Common Features of Extraordinary Rate Theories

Comparing brute force to transition path sampling for gas hydrate nucleation with a flat interface: comments on time reversal symmetry

Fluid to solid nucleation is often investigated with the rare event method transition path sampling (TPS). I claim that the inherent irreversibility of solid nucleation, even at stationary conditions, calls into question TPS's applicability for determining solid nucleation mechanisms, especially for pre-critical behavior. Even when applied to a phenomenon which displays time reversal asymmetry like solid nucleation, TPS is a good means of exploring phase space and giving trends in post-critical structure, and its ability to facilitate nucleation rate and free energy calculations remains outstanding. Forward-only splitting and ratcheting methods such as forward flux sampling are more attractive for understanding nucleation mechanisms as they do not require time reversal symmetry, but at low driving forces may suffer from the same limitations as brute force: they may never make it to the first ratchet. Here I briefly summarize the TPS method and gas hydrate nucleation simulation literature, focusing on topics within both to facilitate a comparison of brute force hydrate nucleation to transition path sampling of hydrate nucleation. Perhaps anecdotally, the brute force technique results in more crystalline trajectories despite having higher driving forces than TPS. I maintain this difference is because of the inherent irreversibility of hydrate nucleation, meaning its pre-critical behavior cannot accurately be determined by the melting trajectories that comprise approximately half of the configurations in TPS's path ensemble.

Anti-Arrhenius behavior of electron transfer reactions in molecular dimers

Rates of chemical reactions typically accelerate as the temperature rises, following the Arrhenius law. However, electron transfer reactions may exhibit weak temperature dependence or counterintuitive behavior, known as anti-Arrhenius behavior, wherein reaction rates decrease as temperature increases. Solvent reorganization energy and torsion-induced changes in electronic couplings could contribute to this unusual behavior, but how each contributes to the overall temperature dependence is unclear. One can decelerate the charge recombination process in photogenerated radical pairs or charge-separated states by harnessing this often-overlooked phenomenon. This means that we could achieve long-lived radical pairs without relying on conventional cooling. Using a series of homo molecular dimers, we showed that the degree of torsional hindrance dictates temperature-dependent torsion-induced changes in electronic coupling and, therefore, charge recombination rates. The overall temperature dependence is controlled by how changes in electronic coupling and the temperature-dependent solvent reorganization energy contribute to the rates of charge recombination. Our findings pave the way for rationally designing molecules that exhibit anti-Arrhenius behavior to slow down charge recombination, opening possibilities for applications in energy-related and quantum information technologies.

An exploration of machine learning models for the determination of reaction coordinates associated with conformational transitions

Determining collective variables (CVs) for conformational transitions is crucial to understanding their dynamics and targeting them in enhanced sampling simulations. Often, CVs are proposed based on intuition or prior knowledge of a system. However, the problem of systematically determining a proper reaction coordinate (RC) for a specific process in terms of a set of putative CVs can be achieved using committor analysis (CA). Identifying essential degrees of freedom that govern such transitions using CA remains elusive because of the high dimensionality of the conformational space. Various schemes exist to leverage the power of machine learning (ML) to extract an RC from CA. Here, we extend these studies and compare the ability of 17 different ML schemes to identify accurate RCs associated with conformational transitions. We tested these methods on an alanine dipeptide in vacuum and on a sarcosine dipeptoid in an implicit solvent. Our comparison revealed that the light gradient boosting machine method outperforms other methods. In order to extract key features from the models, we employed Shapley Additive exPlanations analysis and compared its interpretation with the "feature importance" approach. For the alanine dipeptide, our methodology identifies ϕ and θ dihedrals as essential degrees of freedom in the C7ax to C7eq transition. For the sarcosine dipeptoid system, the dihedrals ψ and ω are the most important for the cisαD to transαD transition. We further argue that analysis of the full dynamical pathway, and not just endpoint states, is essential for identifying key degrees of freedom governing transitions.

Site-Averaged Ab Initio Kinetics: Importance Learning for Multistep Reactions on Amorphous Supports

Single-atom centers on amorphous supports include catalysts for polymerization, partial oxidation, metathesis, hydrogenolysis, and more. The disordered environment makes each site different, and the kinetics exponentially magnifies these differences to make ab initio site-averaged kinetics calculations extremely difficult. This work extends the importance learning algorithm for efficient and precise site-averaged kinetics estimates to ab initio calculations and multistep reaction mechanisms. Specifically, we calculate site-averaged proton transfer relaxation rates on an ensemble of cluster models representing Brønsted acid sites on silica-alumina. We include direct and water-assisted proton transfer pathways and simultaneously estimate the water adsorption and activation enthalpies for forward and backward proton transfers. We use density functional theory (DFT) to obtain a site-averaged rate, somewhat like a turnover frequency, for the proton transfer relaxation rate. Finally, we show that importance learning can provide orders-of-magnitude acceleration over standard sampling methods for site-averaged rate calculations in cases where the rate is dominated by a few highly active sites.

Non-trivial dynamics in a model of glial membrane voltage driven by open potassium pores

Despite the molecular evidence that a nearly linear steady-state current-voltage relationship in mammalian astrocytes reflects a total current resulting from more than one differentially regulated K+ conductance, detailed ordinary differential equation (ODE) models of membrane voltage Vm are still lacking. Various experimental results reporting altered rectification of the major Kir currents in glia, dominated by Kir4.1, have motivated us to develop a detailed model of Vm dynamics incorporating the weaker potassium K2P-TREK1 current in addition to Kir4.1, and study the stability of the resting state Vr. The main question is whether, with the loss of monotonicity in glial I-V curve resulting from altered Kir rectification, the nominal resting state Vr remains stable, and the cell retains the trivial, potassium electrode behavior with Vm after EK. The minimal two-dimensional model of Vm near Vr showed that an N-shape deformed Kir I-V curve induces multistability of Vm in a model that incorporates K2P activation kinetics, and nonspecific K+ leak currents. More specifically, an asymmetrical, nonlinear decrease of outward Kir4.1 conductance, turning the channels into inward rectifiers, introduces instability of Vr. That happens through a robust bifurcation giving birth to a second, more depolarized stable resting state Vdr > -10 mV. Realistic recordings from electrographic seizures were used to perturb the model. Simulations of the model perturbed by constant current through gap junctions and seizure-like discharges as local field potentials led to depolarization and switching of Vm between the two stable states, in a downstate-upstate manner. In the event of prolonged depolarizations near Vdr, such catastrophic instability would affect all aspects of the glial function, from metabolic support to membrane transport, and practically all neuromodulatory roles assigned to glia.

Oriented attachment kinetics for rod-like particles at a flat surface: Buffon's needle at the nanoscale

The adsorption of large rod-like molecules or crystallites on a flat crystal face, similar to Buffon's needle, requires the rods to "land," with their binding sites in precise orientational alignment with matching sites on the surface. An example is provided by long, helical antifreeze proteins (AFPs), which bind at specific facets and orientations on the ice surface. The alignment constraint for adsorption, in combination with the loss in orientational freedom as the molecule diffuses toward the surface, results in an entropic barrier that hinders the adsorption. Prior kinetic models do not factor in the complete geometry of the molecule, nor explicitly enforce orientational constraints for adsorption. Here, we develop a diffusion-controlled adsorption theory for AFP molecules binding at specific orientations to flat ice surfaces. We formulate the diffusion equation with relevant boundary conditions and present analytical solutions to the attachment rate constant. The resulting rate constant is a function of the length and aspect ratio of the AFP, the distance threshold associated with binding, and solvent conditions such as temperature and viscosity. These results and methods of calculation may also be useful for predicting the kinetics of crystal growth through oriented attachment.

Anomalous (Exergonic) Behavior in the Transfer of Electrons between Donors and Acceptors: Mobility, Energy, Caloric Capacity, and Entropy

Understanding the kinetics of electron transfer reactions involves active research in physics, chemistry, biology, and nano-tech. Here, we propose a model to apply in a broader framework by establishing a connection between thermodynamics and kinetics. From a purely thermodynamic point of view, electronic transfer Marcus' theory is revisited; consequently, calculations of thermodynamic variables such as mobility, energy, and entropy are provided. More significantly, two different regimes are explicitly established. In the anomalous region, an exergonic process associated with negative heat capacity appears. Further, in the same region, mobility, energy, and entropy decrease when the temperature increases.

Significance of atomic-scale defects in flexible surfaces on local solvent and ion behaviour

Many factors can affect the course of heterogeneous nucleation, such as surface chemistry, flexibility and topology, substrate concentration and solubility. Atomic-scale defects are rarely investigated in detail and are often considered to be unimportant surface features. In this work, we set out to investigate the significance of atomic-scale defects in a flexible self-assembled monolayer surface for the behaviour of clusters of Ca²⁺ and CO₃^2- ions in water. To this end, we use molecular dynamics simulations to estimate the diffusion coefficients of ion clusters at different topological surface features and obtain ionic radial distribution functions around features of interest. Well-tempered metadynamics is used to gain insight into the free energy of ions around selected surface defects. We find that certain defects, which we refer to as active defects, can impair ionic surface diffusion, as well as affect the diffusion of ions in close proximity to the surface feature in question. Our findings suggest that this effect can result in an ability of such topological features to promote ion clustering and increase local ionic concentration at specific surface sites. The work reported here shows how the presence of small atomic-scale defects can affect the role of a surface in the process of heterogeneous nucleation and contributes towards a rational definition of surfaces as effective nucleating agents.

Nucleation rates from small scale atomistic simulations and transition state theory

The evaluation of nucleation rates from molecular dynamics trajectories is hampered by the slow nucleation time scale and impact of finite size effects. Here, we show that accurate nucleation rates can be obtained in a very general fashion relying only on the free energy barrier, transition state theory, and a simple dynamical correction for diffusive recrossing. In this setup, the time scale problem is overcome by using enhanced sampling methods, in casu metadynamics, whereas the impact of finite size effects can be naturally circumvented by reconstructing the free energy surface from an appropriate ensemble. Approximations from classical nucleation theory are avoided. We demonstrate the accuracy of the approach by calculating macroscopic rates of droplet nucleation from argon vapor, spanning 16 orders of magnitude and in excellent agreement with literature results, all from simulations of very small (512 atom) systems.

Blind Analysis of Molecular Dynamics

We describe a nonparametric approach for accurate determination of the slowest relaxation eigenvectors of molecular dynamics. The approach is blind as it uses no system specific information. In particular, it does not require a functional form with many parameters to closely approximate eigenvectors, e.g., linear combinations of molecular descriptors or a deep neural network, and thus no extensive expertise with the system. We suggest a rigorous and sensitive validation/optimality criterion for an eigenvector. The criterion uses only eigenvector time series and can be used to validate eigenvectors computed by other approaches. The power of the approach is illustrated on long atomistic protein folding trajectories. The determined eigenvectors pass the validation test at a time scale of 0.2 ns, much shorter than alternative approaches.

A phase diagram for energy flow-limited reactivity

Intramolecular energy flow (also known as intramolecular vibrational redistribution or IVR) is often assumed in Rice-Ramsperger-Kassel-Marcus, transition state, collisional energy transfer, and other rate calculations not to be an impediment to reaction. In contrast, experimental spectroscopy, computational results, and models based on Anderson localization have shown that ergodicity is achieved rather slowly during molecular energy flow. The statistical assumption in rate theories might easily fail due to quantum localization. Here, we develop a simple model for the interplay of IVR and energy transfer and simulate the model with near-exact quantum dynamics for a 10-degree of freedom system composed of two five-mode molecular fragments. The calculations are facilitated by applying the van Vleck transformation to local random matrix models of the vibrational Hamiltonian. We find that there is a rather sharp "phase transition" as a function of molecular anharmonicity "a" between a region of facile energy transfer and a region limited by IVR and incomplete accessibility of the state space (classically, the phase space). The very narrow transition range of the order parameter a happens to lie right in the middle of the range expected for molecular torsion, bending, and stretching vibrations, thus demonstrating that reactive energy transfer dynamics several k_BT above the thermal energy occurs not far from the localization boundary, with implications for controllability of reactions.

Grafting metal complexes onto amorphous supports: from elementary steps to catalyst site populationsviakernel regression

We use transition state theory, kernel regression, and population balance modeling techniques to model the grafting of metal complexes onto amorphous catalyst supports.

Bond Fragility Spectra for the Double Proton-Transfer Reaction in the Formic Acid-Type Dimers

The newly developed method of fragility spectra for observation of bond breaking and formation upon a reaction has been applied to the canonical reaction series of the double proton transfer (DPT). Formic acid and its thio-analogues HCXYH (X, Y = O, S) have been chosen for the analysis. Very accurate linear correlations have been determined between the nondiagonal elements of the connectivity matrix, essential for the method, and the Wiberg bond orders for the corresponding bonds. Relation of the slope of this correlation to the global softness and to the atomic numbers of the bonded atoms has been proved, thus corroborating the c-DFT formula describing the fragility spectra. The electron density changes in bonds, as observed by the fragility spectra, are in harmony with the curvature diagrams reported by other authors.

Upside/Downside statistical mechanics of nonequilibrium Brownian motion. II. Heat transfer and energy partitioning of a free particle

The energy partitioning during activation and relaxation events under steady-state conditions for a Brownian particle driven by multiple thermal reservoirs of different local temperatures is investigated. Specifically, we apply the formalism derived in Paper I [G. T. Craven and A. Nitzan, J. Chem. Phys. 148, 044101 (2018)] to examine the thermal transport properties of two sub-ensembles of Brownian processes, distinguished at any given time by the specification that all the trajectories in each group have, at that time, energy either above (upside) or below (downside) a preselected energy threshold. Dynamical properties describing energy accumulation and release during activation/relaxation events and relations for upside/downside energy partitioning between thermal reservoirs are derived. The implications for heat transport induced by upside and downside events are discussed.

Upside/Downside statistical mechanics of nonequilibrium Brownian motion. I. Distributions, moments, and correlation functions of a free particle

Statistical properties of Brownian motion that arise by analyzing, separately, trajectories over which the system energy increases (upside) or decreases (downside) with respect to a threshold energy level are derived. This selective analysis is applied to examine transport properties of a nonequilibrium Brownian process that is coupled to multiple thermal sources characterized by different temperatures. Distributions, moments, and correlation functions of a free particle that occur during upside and downside events are investigated for energy activation and energy relaxation processes and also for positive and negative energy fluctuations from the average energy. The presented results are sufficiently general and can be applied without modification to the standard Brownian motion. This article focuses on the mathematical basis of this selective analysis. In subsequent articles in this series, we apply this general formalism to processes in which heat transfer between thermal reservoirs is mediated by activated rate processes that take place in a system bridging them.

Electrothermal Transistor Effect and Cyclic Electronic Currents in Multithermal Charge Transfer Networks

A theory is developed to describe the coupled transport of energy and charge in networks of electron donor-acceptor sites which are seated in a thermally heterogeneous environment, where the transfer kinetics are dominated by Marcus-type hopping rates. It is found that the coupling of heat and charge transfer in such systems gives rise to exotic transport phenomena which are absent in thermally homogeneous systems and cannot be described by standard thermoelectric relations. Specifically, the directionality and extent of thermal transistor amplification and cyclical electronic currents in a given network can be controlled by tuning the underlying temperature gradient in the system. The application of these findings toward the optimal control of multithermal currents is illustrated on a paradigmatic nanostructure.

Identification of simple reaction coordinates from complex dynamics

Reaction coordinates are widely used throughout chemical physics to model and understand complex chemical transformations. We introduce a definition of the natural reaction coordinate, suitable for condensed phase and biomolecular systems, as a maximally predictive one-dimensional projection. We then show that this criterion is uniquely satisfied by a dominant eigenfunction of an integral operator associated with the ensemble dynamics. We present a new sparse estimator for these eigenfunctions which can search through a large candidate pool of structural order parameters and build simple, interpretable approximations that employ only a small number of these order parameters. Example applications with a small molecule's rotational dynamics and simulations of protein conformational change and folding show that this approach can filter through statistical noise to identify simple reaction coordinates from complex dynamics.

On couplings and excimers: lessons from studies of singlet fission in covalently linked tetracene dimers

Electronic factors controlling singlet fission (SF) rates are investigated in covalently linked dimers of tetracene. Using covalent linkers, relative orientation of the individual chromophores can be controlled, maximizing the rates of SF. Structures with coplanar and staggered arrangements of tetracene moieties are considered. The electronic structure calculations and three-state kinetic model for SF rates provide explanations for experimentally observed low SF yields in coplanar dimers and efficient SF in staggered dimers. The calculations illuminate the role of the excimer formation in SF process. The structural relaxation in the S1 state leads to the increased rate of the multi-exciton (ME) state formation, but impedes the second step, separation of the ME state into independent triplets. The slower second step reduces SF yield by allowing other processes, such as radiationless relaxation, to compete with triplet generation. The calculations of electronic couplings also suggest an increased rate of radiationless relaxation at the excimer geometries. Thus, the excimer serves as a trap of the ME state. The effect of covalent linkers on the electronic factors and SF rates is investigated. In all considered structures, the presence of the linker leads to larger couplings, however, the effect on the overall rate is less straightforward, since the linkers generally result in less favorable energetics. This complex behavior once again illustrates the importance of integrative approaches that evaluate the overall rate, rather than focusing on specific electronic factors such as energies or couplings.

Electron transfer across a thermal gradient

Charge transfer is a fundamental process that underlies a multitude of phenomena in chemistry and biology. Recent advances in observing and manipulating charge and heat transport at the nanoscale, and recently developed techniques for monitoring temperature at high temporal and spatial resolution, imply the need for considering electron transfer across thermal gradients. Here, a theory is developed for the rate of electron transfer and the associated heat transport between donor-acceptor pairs located at sites of different temperatures. To this end, through application of a generalized multidimensional transition state theory, the traditional Arrhenius picture of activation energy as a single point on a free energy surface is replaced with a bithermal property that is derived from statistical weighting over all configurations where the reactant and product states are equienergetic. The flow of energy associated with the electron transfer process is also examined, leading to relations between the rate of heat exchange among the donor and acceptor sites as functions of the temperature difference and the electronic driving bias. In particular, we find that an open electron transfer channel contributes to enhanced heat transport between sites even when they are in electronic equilibrium. The presented results provide a unified theory for charge transport and the associated heat conduction between sites at different temperatures.

The reaction fragility spectrum

We report an original method that provides a new insight into the reaction mechanism by direct observation of bond breaking and formation. Example: HONS → ONSH.

Protein electron transfer: is biology (thermo)dynamic?

Simple physical mechanisms are behind the flow of energy in all forms of life. Energy comes to living systems through electrons occupying high-energy states, either from food (respiratory chains) or from light (photosynthesis). This energy is transformed into the cross-membrane proton-motive force that eventually drives all biochemistry of the cell. Life's ability to transfer electrons over large distances with nearly zero loss of free energy is puzzling and has not been accomplished in synthetic systems. The focus of this review is on how this energetic efficiency is realized. General physical mechanisms and interactions that allow proteins to fold into compact water-soluble structures are also responsible for a rugged landscape of energy states and a broad distribution of relaxation times. Specific to a protein as a fluctuating thermal bath is the protein-water interface, which is heterogeneous both dynamically and structurally. The spectrum of interfacial fluctuations is a consequence of protein's elastic flexibility combined with a high density of surface charges polarizing water dipoles into surface nanodomains. Electrostatics is critical to the protein function and the relevant questions are: (i) What is the spectrum of interfacial electrostatic fluctuations? (ii) Does the interfacial biological water produce electrostatic signatures specific to proteins? (iii) How is protein-mediated chemistry affected by electrostatics? These questions connect the fluctuation spectrum to the dynamical control of chemical reactivity, i.e. the dependence of the activation free energy of the reaction on the dynamics of the bath. Ergodicity is often broken in protein-driven reactions and thermodynamic free energies become irrelevant. Continuous ergodicity breaking in a dense spectrum of relaxation times requires using dynamically restricted ensembles to calculate statistical averages. When applied to the calculation of the rates, this formalism leads to the nonergodic activated kinetics, which extends the transition-state theory to dynamically dispersive media. Releasing the grip of thermodynamics in kinetic calculations through nonergodicity provides the mechanism for an efficient optimization between reaction rates and the spectrum of relaxation times of the protein-water thermal bath. Bath dynamics, it appears, play as important role as the free energy in optimizing biology's performance.