Gaussian Process Model for Collision Dynamics of Complex Molecules

A Gaussian Process Based Δ-Machine Learning Approach to Reactive Potential Energy Surfaces

The Gaussian process (GP) is an efficient nonparametric machine learning (ML) method. A distinct advantage of the GP is its ability to provide an estimate of statistical uncertainties. This is particularly useful in constructing high-dimensional potential energy surfaces (PESs) from ab initio data as it offers an optimal way to add new geometries to reduce the overall error. In this work, GP is employed in the context of Δ-machine learning (Δ-ML), in which a correction PES to an inaccurate low-level PES is constructed using a small number of high-level ab initio calculations. This new method is tested in three prototypical reactive systems, namely, the H + H2 → H + H2, OH + H2 → H2O + H, and H + CH4 → H2 + CH3 reactions. The results show that the GP-based Δ-ML approach is more efficient than its direct application in constructing high-level PESs. We also compare the new method to a previously proposed neural-network-based Δ-ML approach [Liu and Li J. Phys. Chem. Lett. 2022, 13, 4729-4738]. The results indicate that the two Δ-ML methods have comparable efficiencies in constructing accurate PESs.

Molecular dynamics-driven global potential energy surfaces: Application to the AlF dimer

In this work, we present a full-dimensional potential energy surface for AlF-AlF. We apply a general machine learning approach for full-dimensional potential energy surfaces, employing an active learning scheme trained on ab initio points, whose size grows based on the accuracy required. The training points are selected based on molecular dynamics simulations, choosing the most suitable configurations for different collision energy and mapping the most relevant part of the potential energy landscape of the system. The present approach does not require long-range information and is entirely general. As a result, it is possible to provide the full-dimensional AlF-AlF potential energy surface, requiring ≲0.01% of the configurations to be calculated ab initio. Furthermore, we analyze the general properties of the AlF-AlF system, finding critical differences with other reported results on CaF or bi-alkali dimers.

Adaptive regularized Gaussian process regression for application in the context of hydrogen adsorption on graphene sheets

We present a Gaussian process regression (GPR) scheme with an adaptive regularization scheme applied to the QM7 and QM9 test set, several protonated water clusters and specifically to the problem of atomic hydrogen adsorption on graphene sheets. For the last system our goal is to achieve good predictive accuracy with only a few training points. Therefore, we assess for these systems a self-correcting multilayer GPR model, in which the prediction is corrected by a chain of additional GPR models. In our adaptive regularization scheme, we impose no noise on the training data, but use an approach based on the data itself to account for its impurity. The strength of this strategy is that the data points are treated differently based on their importance and that the regularization can still be controlled by a single parameter. We assess how the accuracy of the prediction depends on this parameter. We can show that the new regularization scheme as well as the multilayer approach results in more robust predictors. Furthermore, we demonstrate that the predictor can be in good agreement with the density-functional theory results.

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

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

Vibrational energy pooling via collisions between asymmetric stretching excited CO2: a quasi-classical trajectory study on an accurate full-dimensional potential energy surface

In low temperature plasmas, energy transfer between asymmetric stretching excited CO₂ molecules can be highly efficient, which leads to further excitation (and de-excitation) of the CO₂ molecules: CO₂(v_as) + CO₂(v_as) → CO₂(v_as + 1) + CO₂(v_as - 1). Through such a vibrational ladder climbing mechanism, CO₂ can be activated and eventually dissociates. To gain mechanistic insight of such processes, a full-dimensional accurate potential energy surface (PES) for the CO₂ + CO₂ system is developed using the permutational invariant polynomial-neural network method based on CCSD(T)-F12a/AVTZ energies at about 39 000 geometries. This PES is used in quasi-classical trajectory (QCT) studies of the vibrational energy transfer between CO₂ molecules excited in the asymmetric stretching mode. A machine learning algorithm is used to determine state-specific rate coefficients for the vibrational transfer processes from a limited data set. In addition to the CO₂(v_as + 1) + CO₂(v_as - 1) channel, the QCT simulations revealed significant contributions from the CO₂(v_as + 2,3) + CO₂(v_as - 2,3) channels, particularly at low collision energies/temperatures. These multi-vibrational-quantum processes are attributed to enhanced energy flow in the collisional complex formed by enhanced dipole-dipole interaction between asymmetric stretching excited CO₂ molecules.

Reduced-dimensional surface hopping with offline-online computations

Molecular dynamics simulations often classically evolve the nuclear geometry on adiabatic potential energy surfaces (PESs), punctuated by random hops between energy levels in regions of strong coupling, in an algorithm known as surface hopping. However, the computational expense of integrating the geometry on a full-dimensional PES and computing the required couplings can quickly become prohibitive as the number of atoms increases. In this work, we describe a method for surface hopping that uses only important reaction coordinates, performs all expensive evaluations of the true PESs and couplings only once before simulating dynamics (offline), and then queries the stored values during the surface hopping simulation (online). Our Python codes are freely available on GitHub. Using photodissociation of azomethane as a test case, this method is able to reproduce experimental results that have thus far eluded ab initio surface hopping studies.

Efficient Approximation of Potential Energy Surfaces with Mixed-Basis Interpolation

The potential energy surface (PES) describes the energy of a chemical system as a function of its geometry and is a fundamental concept in modern chemistry. A PES provides much useful information about the system, including the structures and energies of various stationary points, such as stable conformers (local minima) and transition states (first-order saddle points) connected by a minimum-energy path. Our group has previously produced surrogate reduced-dimensional PESs using sparse interpolation along chemically significant reaction coordinates, such as bond lengths, bond angles, and torsion angles. These surrogates used a single interpolation basis, either polynomials or trigonometric functions, in every dimension. However, relevant molecular dynamics (MD) simulations often involve some combination of both periodic and nonperiodic coordinates. Using a trigonometric basis on nonperiodic coordinates, such as bond lengths, leads to inaccuracies near the domain boundary. Conversely, polynomial interpolation on the periodic coordinates does not enforce the periodicity of the surrogate PES gradient, leading to nonconservation of total energy even in a microcanonical ensemble. In this work, we present an interpolation method that uses trigonometric interpolation on the periodic reaction coordinates and polynomial interpolation on the nonperiodic coordinates. We apply this method to MD simulations of possible isomerization pathways of azomethane between cis and trans conformers. This method is the only known interpolative method that appropriately conserves total energy in systems with both periodic and nonperiodic reaction coordinates. In addition, compared to all-polynomial interpolation, the mixed basis requires fewer electronic structure calculations to obtain a given level of accuracy, is an order of magnitude faster, and is freely available on GitHub.

Ab Initio Machine Learning in Chemical Compound Space

Chemical compound space (CCS), the set of all theoretically conceivable combinations of chemical elements and (meta-)stable geometries that make up matter, is colossal. The first-principles based virtual sampling of this space, for example, in search of novel molecules or materials which exhibit desirable properties, is therefore prohibitive for all but the smallest subsets and simplest properties. We review studies aimed at tackling this challenge using modern machine learning techniques based on (i) synthetic data, typically generated using quantum mechanics based methods, and (ii) model architectures inspired by quantum mechanics. Such Quantum mechanics based Machine Learning (QML) approaches combine the numerical efficiency of statistical surrogate models with an ab initio view on matter. They rigorously reflect the underlying physics in order to reach universality and transferability across CCS. While state-of-the-art approximations to quantum problems impose severe computational bottlenecks, recent QML based developments indicate the possibility of substantial acceleration without sacrificing the predictive power of quantum mechanics.

Neural network approach for the dynamics on the normally hyperbolic invariant manifold of periodically driven systems

Chemical reactions in multidimensional systems are often described by a rank-1 saddle, whose stable and unstable manifolds intersect in the normally hyperbolic invariant manifold (NHIM). Trajectories started on the NHIM in principle never leave this manifold when propagated forward or backward in time. However, the numerical investigation of the dynamics on the NHIM is difficult because of the instability of the motion. We apply a neural network to describe time-dependent NHIMs and use this network to stabilize the motion on the NHIM for a periodically driven model system with two degrees of freedom. The method allows us to analyze the dynamics on the NHIM via Poincaré surfaces of section (PSOS) and to determine the transition-state (TS) trajectory as a periodic orbit with the same periodicity as the driving saddle, viz. a fixed point of the PSOS surrounded by near-integrable tori. Based on transition state theory and a Floquet analysis of a periodic TS trajectory we compute the rate constant of the reaction with significantly reduced numerical effort compared to the propagation of a large trajectory ensemble.

Approximating Periodic Potential Energy Surfaces with Sparse Trigonometric Interpolation

The potential energy surface (PES) describes the energy of a chemical system as a function of its geometry and is a fundamental concept in computational chemistry. A PES provides much useful information about the system, including the structures and energies of various stationary points, such as local minima, maxima, and transition states. Construction of full-dimensional PESs for molecules with more than 10 atoms is computationally expensive and often not feasible. Previous work in our group used sparse interpolation with polynomial basis functions to construct a surrogate reduced-dimensional PESs along chemically significant reaction coordinates, such as bond lengths, bond angles, and torsion angles. However, polynomial interpolation does not preserve the periodicity of the PES gradient with respect to angular components of geometry, such as torsion angles, which can lead to nonphysical phenomena. In this work, we construct a surrogate PES using trigonometric basis functions, for a system where the selected reaction coordinates all correspond to the torsion angles, resulting in a periodically repeating PES. We find that a trigonometric interpolation basis not only guarantees periodicity of the gradient but also results in slightly lower approximation error than polynomial interpolation.

Molecular Dynamics Simulations on Relaxed Reduced-Dimensional Potential Energy Surfaces

Molecular dynamics (MD) simulations with full-dimensional potential energy surfaces (PESs) obtained from high-level ab initio calculations are frequently used to model reaction dynamics of small molecules (i.e., molecules with up to 10 atoms). Construction of full-dimensional PESs for larger molecules is, however, not feasible since the number of ab initio calculations required grows rapidly with the increase of dimension. Only a small number of coordinates are often essential for describing the reactivity of even very large systems, and reduced-dimensional PESs with these coordinates can be built for reaction dynamics studies. While analytical methods based on transition-state theory framework are well established for analyzing the reduced-dimensional PESs, MD simulation algorithms capable of generating trajectories on such surfaces are more rare. In this work, we present a new MD implementation that utilizes the relaxed reduced-dimensional PES for standard microcanonical (NVE) and canonical (NVT) MD simulations. The method is applied to the pyramidal inversion of a NH₃ molecule. The results from the MD simulations on a reduced, three-dimensional PES are validated against the ab initio MD simulations, as well as MD simulations on full-dimensional PES and experimental data.

Bayesian machine learning for quantum molecular dynamics

This article discusses applications of Bayesian machine learning for quantum molecular dynamics.

Neural network potential energy surface and dynamical isotope effects for the N+(3P) + H2 → NH+ + H reaction

Dynamical isotope effects are calculated for the N⁺(³P) + H₂ → NH⁺ + H reaction on a new neural network potential energy surface.

Neural network approach to time-dependent dividing surfaces in classical reaction dynamics

In a dynamical system, the transition between reactants and products is typically mediated by an energy barrier whose properties determine the corresponding pathways and rates. The latter is the flux through a dividing surface (DS) between the two corresponding regions, and it is exact only if it is free of recrossings. For time-independent barriers, the DS can be attached to the top of the corresponding saddle point of the potential energy surface, and in time-dependent systems, the DS is a moving object. The precise determination of these direct reaction rates, e.g., using transition state theory, requires the actual construction of a DS for a given saddle geometry, which is in general a demanding methodical and computational task, especially in high-dimensional systems. In this paper, we demonstrate how such time-dependent, global, and recrossing-free DSs can be constructed using neural networks. In our approach, the neural network uses the bath coordinates and time as input, and it is trained in a way that its output provides the position of the DS along the reaction coordinate. An advantage of this procedure is that, once the neural network is trained, the complete information about the dynamical phase space separation is stored in the network's parameters, and a precise distinction between reactants and products can be made for all possible system configurations, all times, and with little computational effort. We demonstrate this general method for two- and three-dimensional systems and explain its straightforward extension to even more degrees of freedom.

Estimating Parameter Uncertainty in Binding-Energy Models by the Frequency-Domain Bootstrap

We propose using the frequency-domain bootstrap (FDB) to estimate errors of modeling parameters when the modeling error is itself a major source of uncertainty. Unlike the usual bootstrap or the simple χ^{2} analysis, the FDB can take into account correlations between errors. It is also very fast compared to the Gaussian process Bayesian estimate as often implemented for computer model calibration. The method is illustrated with a simple example, the liquid drop model of nuclear binding energies. We find that the FDB gives a more conservative estimate of the uncertainty in liquid drop parameters than the χ^{2} method, and is in fair accord with more empirical estimates. For the nuclear physics application, there are no apparent obstacles to apply the method to the more accurate and detailed models based on density-functional theory.

Cold molecules: Progress in quantum engineering of chemistry and quantum matter

Cooling atoms to ultralow temperatures has produced a wealth of opportunities in fundamental physics, precision metrology, and quantum science. The more recent application of sophisticated cooling techniques to molecules, which has been more challenging to implement owing to the complexity of molecular structures, has now opened the door to the longstanding goal of precisely controlling molecular internal and external degrees of freedom and the resulting interaction processes. This line of research can leverage fundamental insights into how molecules interact and evolve to enable the control of reaction chemistry and the design and realization of a range of advanced quantum materials.

A general method for controlling and resolving rotational orientation of molecules in molecule-surface collisions

The outcome of molecule–surface collisions can be modified by pre-aligning the molecule; however, experiments accomplishing this are rare because of the difficulty of preparing molecules in aligned quantum states. Here we present a general solution to this problem based on magnetic manipulation of the rotational magnetic moment of the incident molecule. We apply the technique to the scattering of H₂ from flat and stepped copper surfaces. We demonstrate control of the molecule's initial quantum state, allowing a direct comparison of differences in the stereodynamic scattering from the two surfaces. Our results show that a stepped surface exhibits a much larger dependence of the corrugation of the interaction on the alignment of the molecule than the low-index surface. We also demonstrate an extension of the technique that transforms the set-up into an interferometer, which is sensitive to molecular quantum states both before and after the scattering event.

The rotational orientation of a molecule plays a fundamental role in molecule-surface collisions, yet is difficult to study. Here, the authors present a general approach for controlling and resolving molecular rotational orientation and apply it to study H₂ scattering from flat and stepped copper surfaces.

High-Dimensional Atomistic Neural Network Potentials for Molecule-Surface Interactions: HCl Scattering from Au(111)

Ab initio molecular dynamics (AIMD) simulations of molecule-surface scattering allow first-principles characterization of the dynamics. However, the large number of density functional theory calculations along the trajectories is very costly, limiting simulations of long-time events and giving rise to poor statistics. To avoid this computational bottleneck, we report here the development of a high-dimensional molecule-surface interaction potential energy surface (PES) with movable surface atoms, using a machine learning approach. With 60 degrees of freedom, this PES allows energy transfer between the energetic impinging molecule and thermal surface atoms. Classical trajectory calculations for the scattering of DCl from Au(111) on this PES are found to agree well with AIMD simulations, with ∼10⁵-fold acceleration. Scattering of HCl from Au(111) is further investigated and compared with available experimental results.