Minimum Energy Path Calculations Research Articles

Insights into the variations of kinetic and potential energies in a multi-bond reaction: the reaction electronic flux perspective.

The debate over whether kinetic energy (KE) or potential energy (PE) are the fundamental energy components that contribute to forming covalent bonds has been enduring and stimulating over time. However, the supremacy of these energy components in reactions where multiple bonds are simultaneously formed or broken has yet to be explored. In this study, we use the reaction electronic flux (REF), an effective tool for investigating changes in driving electronic activity when bond formation or dissociation occurs in a chemical reaction, to examine the fluctuations in the KE and PE in a multi-bond reaction. To that end, the activation of CO by low-valent group 14 catalysts through a concerted -bond metathesis mechanism is analyzed. The findings of this preliminary study suggest that the REF can be utilized as a tool to rationalize alterations in the KE and PE in a multi-bond reaction. Specifically, analyses across the reaction coordinate reveal that changes in the KE and PE precede activation in the REF, stimulating the electronic activity where bond formation or dissociation processes dominate. The activation of CO by the low-valent LEH catalysts (L = N,N'-bis(2,6-diisopropyl phenyl)- -diketiminate; E = Si, Ge, Sn, and Pb) was studied along the reaction coordinate at the M06-2X/6-31G(d,p)-LANL2DZ(E) level of theory. The respective minimum energy path calculations were obtained using the intrinsic reaction coordinate (IRC) procedure. The reaction electronic flux (REF) was calculated through the computation of the electronic chemical potential using the frontier molecular orbital approximation. Mayer bond orders along the reaction coordinate have been determined using the NBO 3.1 program in Gaussian16. Most of the reaction coordinate quantities reported in this study (REF, KE, PE, among others) have been determined using the Kudi program and custom Python scripts.

Finding transition state and minimum energy path of bistable elastic continua through energy landscape explorations

Mechanical bistable structures have two stable equilibria and can transit between them under external stimuli. Due to their unique behaviors such as snap-through and substantial shape changes, bistable structures exhibit unprecedented properties compared to conventional structures and thus have found applications in various fields such as soft robots, morphing wings and logic units. To quantitatively predict the performance of bistable structures in these applications, it is desirable to acquire information about the minimum energy barrier and an energy-efficient transition path between the two stable states. However, there is still a general lack of efficient methodologies to obtain this information, particularly for elastic continua with complicated, unintuitive transition paths. To overcome this challenge, here we integrate energy landscape exploration algorithms into finite element method (FEM). We first utilize the binary image transition state search (BITSS) method to identify the saddle point and then perform nudged elastic band (NEB) calculations with an initial guess based on the BITSS results to find the minimum energy path (MEP). This integrated strategy greatly helps the convergence of MEP calculations, which are highly nonlinear. Two representative cases are studied, including bistable buckled beams and a bistable unit of mechanical metamaterials, and the numerical results agree well with the previous works. Importantly, we numerically predict the complicated MEP of an asymmetric bistable unit of mechanical metamaterials and use experiments to demonstrate that following this MEP leads to successful transition between stable states while intuitive uniaxial compression fails to do so. Our work provides an effective numerical platform for identifying the minimum energy barrier and energy-efficient transition path of a bistable continuum, which can offer valuable guidance to the design of actuators, damping structures, energy harvesters, and mechanical metamaterials.

Norbornadiene/Quadricyclane System in the Spotlight: The Role of Rydberg States and Dynamic Electronic Correlation in a Solar‐Thermal Building Block

AbstractIn this contribution, we studied the photophysics and photochemistry of an archetypal molecular switch in terms of solar energy storage and release. In detail, we characterized the valence and Rydberg states of norbornadiene (NBD) and quadricyclane (QC) by means of CASPT2//CASSCF theory level, finding a good agreement of the calculated vertical excitation energies with the experimental counterparts. The NBD↔QC thermal and photochemical valence isomerization [2π+2π] reactions have been addressed and investigated. Low energy crossing points between excited states were identified for both the NBD↔QC photochemical reactions, through the calculation of minimum energy paths which revealed that the photochemistry is ruled by the deformation of two coupled reaction coordinates. Such coordinates were also used to build potential energy surfaces through relaxed energy scans with the goal of building a simplified, still accurate, model system able to catch the NBD↔QC photorelaxation. Also, the S1/S0 respective conical intersection were characterized and related to the reaction quantum yield. Interestingly, we found that reverse photoisomerization from QC to the NBD progenitor could in principle occur by direct excitation of σ‐3s and σ‐3p Rydberg excited states in the UV spectral range. In both NBD and QC photoreactivity, the doubly excited valence state has shown to play a crucial role for reaching crossing points leading to nonadiabatic population transfers.

Ultrafast photoisomerization mechanism of azaborine revealed by nonadiabatic molecular dynamics simulations.

1,2-Dihydro-1,2-azaborine is an isoelectronic analog of benzene with a B-N substitution, and its unique photoisomerization behavior, which is distinct from that of benzene, has drawn significant attention. To understand the detailed mechanism of azaborine photochemistry considering the dynamical effect and gain a comprehensive understanding of photochemical reactions, we investigated the photoisomerization dynamics of azaborine using nonadiabatic molecular dynamics simulations with Tully's surface hopping algorithm. Herein, the structural and energetic analyses of the trajectories revealed three different paths: direct relaxation (path 1), relaxation via a prefulvene-like intermediate (path 2), and formation of the Dewar isomer as a photoproduct (path 3). Our results confirmed that the photoisomerization of azaborine follows the energetically favored pathway predicted by the previous minimum energy path (MEP) calculations, exclusively forming the Dewar isomer, which is consistent with the experimental observations. Additionally, despite the low quantum yield found in our simulations, the high-level excitation energy calculations support the complete conversion observed in the experiments.

Nucleation and manipulation of single skyrmions using spin-polarized currents in antiferromagnetic skyrmion-based racetrack memories

In this work, an ultrafast nucleation of an isolated anti-ferromagnetic (AFM) skyrmion was reported in an AFM layer with DMi strengths of 0.47-0.32 mathrm{mJ}/{mathrm{m}}^{2} using spin-transfer torque by locally injecting pure spin currents into magnetic tracks. Besides, we revealed the key advantages of AFM skyrmion-based racetrack memories by comparing the motion of AFM and FM skyrmions driven by spin–orbit torques (SOTs) for different skyrmion sizes along racetrack memories with various notch sizes. Our results indicate that for AFM skyrmion, the skyrmion Hall effect does not exist during the skyrmion motion, therefore at small skyrmion sizes, we succeeded to overcome the repulsive forces developed in the notch area for low and large SOTs. The obtained findings were carefully analyzed by computing the variation of energy barriers associated with the notch for different skyrmion sizes using minimum energy path (MEP) calculations. We showed that the larger the skyrmion size, the harder it is to shrink the skyrmion in the notch which produces a high energy barrier (Eb) for large skyrmion sizes. Moreover, as the notch size increases, the skyrmion size shrinks further, and hence Eb increases proportionally. Nevertheless, we proved that AFM skyrmions are more efficient and flexible than FM skyrmions against boundary forces.

Carbon-induced strengthening of bcc iron at the atomic scale

In steels, the interaction between screw dislocations and carbon solutes has a great influence on the yield strength. Fe-C potentials used in molecular dynamics (MD) simulations yield a poor description of screw dislocation properties---their core structure and Peierls barrier---compared to ab initio calculations. Here we combine two EAM potentials from the literature, which greatly improves dislocation property accuracy in FeC alloys. Using this hybrid potential, MD simulations of dislocation glide in random solid solutions confirm a powerful solute strengthening, caused by complex interaction processes. We analyze these processes in a model geometry, where a row of carbon atoms is inserted in the dislocation core with varying separations. We use a combination of MD simulations, minimum-energy path calculations, and a statistical model based on the harmonic transition state theory to explain the strengthening induced by carbon. We unveil that carbon disrupts the glide process, as unpinning requires the successive nucleation of two kink pairs. When solute separation is below about 100 Burgers vectors, the activation enthalpy of both kink pairs are markedly increased compared to pure iron, resulting in a strong dependence of the unpinning stress on solute spacing. Our simulations also suggest an effect of carbon spacing on the kink-pair activation entropy. This work provides elementary processes and parameters that will be useful for larger-scale models and, in particular, kinetic Monte Carlo simulations.

Mechanism of Oxygen Evolution and Mn4CaO5 Cluster Restoration in the Natural Water-Oxidizing Catalyst.

Water oxidation occurring in the first steps of natural oxygenic photosynthesis is catalyzed by the pigment/protein complex Photosystem II. This process takes place on the Mn4Ca cluster located in the core of Photosystem II and proceeds along the five steps (S0-S4) of the so-called Kok-Joliot cycle until the release of molecular oxygen. The catalytic cycle can therefore be started afresh through insertion of a new water molecule. Here, combining quantum mechanics/molecular mechanics simulations and minimum energy path calculations, we characterized on different spin surfaces the events occurring in the last sector of the catalytic cycle from structural, electronic, and thermodynamic points of view. We found that the process of oxygen evolution and water insertion can be described well by a two-step mechanism, with oxygen release being the rate-limiting step of the process. Moreover, our results allow us to identify the upcoming water molecule required to regenerate the initial structure of the Mn4Ca cluster in the S0 state. The insertion of the water molecule was found to be coupled with the transfer of a proton to a neighboring hydroxide ion, thus resulting in the reconstitution of the most widely accepted model of the S0 state.

Resolving atomic-scale phase transformation and oxygen loss mechanism in ultrahigh-nickel layered cathodes for cobalt-free lithium-ion batteries

Summary Doped LiNiO2 has recently become one of the most promising cathode materials for its high specific energy, long cycle life, and reduced cobalt content. Despite this, the degradation mechanism of LiNiO2 and its derivatives still remains elusive. Here, by combining in situ electron microscopy and first-principles calculations, we elucidate the atomic-level chemomechanical degradation pathway of LiNiO2-derived cathodes. We uncover that the O1 phase formed at high voltages acts as a preferential site for rock-salt transformation via a two-step pathway involving cation mixing and shear along (003) planes. Moreover, electron tomography reveals that planar cracks nucleated simultaneously from particle interior and surface propagate along the [100] direction on (003) planes, accompanied by concurrent structural degradation in a discrete manner. Our results provide an in-depth understanding of the degradation mechanism of LiNiO2-derived cathodes, pointing out the concept that suppressing the O1 phase and oxygen loss is key to stabilizing LiNiO2 for developing next-generation high-energy cathode materials.

Simple position and orientation preconditioning scheme for minimum energy path calculations.

Minimum-energy path (MEP) calculations, such as those typified by the nudged elastic band method, require input of reactant and product molecular configurations at initialization. In the case of reactions involving more than one molecule, generating initial reactant and product configurations requires careful consideration of the relative position and orientations of the reactive molecules in order to ensure that the resulting MEP calculation proceeds without converging on an alternative reaction-path, and without requiring excessive numbers of optimization iterations; as such, this initial system set-up is most commonly performed "by hand," with an expert user arranging reactive molecules in space to ensure that the following MEP calculation runs smoothly. In this Article, we introduce a simple preconditioning scheme which replaces this labor-intensive, human-knowledge-based step with an automated deterministic computational scheme. In our approach, initial reactant and product configurations are generated such that steric hindrance between reactive molecules is minimized in the reactant and product configurations, while also simultaneously requiring minimal structural differences between the reactants and products. The method is demonstrated using a benchmark test-set of >3400 organic molecular reactions, where comparison of the reactant/product configurations generated using our approach compare very well to initial configurations which were generated on an ad hoc basis.

A Theoretical Stereoselectivity Model of Photochemical Denitrogenations of Diazoalkanes Toward Strained 1,3-Dihalogenated Bicyclobutanes.

Photochemical reactions exemplify "green" chemistry and are an essential tool for synthesizing highly strained molecules under mild conditions with light. The light-promoted denitrogenation of bicyclic azoalkanes affords functionalized, stereoenriched bicyclo[1.1.0]butanes. These reactions were revisited with multireference calculations and non-adiabatic molecular dynamics (NAMD) simulations to provide a detailed analysis of the photophysics, reactivities, and unexplained stereoselectivity of a series of diazabicyclo[2.1.1]hexenes. We used complete active space self-consistent field (CASSCF) calculations with an (8,8) active space and ANO-S-VDZP basis set; the CASSCF energies were corrected with CASPT2 (8,8)/ANO-S-VDZP. The nature of the electronic excitation is n → π* and ranges from 3.77 to 3.91 eV for the diazabicyclo[2.1.1]hexenes reported here. Minimum energy path calculations showed stepwise C-N bond breaking and led directly to a minimum energy crossing point, corresponding to a stereochemical "double inversion" product. Wigner sampling of diazabicyclo[2.1.1]hexene provided initial conditions for 692 NAMD trajectories. We identified competing complete stereoselective and stereochemical scrambling pathways. The stereoselective pathways feature concerted bicyclobutane inversion and N2 extrusion. The stereochemical scrambling pathways involve N2 extrusion followed by bicyclobutane planarization, leading to stereochemical scrambling. The predicted diastereomeric excess (d.e.) almost exactly matches the experiment (calc.d.e. = 46% vs exp.d.e. = 47%). Our NAMD simulations with 672, 568, and 596 trajectories for 1-F, 1-Cl, and 1-Br predicted a d.e. of 94-97% for the double inversion products. Halogenation significantly perturbs the potential energy surface (PES) toward the retention products due to hyperconjugative interactions. The nC → σ*C-X,X=F,Cl,Br hyperconjugative effect leads to a broader shoulder region on the PES for double inversion.

Deep learning-based initial guess for minimum energy path calculations

An autoencoder that automatically generates an initial guess for the minimum energy pathway (MEP) calculations has been designed. Specifically, our autoencoder takes in the trajectories of molecular dynamics simulations as its input and facilitates the generation of feasible molecular coordinates. Two molecules (acetonitrile and alanine dipeptide) were tested using the nudged elastic band calculations and the results provided improvements over linear interpolation and image dependent pair potential methods in terms of the number of SCF iterations, demonstrating the utility of using an autoencoder type of an approach for MEP calculations.

Efficient optimization method for finding minimum energy paths of magnetic transitions

Efficient algorithms for the calculation of minimum energy paths of magnetic transitions are implemented within the geodesic nudged elastic band (GNEB) approach. While an objective function is not available for GNEB and a traditional line search can, therefore, not be performed, the use of limited memory Broyden–Fletcher–Goldfarb–Shanno (LBFGS) and conjugate gradient algorithms in conjunction with orthogonal spin optimization (OSO) approach is shown to greatly outperform the previously used velocity projection and dissipative Landau–Lifschitz dynamics optimization methods. The implementation makes use of energy weighted springs for the distribution of the discretization points along the path and this is found to improve performance significantly. The various methods are applied to several test problems using a Heisenberg-type Hamiltonian, extended in some cases to include Dzyaloshinskii–Moriya and exchange interactions beyond nearest neighbours. Minimum energy paths are found for magnetization reversals in a nano-island, collapse of skyrmions in two-dimensional layers and annihilation of a chiral bobber near the surface of a three-dimensional magnet. The LBFGS-OSO method is found to outperform the dynamics based approaches by up to a factor of 8 in some cases.

Steric Hindrance Effect in High-Temperature Reactions

High-temperature reactions widely exist in nature. However, they are difficult to be characterized either experimentally or computationally. The routinely used minimum energy path (MEP) model in computational modeling of chemical reactions is not justified to describe high-temperature reactions since high-energy structures are actively involved there. In this study, using CH4 decomposition on the Cu(111) surface as an example, we systematically compare MEP results with those obtained by explicitly sampling all relevant structures via ab initio molecular dynamics (AIMD) simulations at different temperatures. Interestingly, we find that, for reactions protected by a strong steric hindrance effect, the MEP is still effectively followed even at a temperature close to the Cu melting point. In contrast, without such a protection, the flexibility of surface Cu atoms can lead to a significant free energy barrier reduction at a high temperature. Accordingly, some conclusions about graphene growth mechanisms based on MEP calculations should be revisited. Physical insights provided by this study can deepen our understanding on high-temperature surface reactions.

Nudged Elastic Band Calculations Accelerated with Gaussian Process Regression Based on Inverse Interatomic Distances.

Calculations of minimum energy paths for atomic rearrangements using the nudged elastic band method can be accelerated with Gaussian process regression to reduce the number of energy and atomic force evaluations needed for convergence. Problems can arise, however, when configurations with large forces due to short distance between atoms are included in the data set. Here, a significant improvement to the Gaussian process regression approach is obtained by basing the difference measure between two atomic configurations in the covariance function on the inverted interatomic distances and by adding a new early stopping criterion for the path relaxation phase. This greatly improves the performance of the method in two applications where the original formulation does not work well: a dissociative adsorption of an H2 molecule on a Cu(110) surface and a diffusion hop of an H2O molecule on an ice Ih(0001) surface. Also, the revised method works better in the previously analyzed benchmark application to rearrangement transitions of a heptamer island on a surface, requiring fewer energy and force evaluations for convergence to the minimum energy path.

Ab Initio Molecular Dynamics Simulation and Energetics of the Ribulose-1,5-biphosphate Carboxylation Reaction Catalyzed by Rubisco: Toward Elucidating the Stereospecific Protonation Mechanism.

In the carboxylation reaction catalyzed by ribulose 1,5-bisphosphate (RuBP) carboxylase-oxygenase (Rubisco), which is fundamental to photosynthesis, scission of a C-C bond in the six-carbon gemdiolate intermediate forms a carbanion that must be protonated stereospecifically to form product. It is thought that a conserved lysine side chain (LYS175 in spinach Rubisco), in the immediate vicinity of the carbanion, provides the necessary proton. Here, we endeavor to determine from the electronic-structure calculations whether protonation via this route is energetically possible. The two-dimensional energy surface was mapped to determine the minimum energy path (MEP) using density functional theory (B3LYP) and incorporating basis set superposition and classical (London) dispersion corrections. The potential of mean force (free energy) was then calculated from ab initio molecular dynamics simulations with umbrella sampling in the vicinity of the MEP on the scission-protonation reaction coordinate. MEP calculations were also carried out to evaluate the possibility of an active-site water near the phosphate (P1) of RuBP, with an excess proton positioned at P1, as an alternative facilitator of stereospecific protonation via a classical Grotthuss mechanism. In both cases, the C-C bond scission in the six-carbon intermediate and proton transfer from the donor was found to be concerted and highly asynchronous, without a stable carbanion intermediate. However, the free energy change was unfavorable for direct protonation by the LYS175 side chain. In contrast, the Grotthuss mechanism yielded stable products and an activation energy in good agreement with experiment. It also provides a plausible mechanism for alternative product formed in enzyme mutations at the LYS175 position and is consistent with the observed deuterium isotope effects.

Synthesis and Computational Study of a Pyridylcarbene Fe(II) Complex: Unexpected Effects of fac/ mer Isomerism in Metal-to-Ligand Triplet Potential Energy Surfaces.

The synthesis and the steady-state absorption spectrum of a new pyridine-imidazolylidene Fe(II) complex (Fe-NHC) are presented. A detailed mechanism of the triplet metal-to-ligand charge-transfer states decay is provided on the basis of minimum energy path (MEP) calculations used to connect the lowest-lying singlet, triplet, and quintet state minima. The competition between the different decay pathways involved in the photoresponse is assessed by analyzing the shapes of the obtained potential energy surfaces. A qualitative difference between facial ( fac) and meridional ( mer) isomers' potential energy surface (PES) topologies is evidenced for the first time in iron-based complexes. Indeed, the mer complex shows a steeper triplet path toward the corresponding 3MC minimum, which lies at a lower energy as compared to the fac isomer, thus pointing to a faster triplet decay of the former. Furthermore, while a major role of the metal-centered quintet state population from the triplet 3MC region is excluded, we identify the enlargement of iron-nitrogen bonds as the main normal modes driving the excited-state decay.

Exploration of Uncharted 3PES Territory for [Ru(bpy)3]2+: A New 3MC Minimum Prone to Ligand Loss Photochemistry.

We have identified a new 3MC state bearing two elongated Ru-N bonds to the same ligand in [Ru(bpy)3]2+. This DFT-optimized structure is a local minimum on the 3PES. This distal MC state (3MCcis) is destabilized by less than 2 kcal/mol with respect to the classical MC state (3MCtrans), and energy barriers to populate 3MCcis and 3MCtrans from the 3MLCT state are similar according to nudged elastic band minimum energy path calculations. Distortions in the classical 3MCtrans, that is, elongation of two Ru-N bonds toward two different bpy ligands, are not expected to favor the formation of ligand-loss photoproducts. On the contrary, the new 3MCcis could be particularly relevant in the photodegradation of Ru(II) polypyridine complexes.

Nudged elastic band calculations accelerated with Gaussian process regression.

Minimum energy paths for transitions such as atomic and/or spin rearrangements in thermalized systems are the transition paths of largest statistical weight. Such paths are frequently calculated using the nudged elastic band method, where an initial path is iteratively shifted to the nearest minimum energy path. The computational effort can be large, especially when ab initio or electron density functional calculations are used to evaluate the energy and atomic forces. Here, we show how the number of such evaluations can be reduced by an order of magnitude using a Gaussian process regression approach where an approximate energy surface is generated and refined in each iteration. When the goal is to evaluate the transition rate within harmonic transition state theory, the evaluation of the Hessian matrix at the initial and final state minima can be carried out beforehand and used as input in the minimum energy path calculation, thereby improving stability and reducing the number of iterations needed for convergence. A Gaussian process model also provides an uncertainty estimate for the approximate energy surface, and this can be used to focus the calculations on the lesser-known part of the path, thereby reducing the number of needed energy and force evaluations to a half in the present calculations. The methodology is illustrated using the two-dimensional Müller-Brown potential surface and performance assessed on an established benchmark involving 13 rearrangement transitions of a heptamer island on a solid surface.

Rock climbing: A local-global algorithm to compute minimum energy and minimum free energy pathways.

The calculation of minimum energy or minimum free energy paths is an important step in the quantitative and qualitative studies of chemical and physical processes. The computations of these coordinates present a significant challenge and have attracted considerable theoretical and computational interest. Here we present a new local-global approach to study reaction coordinates, based on a gradual optimization of an action. Like other global algorithms, it provides a path between known reactants and products, but it uses a local algorithm to extend the current path in small steps. The local-global approach does not require an initial guess to the path, a major challenge for global pathway finders. Finally, it provides an exact answer (the steepest descent path) at the end of the calculations. Numerical examples are provided for the Mueller potential and for a conformational transition in a solvated ring system.

Magnetic exchange force microscopy: theoretical analysis of induced magnetization reversals.

In magnetic exchange force microscopy a magnetic tip is scanned over the surface of a solid and an image representing the exchange interaction recorded. Sudden changes in the image corresponding to magnetization switching can be monitored as a function of the tip-surface distance thereby giving important information about the lifetime of metastable magnetic states and how it is affected by the exchange interaction. Here, theoretical calculations are carried out to study the tip-surface interaction and determine the mechanism and rate of transitions in a magnetic exchange force microscopy experiment, and comparison made with reported experimental data on an Fe cluster interacting with an antiferromagnetic Fe overlayer on a W(001) surface. The activation energy was determined from calculations of minimum energy paths and the pre-exponential factor in the Arrhenius rate expression evaluated from harmonic transition state theory, extended to account for zero modes. A noncollinear extension of the Alexander-Anderson model was used to describe the magnetic properties of an atomic scale representation of the system. The calculations reveal how the tip size, tip-surface distance and magnetic field affect the lifetime of the magnetic states.