Full and fragmented permutationally invariant polynomial potential energy surfaces for trans and cis N-methyl acetamide and isomerization saddle points

DFT-Based Permutationally Invariant Polynomial Potentials Capture the Twists and Turns of C14H30

Hydrocarbons are ubiquitous as fuels, solvents, lubricants, and as the principal components of plastics and fibers, yet our ability to predict their dynamical properties is limited to force-field mechanics. Here, we report two machine-learned potential energy surfaces (PESs) for the linear 44-atom hydrocarbon C14H30 using an extensive data set of roughly 250,000 density functional theory (DFT) (B3LYP) energies for a large variety of configurations, obtained using MM3 direct-dynamics calculations at 500, 1000, and 2500 K. The surfaces, based on Permutationally Invariant Polynomials (PIPs) and using both a many-body expansion approach and a fragmented-basis approach, produce precise fits for energies and forces and also produce excellent out-of-sample agreement with direct DFT calculations for torsional and dihedral angle potentials. Going beyond precision, the PESs are used in molecular dynamics calculations that demonstrate the robustness of the PESs for a large range of conformations. The many-body PIPs PES, although more compute intensive than the fragmented-basis one, is directly transferable for other linear hydrocarbons.

Force Fields for Deep Eutectic Mixtures: Application to Structure, Thermodynamics and 2D-Infrared Spectroscopy

Parametrizing energy functions for ionic systems can be challenging. Here, the total energy function for an eutectic system consisting of water, SCN-, K+ and acetamide is improved vis-a-vis experimentally measured properties. Given the importance of electrostatic interactions, two different types of models are considered: the first (model M0) uses atom-centered multipole whereas the other two (models M1 and M2) are based on fluctuating minimal distributed charges (fMDCM) that respond to geometrical changes of SCN-. The Lennard-Jones parameters of the anion are adjusted to best reproduce experimentally known hydration free energies and densities, which are matched to within a few percent for the final models irrespective of the electrostatic model. Molecular dynamics simulations of the eutectic mixtures with varying water content (between 0 and 100%) yield radial distribution functions and frequency correlation functions for the CN-stretch vibration. Comparison with experiments indicates that models based on fMDCM are considerably more consistent than those using multipoles. Computed viscosities from models M1 and M2 are within 30% of measured values and their change with increasing water content is consistent with experiments. This is not the case for model M0.

Evaluation of Rate Coefficients in the Gas Phase Using Machine-Learned Potentials

We assess the capability of machine-learned potentials to compute rate coefficients by training a neural network (NN) model and applying it to describe the chemical landscape on the C5H5 potential energy surface, which is relevant to molecular weight growth in combustion and interstellar media. We coupled the resulting NN with an automated kinetics workflow code, KinBot, to perform all necessary calculations to compute the rate coefficients. The NN is benchmarked exhaustively by evaluating its performance at the various stages of the kinetics calculations: from the electronic energy through the computation of zero point energy, barrier heights, entropic contributions, the portion of the PES explored, and finally the overall rate coefficients as formulated by transition state theory.

MBX: A many-body energy and force calculator for data-driven many-body simulations

Many-Body eXpansion (MBX) is a C++ library that implements many-body potential energy functions (PEFs) within the "many-body energy" (MB-nrg) formalism. MB-nrg PEFs integrate an underlying polarizable model with explicit machine-learned representations of many-body interactions to achieve chemical accuracy from the gas to the condensed phases. MBX can be employed either as a stand-alone package or as an energy/force engine that can be integrated with generic software for molecular dynamics and Monte Carlo simulations. MBX is parallelized internally using Open Multi-Processing and can utilize Message Passing Interface when available in interfaced molecular simulation software. MBX enables classical and quantum molecular simulations with MB-nrg PEFs, as well as hybrid simulations that combine conventional force fields and MB-nrg PEFs, for diverse systems ranging from small gas-phase clusters to aqueous solutions and molecular fluids to biomolecular systems and metal-organic frameworks.

Accurate and Interpretable Dipole Interaction Model-Based Machine Learning for Molecular Polarizability

Polarizabilities play significant roles in describing dispersive and inductive interactions of the atom and molecular systems. However, an accurate prediction of molecular polarizabilities from first principles is computationally prohibitive. Although physical models or statistical machine learning models have been proposed, either a lack of accurate description of local chemical environments or demanding a large number of samples for training has limited their practical applications. In this study, we combine a physically inspired dipole interaction model and an accurate neural network method for predicting the polarizability tensors of molecules. With the local chemical environment precisely described and the requirement of rotational covariance naturally fulfilled, this hybrid model is proven to give an accurate molecular polarizability prediction, essentially reducing the number of training samples. The atomic polarizabilities are physically interpretable and transferable to larger molecules unseen in the training set. This promising method may find its wide range of applications, such as spectroscopic simulations and the construction of polarizable force fields.

PESPIP: Software to fit complex molecular and many-body potential energy surfaces with permutationally invariant polynomials

We wish to describe a potential energy surface by using a basis of permutationally invariant polynomials whose coefficients will be determined by numerical regression so as to smoothly fit a dataset of electronic energies as well as, perhaps, gradients. The polynomials will be powers of transformed internuclear distances, usually either Morse variables, exp(-r_i,j/λ), where λ is a constant range hyperparameter, or reciprocals of the distances, 1/r_i,j. The question we address is how to create the most efficient basis, including (a) which polynomials to keep or discard, (b) how many polynomials will be needed, (c) how to make sure the polynomials correctly reproduce the zero interaction at a large distance, (d) how to ensure special symmetries, and (e) how to calculate gradients efficiently. This article discusses how these questions can be answered by using a set of programs to choose and manipulate the polynomials as well as to write efficient Fortran programs for the calculation of energies and gradients. A user-friendly interface for access to monomial symmetrization approach results is also described. The software for these programs is now publicly available.

Δ-Machine Learned Potential Energy Surfaces and Force Fields

There has been great progress in developing machine-learned potential energy surfaces (PESs) for molecules and clusters with more than 10 atoms. Unfortunately, this number of atoms generally limits the level of electronic structure theory to less than the "gold standard" CCSD(T) level. Indeed, for the well-known MD17 dataset for molecules with 9-20 atoms, all of the energies and forces were obtained with DFT calculations (PBE). This Perspective is focused on a Δ-machine learning method that we recently proposed and applied to bring DFT-based PESs to close to CCSD(T) accuracy. This is demonstrated for hydronium, N-methylacetamide, acetyl acetone, and ethanol. For 15-atom tropolone, it appears that special approaches (e.g., molecular tailoring, local CCSD(T)) are needed to obtain the CCSD(T) energies. A new aspect of this approach is the extension of Δ-machine learning to force fields. The approach is based on many-body corrections to polarizable force field potentials. This is examined in detail using the TTM2.1 water potential. The corrections make use of our recent CCSD(T) datasets for 2-b, 3-b, and 4-b interactions for water. These datasets were used to develop a new fully ab initio potential for water, termed q-AQUA.

Semiclassical and VSCF/VCI Calculations of the Vibrational Energies of trans- and gauche-Ethanol Using a CCSD(T) Potential Energy Surface

A recent full-dimensional Δ-Machine learning potential energy surface (PES) for ethanol is employed in semiclassical and vibrational self-consistent field (VSCF) and virtual-state configuration interaction (VCI) calculations, using MULTIMODE, to determine the anharmonic vibrational frequencies of vibration for both the trans and gauche conformers of ethanol. Both semiclassical and VSCF/VCI energies agree well with the experimental data. We find significant mixing between the VSCF basis states due to Fermi resonances between bending and stretching modes. The same effects are also accurately described by the full-dimensional semiclassical calculations. These are the first high-level anharmonic calculations using a PES, in particular a "gold-standard" CCSD(T) one.

Quantum Calculations on a New CCSD(T) Machine-Learned Potential Energy Surface Reveal the Leaky Nature of Gas-Phase Trans and Gauche Ethanol Conformers

Ethanol is a molecule of fundamental interest in combustion, astrochemistry, and condensed phase as a solvent. It is characterized by two methyl rotors and trans (anti) and gauche conformers, which are known to be very close in energy. Here we show that based on rigorous quantum calculations of the vibrational zero-point state, using a new ab initio potential energy surface (PES), the ground state resembles the trans conformer, but substantial delocalization to the gauche conformer is present. This explains experimental issues about identification and isolation of the two conformers. This “leak” effect is partially quenched when deuterating the OH group, which further demonstrates the need for a quantum mechanical approach. Diffusion Monte Carlo and full-dimensional semiclassical dynamics calculations are employed. The new PES is obtained by means of a Δ-machine learning approach starting from a pre-existing low level density functional theory surface. This surface is brought to the CCSD(T) level of theory using a relatively small number of ab initio CCSD(T) energies. Agreement between the corrected PES and direct ab initio results for standard tests is excellent. One- and two-dimensional discrete variable representation calculations focusing on the trans–gauche torsional motion are also reported, in reasonable agreement with experiment.

Quantitative molecular simulations

All-atom simulations can provide molecular-level insights into the dynamics of gas-phase, condensed-phase and surface processes. One important requirement is a sufficiently realistic and detailed description of the underlying intermolecular interactions. The present perspective provides an overview of the present status of quantitative atomistic simulations from colleagues' and our own efforts for gas- and solution-phase processes and for the dynamics on surfaces. Particular attention is paid to direct comparison with experiment. An outlook discusses present challenges and future extensions to bring such dynamics simulations even closer to reality.

REANN: A PyTorch-based end-to-end multi-functional deep neural network package for molecular, reactive, and periodic systems

In this work, we present a general purpose deep neural network package for representing energies, forces, dipole moments, and polarizabilities of atomistic systems. This so-called recursively embedded atom neural network model takes advantages of both the physically inspired atomic descriptor based neural networks and the message-passing based neural networks. Implemented in the PyTorch framework, the training process is parallelized on both the central processing unit and the graphics processing unit with high efficiency and low memory in which all hyperparameters can be optimized automatically. We demonstrate the state-of-the-art accuracy, high efficiency, scalability, and universality of this package by learning not only energies (with or without forces) but also dipole moment vectors and polarizability tensors in various molecular, reactive, and periodic systems. An interface between a trained model and LAMMPs is provided for large scale molecular dynamics simulations. We hope that this open-source toolbox will allow for future method development and applications of machine learned potential energy surfaces and quantum-chemical properties of molecules, reactions, and materials.

Enabling Rapid and Accurate Construction of CCSD(T)-Level Potential Energy Surface of Large Molecules Using Molecular Tailoring Approach

The construction of a potential energy surface (PES) of even a medium-sized molecule employing correlated theory, such as CCSD(T), is arduous due to the high computational cost involved. The present study reports the possibility of efficiently constructing such a PES of molecules containing up to 15 atoms and 550 basis functions by employing the fragment-based molecular tailoring approach (MTA) on off-the-shelf hardware. The MTA energies at the CCSD(T)/aug-cc-pVTZ level for several geometries of three test molecules, viz., acetylacetone, N-methylacetamide, and tropolone, are reported. These energies are in excellent agreement with their full calculation counterparts with a time advantage factor of 3-5. The energy barrier from the ground to transition state is also accurately captured. Further, we demonstrate the accuracy and efficiency of MTA for estimating the energy gradients at the CCSD(T) level. As a further application of our MTA methodology, the energies of acetylacetone at ∼430 geometries are computed at the CCSD(T)/aug-cc-pVTZ level and used for generating a Δ-machine learning (Δ-ML) PES. This leads to the H-transfer barrier of 3.02 kcal/mol, well in agreement with the benchmarked barrier of 3.19 kcal/mol. The fidelity of this Δ-ML PES is examined by geometry optimization and normal mode frequency calculations of global minima and saddle point geometries. We trust that the present work is a major development for the rapid and accurate construction of PES at the CCSD(T) level for molecules containing up to 20 atoms and 600 basis functions using off-the-shelf hardware.

Unsupervised Machine Learning Neural Gas Algorithm for Accurate Evaluations of the Hessian Matrix in Molecular Dynamics

The Hessian matrix of the potential energy of molecular systems is employed not only in geometry optimizations or high-order molecular dynamics integrators but also in many other molecular procedures, such as instantaneous normal mode analysis, force field construction, instanton calculations, and semiclassical initial value representation molecular dynamics, to name a few. Here, we present an algorithm for the calculation of the approximated Hessian in molecular dynamics. The algorithm belongs to the family of unsupervised machine learning methods, and it is based on the neural gas idea, where neurons are molecular configurations whose Hessians are adopted for groups of molecular dynamics configurations with similar geometries. The method is tested on several molecular systems of different dimensionalities both in terms of accuracy and computational time versus calculating the Hessian matrix at each time-step, that is, without any approximation, and other Hessian approximation schemes. Finally, the method is applied to the on-the-fly, full-dimensional simulation of a small synthetic peptide (the 46 atom N-acetyl-l-phenylalaninyl-l-methionine amide) at the level of DFT-B3LYP-D/6-31G* theory, from which the semiclassical vibrational power spectrum is calculated.

MB-Fit: Software infrastructure for data-driven many-body potential energy functions

Many-body potential energy functions (MB-PEFs), which integrate data-driven representations of many-body short-range quantum mechanical interactions with physics-based representations of many-body polarization and long-range interactions, have recently been shown to provide high accuracy in the description of molecular interactions from the gas to the condensed phase. Here, we present MB-Fit, a software infrastructure for the automated development of MB-PEFs for generic molecules within the TTM-nrg (Thole-type model energy) and MB-nrg (many-body energy) theoretical frameworks. Besides providing all the necessary computational tools for generating TTM-nrg and MB-nrg PEFs, MB-Fit provides a seamless interface with the MBX software, a many-body energy and force calculator for computer simulations. Given the demonstrated accuracy of the MB-PEFs, particularly within the MB-nrg framework, we believe that MB-Fit will enable routine predictive computer simulations of generic (small) molecules in the gas, liquid, and solid phases, including, but not limited to, the modeling of quantum isomeric equilibria in molecular clusters, solvation processes, molecular crystals, and phase diagrams.

Parsimonious Potential Energy Surface Expansions Using Dictionary Learning with Multipass Greedy Selection

Potential energy surfaces fit with basis set expansions have been shown to provide accurate representations of electronic energies and have enabled a variety of high-accuracy dynamics, kinetics, and spectroscopy applications. The number of terms in these expansions scales poorly with system size, a drawback that challenges their use for systems with more than ∼10 atoms. A solution is presented here using dictionary learning. Subsets of the full set of conventional basis functions are optimized using a newly developed multipass greedy regression method inspired by forward and backward selection methods from the statistics, signal processing, and machine learning literatures. The optimized representations have accuracies comparable to the full set but are 1 or more orders of magnitude smaller, and notably, the number of terms in the optimized multipass greedy expansions scales approximately linearly with the number of atoms.

Permutationally Invariant Polynomial Expansions with Unrestricted Complexity

A general strategy is presented for constructing and validating permutationally invariant polynomial (PIP) expansions for chemical systems of any stoichiometry. Demonstrations are made for three categories of gas-phase dynamics and kinetics: collisional energy-transfer trajectories for predicting pressure-dependent kinetics, three-body collisions for describing transient van der Waals adducts relevant to atmospheric chemistry, and nonthermal reactivity via quasiclassical trajectories. In total, 30 systems are considered with up to 15 atoms and 39 degrees of freedom. Permutational invariance is enforced in PIP expansions with as many as 13 million terms and 13 permutationally distinct atom types by taking advantage of petascale computational resources. The quality of the PIP expansions is demonstrated through the systematic convergence of in-sample and out-of-sample errors with respect to both the number of training data and the order of the expansion, and these errors are shown to predict errors in the dynamics for both reactive and nonreactive applications. The parallelized code distributed as part of this work enables the automation of PIP generation for complex systems with multiple channels and flexible user-defined symmetry constraints and for automatically removing unphysical unconnected terms from the basis set expansions, all of which are required for simulating complex reactive systems.

Breaking the Coupled Cluster Barrier for Machine-Learned Potentials of Large Molecules: The Case of 15-Atom Acetylacetone

Machine-learned potential energy surfaces (PESs) for molecules with more than 10 atoms are typically forced to use lower-level electronic structure methods such as density functional theory (DFT) and second-order Møller-Plesset perturbation theory (MP2). While these are efficient and realistic, they fall short of the accuracy of the "gold standard" coupled-cluster method, especially with respect to reaction and isomerization barriers. We report a major step forward in applying a Δ-machine learning method to the challenging case of acetylacetone, whose MP2 barrier height for H-atom transfer is low by roughly 1.1 kcal/mol relative to the benchmark CCSD(T) barrier of 3.2 kcal/mol. From a database of 2151 local CCSD(T) energies and training with as few as 430 energies, we obtain a new PES with a barrier of 3.5 kcal/mol in agreement with the LCCSD(T) barrier of 3.5 kcal/mol and close to the benchmark value. Tunneling splittings due to H-atom transfer are calculated using this new PES, providing improved estimates over previous ones obtained using an MP2-based PES.

Application of Reaction Path Search Calculations to Potential Energy Surface Fits

There has been significant progress in recent years in the use of machine learning techniques to model high-dimensional reactive potential energy surfaces using large-scale data obtained from ab initio electronic structure calculations. In these methods, the strategy used to gather data becomes a key issue as the molecular size increases. In this work, we examine the applicability of the reaction path search algorithm implemented in the Global Reaction Route Mapping (GRRM) code as a data-gathering approach. The electronic energies and gradients sampled by using the GRRM calculation are directly used in potential energy surface fitting to a permutationally invariant polynomial function. This simple approach was applied to the HNS and HCNO reaction systems, and we found that the fitted potential energy surfaces reasonably reproduce the features of the electronic structure calculations used in the GRRM calculations. This suggests that the GRRM sampling scheme can be used to construct an initial potential energy surface.

Full-dimensional potential energy surface for acetylacetone and tunneling splittings

New, full-dimensional potential energy surface for acetylacetone allows for description of H-tunneling dynamics and characterization of stationary points.

Efficient and Accurate Simulations of Vibrational and Electronic Spectra with Symmetry-Preserving Neural Network Models for Tensorial Properties

Machine learning has revolutionized the high-dimensional representations for molecular properties such as potential energy. However, there are scarce machine learning models targeting tensorial properties, which are rotationally covariant. Here, we propose tensorial neural network (NN) models to learn both tensorial response and transition properties in which atomic coordinate vectors are multiplied with scalar NN outputs or their derivatives to preserve the rotationally covariant symmetry. This strategy keeps structural descriptors symmetry invariant so that the resulting tensorial NN models are as efficient as their scalar counterparts. We validate the performance and universality of this approach by learning response properties of water oligomers and liquid water and transition dipole moment of a model structural unit of proteins. Machine-learned tensorial models have enabled efficient simulations of vibrational spectra of liquid water and ultraviolet spectra of realistic proteins, promising feasible and accurate spectroscopic simulations for biomolecules and materials.

High-Fidelity Potential Energy Surfaces for Gas-Phase and Gas-Surface Scattering Processes from Machine Learning

In this Perspective, we review recent advances in constructing high-fidelity potential energy surfaces (PESs) from discrete ab initio points, using machine learning tools. Such PESs, albeit with substantial initial investments, provide significantly higher efficiency than direct dynamics methods and/or high accuracy at a level that is not affordable by on-the-fly approaches. These PESs not only are a necessity for quantum dynamical studies because of delocalization of wave packets but also enable the study of low-probability and long-time events in (quasi-)classical treatments. Our focus here is on inelastic and reactive scattering processes, which are more challenging than bound systems because of the involvement of continua. Relevant applications and developments for dynamical processes in both the gas phase and at gas-surface interfaces are discussed.

Fitting potential energy surfaces with fundamental invariant neural network. II. Generating fundamental invariants for molecular systems with up to ten atoms

Symmetry adaptation is crucial in representing a permutationally invariant potential energy surface (PES). Due to the rapid increase in computational time with respect to the molecular size, as well as the reliance on the algebra software, the previous neural network (NN) fitting with inputs of fundamental invariants (FIs) has practical limits. Here, we report an improved and efficient generation scheme of FIs based on the computational invariant theory and parallel program, which can be readily used as the input vector of NNs in fitting high-dimensional PESs with permutation symmetry. The newly developed method significantly reduces the evaluation time of FIs, thereby extending the FI-NN method for constructing highly accurate PESs to larger systems beyond five atoms. Because of the minimum size of invariants used in the inputs of the NN, the NN structure can be very flexible for FI-NN, which leads to small fitting errors. The resulting FI-NN PES is much faster on evaluating than the corresponding permutationally invariant polynomial-NN PES.

Efficient Generation of Permutationally Invariant Potential Energy Surfaces for Large Molecules

An efficient method is described for generating a fragmented, permutationally invariant polynomial basis to fit electronic energies and, if available, gradients for large molecules. The method presented rests on the fragmentation of a large molecule into any number of fragments while maintaining the permutational invariance and uniqueness of the polynomials. The new approach improves on a previous one reported by Qu and Bowman by avoiding repetition of polynomials in the fitting basis set and speeding up gradient evaluations while keeping the accuracy of the PES. The method is demonstrated for CH₃–NH–CO–CH₃ (N-methylacetamide) and NH₂–CH₂–COOH (glycine).