Theoretical investigation of the dissociation dynamics of vibrationally excited vinyl bromide on an ab initio potential-energy surface obtained using modified novelty sampling and feedforward neural networks. II. Numerical application of the method

Transition1x - a dataset for building generalizable reactive machine learning potentials

Machine Learning (ML) models have, in contrast to their usefulness in molecular dynamics studies, had limited success as surrogate potentials for reaction barrier search. This is primarily because available datasets for training ML models on small molecular systems almost exclusively contain configurations at or near equilibrium. In this work, we present the dataset Transition1x containing 9.6 million Density Functional Theory (DFT) calculations of forces and energies of molecular configurations on and around reaction pathways at the ωB97x/6-31 G(d) level of theory. The data was generated by running Nudged Elastic Band (NEB) with DFT on 10k organic reactions of various types while saving intermediate calculations. We train equivariant graph message-passing neural network models on Transition1x and cross-validate on the popular ANI1x and QM9 datasets. We show that ML models cannot learn features in transition state regions solely by training on hitherto popular benchmark datasets. Transition1x is a new challenging benchmark that will provide an important step towards developing next-generation ML force fields that also work far away from equilibrium configurations and reactive systems.

Dissociation of hydrogen peroxide in water and methanol through a biased molecular dynamics investigation

The two dissociation channels of HOOH, namely, HOOH and HOOH, in water and methanol are investigated using umbrella-sampling ab initio molecular dynamics. Our potential of mean force calculations reveals the HOOH dissociation to be more favorable in methanol with a free energy barrier of 7.56 kcal/mol, while the HOOH dissociation possesses a free energy barrier of 11.46 kcal/mol. In water, the HOOH dissociation channel is more favorable (8.25 kcal/mol), while the HOOH dissociation process requires a higher free energy (11.28 kcal/mol). Such reaction favorability can be explained by inspecting the formation of secondary radical species during the course of multiple hydrogen donating-accepting processes in each reaction channel. The radical species, that is, H₃ O^• (observed in water) and CH₃ OH₂ ^• (observed in methanol), are the first subordinate species upon the HOOH dissociation. For the HOOH dissociation channel in methanol, the secondary species such as water and formaldehyde can be observed, while the re-generation of HOOH in water can be spotted.

Neural Network Potential Energy Surfaces for Small Molecules and Reactions

We review progress in neural network (NN)-based methods for the construction of interatomic potentials from discrete samples (such as ab initio energies) for applications in classical and quantum dynamics including reaction dynamics and computational spectroscopy. The main focus is on methods for building molecular potential energy surfaces (PES) in internal coordinates that explicitly include all many-body contributions, even though some of the methods we review limit the degree of coupling, due either to a desire to limit computational cost or to limited data. Explicit and direct treatment of all many-body contributions is only practical for sufficiently small molecules, which are therefore our primary focus. This includes small molecules on surfaces. We consider direct, single NN PES fitting as well as more complex methods that impose structure (such as a multibody representation) on the PES function, either through the architecture of one NN or by using multiple NNs. We show how NNs are effective in building representations with low-dimensional functions including dimensionality reduction. We consider NN-based approaches to build PESs in the sums-of-product form important for quantum dynamics, ways to treat symmetry, and issues related to sampling data distributions and the relation between PES errors and errors in observables. We highlight combinations of NNs with other ideas such as permutationally invariant polynomials or sums of environment-dependent atomic contributions, which have recently emerged as powerful tools for building highly accurate PESs for relatively large molecular and reactive systems.

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.

Instanton rate constant calculations using interpolated potential energy surfaces in nonredundant, rotationally and translationally invariant coordinates

A trivial flaw in the utilization of artificial neural networks in interpolating chemical potential energy surfaces (PES) whose descriptors are Cartesian coordinates is their dependence on simple translations and rotations of the molecule under consideration. A different set of descriptors can be chosen to circumvent this problem, internuclear distances, inverse internuclear distances or z-matrix coordinates are three such descriptors. The objective is to use an interpolated PES in instanton rate constant calculations, hence information on the energy, gradient, and Hessian is required at coordinates in the vicinity of the tunneling path. Instanton theory relies on smoothly fitted Hessians, therefore we use energy, gradients, and Hessians in the training procedure. A major challenge is presented in the proper back-transformation of the output gradients and Hessians from internal coordinates to Cartesian coordinates. We perform comparisons between our method, a previous approach and on-the-fly rate constant calcuations on the hydrogen abstraction from methanol and on the hydrogen addition to isocyanic acid. © 2018Wiley Periodicals, Inc.

Machine learning prediction of interaction energies in rigid water clusters

Classical force fields form a computationally efficient avenue for calculating the energetics of large systems. However, due to the constraints of the underlying analytical form, it is sometimes not accurate enough. Quantum mechanical (QM) methods, although accurate, are computationally prohibitive for large systems. In order to circumvent the bottle-neck of interaction energy estimation of large systems, data driven approaches based on machine learning (ML) have been employed in recent years. In most of these studies, the method of choice is artificial neural networks (ANN). In this work, we have shown an alternative ML method, support vector regression (SVR), that provides comparable accuracy with better computational efficiency. We have further used many body expansion (MBE) along with SVR to predict interaction energies in water clusters (decamers). In the case of dimer and trimer interaction energies, the root mean square errors (RMSEs) of the SVR based scheme are 0.12 kcal mol-1 and 0.34 kcal mol-1, respectively. We show that the SVR and MBE based scheme has a RMSE of 2.78% in the estimation of decamer interaction energy against the parent QM method in a computationally efficient way.

Fitting potential energy surfaces to sum-of-products form with neural networks using exponential neurons

In this paper, the use of the neural network (NN) method with exponential neurons for directly fitting ab initio data to generate potential energy surfaces (PESs) in sum-of-product form will be discussed. The utility of the approach will be highlighted using fits of CS2, HFCO, and HONO ground state PESs based upon high-level ab initio data. Using a generic interface between the neural network PES fitting, which is performed in MATLAB, and the Heidelberg multi-configuration time-dependent Hartree (MCTDH) software package, the PESs have been tested via comparison of vibrational energies to experimental measurements. The review demonstrates the potential of the PES fitting method, combined with MCTDH, to tackle high-dimensional quantum dynamics problems.

Ab Initio Investigation of O-H Dissociation from the Al-OH2 Complex Using Molecular Dynamics and Neural Network Fitting

The dissociation dynamics of the O-H bond in Al-OH2 is investigated on an approximated ab initio potential energy surface (PES). By adopting a dynamic sampling method, we obtain a database of 92 834 configurations. The potential energy for each point is calculated using MP2/6-311G (3df, 2p) calculations; then, a 60-neuron feed-forward neural network is utilized to fit the data to construct an analytic PES. The root-mean-square error (rmse) for the training set is reported as 0.0036 eV, while the rmse for the independent testing set is 0.0034 eV. Such excellent fitting accuracy indeed confirms the reliability of the constructed PES. Subsequently, quasi-classical molecular dynamics (MD) trajectories are performed on the constructed PES at various levels of vibrational excitation in the range of 1.03 to 2.23 eV to investigate the probability of O-H bond dissociation. The results indicate a linear relationship between reaction probability and internal energy, from which we can determine the minimum activation internal energy required for the dissociation as 0.62 eV. Moreover, the O-H bond rupture is shown to be highly correlated with the formation of Al-O bond.

Two new methods for computing vibrational energy levels

I review two new ideas for coping with the size of large product basis sets and large product grids when one computes vibrational energy levels. The first is based on a tensor reduction scheme. It exploits advantages of a sum-of-products potential. The key idea is to use a basis each of whose function is a sum of optimized products and to compress the number of terms in each basis function. When the potential does not have the sum-of-products form, it is usually necessary to use quadrature. The second idea uses a nondirect product grid that has structure and is therefore compatible with efficient matrix–vector products.

High-Dimensional Neural Network Potentials for Organic Reactions and an Improved Training Algorithm

Artificial neural networks (NNs) represent a relatively recent approach for the prediction of molecular potential energies, suitable for simulations of large molecules and long time scales. By using NNs to fit electronic structure data, it is possible to obtain empirical potentials of high accuracy combined with the computational efficiency of conventional force fields. However, as opposed to the latter, changing bonding patterns and unusual coordination geometries can be described due to the underlying flexible functional form of the NNs. One of the most promising approaches in this field is the high-dimensional neural network (HDNN) method, which is especially adapted to the prediction of molecular properties. While HDNNs have been mostly used to model solid state systems and surface interactions, we present here the first application of the HDNN approach to an organic reaction, the Claisen rearrangement of allyl vinyl ether to 4-pentenal. To construct the corresponding HDNN potential, a new training algorithm is introduced. This algorithm is termed "element-decoupled" global extended Kalman filter (ED-GEKF) and is based on the decoupled Kalman filter. Using a metadynamics trajectory computed with density functional theory as reference data, we show that the ED-GEKF exhibits superior performance - both in terms of accuracy and training speed - compared to other variants of the Kalman filter hitherto employed in HDNN training. In addition, the effect of including forces during ED-GEKF training on the resulting potentials was studied.

Representing potential energy surfaces by high-dimensional neural network potentials

The development of interatomic potentials employing artificial neural networks has seen tremendous progress in recent years. While until recently the applicability of neural network potentials (NNPs) has been restricted to low-dimensional systems, this limitation has now been overcome and high-dimensional NNPs can be used in large-scale molecular dynamics simulations of thousands of atoms. NNPs are constructed by adjusting a set of parameters using data from electronic structure calculations, and in many cases energies and forces can be obtained with very high accuracy. Therefore, NNP-based simulation results are often very close to those gained by a direct application of first-principles methods. In this review, the basic methodology of high-dimensional NNPs will be presented with a special focus on the scope and the remaining limitations of this approach. The development of NNPs requires substantial computational effort as typically thousands of reference calculations are required. Still, if the problem to be studied involves very large systems or long simulation times this overhead is regained quickly. Further, the method is still limited to systems containing about three or four chemical elements due to the rapidly increasing complexity of the configuration space, although many atoms of each species can be present. Due to the ability of NNPs to describe even extremely complex atomic configurations with excellent accuracy irrespective of the nature of the atomic interactions, they represent a general and therefore widely applicable technique, e.g. for addressing problems in materials science, for investigating properties of interfaces, and for studying solvation processes.

Construction of high-dimensional neural network potentials using environment-dependent atom pairs

An accurate determination of the potential energy is the crucial step in computer simulations of chemical processes, but using electronic structure methods on-the-fly in molecular dynamics (MD) is computationally too demanding for many systems. Constructing more efficient interatomic potentials becomes intricate with increasing dimensionality of the potential-energy surface (PES), and for numerous systems the accuracy that can be achieved is still not satisfying and far from the reliability of first-principles calculations. Feed-forward neural networks (NNs) have a very flexible functional form, and in recent years they have been shown to be an accurate tool to construct efficient PESs. High-dimensional NN potentials based on environment-dependent atomic energy contributions have been presented for a number of materials. Still, these potentials may be improved by a more detailed structural description, e.g., in form of atom pairs, which directly reflect the atomic interactions and take the chemical environment into account. We present an implementation of an NN method based on atom pairs, and its accuracy and performance are compared to the atom-based NN approach using two very different systems, the methanol molecule and metallic copper. We find that both types of NN potentials provide an excellent description of both PESs, with the pair-based method yielding a slightly higher accuracy making it a competitive alternative for addressing complex systems in MD simulations.

A neural network potential-energy surface for the water dimer based on environment-dependent atomic energies and charges

Understanding the unique properties of water still represents a significant challenge for theory and experiment. Computer simulations by molecular dynamics require a reliable description of the atomic interactions, and in recent decades countless water potentials have been reported in the literature. Still, most of these potentials contain significant approximations, for instance a frozen internal structure of the individual water monomers. Artificial neural networks (NNs) offer a promising way for the construction of very accurate potential-energy surfaces taking all degrees of freedom explicitly into account. These potentials are based on electronic structure calculations for representative configurations, which are then interpolated to a continuous energy surface that can be evaluated many orders of magnitude faster. We present a full-dimensional NN potential for the water dimer as a first step towards the construction of a NN potential for liquid water. This many-body potential is based on environment-dependent atomic energy contributions, and long-range electrostatic interactions are incorporated employing environment-dependent atomic charges. We show that the potential and derived properties like vibrational frequencies are in excellent agreement with the underlying reference density-functional theory calculations.

Empirical valence bond models for reactive potential energy surfaces: a parallel multilevel genetic program approach

We describe a new method for constructing empirical valence bond potential energy surfaces using a parallel multilevel genetic program (PMLGP). Genetic programs can be used to perform an efficient search through function space and parameter space to find the best functions and sets of parameters that fit energies obtained by ab initio electronic structure calculations. Building on the traditional genetic program approach, the PMLGP utilizes a hierarchy of genetic programming on two different levels. The lower level genetic programs are used to optimize coevolving populations in parallel while the higher level genetic program (HLGP) is used to optimize the genetic operator probabilities of the lower level genetic programs. The HLGP allows the algorithm to dynamically learn the mutation or combination of mutations that most effectively increase the fitness of the populations, causing a significant increase in the algorithm's accuracy and efficiency. The algorithm's accuracy and efficiency is tested against a standard parallel genetic program with a variety of one-dimensional test cases. Subsequently, the PMLGP is utilized to obtain an accurate empirical valence bond model for proton transfer in 3-hydroxy-gamma-pyrone in gas phase and protic solvent.

Input vector optimization of feed-forward neural networks for fitting ab initio potential-energy databases

The variation in the fitting accuracy of neural networks (NNs) when used to fit databases comprising potential energies obtained from ab initio electronic structure calculations is investigated as a function of the number and nature of the elements employed in the input vector to the NN. Ab initio databases for H(2)O(2), HONO, Si(5), and H(2)C[Double Bond]CHBr were employed in the investigations. These systems were chosen so as to include four-, five-, and six-body systems containing first, second, third, and fourth row elements with a wide variety of chemical bonding and whose conformations cover a wide range of structures that occur under high-energy machining conditions and in chemical reactions involving cis-trans isomerizations, six different types of two-center bond ruptures, and two different three-center dissociation reactions. The ab initio databases for these systems were obtained using density functional theory/B3LYP, MP2, and MP4 methods with extended basis sets. A total of 31 input vectors were investigated. In each case, the elements of the input vector were chosen from interatomic distances, inverse powers of the interatomic distance, three-body angles, and dihedral angles. Both redundant and nonredundant input vectors were investigated. The results show that among all the input vectors investigated, the set employed in the Z-matrix specification of the molecular configurations in the electronic structure calculations gave the lowest NN fitting accuracy for both Si(5) and vinyl bromide. The underlying reason for this result appears to be the discontinuity present in the dihedral angle for planar geometries. The use of trigometric functions of the angles as input elements produced significantly improved fitting accuracy as this choice eliminates the discontinuity. The most accurate fitting was obtained when the elements of the input vector were taken to have the form R(ij) (-n), where the R(ij) are the interatomic distances. When the Levenberg-Marquardt procedure was modified to permit error minimization with respect to n as well as the weights and biases of the NN, the optimum powers were all found to lie in the range of 1.625-2.38 for the four systems studied. No statistically significant increase in fitting accuracy was achieved for vinyl bromide when a different value of n was employed and optimized for each bond type. The rate of change in the fitting error with n is found to be very small when n is near its optimum value. Consequently, good fitting accuracy can be achieved by employing a value of n in the middle of the above range. The use of interparticle distances as elements of the input vector rather than the Z-matrix variables employed in the electronic structure calculations is found to reduce the rms fitting errors by factors of 8.86 and 1.67 for Si(5) and vinyl bromide, respectively. If the interparticle distances are replaced with input elements of the form R(ij) (-n) with n optimized, further reductions in the rms error by a factor of 1.31 to 2.83 for the four systems investigated are obtained. A major advantage of using this procedure to increase NN fitting accuracy rather than increasing the number of neurons or the size of the database is that the required increase in computational effort is very small.

Development of generalized potential-energy surfaces using many-body expansions, neural networks, and moiety energy approximations

A general method for the development of potential-energy hypersurfaces is presented. The method combines a many-body expansion to represent the potential-energy surface with two-layer neural networks (NN) for each M-body term in the summations. The total number of NNs required is significantly reduced by employing a moiety energy approximation. An algorithm is presented that efficiently adjusts all the coupled NN parameters to the database for the surface. Application of the method to four different systems of increasing complexity shows that the fitting accuracy of the method is good to excellent. For some cases, it exceeds that available by other methods currently in literature. The method is illustrated by fitting large databases of ab initio energies for Si(n) (n=3,4,...,7) clusters obtained from density functional theory calculations and for vinyl bromide (C(2)H(3)Br) and all products for dissociation into six open reaction channels (12 if the reverse reactions are counted as separate open channels) that include C-H and C-Br bond scissions, three-center HBr dissociation, and three-center H(2) dissociation. The vinyl bromide database comprises the ab initio energies of 71 969 configurations computed at MP4(SDQ) level with a 6-31G(d,p) basis set for the carbon and hydrogen atoms and Huzinaga's (4333/433/4) basis set augmented with split outer s and p orbitals (43321/4321/4) and a polarization f orbital with an exponent of 0.5 for the bromine atom. It is found that an expansion truncated after the three-body terms is sufficient to fit the Si(5) system with a mean absolute testing set error of 5.693x10(-4) eV. Expansions truncated after the four-body terms for Si(n) (n=3,4,5) and Si(n) (n=3,4,...,7) provide fits whose mean absolute testing set errors are 0.0056 and 0.0212 eV, respectively. For vinyl bromide, a many-body expansion truncated after the four-body terms provides fitting accuracy with mean absolute testing set errors that range between 0.0782 and 0.0808 eV. These errors correspond to mean percent errors that fall in the range 0.98%-1.01%. Our best result using the present method truncated after the four-body summation with 16 NNs yields a testing set error that is 20.3% higher than that obtained using a 15-dimensional (15-140-1) NN to fit the vinyl bromide database. This appears to be the price of the added simplicity of the many-body expansion procedure.

Using neural networks, optimized coordinates, and high-dimensional model representations to obtain a vinyl bromide potential surface

We demonstrate that it is possible to obtain good potentials using high-dimensional model representations (HDMRs) fitted with neural networks (NNs) from data in 12 dimensions and 15 dimensions. The HDMR represents the potential as a sum of lower-dimensional functions and our NN-based approach makes it possible to obtain all of these functions from one set of fitting points. To reduce the number of terms in the HDMR, we use optimized redundant coordinates. By using exponential neurons, one obtains a potential in sum-of-products form, which greatly facilitates quantum dynamics calculations. A 12-dimensional (reference) potential surface for vinyl bromide is first refitted to show that it can be represented as a sum of two-dimensional functions. To fit 3d functions of the original coordinates, to improve the potential, a huge amount of data would be required. Redundant coordinates avoid this problem. They enable us to bypass the combinatorial explosion of the number of terms which plagues all HDMR and multimode-type methods. We also fit to a set of approximately 70,000 ab initio points for vinyl bromide in 15 dimensions [M. Malshe et al., J. Chem. Phys. 127, 134105 (2007)] and show that it is possible to obtain a surface in sum-of-products form of quality similar to the quality of the full-dimensional fit. Although we obtain a full-dimensional surface, we limit the cost of the fitting by building it from fits of six-dimensional functions, each of which requires only a small NN.