This article reviews the concepts and methods of transition path sampling. These methods allow computational studies of rare events without requiring prior knowledge of mechanisms, reaction coordinates, and transition states. Based upon a statistical mechanics of trajectory space, they provide a perspective with which time dependent phenomena, even for systems driven far from equilibrium, can be examined with the same types of importance sampling tools that in the past have been applied so successfully to static equilibrium properties.

We have developed a method to study transition pathways for rare events in complex systems. The method can be used to determine rate constants for transitions between stable states by turning the calculation of reactive flux correlation functions into the computation of an isomorphic reversible work. In contrast to previous dynamical approaches, the method relies neither on prior knowledge nor on explicit specification of transition states. Rather, it provides an importance sampling from which transition states can be characterized statistically. A simple model is analyzed to illustrate the methodology.

Local bond order parameters based on spherical harmonics, also known as Steinhardt order parameters, are often used to determine crystal structures in molecular simulations. Here we propose a modification of this method in which the complex bond order vectors are averaged over the first neighbor shell of a given particle and the particle itself. As demonstrated using soft particle systems, this averaging procedure considerably improves the accuracy with which different crystal structures can be distinguished.

The dissociation of a water molecule in liquid water is the fundamental event in acid-base chemistry, determining the pH of water. Because of the short time scales and microscopic length scales involved, the dynamics of this autoionization have not been directly probed by experiment. Here, the autoionization mechanism is revealed by sampling and analyzing ab initio molecular dynamics trajectories. We identify the rare fluctuations in solvation energies that destabilize an oxygen-hydrogen bond. Through the transfer of protons along a hydrogen bond "wire," the nascent ions separate by three or more neighbors. If the hydrogen bond wire connecting the two ions is subsequently broken, a metastable charge-separated state is visited. The ions may then diffuse to large separations. If, however, the hydrogen bond wire remains unbroken, the ions recombine rapidly. Because of their concomitant large electric fields, the transient ionic species produced in this case may provide an experimentally detectable signal of the dynamics we report.

Transition path sampling has been applied to the molecular dynamics of the alanine dipeptide in vacuum and in aqueous solution. The analysis shows that more degrees of freedom than the traditional dihedral angles, and , are necessary to describe the reaction coordinates for isomerization of this molecule. In vacuum, an additional dihedral angle is identified as significant. In solution, solvent variables are shown to play a significant role, and this role appears to be more specific than can be captured by friction models. Implications for larger molecules are discussed.T his paper concerns the dynamical variables relevant to kinetics of biomolecular isomerization. The backbone dihedral angles and do serve well as order parameters characterizing the stable isomeric states of polypeptides (1). This fact does not, however, imply that and are satisfactory for describing the dynamics of transitions between these states. Indeed, for large enough molecules, clustering of hydrophobic units can lead to drying (2, 3). The nucleation of this phenomenon manifestly involves motion of solvent. On the smaller length scales characterizing side-chain motions, effects from solvent are apparent as well (4). On these scales, it often is assumed that the dynamical role of secondary variables can be captured implicitly, through the mean forces on angles like and and through friction and random local forces (5). We test this idea in this paper through the analysis of trajectories of the alanine dipeptide in vacuum and aqueous solution. We establish that other variables in addition to and are important. Most significantly, we find that solvent degrees of freedom are relevant components to the reaction coordinates for isomerization. Although the dipeptide is a relatively simple molecule, our findings would seem to carry over to the small-length scale motions of more complex biopolymers. Specifically, our results suggest the inadequacy of implicit solvent models for treating the dynamical effects of solvation. Although implicit solvent models can be used to calculate averaged solvent effects, such models are not able to describe a possibly important rearrangement of individual solvent molecules occurring during a transition correctly. Fig. 1 illustrates the ball and stick model and the important dihedral angles of the alanine dipeptide. This molecule often is studied in theoretical work (6-13) because it is among the simplest systems to exhibit some of the important features common to biomolecules. It has the basic elements of a polypeptide backbone with more than one long-lived conformational state. It forms hydrogen bonds with water in aqueous solution. This bonding or solvation in general is responsible for the stability of one of the conformers. The molecule in vacuum has two stable conformers: the C 7eq state with Ϸ Ϫ86°and Ϸ 68°a nd the C ax state with Ϸ 50°and Ϸ Ϫ50°. In solution, these positions shift slightly, for instance, the C 7eq state is located around Ϸ Ϫ80°and Ϸ 160°. In addition, two other states, ␣ R and ␣ L , become s...

Whereas the interactions between water molecules are dominated by strongly directional hydrogen bonds (HBs), it was recently proposed that relatively weak, isotropic van der Waals (vdW) forces are essential for understanding the properties of liquid water and ice. This insight was derived from ab initio computer simulations, which provide an unbiased description of water at the atomic level and yield information on the underlying molecular forces. However, the high computational cost of such simulations prevents the systematic investigation of the influence of vdW forces on the thermodynamic anomalies of water. Here, we develop efficient ab initio-quality neural network potentials and use them to demonstrate that vdW interactions are crucial for the formation of water's density maximum and its negative volume of melting. Both phenomena can be explained by the flexibility of the HB network, which is the result of a delicate balance of weak vdW forces, causing, e.g., a pronounced expansion of the second solvation shell upon cooling that induces the density maximum.water structure | van der Waals interactions | neural network potentials | ab initio liquid water | density-functional theory W ater is an exceptional liquid exhibiting several anomalies, of which the density maximum at 4°C is the most prominent (1). Together with the negative volume of melting, it is responsible for the fact that water freezes from the top down and ice floats on water. The unusual behavior of water can be directly related to its ability to form hydrogen bonds (HBs) which are of strongly directional nature and determine the microscopic structure of water (2, 3). To investigate the anomalies of water at the molecular level, atomistic computer simulations have become an essential tool. Important contributions have been made by simulations using simple empirical water models (3-8).Simulations based on ab initio molecular dynamics (AIMD) (9-11) allow determination of the properties of water with high predictive power and enable a detailed analysis of their underlying microscopic mechanisms. In contrast to empirical water models (5), which depend on experimental data resulting in a limited transferability, in AIMD the atomic forces that govern the molecular dynamics are obtained directly from quantum mechanics. Although this approach is in principle exact [in combination with methods that account for the quantum nature of the nuclei (12, 13)], ab initio simulations of condensed matter systems are feasible only if approximate but efficient methods such as density-functional theory (DFT) are used. Even then, however, simulations are restricted to short times and small systems. AIMD simulations have been used to a limited extent to investigate the phase behavior of water, for instance by estimating melting temperatures (14, 15) and vapor-liquid coexistence curves (16, 17). However, many fundamental thermodynamic properties of water have not been evaluated to date. To circumvent the limitations of on-the-fly AIMD, various efficient water potentia...

The momentum transfer between a photon and an object defines a fundamental limit for the precision with which the object can be measured. If the object oscillates at a frequency Ω0, this measurement back-action adds quantahΩ0 to the oscillator's energy at a rate Γ recoil , a process called photon recoil heating, and sets bounds to coherence times in cavity optomechanical systems. Here, we use an optically levitated nanoparticle in ultrahigh vacuum to directly measure Γ recoil . By means of a phase-sensitive feedback scheme, we cool the harmonic motion of the nanoparticle from ambient to micro-Kelvin temperatures and measure its reheating rate under the influence of the radiation field. The recoil heating rate is measured for different particle sizes and for different excitation powers, without the need for cavity optics or cryogenic environments. The measurements are in quantitative agreement with theoretical predictions and provide valuable guidance for the realization of quantum ground-state cooling protocols and the measurement of ultrasmall forces.

We develop an efficient Monte-Carlo algorithm to sample an ensemble of stochastic transition paths between stable states. In our description, paths are represented by chains of states linked by Markovian transition probabilities. Rate constants and mechanisms characterizing the transition may be determined from the path ensemble. We have previously devised several algorithms for sampling the path ensemble. For these algorithms, the numerical effort scales with the square of the path length. In the new simulation scheme, the required computation scales linearly with the length of the transition path. This improved efficiency allows the calculation of rate constants in complex molecular systems. As an example, we study rearrangement processes in a cluster consisting of seven Lennard-Jones particles in two dimensions. Using a quenching technique we are able to identify the relevant transition mechanisms and to locate the related transition states. We furthermore calculate transition rate constants for various isomerization processes.