Score Dynamics: Scaling Molecular Dynamics with Picoseconds Time Steps via Conditional Diffusion Model.

We propose score dynamics (SD), a general framework for learning accelerated evolution operators with large timesteps from molecular dynamics (MD) simulations. SD is centered around scores or derivatives of the transition log-probability with respect to the dynamical degrees of freedom. The latter play the same role as force fields in MD but are used in denoising diffusion probability models to generate discrete transitions of the dynamical variables in an SD time step, which can be orders of magnitude larger than a typical MD time step. In this work, we construct graph neural network-based SD models of realistic molecular systems that are evolved with 10 ps timesteps. We demonstrate the efficacy of SD with case studies of the alanine dipeptide and short alkanes in aqueous solution. Both equilibrium predictions derived from the stationary distributions of the conditional probability and kinetic predictions for the transition rates and transition paths are in good agreement with MD. Our current SD implementation is about 2 orders of magnitude faster than the MD counterpart for the systems studied in this work. Open challenges and possible future remedies to improve SD are also discussed.

Spectral-partitioned Kohn-Sham density functional theory.

We introduce a general, variational scheme for systematic approximation of a given Kohn-Sham free-energy functional by partitioning the density matrix into distinct spectral domains, each of which may be spanned by an independent diagonal representation without requirement of mutual orthogonality. It is shown that by generalizing the entropic contribution to the free energy to allow for independent representations in each spectral domain, the free energy becomes an upper bound to the exact (unpartitioned) Kohn-Sham free energy, attaining this limit as the representations approach Kohn-Sham eigenfunctions. A numerical procedure is devised for calculation of the generalized entropy associated with spectral partitioning of the density matrix. The result is a powerful framework for Kohn-Sham calculations of systems whose occupied subspaces span multiple energy regimes. As a case in point, we apply the proposed framework to warm- and hot-dense matter described by finite-temperature density functional theory, where at high energies the density matrix is represented by that of the free-electron gas, while at low energies it is variationally optimized. We derive expressions for the spectral-partitioned Kohn-Sham Hamiltonian, atomic forces, and macroscopic stresses within the projector-augmented wave (PAW) and the norm-conserving pseudopotential methods. It is demonstrated that at high temperatures, spectral partitioning facilitates accurate calculations at dramatically reduced computational cost. Moreover, as temperature is increased, fewer exact Kohn-Sham states are required for a given accuracy, leading to further reductions in computational cost. Finally, it is shown that standard multiprojector expansions of electronic orbitals within atomic spheres in the PAW method lack sufficient completeness at high temperatures. Spectral partitioning provides a systematic solution for this fundamental problem.

Properties of carbon up to 10 million kelvin from Kohn-Sham density functional theory molecular dynamics.

Accurately modeling dense plasmas over wide-ranging conditions of pressure and temperature is a grand challenge critically important to our understanding of stellar and planetary physics as well as inertial confinement fusion. In this work, we employ Kohn-Sham density functional theory (DFT) molecular dynamics (MD) to compute the properties of carbon at warm and hot dense matter conditions in the vicinity of the principal Hugoniot. In particular, we calculate the equation of state (EOS), Hugoniot, pair distribution functions, and diffusion coefficients for carbon at densities spanning 8 g/cm^{3} to 16 g/cm^{3} and temperatures ranging from 100 kK to 10 MK using the Spectral Quadrature method. We find that the computed EOS and Hugoniot are in good agreement with path integral Monte Carlo results and the sesame database. Additionally, we calculate the ion-ion structure factor and viscosity for selected points. All results presented are at the level of full Kohn-Sham DFT-MD, free of empirical parameters, average-atom, and orbital-free approximations employed previously at such conditions.

Accurate parameterization of the kinetic energy functional for calculations using exact-exchange.

Electronic structure calculations based on Kohn-Sham density functional theory (KSDFT) that incorporate exact-exchange or hybrid functionals are associated with a large computational expense, a consequence of the inherent cubic scaling bottleneck and large associated prefactor, which limits the length and time scales that can be accessed. Although orbital-free density functional theory (OFDFT) calculations scale linearly with system size and are associated with a significantly smaller prefactor, they are limited by the absence of accurate density-dependent kinetic energy functionals. Therefore, the development of accurate density-dependent kinetic energy functionals is important for OFDFT calculations of large realistic systems. To this end, we propose a method to train kinetic energy functional models at the exact-exchange level of theory by using a dictionary of physically relevant terms that have been proposed in the literature in conjunction with linear or nonlinear regression methods to obtain the fitting coefficients. For our dictionary, we use a gradient expansion of the kinetic energy nonlocal models proposed in the literature and their nonlinear combinations, such as a model that incorporates spatial correlations between higher order derivatives of electron density at two points. The predictive capabilities of these models are assessed by using a variety of model one-dimensional (1D) systems that exhibit diverse bonding characteristics, such as a chain of eight hydrogens, LiF, LiH, C4H2, C4N2, and C3O2. We show that by using the data from model 1D KSDFT calculations performed using the exact-exchange functional for only a few neutral structures, it is possible to generate models with high accuracy for charged systems and electron and kinetic energy densities during self-consistent field iterations. In addition, we show that it is possible to learn both the orbital dependent terms, i.e., the kinetic energy and the exact-exchange energy, and models that incorporate additional nonlinearities in spatial correlations, such as a quadratic model, are needed to capture subtle features of the kinetic energy density that are present in exact-exchange-based KSDFT calculations.

Accurate parameterization of the kinetic energy functional.

The absence of a reliable formulation of the kinetic energy density functional has hindered the development of orbital free density functional theory. Using the data-aided learning paradigm, we propose a simple prescription to accurately model the kinetic energy density of any system. Our method relies on a dictionary of functional forms for local and nonlocal contributions, which have been proposed in the literature, and the appropriate coefficients are calculated via a linear regression framework. To model the nonlocal contributions, we explore two new nonlocal functionals-a functional that captures fluctuations in electronic density and a functional that incorporates gradient information. Since the analytical functional forms of the kernels present in these nonlocal terms are not known from theory, we propose a basis function expansion to model these seemingly difficult nonlocal quantities. This allows us to easily reconstruct kernels for any system using only a few structures. The proposed method is able to learn kinetic energy densities and total kinetic energies of molecular and periodic systems, such as H2, LiH, LiF, and a one-dimensional chain of eight hydrogens using data from Kohn-Sham density functional theory calculations for only a few structures.

Semi-Automated Creation of Density Functional Tight Binding Models through Leveraging Chebyshev Polynomial-Based Force Fields.

Density functional tight binding (DFTB) is an attractive method for accelerated quantum simulations of condensed matter due to its enhanced computational efficiency over standard density functional theory (DFT) approaches. However, DFTB models can be challenging to determine for individual systems of interest, especially for metallic and interfacial systems where different bonding arrangements can lead to significant changes in electronic states. In this regard, we have created a rapid-screening approach for determining systematically improvable DFTB interaction potentials that can yield transferable models for a variety of conditions. Our method leverages a recent reactive molecular dynamics force field where many-body interactions are represented by linear combinations of Chebyshev polynomials. This allows for the efficient creation of multi-center representations with relative ease, requiring only a small investment in initial DFT calculations. We have focused our workflow on TiH2 as a model system and show that a relatively small training set based on unit-cell-sized calculations yields a model accurate for both bulk and surface properties. Our approach is easy to implement and can yield reliable DFTB models over a broad range of thermodynamic conditions, where physical and chemical properties can be difficult to interrogate directly and there is historically a significant reliance on theoretical approaches for interpretation and validation of experimental results.

Self-supervised learning and prediction of microstructure evolution with convolutional recurrent neural networks.

Microstructural evolution is a key aspect of understanding and exploiting the processing-structure-property relationship of materials. Modeling microstructure evolution usually relies on coarse-grained simulations with evolution principles described by partial differential equations (PDEs). Here we demonstrate that convolutional recurrent neural networks can learn the underlying physical rules and replace PDE-based simulations in the prediction of microstructure phenomena. Neural nets are trained by self-supervised learning with image sequences from simulations of several common processes, including plane-wave propagation, grain growth, spinodal decomposition, and dendritic crystal growth. The trained networks can accurately predict both short-term local dynamics and long-term statistical properties of microstructures assessed herein and are capable of extrapolating beyond the training datasets in spatiotemporal domains and configurational and parametric spaces. Such a data-driven approach offers significant advantages over PDE-based simulations in time-stepping efficiency and offers a useful alternative, especially when the material parameters or governing PDEs are not well determined.

Metastable-solid phase diagrams derived from polymorphic solidification kinetics.

Nonequilibrium processes during solidification can lead to kinetic stabilization of metastable crystal phases. A general framework for predicting the solidification conditions that lead to metastable-phase growth is developed and applied to a model face-centered cubic (fcc) metal that undergoes phase transitions to the body-centered cubic (bcc) as well as the hexagonal close-packed phases at high temperatures and pressures. Large-scale molecular dynamics simulations of ultrarapid freezing show that bcc nucleates and grows well outside of the region of its thermodynamic stability. An extensive study of crystal-liquid equilibria confirms that at any given pressure, there is a multitude of metastable solid phases that can coexist with the liquid phase. We define for every crystal phase, a solid cluster in liquid (SCL) basin, which contains all solid clusters of that phase coexisting with the liquid. A rigorous methodology is developed that allows for practical calculations of nucleation rates into arbitrary SCL basins from the undercooled melt. It is demonstrated that at large undercoolings, phase selections made during the nucleation stage can be undone by kinetic instabilities amid the growth stage. On these bases, a solidification-kinetic phase diagram is drawn for the model fcc system that delimits the conditions for macroscopic grains of metastable bcc phase to grow from the melt. We conclude with a study of unconventional interfacial kinetics at special interfaces, which can bring about heterogeneous multiphase crystal growth. A first-order interfacial phase transformation accompanied by a growth-mode transition is examined.

Space- and time-resolved small angle X-ray scattering to probe assembly of silver nanocrystal superlattices.

The structure of nanocrystal superlattices has been extensively studied and well documented, however, their assembly process is poorly understood. In this work, we demonstrate an in situ space- and time-resolved small angle X-ray scattering measurement that we use to probe the assembly of silver nanocrystal superlattices driven by electric fields. The electric field creates a nanocrystal flux to the surface, providing a systematic means to vary the nanocrystal concentration near the electrode and thereby to initiate nucleation and growth of superlattices in several minutes. Using this approach, we measure the space- and time-resolved concentration and polydispersity gradients during deposition and show how they affect the superlattice constant and degree of order. We find that the field induces a size-selection effect that can reduce the polydispersity near the substrate by 21% leading to better quality crystals and resulting in field strength-dependent superlattice lattice constants.

Nanosecond Freezing of Water at High Pressures: Nucleation and Growth near the Metastability Limit.

The fundamental study of phase transition kinetics has motivated experimental methods toward achieving the largest degree of undercooling possible, more recently culminating in the technique of rapid, quasi-isentropic compression. This approach has been demonstrated to freeze water into the high-pressure ice VII phase on nanosecond timescales, with some experiments undergoing heterogeneous nucleation while others, in apparent contradiction, suggest a homogeneous nucleation mode. In this study, we show through a combination of theory, simulation, and analysis of experiments that these seemingly contradictory results are in agreement when viewed from the perspective of classical nucleation theory. We find that, perhaps surprisingly, classical nucleation theory is capable of accurately predicting the solidification kinetics of ice VII formation under an extremely high driving force (|Δµ/k_{B}T|≈1) but only if amended by two important considerations: (i) transient nucleation and (ii) separate liquid and solid temperatures. This is the first demonstration of a model that is able to reproduce the experimentally observed rapid freezing kinetics.

Magnetostructural Transition Kinetics in Shocked Iron.

A generalized Heisenberg model is implemented to study the effect of thermal magnetic disorder on kinetics of the Fe α-ε transition. The barrier to bulk martensitic displacement remains large in α-Fe shocked well past the phase line but is much reduced in the [001] α-ε boundary. The first result is consistent with observed overdriving to metastable α, while the second suggests structural instability, as implied by observation of a [001] shock transformation front without plastic relaxation. Reconciling both behaviors may require concurrent treatment of magnetic and structural order.

Low-temperature criticality of martensitic transformations of Cu nanoprecipitates in α-Fe.

Nanoprecipitates form during nucleation of multiphase equilibria in phase segregating multicomponent systems. In spite of their ubiquity, their size-dependent physical chemistry, in particular, at the boundary between phases with incompatible topologies, is still rather arcane. Here, we use extensive atomistic simulations to map out the size-temperature phase diagram of Cu nanoprecipitates in α-Fe. The growing precipitates undergo martensitic transformations from the body-centered cubic (bcc) phase to multiply twinned 9R structures. At high temperatures, the transitions exhibit strong first-order character and prominent hysteresis. Upon cooling, the discontinuities become less pronounced and the transitions occur at ever smaller cluster sizes. Below 300 K, the hysteresis vanishes while the transition remains discontinuous with a finite but diminishing latent heat. This unusual size-temperature phase diagram results from the entropy generated by the soft modes of the bcc-Cu phase, which are stabilized through confinement by the α-Fe lattice.

First-passage kinetic Monte Carlo method.

We present an efficient method for Monte Carlo simulations of diffusion-reaction processes. Introduced by us in a previous paper [Phys. Rev. Lett. 97, 230602 (2006)], our algorithm skips the traditional small diffusion hops and propagates the diffusing particles over long distances through a sequence of superhops, one particle at a time. By partitioning the simulation space into nonoverlapping protecting domains each containing only one or two particles, the algorithm factorizes the N -body problem of collisions among multiple Brownian particles into a set of much simpler single-body and two-body problems. Efficient propagation of particles inside their protective domains is enabled through the use of time-dependent Green's functions (propagators) obtained as solutions for the first-passage statistics of random walks. The resulting Monte Carlo algorithm is event-driven and asynchronous; each Brownian particle propagates inside its own protective domain and on its own time clock. The algorithm reproduces the statistics of the underlying Monte Carlo model exactly. Extensive numerical examples demonstrate that for an important class of diffusion-reaction models the algorithm is efficient at low particle densities, where other existing algorithms slow down severely.

Thermodynamic ground states of platinum metal nitrides.

The thermodynamic stabilities of various phases of the nitrides of the platinum-metal elements are systematically studied using density functional theory. It is shown that for the nitrides of Rh, Pd, Ir, and Pt two new crystal structures, in which the metal ions occupy simple tetragonal lattice sites, have lower formation enthalpies at ambient conditions than any previously proposed structures. The region of stability with respect to those structures extends to 17 GPa for PtN2. Calculations show that the PtN2 simple tetragonal structures at this pressure are thermodynamically stable also with respect to phase separation. The fact that the local density and generalized gradient approximations predict different values of the absolute formation enthalpies as well different relative stabilities between simple tetragonal and the pyrite or marcasite structures are further discussed.

Synthesis and characterization of the nitrides of platinum and iridium.

Transition metal nitrides are of great technological and fundamental importance because of their strength and durability and because of their useful optical, electronic, and magnetic properties. We have evaluated a recently synthesized platinum nitride (PtN) that was shown to have a large bulk modulus, and we propose a structure that is isostructural with pyrite and has the stoichiometry PtN2. We have also synthesized a recoverable nitride of iridium under nearly the same conditions of pressure and temperature as PtN2. Although it has the same stoichiometry, it exhibits much lower structural symmetry. Preliminary results suggest that the bulk modulus of this material is also very large.

First-passage Monte Carlo algorithm: diffusion without all the hops.

We present a novel Monte Carlo algorithm for N diffusing finite particles that react on collisions. Using the theory of first-passage processes and time dependent Green's functions, we break the difficult N-body problem into independent single- and two-body propagations circumventing numerous diffusion hops used in standard Monte Carlo simulations. The new algorithm is exact, extremely efficient, and applicable to many important physical situations in arbitrary integer dimensions.

Adaptive importance sampling Monte Carlo simulation of rare transition events.

We develop a general theoretical framework for the recently proposed importance sampling method for enhancing the efficiency of rare-event simulations [W. Cai, M. H. Kalos, M. de Koning, and V. V. Bulatov, Phys. Rev. E 66, 046703 (2002)], and discuss practical aspects of its application. We define the success/fail ensemble of all possible successful and failed transition paths of any duration and demonstrate that in this formulation the rare-event problem can be interpreted as a "hit-or-miss" Monte Carlo quadrature calculation of a path integral. The fact that the integrand contributes significantly only for a very tiny fraction of all possible paths then naturally leads to a "standard" importance sampling approach to Monte Carlo (MC) quadrature and the existence of an optimal importance function. In addition to showing that the approach is general and expected to be applicable beyond the realm of Markovian path simulations, for which the method was originally proposed, the formulation reveals a conceptual analogy with the variational MC (VMC) method. The search for the optimal importance function in the former is analogous to finding the ground-state wave function in the latter. In two model problems we discuss practical aspects of finding a suitable approximation for the optimal importance function. For this purpose we follow the strategy that is typically adopted in VMC calculations: the selection of a trial functional form for the optimal importance function, followed by the optimization of its adjustable parameters. The latter is accomplished by means of an adaptive optimization procedure based on a combination of steepest-descent and genetic algorithms.

Density-functional calculations of alpha, beta, gamma, delta, delta', and epsilon plutonium.

Total energies for the six known polymorphs of plutonium metal have been calculated within spin and orbital polarized density-functional theory as a function of lattice constant. Theoretical equilibrium volumes and bulk moduli correspond well with experimental data and the calculated total energies are consistent with the known phase diagram of Pu. It is shown that a preference for the formation of magnetic moments, increasing through the alpha-->beta-->gamma phases, explains their position in the ambient pressure phase diagram and their anomalous variation of atomic density. A simple model is presented that establishes a relationship between atomic density, crystal symmetry, and magnetic moments which is universally valid for all Pu phases.

Transformations in the medium-range order of fused silica under high pressure.

Molecular dynamics simulations of fused silica at shock pressures reproduce the experimental equation of state of this material and explain its characteristic shape. We demonstrate that shock waves modify the medium-range order of this amorphous system, producing changes that are only clearly revealed by its ring size distribution. The ring size distribution remains practically unchanged during elastic compression but varies continuously after the transition to the plastic regime.