MLPs) in multiscale ONIOM(QM:MM) schemes to describe the core parts (e.g., drugs/inhibitors), replacing the expensive QM method. Additionally, two levels of MLPs are combined for the first time to overcome MLP limitations. Our unique MLPs+ONIOM-based QR methods achieve QM-level accuracy with significantly higher efficiency. Furthermore, our refinements provide computational evidence for the existence of bonded and nonbonded forms of the Food and Drug Administration (FDA)-approved drug nirmatrelvir in one SARS-CoV-2 main protease structure. This study highlights that powerful MLPs accelerate QRs for reliable protein-drug complexes, promote broader QR applications and provide more atomistic insights into drug development.

Training machine learning potentials for reactive systems: A Colab tutorial on basic models

In the last several years, there has been a surge in the development of machine learning potential (MLP) models for describing molecular systems. We are interested in a particular area of this field - the training of system-specific MLPs for reactive systems - with the goal of using these MLPs to accelerate free energy simulations of chemical and enzyme reactions. To help new members in our labs become familiar with the basic techniques, we have put together a self-guided Colab tutorial (https://cc-ats.github.io/mlp_tutorial/), which we expect to be also useful to other young researchers in the community. Our tutorial begins with the introduction of simple feedforward neural network (FNN) and kernel-based (using Gaussian process regression, GPR) models by fitting the two-dimensional Müller-Brown potential. Subsequently, two simple descriptors are presented for extracting features of molecular systems: symmetry functions (including the ANI variant) and embedding neural networks (such as DeepPot-SE). Lastly, these features will be fed into FNN and GPR models to reproduce the energies and forces for the molecular configurations in a Claisen rearrangement reaction.

Perspectives on Computational Enzyme Modeling: From Mechanisms to Design and Drug Development

Understanding enzyme mechanisms is essential for unraveling the complex molecular machinery of life. In this review, we survey the field of computational enzymology, highlighting key principles governing enzyme mechanisms and discussing ongoing challenges and promising advances. Over the years, computer simulations have become indispensable in the study of enzyme mechanisms, with the integration of experimental and computational exploration now established as a holistic approach to gain deep insights into enzymatic catalysis. Numerous studies have demonstrated the power of computer simulations in characterizing reaction pathways, transition states, substrate selectivity, product distribution, and dynamic conformational changes for various enzymes. Nevertheless, significant challenges remain in investigating the mechanisms of complex multistep reactions, large-scale conformational changes, and allosteric regulation. Beyond mechanistic studies, computational enzyme modeling has emerged as an essential tool for computer-aided enzyme design and the rational discovery of covalent drugs for targeted therapies. Overall, enzyme design/engineering and covalent drug development can greatly benefit from our understanding of the detailed mechanisms of enzymes, such as protein dynamics, entropy contributions, and allostery, as revealed by computational studies. Such a convergence of different research approaches is expected to continue, creating synergies in enzyme research. This review, by outlining the ever-expanding field of enzyme research, aims to provide guidance for future research directions and facilitate new developments in this important and evolving field.

Reference-Quality Free Energy Barriers in Catalysis from Machine Learning Thermodynamic Perturbation Theory

For the first time, we report calculations of the free energies of activation of cracking and isomerization reactions of alkenes that combine several different electronic structure methods with molecular dynamics simulations. We demonstrate that the use of a high level of theory (here Random Phase Approximation-RPA) is necessary to bridge the gap between experimental and computed values. These transformations, catalyzed by zeolites and proceeding via cationic intermediates and transition states, are building blocks of many chemical transformations for valorization of long chain paraffins originating, e.g., from plastic waste, vegetable oils, Fischer-Tropsch waxes or crude oils. Compared with the free energy barriers computed at the PBE+D2 production level of theory via constrained ab initio molecular dynamics, the barriers computed at the RPA level by the application of Machine Learning thermodynamic Perturbation Theory (MLPT) show a significant decrease for isomerization reaction and an increase of a similar magnitude for cracking, yielding an unprecedented agreement with the results obtained by experiments and kinetic modeling.

Development of a machine learning finite-range nonlocal density functional

Kohn-Sham density functional theory has been the most popular method in electronic structure calculations. To fulfill the increasing accuracy requirements, new approximate functionals are needed to address key issues in existing approximations. It is well known that nonlocal components are crucial. Current nonlocal functionals mostly require orbital dependence such as in Hartree-Fock exchange and many-body perturbation correlation energy, which, however, leads to higher computational costs. Deviating from this pathway, we describe functional nonlocality in a new approach. By partitioning the total density to atom-centered local densities, a many-body expansion is proposed. This many-body expansion can be truncated at one-body contributions, if a base functional is used and an energy correction is approximated. The contribution from each atom-centered local density is a single finite-range nonlocal functional that is universal for all atoms. We then use machine learning to develop this universal atom-centered functional. Parameters in this functional are determined by fitting to data that are produced by high-level theories. Extensive tests on several different test sets, which include reaction energies, reaction barrier heights, and non-covalent interaction energies, show that the new functional, with only the density as the basic variable, can produce results comparable to the best-performing double-hybrid functionals, (for example, for the thermochemistry test set selected from the GMTKN55 database, BLYP based machine learning functional gives a weighted total mean absolute deviations of 3.33 kcal/mol, while DSD-BLYP-D3(BJ) gives 3.28 kcal/mol) with a lower computational cost. This opens a new pathway to nonlocal functional development and applications.

Free energies at QM accuracy from force fields via multimap targeted estimation

Accurate predictions of ligand binding affinities would greatly accelerate the first stages of drug discovery campaigns. However, using highly accurate interatomic potentials based on quantum mechanics (QM) in free energy methods has been so far largely unfeasible due to their prohibitive computational cost. Here, we present an efficient method to compute QM free energies from simulations using cheap reference potentials, such as force fields (FFs). This task has traditionally been out of reach due to the slow convergence of computing the correction from the FF to the QM potential. To overcome this bottleneck, we generalize targeted free energy methods to employ multiple maps-implemented with normalizing flow neural networks (NNs)-that maximize the overlap between the distributions. Critically, the method requires neither a separate expensive training phase for the NNs nor samples from the QM potential. We further propose a one-epoch learning policy to efficiently avoid overfitting, and we combine our approach with enhanced sampling strategies to overcome the pervasive problem of poor convergence due to slow degrees of freedom. On the drug-like molecules in the HiPen dataset, the method accelerates the calculation of the free energy difference of switching from an FF to a DFTB3 potential by three orders of magnitude compared to standard free energy perturbation and by a factor of eight compared to previously published nonequilibrium calculations. Our results suggest that our method, in combination with efficient QM/MM calculations, may be used in lead optimization campaigns in drug discovery and to study protein-ligand molecular recognition processes.

Accurate and Efficient Multilevel Free Energy Simulations with Neural Network-Assisted Enhanced Sampling

Free energy differences (ΔF) are essential to quantitative characterization and understanding of chemical and biological processes. Their direct estimation with an accurate quantum mechanical potential is of great interest and yet impractical due to high computational cost and incompatibility with typical alchemical free energy protocols. One promising solution is the multilevel free energy simulation in which the estimate of ΔF at an inexpensive low level of theory is combined with the correction toward a higher level of theory. The poor configurational overlap generally expected between the two levels of theory, however, presents a major challenge. We overcome this challenge by using a deep neural network model and enhanced sampling simulations. An adversarial autoencoder is used to identify a low-dimensional (latent) space that compactly represents the degrees of freedom that encode the distinct distributions at the two levels of theory. Enhanced sampling in this latent space is then used to drive the sampling of configurations that predominantly contribute to the free energy correction. Results for both gas phase and condensed phase systems demonstrate that this data-driven approach offers high accuracy and efficiency with great potential for scalability to complex systems.

Bridging semiempirical and ab initio QM/MM potentials by Gaussian process regression and its sparse variants for free energy simulation

Free energy simulations that employ combined quantum mechanical and molecular mechanical (QM/MM) potentials at ab initio QM (AI) levels are computationally highly demanding. Here, we present a machine-learning-facilitated approach for obtaining AI/MM-quality free energy profiles at the cost of efficient semiempirical QM/MM (SE/MM) methods. Specifically, we use Gaussian process regression (GPR) to learn the potential energy corrections needed for an SE/MM level to match an AI/MM target along the minimum free energy path (MFEP). Force modification using gradients of the GPR potential allows us to improve configurational sampling and update the MFEP. To adaptively train our model, we further employ the sparse variational GP (SVGP) and streaming sparse GPR (SSGPR) methods, which efficiently incorporate previous sample information without significantly increasing the training data size. We applied the QM-(SS)GPR/MM method to the solution-phase SN2 Menshutkin reaction, NH3+CH3Cl→CH3NH3++Cl-, using AM1/MM and B3LYP/6-31+G(d,p)/MM as the base and target levels, respectively. For 4000 configurations sampled along the MFEP, the iteratively optimized AM1-SSGPR-4/MM model reduces the energy error in AM1/MM from 18.2 to 4.4 kcal/mol. Although not explicitly fitting forces, our method also reduces the key internal force errors from 25.5 to 11.1 kcal/mol/Å and from 30.2 to 10.3 kcal/mol/Å for the N-C and C-Cl bonds, respectively. Compared to the uncorrected simulations, the AM1-SSGPR-4/MM method lowers the predicted free energy barrier from 28.7 to 11.7 kcal/mol and decreases the reaction free energy from -12.4 to -41.9 kcal/mol, bringing these results into closer agreement with their AI/MM and experimental benchmarks.

Toward a general neural network force field for protein simulations: Refining the intramolecular interaction in protein

Molecular dynamics (MD) is an extremely powerful, highly effective, and widely used approach to understanding the nature of chemical processes in atomic details for proteins. The accuracy of results from MD simulations is highly dependent on force fields. Currently, molecular mechanical (MM) force fields are mainly utilized in MD simulations because of their low computational cost. Quantum mechanical (QM) calculation has high accuracy, but it is exceedingly time consuming for protein simulations. Machine learning (ML) provides the capability for generating accurate potential at the QM level without increasing much computational effort for specific systems that can be studied at the QM level. However, the construction of general machine learned force fields, needed for broad applications and large and complex systems, is still challenging. Here, general and transferable neural network (NN) force fields based on CHARMM force fields, named CHARMM-NN, are constructed for proteins by training NN models on 27 fragments partitioned from the residue-based systematic molecular fragmentation (rSMF) method. The NN for each fragment is based on atom types and uses new input features that are similar to MM inputs, including bonds, angles, dihedrals, and non-bonded terms, which enhance the compatibility of CHARMM-NN to MM MD and enable the implementation of CHARMM-NN force fields in different MD programs. While the main part of the energy of the protein is based on rSMF and NN, the nonbonded interactions between the fragments and with water are taken from the CHARMM force field through mechanical embedding. The validations of the method for dipeptides on geometric data, relative potential energies, and structural reorganization energies demonstrate that the CHARMM-NN local minima on the potential energy surface are very accurate approximations to QM, showing the success of CHARMM-NN for bonded interactions. However, the MD simulations on peptides and proteins indicate that more accurate methods to represent protein-water interactions in fragments and non-bonded interactions between fragments should be considered in the future improvement of CHARMM-NN, which can increase the accuracy of approximation beyond the current mechanical embedding QM/MM level.

The confluence of machine learning and multiscale simulations

Multiscale modeling has a long history of use in structural biology, as computational biologists strive to overcome the time- and length-scale limits of atomistic molecular dynamics. Contemporary machine learning techniques, such as deep learning, have promoted advances in virtually every field of science and engineering and are revitalizing the traditional notions of multiscale modeling. Deep learning has found success in various approaches for distilling information from fine-scale models, such as building surrogate models and guiding the development of coarse-grained potentials. However, perhaps its most powerful use in multiscale modeling is in defining latent spaces that enable efficient exploration of conformational space. This confluence of machine learning and multiscale simulation with modern high-performance computing promises a new era of discovery and innovation in structural biology.

The application of QM/MM simulations in heterogeneous catalysis

The QM/MM simulation method is provenly efficient for the simulation of biological systems, where an interplay of extensive environment and delicate local interactions drives a process of interest through a funnel on a complex energy landscape. Recent advances in quantum chemistry and force-field methods present opportunities for the adoption of QM/MM to simulate heterogeneous catalytic processes, and their related systems, where similar intricacies exist on the energy landscape. Herein, the fundamental theoretical considerations for performing QM/MM simulations, and the practical considerations for setting up QM/MM simulations of catalytic systems, are introduced; then, areas of heterogeneous catalysis are explored where QM/MM methods have been most fruitfully applied. The discussion includes simulations performed for adsorption processes in solvent at metallic interfaces, reaction mechanisms within zeolitic systems, nanoparticles, and defect chemistry within ionic solids. We conclude with a perspective on the current state of the field and areas where future opportunities for development and application exist.

Accelerated Quantum Mechanics/Molecular Mechanics Simulations via Neural Networks Incorporated with Mechanical Embedding Scheme

A powerful tool to study the mechanism of reactions in solutions or enzymes is to perform the ab initio quantum mechanical/molecular mechanical (QM/MM) molecular dynamics (MD) simulations. However, the computational cost is too high due to the explicit electronic structure calculations at every time step of the simulation. A neural network (NN) method can accelerate the QM/MM-MD simulations, but it has long been a problem to accurately describe the QM/MM electrostatic coupling by NN in the electrostatic embedding (EE) scheme. In this work, we developed a new method to accelerate QM/MM calculations in the mechanic embedding (ME) scheme. The potentials and partial point charges of QM atoms are first learned in vacuo by the embedded atom neural networks (EANN) approach. MD simulations are then performed on this EANN/MM potential energy surface (PES) to obtain free energy (FE) profiles for reactions, in which the QM/MM electrostatic coupling is treated in the mechanic embedding (ME) scheme. Finally, a weighted thermodynamic perturbation (wTP) corrects the FE profiles in the ME scheme to the EE scheme. For two reactions in water and one in methanol, our simulations reproduced the B3LYP/MM free energy profiles within 0.5 kcal/mol with a speed-up of 30-60-fold. The results show that the strategy of combining EANN potential in the ME scheme with the wTP correction is efficient and reliable for chemical reaction simulations in liquid. Another advantage of our method is that the QM PES is independent of the MM subsystem, so it can be applied to various MM environments as demonstrated by an S_N2 reaction studied in water and methanol individually, which used the same EANN PES. The free energy profiles are in excellent accordance with the results obtained from B3LYP/MM-MD simulations. In future, this method will be applied to the reactions of enzymes and their variants.

Machine Learning Exciton Hamiltonians in Light-Harvesting Complexes

We propose a machine learning (ML)-based strategy for an inexpensive calculation of excitonic properties of light-harvesting complexes (LHCs). The strategy uses classical molecular dynamics simulations of LHCs in their natural environment in combination with ML prediction of the excitonic Hamiltonian of the embedded aggregate of pigments. The proposed ML model can reproduce the effects of geometrical fluctuations together with those due to electrostatic and polarization interactions between the pigments and the protein. The training is performed on the chlorophylls of the major LHC of plants, but we demonstrate that the model is able to extrapolate well beyond the initial training set. Moreover, the accuracy in predicting the effects of the environment is tested on the simulation of the small changes observed in the absorption spectra of the wild-type and a mutant of a minor LHC.

Protein dynamics: The future is bright and complicated!

Biological life depends on motion, and this manifests itself in proteins that display motion over a formidable range of time scales spanning from femtoseconds vibrations of atoms at enzymatic transition states, all the way to slow domain motions occurring on micro to milliseconds. An outstanding challenge in contemporary biophysics and structural biology is a quantitative understanding of the linkages among protein structure, dynamics, and function. These linkages are becoming increasingly explorable due to conceptual and methodological advances. In this Perspective article, we will point toward future directions of the field of protein dynamics with an emphasis on enzymes. Research questions in the field are becoming increasingly complex such as the mechanistic understanding of high-order interaction networks in allosteric signal propagation through a protein matrix, or the connection between local and collective motions. In analogy to the solution to the "protein folding problem," we argue that the way forward to understanding these and other important questions lies in the successful integration of experiment and computation, while utilizing the present rapid expansion of sequence and structure space. Looking forward, the future is bright, and we are in a period where we are on the doorstep to, at least in part, comprehend the importance of dynamics for biological function.

Computer-aided drug design, quantum-mechanical methods for biological problems

Quantum chemistry enables to study systems with chemical accuracy (<1 kcal/mol from experiment) but is restricted to a handful of atoms due to its computational expense. This has led to ongoing interest to optimize and simplify these methods while retaining accuracy. Implementing quantum mechanical (QM) methods on modern hardware such as multiple-GPUs is one example of how the field is optimizing performance. Multiscale approaches like the so-called QM/molecular mechanical method are gaining popularity in drug discovery because they focus the application of QM methods on the region of choice (e.g., the binding site), while using efficient MM models to represent less relevant areas. The creation of simplified QM methods is another example, including the use of machine learning to create ultra-fast and accurate QM models. Herein, we summarize recent advancements in the development of optimized QM methods that enhance our ability to use these methods in computer aided drug discovery.

BuRNN: Buffer Region Neural Network Approach for Polarizable-Embedding Neural Network/Molecular Mechanics Simulations

Hybrid quantum mechanics/molecular mechanics (QM/MM) simulations have advanced the field of computational chemistry tremendously. However, they require the partitioning of a system into two different regions that are treated at different levels of theory, which can cause artifacts at the interface. Furthermore, they are still limited by high computational costs of quantum chemical calculations. In this work, we develop the buffer region neural network (BuRNN), an alternative approach to existing QM/MM schemes, which introduces a buffer region that experiences full electronic polarization by the inner QM region to minimize artifacts. The interactions between the QM and the buffer region are described by deep neural networks (NNs), which leads to the high computational efficiency of this hybrid NN/MM scheme while retaining quantum chemical accuracy. We demonstrate the BuRNN approach by performing NN/MM simulations of the hexa-aqua iron complex.

Constructing Features Using a Hybrid Genetic Algorithm

A hybrid procedure that incorporates grammatical evolution and a weight decaying technique is proposed here for various classification and regression problems. The proposed method has two main phases: the creation of features and the evaluation of these features. During the first phase, using grammatical evolution, new features are created as non-linear combinations of the original features of the datasets. In the second phase, based on the characteristics of the first phase, the original dataset is modified and a neural network trained with a genetic algorithm is applied to this dataset. The proposed method was applied to an extremely wide set of datasets from the relevant literature and the experimental results were compared with four other techniques.

Assessing the persistence of chalcogen bonds in solution with neural network potentials

Non-covalent bonding patterns are commonly harvested as a design principle in the field of catalysis, supramolecular chemistry and functional materials to name a few. Yet, their computational description generally neglects finite temperature and environment effects, which promote competing interactions and alter their static gas-phase properties. Recently, neural network potentials (NNPs) trained on Density Functional Theory (DFT) data have become increasingly popular to simulate molecular phenomena in condensed phase with an accuracy comparable to ab initio methods. To date, most applications have centered on solid-state materials or fairly simple molecules made of a limited number of elements. Herein, we focus on the persistence and strength of chalcogen bonds involving a benzotelluradiazole in condensed phase. While the tellurium-containing heteroaromatic molecules are known to exhibit pronounced interactions with anions and lone pairs of different atoms, the relevance of competing intermolecular interactions, notably with the solvent, is complicated to monitor experimentally but also challenging to model at an accurate electronic structure level. Here, we train direct and baselined NNPs to reproduce hybrid DFT energies and forces in order to identify what are the most prevalent non-covalent interactions occurring in a solute-Cl$^-$-THF mixture. The simulations in explicit solvent highlight competition with chalcogen bonds formed with the solvent and the short-range directionality of the interaction with direct consequences for the molecular properties in the solution. The comparison with other potentials (e.g., AMOEBA, direct NNP and continuum solvent model) also demonstrates that baselined NNPs offer a reliable picture of the non-covalent interaction interplay occurring in solution.

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.

Progress on Exploring the Luminescent Properties of Organic Molecular Aggregates by Multiscale Modeling

Luminescent molecular aggregates have attracted worldwide attention because of their potential applications in many fields. The luminescent properties of organic aggregates are complicated and highly morphology-dependent, unraveling the intrinsic mechanism behind is urgent. This review summarizes recent works on investigating the structure-property relationships of organic molecular aggregates at different environments, including crystal, cocrystal, amorphous aggregate, and doped systems by multiscale modeling protocol. We aim to explore the influence of intermolecular non-covalent interactions on molecular packing and their photophysical properties and then pave the effective way to design, synthesize, and develop advanced organic luminescent materials.

Affordable Ab Initio Path Integral for Thermodynamic Properties via Molecular Dynamics Simulations Using Semiempirical Reference Potential

Path integral molecular dynamics (PIMD) is becoming a routinely applied method for incorporating the nuclear quantum effect in computer simulations. However, direct PIMD simulations at an ab initio level of theory are formidably expensive. Using the protonated 1,8-bis(dimethylamino)naphthalene molecule as an example, we show in this work that the computational expense for the intramolecular proton transfer between the two nitrogen atoms can be remarkably reduced by implementing the idea of reference-potential methods. The simulation time can be easily extended to a scale of nanoseconds while maintaining the accuracy on an ab initio level of theory for thermodynamic properties. In addition, postprocessing can be carried out in parallel on massive computer nodes. A 545-fold reduction in the total CPU time can be achieved in this way as compared to a direct PIMD simulation at the same ab initio level of theory.

Doubly Polarized QM/MM with Machine Learning Chaperone Polarizability

A major shortcoming of semiempirical (SE) molecular orbital methods is their severe underestimation of molecular polarizability compared with experimental and ab initio (AI) benchmark data. In a combined quantum mechanical and molecular mechanical (QM/MM) treatment of solution-phase reactions, solute described by SE methods therefore tends to generate inadequate electronic polarization response to solvent electric fields, which often leads to large errors in free energy profiles. To address this problem, here we present a hybrid framework that improves the response property of SE/MM methods through high-level molecular-polarizability fitting. Specifically, we place on QM atoms a set of corrective polarizabilities (referred to as chaperone polarizabilities), whose magnitudes are determined from machine learning (ML) to reproduce the condensed-phase AI molecular polarizability along the minimum free energy path. These chaperone polarizabilities are then used in a machinery similar to a polarizable force field calculation to compensate for the missing polarization energy in the conventional SE/MM simulations. Because QM atoms in this treatment host SE wave functions as well as classical polarizabilities, both polarized by MM electric fields, we name this method doubly polarized QM/MM (dp-QM/MM). We demonstrate the new method on the free energy simulations of the Menshutkin reaction in water. Using AM1/MM as a base method, we show that ML chaperones greatly reduce the error in the solute molecular polarizability from 6.78 to 0.03 ų with respect to the density functional theory benchmark. The chaperone correction leads to ∼10 kcal/mol of additional polarization energy in the product region, bringing the simulated free energy profiles to closer agreement with the experimental results. Furthermore, the solute-solvent radial distribution functions show that the chaperone polarizabilities modify the free energy profiles through enhanced solvation corrections when the system evolves from the charge-neutral reactant state to the charge-separated transition and product states. These results suggest that the dp-QM/MM method, enabled by ML chaperone polarizabilities, provides a very physical remedy for the underpolarization problem in SE/MM-based free energy simulations.

Development of Range-Corrected Deep Learning Potentials for Fast, Accurate Quantum Mechanical/Molecular Mechanical Simulations of Chemical Reactions in Solution

We develop a new deep potential─range correction (DPRc) machine learning potential for combined quantum mechanical/molecular mechanical (QM/MM) simulations of chemical reactions in the condensed phase. The new range correction enables short-ranged QM/MM interactions to be tuned for higher accuracy, and the correction smoothly vanishes within a specified cutoff. We further develop an active learning procedure for robust neural network training. We test the DPRc model and training procedure against a series of six nonenzymatic phosphoryl transfer reactions in solution that are important in mechanistic studies of RNA-cleaving enzymes. Specifically, we apply DPRc corrections to a base QM model and test its ability to reproduce free-energy profiles generated from a target QM model. We perform these comparisons using the MNDO/d and DFTB2 semiempirical models because they differ in the way they treat orbital orthogonalization and electrostatics and produce free-energy profiles which differ significantly from each other, thereby providing us a rigorous stress test for the DPRc model and training procedure. The comparisons show that accurate reproduction of the free-energy profiles requires correction of the QM/MM interactions out to 6 Å. We further find that the model's initial training benefits from generating data from temperature replica exchange simulations and including high-temperature configurations into the fitting procedure, so the resulting models are trained to properly avoid high-energy regions. A single DPRc model was trained to reproduce four different reactions and yielded good agreement with the free-energy profiles made from the target QM/MM simulations. The DPRc model was further demonstrated to be transferable to 2D free-energy surfaces and 1D free-energy profiles that were not explicitly considered in the training. Examination of the computational performance of the DPRc model showed that it was fairly slow when run on CPUs but was sped up almost 100-fold when using NVIDIA V100 GPUs, resulting in almost negligible overhead. The new DPRc model and training procedure provide a potentially powerful new tool for the creation of next-generation QM/MM potentials for a wide spectrum of free-energy applications ranging from drug discovery to enzyme design.

Targeted Free Energy Perturbation Revisited: Accurate Free Energies from Mapped Reference Potentials

We present an approach that extends the theory of targeted free energy perturbation (TFEP) to calculate free energy differences and free energy surfaces at an accurate quantum mechanical level of theory from a cheaper reference potential. The convergence is accelerated by a mapping function that increases the overlap between the target and the reference distributions. Building on recent work, we show that this map can be learned with a normalizing flow neural network, without requiring simulations with the expensive target potential but only a small number of single-point calculations, and, crucially, avoiding the systematic error that was found previously. We validate the method by numerically evaluating the free energy difference in a system with a double-well potential and by describing the free energy landscape of a simple chemical reaction in the gas phase.

Machine-Learning-Assisted Free Energy Simulation of Solution-Phase and Enzyme Reactions

Despite recent advances in the development of machine learning potentials (MLPs) for biomolecular simulations, there has been limited effort on developing stable and accurate MLPs for enzymatic reactions. Here we report a protocol for performing machine-learning-assisted free energy simulation of solution-phase and enzyme reactions at the ab initio quantum-mechanical/molecular-mechanical (ai-QM/MM) level of accuracy. Within our protocol, the MLP is built to reproduce the ai-QM/MM energy and forces on both QM (reactive) and MM (solvent/enzyme) atoms. As an alternative strategy, a delta machine learning potential (ΔMLP) is trained to reproduce the differences between the ai-QM/MM and semiempirical (se) QM/MM energies and forces. To account for the effect of the condensed-phase environment in both MLP and ΔMLP, the DeePMD representation of a molecular system is extended to incorporate the external electrostatic potential and field on each QM atom. Using the Menshutkin and chorismate mutase reactions as examples, we show that the developed MLP and ΔMLP reproduce the ai-QM/MM energy and forces with errors that on average are less than 1.0 kcal/mol and 1.0 kcal mol-1 Å-1, respectively, for representative configurations along the reaction pathway. For both reactions, MLP/ΔMLP-based simulations yielded free energy profiles that differed by less than 1.0 kcal/mol from the reference ai-QM/MM results at only a fraction of the computational cost.

General Many-Body Framework for Data-Driven Potentials with Arbitrary Quantum Mechanical Accuracy: Water as a Case Study

We present a general framework for the development of data-driven many-body (MB) potential energy functions (MB-QM PEFs) that represent the interactions between small molecules at an arbitrary quantum-mechanical (QM) level of theory. As a demonstration, a family of MB-QM PEFs for water is rigorously derived from density functionals belonging to different rungs across Jacob's ladder of approximations within density functional theory (MB-DFT) and from Møller-Plesset perturbation theory (MB-MP2). Through a systematic analysis of individual MB contributions to the interaction energies of water clusters, we demonstrate that all MB-QM PEFs preserve the same accuracy as the corresponding ab initio calculations, with the exception of those derived from density functionals within the generalized gradient approximation (GGA). The differences between the DFT and MB-DFT results are traced back to density-driven errors that prevent GGA functionals from accurately representing the underlying molecular interactions for different cluster sizes and hydrogen-bonding arrangements. We show that this shortcoming may be overcome, within the MB formalism, by using density-corrected functionals (DC-DFT) that provide a more consistent representation of each individual MB contribution. This is demonstrated through the development of a MB-DFT PEF derived from DC-PBE-D3 data, which more accurately reproduce the corresponding ab initio results.

Liquid-Liquid Critical Point in Phosphorus

The study of liquid-liquid phase transitions has attracted considerable attention. One interesting example of this phenomenon is phosphorus, for which the existence of a first-order phase transition between a low density insulating molecular phase and a conducting polymeric phase has been experimentally established. In this Letter, we model this transition by an ab initio quality molecular dynamics simulation and explore a large portion of the liquid section of the phase diagram. We draw the liquid-liquid coexistence curve and determine that it terminates into a second-order critical point. Close to the critical point, large coupled structure and electronic structure fluctuations are observed.

Reaction Path-Force Matching in Collective Variables: Determining Ab Initio QM/MM Free Energy Profiles by Fitting Mean Force

First-principles determination of free energy profiles for condensed-phase chemical reactions is hampered by the daunting costs associated with configurational sampling on ab initio quantum mechanical/molecular mechanical (AI/MM) potential energy surfaces. Here, we report a new method that enables efficient AI/MM free energy simulations through mean force fitting. In this method, a free energy path in collective variables (CVs) is first determined on an efficient reactive aiding potential. Based on the configurations sampled along the free energy path, correcting forces to reproduce the AI/MM forces on the CVs are determined through force matching. The AI/MM free energy profile is then predicted from simulations on the aiding potential in conjunction with the correcting forces. Such cycles of correction-prediction are repeated until convergence is established. As the instantaneous forces on the CVs sampled in equilibrium ensembles along the free energy path are fitted, this procedure faithfully restores the target free energy profile by reproducing the free energy mean forces. Due to its close connection with the reaction path-force matching (RP-FM) framework recently introduced by us, we designate the new method as RP-FM in collective variables (RP-FM-CV). We demonstrate the effectiveness of this method on a type-II solution-phase SN2 reaction, NH3 + CH3Cl (the Menshutkin reaction), simulated with an explicit water solvent. To obtain the AI/MM free energy profiles, we employed the semiempirical AM1/MM Hamiltonian as the base level for determining the string minimum free energy pathway, along which the free energy mean forces are fitted to various target AI/MM levels using the Hartree-Fock (HF) theory, density functional theory (DFT), and the second-order Møller-Plesset perturbation (MP2) theory as the AI method. The forces on the bond-breaking and bond-forming CVs at both the base and target levels are obtained by force transformation from Cartesian to redundant internal coordinates under the Wilson B-matrix formalism, where the linearized FM is facilitated by the use of spline functions. For the Menshutkin reaction tested, our FM treatment greatly reduces the deviations on the CV forces, originally in the range of 12-33 to ∼2 kcal/mol/Å. Comparisons with the experimental and benchmark AI/MM results, tests of the new method under a variety of simulation protocols, and analyses of the solute-solvent radial distribution functions suggest that RP-FM-CV can be used as an efficient, accurate, and robust method for simulating solution-phase chemical reactions.

Accelerating Metadynamics-Based Free-Energy Calculations with Adaptive Machine Learning Potentials

There is an increasing demand for free-energy calculations using ab initio molecular dynamics these days. Metadynamics (MetaD) is frequently utilized to reconstruct the free-energy surface, but it is often computationally intractable for the first-principles calculations. Machine learning potentials (MLPs) have become popular alternatives. However, the training could be a long and arduous process before using them in practical applications. To accelerate MetaD use with MLPs for the free-energy calculation in an easy manner, we propose the adaptive machine learning potential-accelerated metadynamics (AMLP-MetaD). In this method, the MLP in the form of a Gaussian approximation potential (GAP) can adapt itself based on its uncertainty estimation, which decides whether to accept the model prediction or recalculate it with a reference method (usually density functional theory) for further training during the MetaD simulation. We demonstrate that the free-energy landscape similar to the ab initio one can be obtained using AMLP-MetaD with a 10-time speedup. Moreover, the quality of the free-energy results can be deeply improved using Δ-MLP, which is the GAP-corrected density functional tight binding in our case. We exemplify this novel method with two model systems, CO adsorption on the Pt₁₃ cluster and the Pt(111) surface, which are of vital importance in heterogeneous catalysis. The successful application in these two tests highlights that our proposed method can be used in both cluster and periodic systems and for up to two collective variables.

Biomolecular modeling thrives in the age of technology

The biomolecular modeling field has flourished since its early days in the 1970s due to the rapid adaptation and tailoring of state-of-the-art technology. The resulting dramatic increase in size and timespan of biomolecular simulations has outpaced Moore's law. Here, we discuss the role of knowledge-based versus physics-based methods and hardware versus software advances in propelling the field forward. This rapid adaptation and outreach suggests a bright future for modeling, where theory, experimentation and simulation define three pillars needed to address future scientific and biomedical challenges.

Machine Learning in QM/MM Molecular Dynamics Simulations of Condensed-Phase Systems

Quantum mechanics/molecular mechanics (QM/MM) molecular dynamics (MD) simulations have been developed to simulate molecular systems, where an explicit description of changes in the electronic structure is necessary. However, QM/MM MD simulations are computationally expensive compared to fully classical simulations as all valence electrons are treated explicitly and a self-consistent field (SCF) procedure is required. Recently, approaches have been proposed to replace the QM description with machine-learned (ML) models. However, condensed-phase systems pose a challenge for these approaches due to long-range interactions. Here, we establish a workflow, which incorporates the MM environment as an element type in a high-dimensional neural network potential (HDNNP). The fitted HDNNP describes the potential-energy surface of the QM particles with an electrostatic embedding scheme. Thus, the MM particles feel a force from the polarized QM particles. To achieve chemical accuracy, we find that even simple systems require models with a strong gradient regularization, a large number of data points, and a substantial number of parameters. To address this issue, we extend our approach to a Δ-learning scheme, where the ML model learns the difference between a reference method (density functional theory (DFT)) and a cheaper semiempirical method (density functional tight binding (DFTB)). We show that such a scheme reaches the accuracy of the DFT reference method while requiring significantly less parameters. Furthermore, the Δ-learning scheme is capable of correctly incorporating long-range interactions within a cutoff of 1.4 nm. It is validated by performing MD simulations of retinoic acid in water and the interaction between S-adenoslymethioniat and cytosine in water. The presented results indicate that Δ-learning is a promising approach for (QM)ML/MM MD simulations of condensed-phase systems.

Accelerated Computation of Free Energy Profile at Ab Initio Quantum Mechanical/Molecular Mechanics Accuracy via a Semiempirical Reference Potential. 4. Adaptive QM/MM

Although quantum mechanical/molecular mechanics (QM/MM) methods are now routinely applied to the studies of chemical reactions in condensed phases and enzymatic reactions, they may experience technical difficulties when the reactive region is varying over time. For instance, when the solvent molecules are directly participating in the reaction, the exchange of water molecules between the QM and MM regions may occur on a time scale comparable to the reaction time. To cope with this situation, several adaptive QM/MM schemes have been proposed. However, these methods either add significantly to the computational cost or introduce artificial restraints to the system. In this work, we developed a novel adaptive QM/MM scheme and applied it to the study of a nucleophilic addition reaction. In this scheme, the configuration sampling was performed with a small QM region (without solvent molecules), and the thermodynamic properties under another potential energy function with a larger QM region (with a certain number of solvent molecules and/or different levels of QM theory) are computed via extrapolation using the reference-potential method. Our simulation results show that this adaptive QM/MM scheme is numerically stable, at least for the case studied in this work. Furthermore, this method also offers an inexpensive way to examine the convergence of the QM/MM calculation with respect to the size of the QM region.

What Does the Brønsted Slope Measure in the Phosphoryl Transfer Transition State?

The structural and energetic features of phosphate and phosphonate hydrolysis in Protein Phosphatase-1 (PP1) and water are studied using quantum mechanical (QM) cluster models. The calculations are able to reproduce observed kinetic isotope effects and capture several key trends in the experimental Brønsted plots: the β l g values are rather different for phosphate and phosphonate ester hydrolysis in solution but are similar in PP1. Detailed analyses of structure, charge distribution and bond order of computed transition states support the general conclusion from experimental study that phosphoryl transfer transition states are different for the two classes of substrates in solution but similar in PP1. On the other hand, the microscopic models also highlight notable differences between the phosphate and phosphonate transition states, which are manifested in not only structure but also kinetic isotope effects. Overall, we find that while β l g / β E Q , l g generally correlates with the partial charge on leaving group oxygen and the fractional bond order of the breaking P- O l g bond, the precise mapping between β l g / β E Q , l g and P- O l g bond order in the transition state is difficult due largely to the cross talk between breaking and forming P-O bonds. Therefore, further supporting previous analyses of limitations of free energy relations, our results suggest that while free energy relation is a valuable tool for probing the nature of transition state, a quantitative mapping of β l g and β l g / β E Q , l g values to structure or charge in the transition state should be conducted with great care.

Accelerated Computation of Free Energy Profile at Ab Initio Quantum Mechanical/Molecular Mechanics Accuracy via a Semiempirical Reference Potential. 3. Gaussian Smoothing on Density-of-States

Calculations of the free energy profile, also known as potential of mean force (PMF), along a chosen collective variable (CV) are now routinely applied in the studies of chemical processes, such as enzymatic reactions and chemical reactions in condensed phases. However, if the ab initio quantum mechanical/molecular mechanics (QM/MM) level of accuracy is required for the PMF, it can be formidably demanding even with the most advanced enhanced sampling methods, such as umbrella sampling. To ameliorate this difficulty, we developed a novel method for the computation of the free energy profile based on the reference-potential method recently, in which a low-level reference Hamiltonian is employed for phase space sampling and the free energy profile can be corrected to the level of interest (the target Hamiltonian) by energy reweighting in a nonparametric way. However, when the reference Hamiltonian is very different from the target Hamiltonian, the calculated ensemble averages, including the PMF, often suffer from numerical instability, which mainly comes from the overestimation of the density-of-states (DoS) in the low-energy region. Stochastic samplings of these low-energy configurations are rare events, and some low-energy conformations may get oversampled in simulations of a finite length. In this work, an assumption of Gaussian distribution is applied to the DoS in each CV bin, and the weight of each configuration is rescaled according to the accumulated DoS. The results show that this smoothing process can remarkably reduce the ruggedness of the PMF and increase the reliability of the reference-potential method.

Unraveling the energetic significance of chemical events in enzyme catalysis via machine-learning based regression approach

The bacterial enzyme class of β-lactamases are involved in benzylpenicillin acylation reactions, which are currently being revisited using hybrid quantum mechanical molecular mechanical (QM/MM) chain-of-states pathway optimizations. Minimum energy pathways are sampled by reoptimizing pathway geometry under different representative protein environments obtained through constrained molecular dynamics simulations. Predictive potential energy surface models in the reaction space are trained with machine-learning regression techniques. Herein, using TEM-1/benzylpenicillin acylation reaction as the model system, we introduce two model-independent criteria for delineating the energetic contributions and correlations in the predicted reaction space. Both methods are demonstrated to effectively quantify the energetic contribution of each chemical process and identify the rate limiting step of enzymatic reaction with high degrees of freedom. The consistency of the current workflow is tested under seven levels of quantum chemistry theory and three non-linear machine-learning regression models. The proposed approaches are validated to provide qualitative compliance with experimental mutagenesis studies.

Incorporating Electronic Information into Machine Learning Potential Energy Surfaces via Approaching the Ground-State Electronic Energy as a Function of Atom-Based Electronic Populations

Machine learning (ML) approximations to density functional theory (DFT) potential energy surfaces (PESs) are showing great promise for reducing the computational cost of accurate molecular simulations, but at present, they are not applicable to varying electronic states, and in particular, they are not well suited for molecular systems in which the local electronic structure is sensitive to the medium to long-range electronic environment. With this issue as the focal point, we present a new machine learning approach called "BpopNN" for obtaining efficient approximations to DFT PESs. Conceptually, the methodology is based on approaching the true DFT energy as a function of electron populations on atoms; in practice, this is realized with available density functionals and constrained DFT (CDFT). The new approach creates approximations to this function with neural networks. These approximations thereby incorporate electronic information naturally into a ML approach, and optimizing the model energy with respect to populations allows the electronic terms to self-consistently adapt to the environment, as in DFT. We confirm the effectiveness of this approach with a variety of calculations on Li_nH_n clusters.

DFTB+, a software package for efficient approximate density functional theory based atomistic simulations

DFTB+ is a versatile community developed open source software package offering fast and efficient methods for carrying out atomistic quantum mechanical simulations. By implementing various methods approximating density functional theory (DFT), such as the density functional based tight binding (DFTB) and the extended tight binding method, it enables simulations of large systems and long timescales with reasonable accuracy while being considerably faster for typical simulations than the respective ab initio methods. Based on the DFTB framework, it additionally offers approximated versions of various DFT extensions including hybrid functionals, time dependent formalism for treating excited systems, electron transport using non-equilibrium Green's functions, and many more. DFTB+ can be used as a user-friendly standalone application in addition to being embedded into other software packages as a library or acting as a calculation-server accessed by socket communication. We give an overview of the recently developed capabilities of the DFTB+ code, demonstrating with a few use case examples, discuss the strengths and weaknesses of the various features, and also discuss on-going developments and possible future perspectives.

Quantum Chemistry in the Age of Machine Learning

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