Skipping the Replica Exchange Ladder with Normalizing Flows.

We combine replica exchange (parallel tempering) with normalizing flows, a class of deep generative models. These two sampling strategies complement each other, resulting in an efficient method for sampling molecular systems characterized by rare events, which we call learned replica exchange (LREX). In LREX, a normalizing flow is trained to map the configurations of the fastest-mixing replica into configurations belonging to the target distribution, allowing direct exchanges between the two without the need to simulate intermediate replicas. This can significantly reduce the computational cost compared to standard replica exchange. The proposed method also offers several advantages with respect to Boltzmann generators that directly use normalizing flows to sample the target distribution. We apply LREX to some prototypical molecular dynamics systems, highlighting the improvements over previous methods.

Rare Event Kinetics from Adaptive Bias Enhanced Sampling.

We introduce a novel enhanced sampling approach named on-the-fly probability enhanced sampling (OPES) flooding for calculating the kinetics of rare events from atomistic molecular dynamics simulation. This method is derived from the OPES approach [Invernizzi and Parrinello, J. Phys. Chem. Lett. 2020, 11, 7, 2731-2736], which has been recently developed for calculating converged free energy surfaces for complex systems. In this paper, we describe the theoretical details of the OPES flooding technique and demonstrate the application on three systems of increasing complexity: barrier crossing in a two-dimensional double-well potential, conformational transition in the alanine dipeptide in the gas phase, and the folding and unfolding of the chignolin polypeptide in an aqueous environment. From extensive tests, we show that the calculation of accurate kinetics not only requires the transition state to be bias-free, but the amount of bias deposited should also not exceed the effective barrier height measured along the chosen collective variables. In this vein, the possibility of computing rates from biasing suboptimal order parameters has also been explored. Furthermore, we describe the choice of optimum parameter combinations for obtaining accurate results from limited computational effort.

Exploration vs Convergence Speed in Adaptive-Bias Enhanced Sampling.

In adaptive-bias enhanced sampling methods, a bias potential is added to the system to drive transitions between metastable states. The bias potential is a function of a few collective variables and is gradually modified according to the underlying free energy surface. We show that when the collective variables are suboptimal, there is an exploration-convergence tradeoff, and one must choose between a quickly converging bias that will lead to fewer transitions or a slower to converge bias that can explore the phase space more efficiently but might require a much longer time to produce an accurate free energy estimate. The recently proposed on-the-fly probability enhanced sampling (OPES) method focuses on fast convergence, but there are cases where fast exploration is preferred instead. For this reason, we introduce a new variant of the OPES method that focuses on quickly escaping metastable states at the expense of convergence speed. We illustrate the benefits of this approach in prototypical systems and show that it outperforms the popular metadynamics method.

A combined molecular dynamics and experimental study of two-step process enabling low-temperature formation of phase-pure α-FAPbI3.

It is well established that the lack of understanding the crystallization process in a two-step sequential deposition has a direct impact on efficiency, stability, and reproducibility of perovskite solar cells. Here, we try to understand the solid-solid phase transition occurring during the two-step sequential deposition of methylammonium lead iodide and formamidinium lead iodide. Using metadynamics, x-ray diffraction, and Raman spectroscopy, we reveal the microscopic details of this process. We find that the formation of perovskite proceeds through intermediate structures and report polymorphs found for methylammonium lead iodide and formamidinium lead iodide. From simulations, we discover a possible crystallization pathway for the highly efficient metastable α phase of formamidinium lead iodide. Guided by these simulations, we perform experiments that result in the low-temperature crystallization of phase-pure α-formamidinium lead iodide.

Attenuating the fermion sign problem in path integral Monte Carlo simulations using the Bogoliubov inequality and thermodynamic integration.

Accurate thermodynamic simulations of correlated fermions using path integral Monte Carlo (PIMC) methods are of paramount importance for many applications such as the description of ultracold atoms, electrons in quantum dots, and warm-dense matter. The main obstacle is the fermion sign problem (FSP), which leads to an exponential increase in computation time both with an increase in the system size and with a decrease in the temperature. Very recently, Hirshberg et al. [J. Chem. Phys. 152, 171102 (2020)] have proposed to alleviate the FSP based on the Bogoliubov inequality. In the present work, we extend this approach by adding a parameter that controls the perturbation, allowing for an extrapolation to the exact result. In this way, we can also use thermodynamic integration to obtain an improved estimate of the fermionic energy. As a test system, we choose electrons in 2D and 3D quantum dots and find in some cases a speed-up exceeding 106, as compared to standard PIMC, while retaining a relative accuracy of ∼0.1%. Our approach is quite general and can readily be adapted to other simulation methods.

Path integral molecular dynamics for fermions: Alleviating the sign problem with the Bogoliubov inequality.

We present a method for performing path integral molecular dynamics (PIMD) simulations for fermions and address its sign problem. PIMD simulations are widely used for studying many-body quantum systems at thermal equilibrium. However, they assume that the particles are distinguishable and neglect bosonic and fermionic exchange effects. Interacting fermions play a key role in many chemical and physical systems, such as electrons in quantum dots and ultracold trapped atoms. A direct sampling of the fermionic partition function is impossible using PIMD since its integrand is not positive definite. We show that PIMD simulations for fermions are feasible by employing our recently developed method for bosonic PIMD and reweighting the results to obtain fermionic expectation values. The approach is tested against path integral Monte Carlo (PIMC) simulations for up to seven electrons in a two-dimensional quantum dot for a range of interaction strengths. However, like PIMC, the method suffers from the sign problem at low temperatures. We propose a simple approach for alleviating it by simulating an auxiliary system with a larger average sign and obtaining an upper bound to the energy of the original system using the Bogoliubov inequality. This allows fermions to be studied at temperatures lower than would otherwise have been feasible using PIMD, as demonstrated in the case of a three-electron quantum dot. Our results extend the boundaries of PIMD simulations of fermions and will hopefully stimulate the development of new approaches for tackling the sign problem.

Rethinking Metadynamics: From Bias Potentials to Probability Distributions.

Metadynamics is an enhanced sampling method of great popularity, based on the on-the-fly construction of a bias potential that is a function of a selected number of collective variables. We propose here a change in perspective that shifts the focus from the bias to the probability distribution reconstruction while retaining some of the key characteristics of metadynamics, such as flexible on-the-fly adjustments to the free energy estimate. The result is an enhanced sampling method that presents a drastic improvement in convergence speed, especially when dealing with suboptimal and/or multidimensional sets of collective variables. The method is especially robust and easy to use and in fact requires only a few simple parameters to be set, and it has a straightforward reweighting scheme to recover the statistics of the unbiased ensemble. Furthermore, it gives more control of the desired exploration of the phase space since the deposited bias is not allowed to grow indefinitely and it does not push the simulation to uninteresting high free energy regions. We demonstrate the performance of the method in a number of representative examples.

Making the Best of a Bad Situation: A Multiscale Approach to Free Energy Calculation.

Many enhanced sampling techniques rely on the identification of a number of collective variables that describe all the slow modes of the system. By constructing a bias potential in this reduced space, one is then able to sample efficiently and reconstruct the free energy landscape. In methods such as metadynamics, the quality of these collective variables plays a key role in convergence efficiency. Unfortunately in many systems of interest it is not possible to identify an optimal collective variable, and one must deal with the nonideal situation of a system in which some slow modes are not accelerated. We propose a two-step approach in which, by taking into account the residual multiscale nature of the problem, one is able to significantly speed up convergence. To do so, we combine an exploratory metadynamics run with an optimization of the free energy difference between metastable states, based on the recently proposed variationally enhanced sampling method. This new method is well parallelizable and is especially suited for complex systems, because of its simplicity and clear underlying physical picture.

Enhanced Sampling of Transition States.

The free energy landscapes of several fundamental processes are characterized by high barriers separating long-lived metastable states. In order to explore these type of landscapes, enhanced sampling methods are used. While many such methods are able to obtain sufficient sampling in order to draw the free energy, the transition states are often sparsely sampled. We propose an approach based on the Variationally Enhanced Sampling Method to enhance sampling in the transition region. To this effect, we introduce a dynamic target distribution which uses the derivative of the instantaneous free energy surface to locate the transition regions on the fly and modulate the probability of sampling different regions. Finally, we exemplify the effectiveness of this approach in enriching the number of configurations in the transition state region in the cases of a chemical reaction and of a nucleation process.

Molecular dynamics simulations of liquid silica crystallization.

Silica is one of the most abundant minerals on Earth and is widely used in many fields. Investigating the crystallization of liquid silica by atomic simulations is of great importance to understand the crystallization mechanism; however, the high crystallization barrier and the tendency of silica to form glasses make such simulations very challenging. Here we have studied liquid silica crystallization to ß-cristobalite with metadynamics, using X-ray diffraction (XRD) peak intensities as collective variables. The frequent transitions between solid and liquid of the biased runs demonstrate the highly successful use of the XRD peak intensities as collective variables, which leads to the convergence of the free-energy surface. By calculating the difference in free energy, we have estimated the melting temperature of ß-cristobalite, which is in good agreement with the literature. The nucleation mechanism during the crystallization of liquid silica can be described by classical nucleation theory.

Coarse graining from variationally enhanced sampling applied to the Ginzburg-Landau model.

A powerful way to deal with a complex system is to build a coarse-grained model capable of catching its main physical features, while being computationally affordable. Inevitably, such coarse-grained models introduce a set of phenomenological parameters, which are often not easily deducible from the underlying atomistic system. We present a unique approach to the calculation of these parameters, based on the recently introduced variationally enhanced sampling method. It allows us to obtain the parameters from atomistic simulations, providing thus a direct connection between the microscopic and the mesoscopic scale. The coarse-grained model we consider is that of Ginzburg-Landau, valid around a second-order critical point. In particular, we use it to describe a Lennard-Jones fluid in the region close to the liquid-vapor critical point. The procedure is general and can be adapted to other coarse-grained models.