A Quadratically-Converging Nudged Elastic Band Optimizer

Physical Prior Mean Function-Driven Gaussian Processes Search for Minimum-Energy Reaction Paths with a Climbing-Image Nudged Elastic Band: A General Method for Gas-Phase, Interfacial, and Bulk-Phase Reactions

The climbing-image nudged elastic band (CI-NEB) method serves as an indispensable tool for computational chemists, offering insight into minimum-energy reaction paths (MEPs) by delineating both transition states (TSs) and intermediate nonstationary structures along reaction coordinates. However, executing CI-NEB calculations for reactions with extensive reaction coordinate spans necessitates a large number of images to ensure a reliable convergence of the MEPs and TS structures, presenting a computationally demanding optimization challenge, even with mildly costly electronic-structure methods. In this study, we advocate for the utilization of physically inspired prior mean function-based Gaussian processes (GPs) to expedite MEP exploration and TS optimization via the CI-NEB method. By incorporating reliable prior physical approximations into potential energy surface (PES) modeling, we demonstrate enhanced efficiency in multidimensional CI-NEB optimization with surrogate-based optimizers. Our physically informed GP approach not only outperforms traditional nonsurrogate-based optimizers in optimization efficiency but also on-the-fly learns the reaction path valley during optimization, culminating in significant advancements. The surrogate PES derived from our optimization exhibits high accuracy compared to true PES references, aligning with our emphasis on leveraging reliable physical priors for robust and efficient posterior mean learning in GPs. Through a systematic benchmark study encompassing various reaction pathways, including gas-phase, bulk-phase, and interfacial/surface reactions, our physical GPs consistently demonstrate superior efficiency and reliability. For instance, they outperform the popular fast inertial relaxation engine optimizer by approximately a factor of 10, showcasing their versatility and efficacy in exploring reaction mechanisms and surface reaction PESs.

Path optimization by a variational reaction coordinate method. II. Improved computational efficiency through internal coordinates and surface interpolation

Reaction path optimization is being used more frequently as an alternative to the standard practice of locating a transition state and following the path downhill. The Variational Reaction Coordinate (VRC) method was proposed as an alternative to chain-of-states methods like nudged elastic band and string method. The VRC method represents the path using a linear expansion of continuous basis functions, allowing the path to be optimized variationally by updating the expansion coefficients to minimize the line integral of the potential energy gradient norm, referred to as the Variational Reaction Energy (VRE) of the path. When constraints are used to control the spacing of basis functions and to couple the minimization of the VRE with the optimization of one or more individual points along the path (representing transition states and intermediates), an approximate path as well as the converged geometries of transition states and intermediates along the path are determined in only a few iterations. This algorithmic efficiency comes at a high per-iteration cost due to numerical integration of the VRE derivatives. In the present work, methods for incorporating redundant internal coordinates and potential energy surface interpolation into the VRC method are described. With these methods, the per-iteration cost, in terms of the number of potential energy surface evaluations, of the VRC method is reduced while the high algorithmic efficiency is maintained.

Locating Instantons in Calculations of Tunneling Splittings: The Test Case of Malonaldehyde

The recently developed ring-polymer instanton (RPI) method [J. Chem. Phys. 2011, 134, 054109] is an efficient technique for calculating approximate tunneling splittings in high-dimensional molecular systems. The key step is locating the instanton tunneling-path at zero temperature. Here, we show that techniques previously designed for locating instantons in finite-temperature rate calculations can be adapted to the RPI method, where they become extremely efficient, reducing the number of potential energy calls by 2 orders of magnitude. We investigate one technique that employs variable time steps to minimize the action integral, and two that employ equally spaced position steps to minimize the abbreviated (i.e., Jacobi) action integral, using respectively the nudged elastic band (NEB) and string methods. We recommend use of the latter because it is parameter-free, but all three methods give comparable efficiency savings. Having located the instanton pathway, we then interpolate the instanton path onto a fine grid of imaginary time points, allowing us to compute the fluctuation prefactor. The crucial modification needed to the original finite-temperature algorithms is to allow the end points of the zero-temperature instanton path to describe overall rotations, which is done using a standard quaternion algorithm. These approaches will allow the RPI method to be combined effectively with expensive potential energy surfaces or on-the-fly electronic structure methods.

Path optimization by a variational reaction coordinate method. I. Development of formalism and algorithms

The development of algorithms to optimize reaction pathways between reactants and products is an active area of study. Existing algorithms typically describe the path as a discrete series of images (chain of states) which are moved downhill toward the path, using various reparameterization schemes, constraints, or fictitious forces to maintain a uniform description of the reaction path. The Variational Reaction Coordinate (VRC) method is a novel approach that finds the reaction path by minimizing the variational reaction energy (VRE) of Quapp and Bofill. The VRE is the line integral of the gradient norm along a path between reactants and products and minimization of VRE has been shown to yield the steepest descent reaction path. In the VRC method, we represent the reaction path by a linear expansion in a set of continuous basis functions and find the optimized path by minimizing the VRE with respect to the linear expansion coefficients. Improved convergence is obtained by applying constraints to the spacing of the basis functions and coupling the minimization of the VRE to the minimization of one or more points along the path that correspond to intermediates and transition states. The VRC method is demonstrated by optimizing the reaction path for the Müller-Brown surface and by finding a reaction path passing through 5 transition states and 4 intermediates for a 10 atom Lennard-Jones cluster.

Stepwise catalytic mechanism via short-lived intermediate inferred from combined QM/MM MERP and PES calculations on retaining glycosyltransferase ppGalNAcT2

The glycosylation of cell surface proteins plays a crucial role in a multitude of biological processes, such as cell adhesion and recognition. To understand the process of protein glycosylation, the reaction mechanisms of the participating enzymes need to be known. However, the reaction mechanism of retaining glycosyltransferases has not yet been sufficiently explained. Here we investigated the catalytic mechanism of human isoform 2 of the retaining glycosyltransferase polypeptide UDP-GalNAc transferase by coupling two different QM/MM-based approaches, namely a potential energy surface scan in two distance difference dimensions and a minimum energy reaction path optimisation using the Nudged Elastic Band method. Potential energy scan studies often suffer from inadequate sampling of reactive processes due to a predefined scan coordinate system. At the same time, path optimisation methods enable the sampling of a virtually unlimited number of dimensions, but their results cannot be unambiguously interpreted without knowledge of the potential energy surface. By combining these methods, we have been able to eliminate the most significant sources of potential errors inherent to each of these approaches. The structural model is based on the crystal structure of human isoform 2. In the QM/MM method, the QM region consists of 275 atoms, the remaining 5776 atoms were in the MM region. We found that ppGalNAcT2 catalyzes a same-face nucleophilic substitution with internal return (SNi). The optimized transition state for the reaction is 13.8 kcal/mol higher in energy than the reactant while the energy of the product complex is 6.7 kcal/mol lower. During the process of nucleophilic attack, a proton is synchronously transferred to the leaving phosphate. The presence of a short-lived metastable oxocarbenium intermediate is likely, as indicated by the reaction energy profiles obtained using high-level density functionals.

QM/MM investigation of ATP hydrolysis in aqueous solution

Adenosine-5'-triphosphate (ATP) hydrolysis represents a most important reaction in biology. Despite extensive research efforts, the mechanism for ATP hydrolysis in aqueous solution still remains under debate. Previous theoretical studies often predefined reaction coordinates to characterize the mechanism for ATP hydrolysis in water with Mg(2+) by evaluating free energy profiles through these preassumed reaction paths. In the present work, a nudged elastic band method is applied to identify the minimum energy path calculated with a hybrid quantum mechanics and molecular mechanics approach. Along the reaction path, the free energy profile was obtained to have a single transition state and the activation energy of 32.5 kcal/mol. This transition state bears a four-centered structure that describes a concerted nature of the reaction. In the More-O'Ferrall-Jencks diagram, the results show that the reaction proceeds through a concerted path before the system reaches the transition state and along an associative path after the transition state. In addition, the calculated reaction free energy is -7.0 kcal/mol, in good agreement with experiment, capturing the exothermic feature of MgATP(2-) hydrolysis in aqueous solution, whereas the reaction was often shown to be endothermic in the previous theoretical studies. As Mg(2+) is required for ATP hydrolysis in cells, its role in the reaction is also elucidated.