Electrostatic Forces: Formulas for the First Derivatives of a Polarizable, Anisotropic Electrostatic Potential Energy Function Based on Machine Learning

FFLUX molecular simulations driven by atomic Gaussian process regression models

Machine learning (ML) force fields are revolutionizing molecular dynamics (MD) simulations as they bypass the computational cost associated with ab initio methods but do not sacrifice accuracy in the process. In this work, the GPyTorch library is used to create Gaussian process regression (GPR) models that are interfaced with the next-generation ML force field FFLUX. These models predict atomic properties of different molecular configurations that appear in a progressing MD simulation. An improved kernel function is utilized to correctly capture the periodicity of the input descriptors. The first FFLUX molecular simulations of ammonia, methanol, and malondialdehyde with the updated kernel are performed. Geometry optimizations with the GPR models result in highly accurate final structures with a maximum root-mean-squared deviation of 0.064 Å and sub-kJ mol-1 total energy predictions. Additionally, the models are tested in 298 K MD simulations with FFLUX to benchmark for robustness. The resulting energy and force predictions throughout the simulation are in excellent agreement with ab initio data for ammonia and methanol but decrease in quality for malondialdehyde due to the increased system complexity. GPR model improvements are discussed, which will ensure the future scalability to larger systems.

An Unsupervised Machine Learning Approach for the Automatic Construction of Local Chemical Descriptors

Condensing the many physical variables defining a chemical system into a fixed-size array poses a significant challenge in the development of chemical Machine Learning (ML). Atom Centered Symmetry Functions (ACSFs) offer an intuitive featurization approach by means of a tedious and labor-intensive selection of tunable parameters. In this work, we implement an unsupervised ML strategy relying on a Gaussian Mixture Model (GMM) to automatically optimize the ACSF parameters. GMMs effortlessly decompose the vastness of the chemical and conformational spaces into well-defined radial and angular clusters, which are then used to build tailor-made ACSFs. The unsupervised exploration of the space has demonstrated general applicability across a diverse range of systems, spanning from various unimolecular landscapes to heterogeneous databases. The impact of the sampling technique and temperature on space exploration is also addressed, highlighting the particularly advantageous role of high-temperature Molecular Dynamics (MD) simulations. The reliability of the resulting features is assessed through the estimation of the atomic charges of a prototypical capped amino acid and a heterogeneous collection of CHON molecules. The automatically constructed ACSFs serve as high-quality descriptors, consistently yielding typical prediction errors below 0.010 electrons bound for the reported atomic charges. Altering the spatial distribution of the functions with respect to the cluster highlights the critical role of symmetry rupture in achieving significantly improved features. More specifically, using two separate functions to describe the lower and upper tails of the cluster results in the best performing models with errors as low as 0.006 electrons. Finally, the effectiveness of finely tuned features was checked across different architectures, unveiling the superior performance of Gaussian Process (GP) models over Feed Forward Neural Networks (FFNNs), particularly in low-data regimes, with nearly a 2-fold increase in prediction quality. Altogether, this approach paves the way toward an easier construction of local chemical descriptors, while providing valuable insights into how radial and angular spaces should be mapped. Finally, this work opens the possibility of encoding many-body information beyond angular terms into upcoming ML features.

Gaussian Process Regression Models for Predicting Atomic Energies and Multipole Moments

Developing a force field is a difficult task because its design is typically pulled in opposite directions by speed and accuracy. FFLUX breaks this trend by utilizing Gaussian process regression (GPR) to predict, at ab initio accuracy, atomic energies and multipole moments as obtained from the quantum theory of atoms in molecules (QTAIM). This work demonstrates that the in-house FFLUX training pipeline can generate successful GPR models for six representative molecules: peptide-capped glycine and alanine, glucose, paracetamol, aspirin, and ibuprofen. The molecules were sufficiently distorted to represent configurations from an AMBER-GAFF2 molecular dynamics run. All internal degrees of freedom were covered corresponding to 93 dimensions in the case of the largest molecule ibuprofen (33 atoms). Benefiting from active learning, the GPR models contain only about 2000 training points and return largely sub-kcal mol^-1 prediction errors for the validation sets. A proof of concept has been reached for transferring the model produced through active learning on one atomic property to that of the remaining atomic properties. The prediction of electrostatic interaction can be assessed at the intermolecular level, and the vast majority of interactions have a root-mean-square error of less than 0.1 kJ mol^-1 with a maximum value of ∼1 kJ mol^-1 for a glycine and paracetamol dimer.

Producing chemically accurate atomic Gaussian process regression models by active learning for molecular simulation

Machine learning is becoming increasingly more important in the field of force field development. Never has it been more vital to have chemically accurate machine learning potentials because force fields become more sophisticated and their applications expand. In this study a method for developing chemically accurate Gaussian process regression models is demonstrated for an increasingly complex set of molecules. This work is an extension to previous work showing the progression of the active learning technique in producing more accurate models in much less CPU time than ever before. The per-atom active learning approach has unlocked the potential to generate chemically accurate models for molecules such as peptide-capped glycine.

Quantum Chemical Calculations with Machine Learning for Multipolar Electrostatics Prediction in RNA: An Application to Pentose

To develop a realistic electrostatic model that allows for the anisotropy of the atomic electron density, high-rank atomic multipole moments computed by quantum chemical calculations have been studied extensively. However, it is hard to process huge RNA systems only relying on quantum chemical calculations due to its highly computational cost. In this study, we employ five machine learning methods of Gaussian process regression with automatic relevance determination (ARDGPR), Kriging, radial basis function neural networks, Bagging, and generalized regression neural network to predict atomic multipole moments. Atom-atom electrostatic interaction energies are subsequently computed using the predicted atomic multipole moments in the pilot system pentose of RNA. Here, the performance of the five methods is compared in terms of both the multipole moment prediction errors and the electrostatic energy prediction errors. For the predicted high-rank multipole moments of the four elements (O, C, N, and H) in capped pentose, ARDGPR and Kriging consistently outperform the other three methods. Therefore, the multipole moments predicted by the two best methods of ARDGPR and Kriging are then used to predict electrostatic interaction energy of each pentose. Finally, the absolute average energy errors of ARDGPR and Kriging are 1.83 and 4.33 kJ mol^-1, respectively. Compared to Kriging, the ARDGPR method achieves a 58% decrease in the absolute average energy error. These satisfactory results demonstrated that the ARDGPR method with the strong feature extraction ability can predict the electrostatic interaction energy of pentose in RNA correctly and reliably.

Using atomic charges to model molecular polarization

We review different models for introducing electric polarization in force fields, with special focus on methods where polarization is modelled at the atomic charge level. While electric polarization has been included in several force fields, the common approach has been to focus on atomic dipole polarizability. Several approaches allow modelling electric polarization by using charge-flow between charge sites instead, but this has been less exploited, despite that atomic charges and charge-flow is expected to be more important than atomic dipoles and dipole polarizability. A number of challenges are required to be solved for charge-flow models to be incorporated into polarizable force fields, for example how to parameterize the models and how to make them computational efficient.

DL_FFLUX: A Parallel, Quantum Chemical Topology Force Field

DL_FFLUX is a force field based on quantum chemical topology that can perform molecular dynamics for flexible molecules endowed with polarizable atomic multipole moments (up to hexadecapole). Using the machine learning method kriging (aka Gaussian process regression), DL_FFLUX has access to atomic properties (energy, charge, dipole moment, etc.) with quantum mechanical accuracy. Newly optimized and parallelized using domain decomposition Message Passing Interface (MPI), DL_FFLUX is now able to deliver this rigorous methodology at scale while still in reasonable time frames. DL_FFLUX is delivered as an add-on to the widely distributed molecular dynamics code DL_POLY 4.08. For the systems studied here (10³–10⁵ atoms), DL_FFLUX is shown to add minimal computational cost to the standard DL_POLY package. In fact, the optimization of the electrostatics in DL_FFLUX means that, when high-rank multipole moments are enabled, DL_FFLUX is up to 1.25× faster than standard DL_POLY. The parallel DL_FFLUX preserves the quality of the scaling of MPI implementation in standard DL_POLY. For the first time, it is feasible to use the full capability of DL_FFLUX to study systems that are large enough to be of real-world interest. For example, a fully flexible, high-rank polarized (up to and including quadrupole moments) 1 ns simulation of a system of 10 125 atoms (3375 water molecules) takes 30 h (wall time) on 18 cores.

Machine-learned potentials for next-generation matter simulations

The choice of simulation methods in computational materials science is driven by a fundamental trade-off: bridging large time- and length-scales with highly accurate simulations at an affordable computational cost. Venturing the investigation of complex phenomena on large scales requires fast yet accurate computational methods. We review the emerging field of machine-learned potentials, which promises to reach the accuracy of quantum mechanical computations at a substantially reduced computational cost. This Review will summarize the basic principles of the underlying machine learning methods, the data acquisition process and active learning procedures. We highlight multiple recent applications of machine-learned potentials in various fields, ranging from organic chemistry and biomolecules to inorganic crystal structure predictions and surface science. We furthermore discuss the developments required to promote a broader use of ML potentials, and the possibility of using them to help solve open questions in materials science and facilitate fully computational materials design.

Multipolar Description of Atom-Atom Electrostatic Interaction Energies in Single/Double-Stranded DNAs

Molecular force field simulation is an effective method to explore the properties of DNA molecules in depth. Almost all current popular force fields calculate atom-atom electrostatic interaction energies for DNAs based on the atomic charge and dipole or quadrupole moments, without considering high-rank atomic multipole moments for more accurate electrostatics. Actually, the distribution of electrons around atomic nuclei is not spherically symmetric but is geometry dependent. In this work, a multipole expansion method that allows us to combine polarizability and anisotropy was applied. One single-stranded DNA and one double-stranded DNA were selected as pilot systems. Deoxynucleotides were cut out from pilot systems and capped by mimicking the original DNA environment. Atomic multipole moments were integrated instead of fixed-point charges to calculate atom-atom electrostatic energies to improve the accuracy of force fields for DNA simulations. Also, the applicability of modeling the behavior of both single-stranded and double-stranded DNAs was investigated. The calculation results indicated that the models can be transferred from pilot systems to test systems, which is of great significance for the development of future DNA force fields.

On the many-body nature of intramolecular forces in FFLUX and its implications

FFLUX is a biomolecular force field under construction, based on Quantum Chemical Topology (QCT) and machine learning (kriging), with a minimalistic and physically motivated design. A detailed analysis of the forces within the kriging models as treated in FFLUX is presented, taking as a test example a liquid water model. The energies of topological atoms are modeled as 3N_atoms -6 dimensional potential energy surfaces, using atomic local frames to represent the internal degrees of freedom. As a result, the forces within the kriging models in FFLUX are inherently N-body in nature where N refers to N_atoms . This provides a fuller picture that is closer to a true quantum mechanical representation of interactions between atoms. The presented computational example quantitatively showcases the non-negligible (as much as 9%) three-body nature of bonded forces and angular forces in a water molecule. We discuss the practical impact on the pressure calculation with N-body forces and periodic boundary conditions (PBC) in molecular dynamics, as opposed to classical force fields with two-body forces. The equivalence between the PBC-related correction terms in the general virial equation is shown mathematically.

Atomic Partitioning of the MPn (n = 2, 3, 4) Dynamic Electron Correlation Energy by the Interacting Quantum Atoms Method: A Fast and Accurate Electrostatic Potential Integral Approach

Recently, the quantum topological energy partitioning method called interacting quantum atoms (IQA) has been extended to MPn (n = 2, 3, 4) wave functions. This enables the extraction of chemical insight related to dynamic electron correlation. The large computational expense of the IQA-MPn approach is compensated by the advantages that IQA offers compared to older nontopological energy decomposition schemes. This expense is problematic in the construction of a machine learning training set to create kriging models for topological atoms. However, the algorithm presented here markedly accelerates the calculation of atomically partitioned electron correlation energies. Then again, the algorithm cannot calculate pairwise interatomic energies because it applies analytical integrals over whole space (rather than over atomic volumes). However, these pairwise energies are not needed in the quantum topological force field FFLUX, which only uses the energy of an atom interacting with all remaining atoms of the system that it is part of. Thus, it is now feasible to generate accurate and sizeable training sets at MPn level of theory. © 2019 The Authors. Journal of Computational Chemistry published by Wiley Periodicals, Inc.

Assessing Force Field Potential Energy Function Accuracy via a Multipolar Description of Atomic Electrostatic Interactions in RNA

Computational investigations of RNA properties often rely on a molecular mechanical approach to define molecular potential energy. Force fields for RNA typically employ a point charge model of electrostatics, which does not provide a realistic quantum-mechanical picture. In reality, electron distributions around nuclei are not spherically symmetric and are geometry dependent. A multipole expansion method which allows for incorporation of polarizability and anisotropy in a force field is described, and its applicability to modeling the behavior of RNA molecules is investigated. Transferability of the model, critical for force field development, is also investigated.

Geometry Optimization with Machine Trained Topological Atoms

The geometry optimization of a water molecule with a novel type of energy function called FFLUX is presented, which bypasses the traditional bonded potentials. Instead, topologically-partitioned atomic energies are trained by the machine learning method kriging to predict their IQA atomic energies for a previously unseen molecular geometry. Proof-of-concept that FFLUX’s architecture is suitable for geometry optimization is rigorously demonstrated. It is found that accurate kriging models can optimize 2000 distorted geometries to within 0.28 kJ mol⁻¹ of the corresponding ab initio energy, and 50% of those to within 0.05 kJ mol⁻¹. Kriging models are robust enough to optimize the molecular geometry to sub-noise accuracy, when two thirds of the geometric inputs are outside the training range of that model. Finally, the individual components of the potential energy are analyzed, and chemical intuition is reflected in the independent behavior of the three energy terms \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${E}_{{\rm{intra}}}^{{\rm{A}}}$$\end{document}EintraA(intra-atomic), \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${V}_{{\rm{cl}}}^{\text{AA}\text{'}}$$\end{document}VclAA' (electrostatic) and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${V}_{{\rm{x}}}^{\text{AA}\text{'}}$$\end{document}VxAA' (exchange), in contrast to standard force fields.

Prediction of conformationally dependent atomic multipole moments in carbohydrates

The conformational flexibility of carbohydrates is challenging within the field of computational chemistry. This flexibility causes the electron density to change, which leads to fluctuating atomic multipole moments. Quantum Chemical Topology (QCT) allows for the partitioning of an "atom in a molecule," thus localizing electron density to finite atomic domains, which permits the unambiguous evaluation of atomic multipole moments. By selecting an ensemble of physically realistic conformers of a chemical system, one evaluates the various multipole moments at defined points in configuration space. The subsequent implementation of the machine learning method kriging delivers the evaluation of an analytical function, which smoothly interpolates between these points. This allows for the prediction of atomic multipole moments at new points in conformational space, not trained for but within prediction range. In this work, we demonstrate that the carbohydrates erythrose and threose are amenable to the above methodology. We investigate how kriging models respond when the training ensemble incorporating multiple energy minima and their environment in conformational space. Additionally, we evaluate the gains in predictive capacity of our models as the size of the training ensemble increases. We believe this approach to be entirely novel within the field of carbohydrates. For a modest training set size of 600, more than 90% of the external test configurations have an error in the total (predicted) electrostatic energy (relative to ab initio) of maximum 1 kJ mol(-1) for open chains and just over 90% an error of maximum 4 kJ mol(-1) for rings.

Calculation of Raman optical activity spectra for vibrational analysis

This review provides the necessary knowledge to accurately model ROA spectra of solvated systems and interpret their vibrational characteristics.