Spatial dynamics of steady flames 2. Low-dimensional manifolds and the role of transport processes

The study of the spatial dynamics of steady one-dimensional H 2/O 2 flames is continued. Algorithms for generating low-dimensional manifolds for these systems are presented and used to find low-dimensional manifolds for the flames and the corresponding adiabatic, isobaric chemical-kinetic systems. It is demonstrated that these algorithms generate manifolds that are more accurate than the ILDM algorithm for two-dimensional manifolds of the flames. The manifolds are then employed to study the relationship between the manifolds of the flame and the manifolds of the chemical-kinetic system. It is shown that the one-dimensional manifolds of the flame match well with the composite manifolds of the chemical kinetics, but that for two-dimensional manifolds there are discrepancies between the flame manifolds and the chemical-kinetic manifolds.     

Atomistic simulations of rare events using gentlest ascent dynamics

The dynamics of complex systems often involve thermally activated barrier crossing events that allow these systems to move from one basin of attraction on the high dimensional energy surface to another. Such events are ubiquitous, but challenging to simulate using conventional simulation tools, such as molecular dynamics. Recently, E and Zhou [Nonlinearity 24(6), 1831 (2011)] proposed a set of dynamic equations, the gentlest ascent dynamics (GAD), to describe the escape of a system from a basin of attraction and proved that solutions of GAD converge to index-1 saddle points of the underlying energy. In this paper, we extend GAD to enable finite temperature simulations in which the system hops between different saddle points on the energy surface. An effective strategy to use GAD to sample an ensemble of low barrier saddle points located in the vicinity of a locally stable configuration on the high dimensional energy surface is proposed. The utility of the method is demonstrated by studying the low barrier saddle points associated with point defect activity on a surface. This is done for two representative systems, namely, (a) a surface vacancy and ad-atom pair and (b) a heptamer island on the (111) surface of copper.     

Chaotic dynamics and entropy elastic forces of chain molecules

The dynamic behaviour of a phantom molecular chain, consisting of mass points connected by mass-less rigid rods, is investigated in two dimensions. The end points of the chain are fixed in space. The dynamics of the molecule are determined by Newton's equations of motion. Their numerical solution is represented in form of Poincaré sections and shows deterministic chaos. It is shown that the forces of constraint, acting on the fixed end points, fluctuate in time. The time average of these forces is calculated as a function of the extension of the chain.     

Phase space geometry of dynamics passing through saddle coupled with spatial rotation

Nonlinear reaction dynamics through a rank-one saddle is investigated for many-particle system with spatial rotation. Based on the recently developed theories of the phase space geometry in the saddle region, we present a theoretical framework to incorporate the spatial rotation which is dynamically coupled with the internal vibrational motions through centrifugal and Coriolis interactions. As an illustrative simple example, we apply it to isomerization reaction of HCN with some nonzero total angular momenta. It is found that no-return transition state (TS) and a set of impenetrable reaction boundaries to separate the "past" and "future" of trajectories can be identified analytically under rovibrational couplings. The three components of the angular momentum are found to have distinct effects on the migration of the "anchor" of the TS and the reaction boundaries through rovibrational couplings and anharmonicities in vibrational degrees of freedom. This method provides new insights in understanding the origin of a wide class of reactions with nonzero angular momentum.     

Accelerated molecular dynamics: a promising and efficient simulation method for biomolecules

Many interesting dynamic properties of biological molecules cannot be simulated directly using molecular dynamics because of nanosecond time scale limitations. These systems are trapped in potential energy minima with high free energy barriers for large numbers of computational steps. The dynamic evolution of many molecular systems occurs through a series of rare events as the system moves from one potential energy basin to another. Therefore, we have proposed a robust bias potential function that can be used in an efficient accelerated molecular dynamics approach to simulate the transition of high energy barriers without any advance knowledge of the location of either the potential energy wells or saddle points. In this method, the potential energy landscape is altered by adding a bias potential to the true potential such that the escape rates from potential wells are enhanced, which accelerates and extends the time scale in molecular dynamics simulations. Our definition of the bias potential echoes the underlying shape of the potential energy landscape on the modified surface, thus allowing for the potential energy minima to be well defined, and hence properly sampled during the simulation. We have shown that our approach, which can be extended to biomolecules, samples the conformational space more efficiently than normal molecular dynamics simulations, and converges to the correct canonical distribution.     

Low-dimensional manifolds in reaction-diffusion equations. 1. Fundamental aspects

The approach to equilibrium for systems of reaction-diffusion equations on bounded domains is studied geometrically. It is shown that equilibrium is approached via low-dimensional manifolds in the infinite-dimensional function space for these dissipative, parabolic systems. The fundamental aspects of this process are mapped out in some detail for single species cases and for two-species cases where there is an exact solution. It is shown how the manifolds reduce the dimensionality of the system from infinite dimensions to only a few dimensions.     

Trajectory approach to dissipative quantum phase space dynamics: Application to barrier scattering

The Caldeira-Leggett master equation, expressed in Lindblad form, has been used in the numerical study of the effect of a thermal environment on the dynamics of the scattering of a wave packet from a repulsive Eckart barrier. The dynamics are studied in terms of phase space trajectories associated with the distribution function, W(q,p,t). The equations of motion for the trajectories include quantum terms that introduce nonlocality into the motion, which imply that an ensemble of correlated trajectories needs to be propagated. However, use of the derivative propagation method (DPM) allows each trajectory to be propagated individually. This is achieved by deriving equations of motion for the partial derivatives of W(q,p,t) that appear in the master equation. The effects of dissipation on the trajectories are studied and results are shown for the transmission probability. On short time scales, decoherence is demonstrated by a swelling of trajectories into momentum space. For a nondissipative system, a comparison is made of the DPM with the "exact" transmission probability calculated from a fixed grid calculation.     

Regular dynamics in transition states with flat saddles

In the vicinity of a transition state, the dynamics is constrained by approximate local invariants of the motion even if the potential energy surface is anharmonic. The concept of local regularity near a saddle point is investigated in the framework of classical mechanics. The dynamics along the reaction coordinate decouples locally into a reactive mode and several bounded degrees of freedom. The partial energy stored in the unbounded mode is adiabatically invariant. Starting from a purely harmonic situation at the saddle point, anharmonicity coefficients are observed to come into play in a sequential way in the laws of motion. In most cases, each kind of anharmonic coefficient can be related to a particular feature of the potential energy surface or of the reaction path. These regularities account for previous classical trajectory calculations by Berry and co-workers, who observed that for flat saddles (i.e., those characterized by a low value of the modulus of the imaginary frequency), trajectories become temporarily collimated and less chaotic during passage through the transition state.     

How to find a saddle point

This paper explains a method for finding saddle points on a multidimensional surface and shows how it may be used to define saddle-point seeking curves that have properties similar to well-known orthogonal trajectories. It is shown that a gradient extremal is a special case of one of these curves, and its chemical significance as the path, defined by local criteria, which starts from a stable structure and leads to a transition state, is discussed briefly in relation to the intrinsic reaction coordinate. It is emphasized that this theory gives a natural method for locating points that have Hessians of similar structure to those of transition states.     

Finite-Temperature Dimer Method for Finding Saddle Points on Free Energy Surfaces

The dimer method and its variants have been shown to be efficient in finding saddle points on potential surfaces. In the dimer method, the most unstable direction is approximately obtained by minimizing the total potential energy of the dimer. Then, the force in this direction is reversed to move the dimer toward saddle points. When the finite-temperature effect is important for a high-dimensional system, one usually needs to describe the dynamics in a low-dimensional space of reaction coordinates. In this case, transition states are collected as saddle points on the free energy surface. The traditional dimer method cannot be directly employed to find saddle points on a free energy surface since the surface is not known a priori. Here, we develop a finite-temperature dimer method for searching saddle points on the free energy surface. In this method, a constrained rotation dynamics of the dimer system is used to sample dimer directions and an efficient average method is used to obtain a good approximation of the most unstable direction. This approximated direction is then used in reversing the force component and evolving the dimer toward saddle points. Our numerical results suggest that the new method is efficient in finding saddle points on free energy surfaces. © 2019 Wiley Periodicals, Inc.     

A Smoluchowski model of crystallization dynamics of small colloidal clusters

We investigate the dynamics of colloidal crystallization in a 32-particle system at a fixed value of interparticle depletion attraction that produces coexisting fluid and solid phases. Free energy landscapes (FELs) and diffusivity landscapes (DLs) are obtained as coefficients of 1D Smoluchowski equations using as order parameters either the radius of gyration or the average crystallinity. FELs and DLs are estimated by fitting the Smoluchowski equations to Brownian dynamics (BD) simulations using either linear fits to locally initiated trajectories or global fits to unbiased trajectories using Bayesian inference. The resulting FELs are compared to Monte Carlo Umbrella Sampling results. The accuracy of the FELs and DLs for modeling colloidal crystallization dynamics is evaluated by comparing mean first-passage times from BD simulations with analytical predictions using the FEL and DL models. While the 1D models accurately capture dynamics near the free energy minimum fluid and crystal configurations, predictions near the transition region are not quantitatively accurate. A preliminary investigation of ensemble averaged 2D order parameter trajectories suggests that 2D models are required to capture crystallization dynamics in the transition region.     

Vibronic excitation of single molecules: a new technique for studying low-temperature dynamics.

Herein, we present vibronic excitation and detection of purely electronic zero-phonon lines (ZPL) of single molecules as a new tool for investigating dynamics at cryogenic temperatures. Applications of this technique to study crystalline and amorphous matrix materials are presented. In the crystalline environment, spectrally stable ZPLs are observed at moderate excitation powers. By contrast, investigations at higher excitation intensities reveal the opening of local degrees of freedom and spectral jumps, which we interpret as the observation of elementary steps in the melting of a crystal. We compare these results to spectral single-molecule trajectories recorded in a polymer. The way in which much more complicated spectral features can be analysed is shown. Surprisingly, pronounced spectral shifts on a previously not accessible large energy scale are observed, which are hard to reconcile with the standard two-level model system used to describe low-temperature dynamics in disordered systems.     

Moving boundary truncated grid method: Multidimensional quantum dynamics

The moving boundary truncated grid (TG) method is used to study wave packet dynamics of multidimensional quantum systems. As time evolves, appropriate Eulerian grid points required for propagating a wave packet are activated and deactivated with no advance information about the dynamics. This method is applied to the Henon-Heiles potential and wave packet barrier scattering in two, three, and four dimensions. Computational results demonstrate that the TG method not only leads to a great reduction in the number of grid points needed to perform accurate calculations but also is computationally more efficient than the full grid calculations.     

Identifying reactive trajectories using a moving transition state

A time-dependent no-recrossing dividing surface is shown to lead to a new criterion for identifying reactive trajectories well before they are evolved to infinite time. Numerical dynamics simulations of a dissipative anharmonic two-dimensional system confirm the efficiency of this approach. The results are compared to the standard fixed transition state dividing surface that is well-known to suffer from recrossings and therefore requires trajectories to be evolved over a long time interval before they can reliably be classified as reactive or nonreactive. The moving dividing surface can be used to identify reactive trajectories in harmonic or moderately anharmonic systems with considerably lower numerical effort or even without any simulation at all.     

Ab initio molecular dynamics with dual basis set methods

On-the-fly, ab initio classical molecular dynamics are demonstrated with an underlying dual basis set potential energy surface. Dual-basis self-consistent field (Hartree-Fock and density functional theory) and resolution-of-the-identity second-order M?ller-Plesset perturbation theory (RI-MP2) dynamics are tested for small systems, including the water dimer. The resulting dynamics are shown to be faithful representations of their single-basis analogues for individual trajectories, as well as vibrational spectra. Computational cost savings of 58% are demonstrated for SCF methods, even relative to Fock-extrapolated dynamics, and savings are further increased to 71% with RI-MP2. Notably, these timings outperform an idealized estimate of extended-Lagrangian molecular dynamics. The method is subsequently demonstrated on the vibrational absorption spectrum of two NO(+)(H?O)? isomers and is shown to recover the significant width of the shared-proton bands observed experimentally.     

Computation of correlation functions and wave function projections in the context of quantum trajectory dynamics

The de Broglie-Bohm formulation of the Schrodinger equation implies conservation of the wave function probability density associated with each quantum trajectory in closed systems. This conservation property greatly simplifies numerical implementations of the quantum trajectory dynamics and increases its accuracy. The reconstruction of a wave function, however, becomes expensive or inaccurate as it requires fitting or interpolation procedures. In this paper we present a method of computing wave packet correlation functions and wave function projections, which typically contain all the desired information about dynamics, without the full knowledge of the wave function by making quadratic expansions of the wave function phase and amplitude near each trajectory similar to expansions used in semiclassical methods. Computation of the quantities of interest in this procedure is linear with respect to the number of trajectories. The introduced approximations are consistent with approximate quantum potential dynamics method. The projection technique is applied to model chemical systems and to the H+H(2) exchange reaction in three dimensions.