From transition paths to transition states and rate coefficients

Transition states are defined as points in configuration space with the highest probability that trajectories passing through them are reactive (i.e., form transition paths between reactants and products). In the high-friction (diffusive) limit of Langevin dynamics, the resulting ensemble of transition states is shown to coincide with the separatrix formed by points of equal commitment (or splitting) probabilities for reaching the product and reactant regions. Transition states according to the new criterion can be identified directly from equilibrium trajectories, or indirectly by calculating probability densities in the equilibrium and transition-path ensembles using umbrella and transition-path sampling, respectively. An algorithm is proposed to calculate rate coefficients from the transition-path and equilibrium ensembles by estimating the frequency of transitions between reactants and products.     

A doubly nudged elastic band method for finding transition states

A modification of the nudged elastic band (NEB) method is presented that enables stable optimizations to be run using both the limited-memory Broyden-Fletcher-Goldfarb-Shanno (L-BFGS) quasi-Newton and slow-response quenched velocity Verlet minimizers. The performance of this new "doubly nudged" DNEB method is analyzed in conjunction with both minimizers and compared with previous NEB formulations. We find that the fastest DNEB approach (DNEB/L-BFGS) can be quicker by up to 2 orders of magnitude. Applications to permutational rearrangements of the seven-atom Lennard-Jones cluster (LJ7) and highly cooperative rearrangements of LJ38 and LJ75 are presented. We also outline an updated algorithm for constructing complicated multi-step pathways using successive DNEB runs.     

Optimization methods for finding minimum energy paths

A comparison of chain-of-states based methods for finding minimum energy pathways (MEPs) is presented. In each method, a set of images along an initial pathway between two local minima is relaxed to find a MEP. We compare the nudged elastic band (NEB), doubly nudged elastic band, string, and simplified string methods, each with a set of commonly used optimizers. Our results show that the NEB and string methods are essentially equivalent and the most efficient methods for finding MEPs when coupled with a suitable optimizer. The most efficient optimizer was found to be a form of the limited-memory Broyden-Fletcher-Goldfarb-Shanno method in which the approximate inverse Hessian is constructed globally for all images along the path. The use of a climbing-image allows for finding the saddle point while representing the MEP with as few images as possible. If a highly accurate MEP is desired, it is found to be more efficient to descend from the saddle to the minima than to use a chain-of-states method with many images. Our results are based on a pairwise Morse potential to model rearrangements of a heptamer island on Pt(111), and plane-wave based density functional theory to model a rollover diffusion mechanism of a Pd tetramer on MgO(100) and dissociative adsorption and diffusion of oxygen on Au(111).     

RRKM reaction rate theory for transition states of any looseness

Unimolecular rate theory for various types of reactions is implemented for any looseness of transition state. Quantum states are counted for all but the “transitional” modes, their phase space being counted via Monte Carlo sampling. The rate constant k_EJ is then weighted with the initial E and J distributions.     

Stochastic transition states: reaction geometry amidst noise

Classical transition state theory (TST) is the cornerstone of reaction-rate theory. It postulates a partition of phase space into reactant and product regions, which are separated by a dividing surface that reactive trajectories must cross. In order not to overestimate the reaction rate, the dynamics must be free of recrossings of the dividing surface. This no-recrossing rule is difficult (and sometimes impossible) to enforce, however, when a chemical reaction takes place in a fluctuating environment such as a liquid. High-accuracy approximations to the rate are well known when the solvent forces are treated using stochastic representations, though again, exact no-recrossing surfaces have not been available. To generalize the exact limit of TST to reactive systems driven by noise, we introduce a time-dependent dividing surface that is stochastically moving in phase space, such that it is crossed once and only once by each transition path.     

A dimensionless reaction coordinate for quantifying the lateness of transition states

The Hammond‐Leffler postulate asserts that transition states of exothermic reactions are reactant‐like (early), whereas transition states of endothermic reactions are product‐like (late). Related postulates have been proposed to describe the sensitivity of activation barriers for reactions occurring on catalytic surfaces to the catalyst structure. To evaluate the validity of these postulates for different chemical reactions, a general method for classifying transition states as either early or late is needed. One can envision a dimensionless reaction coordinate that changes continuously and monotonically from 0 to 1 along a minimum energy reaction pathway. The value of the dimensionless reaction coordinate for the transition state (W_TS) classifies transition states as (a) early when W_TS < 0.5, (b) late when W_TS > 0.5, and (c) equidistant between reactants and products when W_TS = 0.5. In this article, we derive such a dimensionless reaction coordinate and illustrate its usefulness for several different chemical reactions. © 2009 Wiley Periodicals, Inc. J Comput Chem, 2010     

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.     

Using redundant internal coordinates to optimize equilibrium geometries and transition states

A redundant internal coordinate system for optimizing molecular geometries is constructed from all bonds, all valence angles between bonded atoms, and all dihedral angles between bonded atoms. Redundancies are removed by using the generalized inverse of the G matrix; constraints can be added by using an appropriate projector. For minimizations, redundant internal coordinates provide substantial improvements in optimization efficiency over Cartesian and nonredundant internal coordinates, especially for flexible and polycyclic systems. Transition structure searches are also improved when redundant coordinates are used and when the initial steps are guided by the quadratic synchronous transit approach. © 1996 by John Wiley & Sons, Inc.     

Solvent-dependent transition states for decarboxylations

The rate constants and kinetic isotope effects for decarboxylation of 4-pyridylacetic acid depend strongly on whether the solvent is water or dioxane, and the present paper interprets this finding. We calculate the solvent dependence of the free energy barrier and of the (13)C and (18)O kinetic isotope effects using a quantum mechanical solvation model based on class IV charges and semiempirical atomic surface tensions. The calculations provide a consistent interpretation of the experimental results, which provides a striking confirmation of the soundness of the solvation modeling. Even more significantly, the agreement of theory and experiment gives us confidence in the physical picture of the reaction provided by the model. This indicates that the location of the transition state, as measured by the length of the breaking C--C bond, is 0.24 A later than the gas phase in dioxane and 0.37 A later than the gas phase in water. Charge development at the transition state also depends strongly on the solvent; in particular the CO(2) moiety is 0.07 electronic charge units more negative at the transition state in dioxane than in water.     

Coordinate reduction for exploring chemical reaction paths

The potential energy surface for the reaction of a typical molecular system composed of N atoms is defined uniquely by 3N-6 coordinates. These coordinates can be defined by the Cartesian coordinates of the atomic centers (minus overall translation and rotation), or a set of internally defined coordinates such as bond stretches, angle bends, and torsions. By applying principal component analysis to the geometries along a reaction path, a reduced set of coordinates, d ≪ 3N-6, can be obtained. This reduced set of coordinates can reproduce the changes in geometry along the reaction path with chemical accuracy and may help improve the efficiency of reaction path optimization algorithms.     

Intrinsic reaction coordinate calculations for reaction paths possessing branching points

A 3-21+G energy surface corresponding to the proton transfer reaction in the hydroperoxyl anion solvated by one water molecule presents interesting topological features. In particular the intrinsic reaction coordinate that begins at the transition state does not lead to a minimum but to a saddle point of second order passing through two branching points. A new strategy to obtain the true reaction path in these cases is proposed.     

A procedure for determining reaction paths and saddle points

To determine reaction paths, a method that does not require the previous location of extrema is presented and illustrated by an example. The procedure is based on a local symmetry property of the potential surface.     

Symmetry calculation for molecules and transition states

The symmetry of molecules and transition states of elementary reactions is an essential property with important implications for computational chemistry. The automated identification of symmetry by computers is a very useful tool for many applications, but often relies on the availability of three‐dimensional coordinates of the atoms in the molecule and hence becomes less useful when these coordinates are a priori unavailable. This article presents a new algorithm that identifies symmetry of molecules and transition states based on an augmented graph representation of the corresponding structures, in which both topology and the presence of stereocenters are accounted for. The automorphism group order of the graph associated with the molecule or transition state is used as a starting point. A novel concept of label‐stereoisomers, that is, stereoisomers that arise after labeling homomorph substituents in the original molecule so that they become distinguishable, is introduced and used to obtain the symmetry number. The algorithm is characterized by its generic nature and avoids the use of heuristic rules that would limit the applicability. The calculated symmetry numbers are in agreement with expected values for a large and diverse set of structures, ranging from asymmetric, small molecules such as fluorochlorobromomethane to highly symmetric structures found in drug discovery assays. The new algorithm opens up new possibilities for the fast screening of the degree of symmetry of large sets of molecules. © 2014 Wiley Periodicals, Inc.     

Bifurcations and transition states

Bifurcations of reaction channels are related to valley-ridge inflection points and it is examined what happens when these do not coincide with transition states. Under such conditions there result bifurcating regions. There exist a number of different prototypes for such regions which are discussed explicitly on the basis of the pertinent Taylor expansions. When bifurcations occur close enough to transition states then there result bifurcating transition regions. An example for a bifurcating transition region is exhibited which is obtained from a quantum mechanical ab initio calculation for the ring opening of cyclopropylidene to aliene. In general there exist no orthogonal trajectory patterns which could serve as simplified models for channel bifurcations.Operated for the U.S. Department of Energy by Iowa State University under Contract No. W-7405-ENG-82. This work was supported by the Office of Basic Energy Sciences     

α-tritium isotope effects in the menschutkin-type reaction with variable transition states

Secondary α-tritium isotope effects in the series of the Menschutkin-type reaction of benzyl benzenesulfonates with N,N-dimethylanilines were all small and varied only slightly; it was concluded that the transition states vary mainly in the parallel direction to the reaction coordinate.     

Survival paths for reaction dynamics in fluctuating environments

We study rate processes in general Gaussian fluctuating environments using a path integral formalism. We derive a variational equation for the dominant survival path when the fluctuations relax exponentially or according to a stretched exponential law. In the case of a slowly varying barrier, the equilibrium regression approximation which is used by Frauenfelder and coworkers emerges. In this approximation, the survival path follows the ordinary law of relaxation to equilibrium. If the rate coefficients vary rapidly with environmental variables, however, the dominant survival paths exhibit more complex behaviour. Many phenomena analogous to geometrical optics occur. These include reflection off of rapid variations in rate constant, as well as refraction, giving paths very different from the equilibrium relaxation properties. A model with a piece-wise linear rate exhibits the basic phenomena, and the survival path equation is exactly solved for the general stretched exponential relaxing environment.     

Efficient methods for finding transition states in chemical reactions: comparison of improved dimer method and partitioned rational function optimization method

A combination of interpolation methods and local saddle-point search algorithms is probably the most efficient way of finding transition states in chemical reactions. Interpolation methods such as the growing-string method and the nudged-elastic band are able to find an approximation to the minimum-energy pathway and thereby provide a good initial guess for a transition state and imaginary mode connecting both reactant and product states. Since interpolation methods employ usually just a small number of configurations and converge slowly close to the minimum-energy pathway, local methods such as partitioned rational function optimization methods using either exact or approximate Hessians or minimum-mode-following methods such as the dimer or the Lanczos method have to be used to converge to the transition state. A modification to the original dimer method proposed by [Henkelman and Jonnson J. Chem. Phys. 111, 7010 (1999)] is presented, reducing the number of gradient calculations per cycle from six to four gradients or three gradients and one energy, and significantly improves the overall performance of the algorithm on quantum-chemical potential-energy surfaces, where forces are subject to numerical noise. A comparison is made between the dimer methods and the well-established partitioned rational function optimization methods for finding transition states after the use of interpolation methods. Results for 24 different small- to medium-sized chemical reactions covering a wide range of structural types demonstrate that the improved dimer method is an efficient alternative saddle-point search algorithm on medium-sized to large systems and is often even able to find transition states when partitioned rational function optimization methods fail to converge.     

Substituent effects and nearly degenerate transition states: rational design of substrates for the tandem Wolff-Cope reaction

The substrate scope for a ketene-assisted Cope (tandem Wolff-Cope) reaction is elucidated from first-principles quantum mechanics. An alternate pathway (trans) leading to an undesired and unstable product lies perilously close ( approximately 2.5 kcal/mol) to the primary (cis) reaction pathway; this near-degeneracy arises from preferential ketene stabilization of a radicaloid trans transition state over an aromatic cis transition state. Normally, substitution at "forbidden" sites causes the alternate pathway to be favored and the reaction to fail, but using simple conformational analysis principles we design substrates that defy this rule.