Incorporating Linear Synchronous Transit Interpolation into the Growing String Method: Algorithm and Applications

Exploration of biochemical reactivity with a QM/MM growing string method

In this work, we have implemented the single-ended growing string method using a hybrid internal/Cartesian coordinate scheme within our in-house QM/MM package, QoMMMa, representing the first implementation of the growing string method in the QM/MM framework. The goal of the implementation was to facilitate generation of QM/MM reaction pathways with minimal user input, and also to improve the quality of the pathways generated as compared to the widely used adiabatic mapping approach. We have validated the algorithm against a reaction which has been studied extensively in previous computational investigations - the Claisen rearrangement catalysed by chorismate mutase. The nature of the transition state and the height of the barrier was predicted well using our algorithm, where more than 88% of the pathways generated were deemed to be of production quality. Directly compared to using adiabatic mapping, we found that while our QM/MM single-ended growing string method is slightly less efficient, it readily produces reaction pathways with fewer discontinuites and thus minimises the need for involved remapping of unsatisfactory energy profiles.

Locating saddle points using gradient extremals on manifolds adaptively revealed as point clouds

Steady states are invaluable in the study of dynamical systems. High-dimensional dynamical systems, due to separation of time scales, often evolve toward a lower dimensional manifold M. We introduce an approach to locate saddle points (and other fixed points) that utilizes gradient extremals on such a priori unknown (Riemannian) manifolds, defined by adaptively sampled point clouds, with local coordinates discovered on-the-fly through manifold learning. The technique, which efficiently biases the dynamical system along a curve (as opposed to exhaustively exploring the state space), requires knowledge of a single minimum and the ability to sample around an arbitrary point. We demonstrate the effectiveness of the technique on the Müller-Brown potential mapped onto an unknown surface (namely, a sphere). Previous work employed a similar algorithmic framework to find saddle points using Newton trajectories and gentlest ascent dynamics; we, therefore, also offer a brief comparison with these methods.

Combined molecular dynamics and coordinate driving method for automatically searching complicated reaction pathways

The combined molecular dynamics and coordinate driving (MD/CD) method is updated and generalized in this work to broaden its applications in automatically searching reaction pathways for complicated reactions. In this updated version, MD simulations are performed with the GFN's family of methods to systematically sample conformers for almost any systems with atomic numbers Z ≤ 86. The improved CD procedure is greatly accelerated by applying a pre-screening stage at the semiempirical GFN2-xTB level. An automatic module based on the Marcus theory and its improved version (the Wolynes theory) is designed to include single electron transfer (SET) processes into reaction pathways. The capabilities of this method are demonstrated by exploring the most possible reaction pathways of three experimentally reported reactions: the organophosphine-catalyzed trans phosphinoboration, the Fe(II) complex-mediated C(sp2)-H borylation reaction, and the SET-triggered deaminative radical cross-coupling reaction. Comprehensive reaction networks are obtained for all three reactions with reasonable computational costs. Detailed mechanisms for these reactions can account for the reported experimental facts.

Constructing Spin-Adiabatic States for the Modeling of Spin-Crossing Reactions. I. A Shared-Orbital Implementation

In the modeling of spin-crossing reactions, it has become popular to directly explore the spin-adiabatic surfaces. Specifically, through constructing spin-adiabatic states from a two-state Hamiltonian (with spin-orbit coupling matrix elements) at each geometry, one can readily employ advanced geometry optimization algorithms to acquire a "transition state" structure, where the spin crossing occurs. In this work, we report the implementation of a fully-variational spin-adiabatic approach based on Kohn-Sham density functional theory spin states (sharing the same set of molecular orbitals) and the Breit-Pauli one-electron spin-orbit operator. For three model spin-crossing reactions [predissociation of N₂O, singlet-triplet conversion in CH₂, and CO addition to Fe(CO)₄], the spin-crossing points were obtained. Our results also indicated the Breit-Pauli one-electron spin-orbit coupling can vary significantly along the reaction pathway on the spin-adiabatic energy surface. On the other hand, due to the restriction that low-spin and high-spin states share the same set of molecular orbitals, the acquired spin-adiabatic energy surface shows a cusp (i.e. a first-order discontinuity) at the crossing point, which prevents the use of standard geometry optimization algorithms to pinpoint the crossing point. An extension with this restriction removed is being developed to achieve the smoothness of spin-adiabatic surfaces.

Tinker 8: Software Tools for Molecular Design

The Tinker software, currently released as version 8, is a modular molecular mechanics and dynamics package written primarily in a standard, easily portable dialect of Fortran 95 with OpenMP extensions. It supports a wide variety of force fields, including polarizable models such as the Atomic Multipole Optimized Energetics for Biomolecular Applications (AMOEBA) force field. The package runs on Linux, macOS, and Windows systems. In addition to canonical Tinker, there are branches, Tinker-HP and Tinker-OpenMM, designed for use on message passing interface (MPI) parallel distributed memory supercomputers and state-of-the-art graphical processing units (GPUs), respectively. The Tinker suite also includes a tightly integrated Java-based graphical user interface called Force Field Explorer (FFE), which provides molecular visualization capabilities as well as the ability to launch and control Tinker calculations.

Generating conformational transition paths with low potential-energy barriers for proteins

The knowledge of conformational transition paths in proteins can be useful for understanding protein mechanisms. Recently, we have introduced the As-Rigid-As-Possible (ARAP) interpolation method, for generating interpolation paths between two protein conformations. The method was shown to preserve well the rigidity of the initial conformation along the path. However, because the method is totally geometry-based, the generated paths may be inconsistent because the atom interactions are ignored. Therefore, in this article, we would like to introduce a new method to generate conformational transition paths with low potential-energy barriers for proteins. The method is composed of three processing stages. First, ARAP interpolation is used for generating an initial path. Then, the path conformations are enhanced by a clash remover. Finally, Nudged Elastic Band, a path-optimization method, is used to produce a low-energy path. Large energy reductions are found in the paths obtained from the method than in those obtained from the ARAP interpolation method alone. The results also show that ARAP interpolation is a good candidate for generating an initial path because it leads to lower potential-energy paths than two other common methods for path interpolation.

Perturbation-Theory and Machine Learning (PTML) Model for High-Throughput Screening of Parham Reactions: Experimental and Theoretical Studies

Machine learning (ML) algorithms are gaining importance in the processing of chemical information and modeling of chemical reactivity problems. In this work, we have developed a perturbation-theory and machine learning (PTML) model combining perturbation theory (PT) and ML algorithms for predicting the yield of a given reaction. For this purpose, we have selected Parham cyclization, which is a general and powerful tool for the synthesis of heterocyclic and carbocyclic compounds. This reaction has both structural (substitution pattern on the substrate, internal electrophile, ring size, etc.) and operational variables (organolithium reagent, solvent, temperature, time, etc.), so predicting the effect of changes on substrate design (internal elelctrophile, halide, etc.) or reaction conditions on the yield is an important task that could help to optimize the reaction design. The PTML model developed uses PT operators to account for perturbations under experimental conditions and/or structural variables of all the molecules involved in a query reaction, compared to a reaction of reference. Thus, a dataset of >100 reactions has been collected for different substrates and internal electrophiles, under different reaction conditions, with a wide range of yields (0-98%). The best PTML model found using General Linear Regression (GLR) has R = 0.88 in training and R = 0.83 in external validation series for 10 000 pairs of query and reference reactions. The PTML model has a final R = 0.95 for all reactions using multiple reactions of reference. We also report a comparative study of linear versus nonlinear PTML models based on artificial neural network (ANN) algorithms. PTML-ANN models (LNN, MLP, RBF) with R ≈ 0.1-0.8 do not outperform the first PMTL model. This result confirms the validity of the linearity of the model. Next, we carried out an experimental and theoretical study of nonreported Parham reactions to illustrate the practical use of the PTML model. A 500 000-point simulation and a Hammett analysis of the reactivity space of Parham reactions are also reported.

Automated Transition State Search and Its Application to Diverse Types of Organic Reactions

Transition state search is at the center of multiple types of computational chemical predictions related to mechanistic investigations, reactivity and regioselectivity predictions, and catalyst design. The process of finding transition states in practice is, however, a laborious multistep operation that requires significant user involvement. Here, we report a highly automated workflow designed to locate transition states for a given elementary reaction with minimal setup overhead. The only essential inputs required from the user are the structures of the separated reactants and products. The seamless workflow combining computational technologies from the fields of cheminformatics, molecular mechanics, and quantum chemistry automatically finds the most probable correspondence between the atoms in the reactants and the products, generates a transition state guess, launches a transition state search through a combined approach involving the relaxing string method and the quadratic synchronous transit, and finally validates the transition state via the analysis of the reactive chemical bonds and imaginary vibrational frequencies as well as by the intrinsic reaction coordinate method. Our approach does not target any specific reaction type, nor does it depend on training data; instead, it is meant to be of general applicability for a wide variety of reaction types. The workflow is highly flexible, permitting modifications such as a choice of accuracy, level of theory, basis set, or solvation treatment. Successfully located transition states can be used for setting up transition state guesses in related reactions, saving computational time and increasing the probability of success. The utility and performance of the method are demonstrated in applications to transition state searches in reactions typical for organic chemistry, medicinal chemistry, and homogeneous catalysis research. In particular, applications of our code to Michael additions, hydrogen abstractions, Diels-Alder cycloadditions, carbene insertions, and an enzyme reaction model involving a molybdenum complex are shown and discussed.

Fast exploration of an optimal path on the multidimensional free energy surface

In a reaction, determination of an optimal path with a high reaction rate (or a low free energy barrier) is important for the study of the reaction mechanism. This is a complicated problem that involves lots of degrees of freedom. For simple models, one can build an initial path in the collective variable space by the interpolation method first and then update the whole path constantly in the optimization. However, such interpolation method could be risky in the high dimensional space for large molecules. On the path, steric clashes between neighboring atoms could cause extremely high energy barriers and thus fail the optimization. Moreover, performing simulations for all the snapshots on the path is also time-consuming. In this paper, we build and optimize the path by a growing method on the free energy surface. The method grows a path from the reactant and extends its length in the collective variable space step by step. The growing direction is determined by both the free energy gradient at the end of the path and the direction vector pointing at the product. With fewer snapshots on the path, this strategy can let the path avoid the high energy states in the growing process and save the precious simulation time at each iteration step. Applications show that the presented method is efficient enough to produce optimal paths on either the two-dimensional or the twelve-dimensional free energy surfaces of different small molecules.

An investigation into the applicability of the semiempirical method PM7 for modeling the catalytic mechanism in the enzyme chymotrypsin

The catalytic cycle for the serine protease α-chymotrypsin was investigated in an attempt to determine the suitability of using the semiempirical method PM7 in the program MOPAC for investigating enzyme-catalyzed reactions. All six classical intermediates were modeled using standard methods, and were characterized as stable minima on the potential energy surface. Using a modified saddle point optimization method, five transition states were located and verified both by vibrational and by intrinsic reaction coordinate analysis. Some individual features, such as the hydrogen bonds in the oxyanion hole, the nature of various electrostatic interactions, and the role of Met192, were examined. This involved designing and running computational experiments to model mutations that would allow features of interest, in particular the energies involved, to be isolated. Three features within the enzyme were examined in detail: the reaction site itself, where covalent bonds were made and broken, the electrostatic effects of the buried aspartate anion, a passive but essential component of the catalytic triad, and the oxyanion hole, where hydrogen bonds help stabilize charged intermediates. With one minor exception, all phenomena investigated agreed with previously-reported descriptions. This result, along with the fact that all the techniques used were relatively straightforward, leads to the recommendation that PM7 and related methods, such as PM6-D3H4, are appropriate for modeling similar enzyme-catalyzed reactions. Graphical abstract Fifth of six transition states, showing water splitting into hydroxyl anion and a proton, to form the second tetrahedral intermediate and histidinium ion. Atoms of the water molecule involved in the hydrolysis are indicated by halos.

Reliable and efficient reaction path and transition state finding for surface reactions with the growing string method

The computational challenge of fast and reliable transition state and reaction path optimization requires new methodological strategies to maintain low cost, high accuracy, and systematic searching capabilities. The growing string method using internal coordinates has proven to be highly effective for the study of molecular, gas phase reactions, but difficulties in choosing a suitable coordinate system for periodic systems has prevented its use for surface chemistry. New developments are therefore needed, and presented herein, to handle surface reactions which include atoms with large coordination numbers that cannot be treated using standard internal coordinates. The double-ended and single-ended growing string methods are implemented using a hybrid coordinate system, then benchmarked for a test set of 43 elementary reactions occurring on surfaces. These results show that the growing string method is at least 45% faster than the widely used climbing image-nudged elastic band method, which also fails to converge in several of the test cases. Additionally, the surface growing string method has a unique single-ended search method which can move outward from an initial structure to find the intermediates, transition states, and reaction paths simultaneously. This powerful explorative feature of single ended-growing string method is demonstrated to uncover, for the first time, the mechanism for atomic layer deposition of TiN on Cu(111) surface. This reaction is found to proceed through multiple hydrogen-transfer and ligand-exchange events, while formation of H-bonds stabilizes intermediates of the reaction. Purging gaseous products out of the reaction environment is the driving force for these reactions. © 2017 Wiley Periodicals, Inc.

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.

Automated Discovery and Refinement of Reactive Molecular Dynamics Pathways

We describe a flexible and broadly applicable energy refinement method, "nebterpolation," for identifying and characterizing the reaction events in a molecular dynamics (MD) simulation. The new method is applicable to ab initio simulations with hundreds of atoms containing complex and multimolecular reaction events. A key aspect of nebterpolation is smoothing of the reactive MD trajectory in internal coordinates to initiate the search for the reaction path on the potential energy surface. We apply nebterpolation to analyze the reaction events in an ab initio nanoreactor simulation that discovers new molecules and mechanisms, including a C-C coupling pathway for glycolaldehyde synthesis. We find that the new method, which incorporates information from the MD trajectory that connects reactants with products, produces a dramatically distinct set of minimum energy paths compared to existing approaches that start from information for the reaction end points alone. The energy refinement method described here represents a key component of an emerging simulation paradigm where molecular dynamics simulations are applied to discover the possible reaction mechanisms.

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.

An automated transition state search using classical trajectories initialized at multiple minima

TS finding using iterative TSSCDS with trajectories initialized at different minima.

Reaction Path Optimization without NEB Springs or Interpolation Algorithms

This letter describes a chain-of-states method that optimizes reaction paths under the sole constraint of equally spaced structures. In contrast to NEB and string methods, it requires no spring forces, interpolation algorithms, or other heuristics to control structure distribution. Rigorous use of a quadratic PES allows calculation of an optimization step with a predefined distribution in Cartesian space. The method is a formal extension of single-structure quasi-Newton methods. An initial guess can be evolved, as in the growing string method.

Automated Transition State Searches without Evaluating the Hessian

Accurate and speedy determination of transition structures (TSs) is essential for computational studies on reaction pathways, particularly when the process involves expensive electronic structure calculations. Many search algorithms require a good initial guess of the TS geometry, as well as a Hessian input that possesses a structure consistent with the desired saddle point. Among the double-ended interpolation methods for generation of the guess for the TS, the freezing string method (FSM) is proven to be far less expensive compared to its predecessor, the growing string method (GSM). In this paper, it is demonstrated that the efficiency of this technique can be improved further by replacing the conjugate gradient optimization step (FSM-CG) with a quasi-Newton line search coupled with a BFGS Hessian update (FSM-BFGS). A second crucial factor that affects the speed with which convergence to the TS is achieved is the quality and cost of the Hessian of the energy for the guessed TS. For electronic structure calculations, the cost of calculating an exact Hessian increases more rapidly with system size than the energy and gradient. Therefore, to sidestep calculation of the exact Hessian, an approximate Hessian is constructed, using the tangent direction and local curvature at the TS guess. It is demonstrated that the partitioned-rational function optimization algorithm for locating TSs with this approximate Hessian input performs at least as well as with an exact Hessian input in most test cases. The two techniques, FSM and approximate Hessian construction, therefore can significantly reduce costs associated with finding TSs.