A unified methodological framework for the simulation of nonisothermal ensembles

Accelerated convergence via adiabatic sampling for adsorption and desorption processes

Under isothermal conditions, phase transitions occur through a nucleation event when conditions are sufficiently close to coexistence. The formation of a nucleus of the new phase requires the system to overcome a free energy barrier of formation, whose height rapidly rises as supersaturation decreases. This phenomenon occurs both in the bulk and under confinement and leads to a very slow kinetics for the transition, ultimately resulting in hysteresis, where the system can remain in a metastable state for a long time. This has broad implications, for instance, when using simulations to predict phase diagrams or screen porous materials for gas storage applications. Here, we leverage simulations in an adiabatic statistical ensemble, known as adiabatic grand-isochoric ensemble (μ, V, L) ensemble, to reach equilibrium states with a greater efficiency than its isothermal counterpart, i.e., simulations in the grand-canonical ensemble. For the bulk, we show that at low supersaturation, isothermal simulations converge slowly, while adiabatic simulations exhibit a fast convergence over a wide range of supersaturation. We then focus on adsorption and desorption processes in nanoporous materials, assess the reliability of (μ, V, L) simulations on the adsorption of argon in IRMOF-1, and demonstrate the efficiency of adiabatic simulations to predict efficiently the equilibrium loading during the adsorption and desorption of argon in MCM-41, a system that exhibits significant hysteresis. We provide quantitative measures of the increased rate of convergence when using adiabatic simulations. Adiabatic simulations explore a wide temperature range, leading to a more efficient exploration of the configuration space.

Classical statistical mechanics in the μVL and μpR ensembles

Molecular expressions for thermodynamic properties and derivatives of the entropy up to third order in the adiabatic grand-isochoric μVL and adiabatic grand-isobaric μpR ensembles are systematically derived using the methodology developed by Lustig for the microcanonical and canonical ensembles [J. Chem. Phys. 100, 3048 (1994)10.1063/1.466446; Mol. Phys. 110, 3041 (2012)10.1080/00268976.2012.695032]. They are expressed by phase-space functions, which represent derivatives of the entropy with respect to the chemical potential, the volume, and the Hill energy L in the μVL ensemble and with respect to the chemical potential, the pressure, and the Ray energy R in the μpR ensemble. The derived expressions are validated for both ensembles by Monte Carlo simulations for the simple Lennard-Jones model fluid at three selected state points.

Rigorous expressions for thermodynamic properties in the NpH ensemble

Molecular expressions for thermodynamic properties of fluids and derivatives of the entropy up to third order in the isoenthalpic-isobaric ensemble are derived by using the methodology developed by Lustig for the microcanonical and canonical ensembles [J. Chem. Phys. 100, 3048 (1994)JCPSA60021-960610.1063/1.466446; Mol. Phys. 110, 3041 (2012)MOPHAM0026-897610.1080/00268976.2012.695032]. They are expressed in a systematic way by phase-space functions, which represent derivatives of the phase-space volume with respect to enthalpy and pressure. The expressions for thermodynamic properties contain only ensemble averages of combinations of the kinetic energy and volume of the system. Thus, the calculation of thermodynamic properties in the isoenthalpic-isobaric ensemble does not require volume derivatives of the potential energy. This is particularly advantageous in Monte Carlo simulations when the interactions between molecules are described by very accurate ab initio pair and nonadditive three-body potentials. The derived expressions are validated by Monte Carlo simulations for the simple Lennard-Jones model fluid as a test case.

On the calculation of free energies over Hamiltonian and order parameters via perturbation and thermodynamic integration

In this work, complementary formulas are presented to compute free-energy differences via perturbation (FEP) methods and thermodynamic integration (TI). These formulas are derived by selecting only the most statistically significant data from the information extractable from the simulated points involved. On the one hand, commonly used FEP techniques based on overlap sampling leverage the full information contained in the overlapping macrostate probability distributions. On the other hand, conventional TI methods only use information on the first moments of those distributions, as embodied by the first derivatives of the free energy. Since the accuracy of simulation data degrades considerably for high-order moments (for FEP) or free-energy derivatives (for TI), it is proposed to consider, consistently for both methods, data up to second-order moments/derivatives. This provides a compromise between the limiting strategies embodied by common FEP and TI and leads to simple, optimized expressions to evaluate free-energy differences. The proposed formulas are validated with an analytically solvable harmonic Hamiltonian (for assessing systematic errors), an atomistic system (for computing the potential of mean force with coordinate-dependent order parameters), and a binary-component coarse-grained model (for tracing a solid-liquid phase diagram in an ensemble sampled through alchemical transformations). It is shown that the proposed FEP and TI formulas are straightforward to implement, perform similarly well, and allow robust estimation of free-energy differences even when the spacing of successive points does not guarantee them to have proper overlapping in phase space.

A comparison of methods for melting point calculation using molecular dynamics simulations

Accurate and efficient prediction of melting points for complex molecules is still a challenging task for molecular simulation, although many methods have been developed. Four melting point computational methods, including one free energy-based method (the pseudo-supercritical path (PSCP) method) and three direct methods (two interface-based methods and the voids method) were applied to argon and a widely studied ionic liquid 1-n-butyl-3-methylimidazolium chloride ([BMIM][Cl]). The performance of each method was compared systematically. All the methods under study reproduce the argon experimental melting point with reasonable accuracy. For [BMIM][Cl], the melting point was computed to be 320 K using a revised PSCP procedure, which agrees with the experimental value 337-339 K very well. However, large errors were observed in the computed results using the direct methods, suggesting that these methods are inappropriate for large molecules with sluggish dynamics. The strengths and weaknesses of each method are discussed.

From multidimensional replica-exchange method to multidimensional multicanonical algorithm and simulated tempering

We discuss multidimensional generalizations of multicanonical algorithm, simulated tempering, and replica-exchange method. We generalize the original potential-energy function E0 by adding any physical quantity V of interest as a new energy term with a coupling constant lambda. We then perform a multidimensional multicanonical simulation where a random walk in E0 and V spaces is realized. We can alternately perform a multidimensional simulated-tempering simulation where a random walk in temperature T and parameter lambda is realized. The results of the multidimensional replica-exchange simulations can be used to determine the weight factors for these multidimensional multicanonical and simulated-tempering simulations.

Optimized expanded ensembles for simulations involving molecular insertions and deletions. II. Open systems

In the Grand Canonical, osmotic, and Gibbs ensembles, chemical potential equilibrium is attained via transfers of molecules between the system and either a reservoir or another subsystem. In this work, the expanded ensemble (EXE) methods described in part I [F. A. Escobedo and F. J. Martinez-Veracoechea, J. Chem. Phys. 127, 174103 (2007)] of this series are extended to these ensembles to overcome the difficulties associated with implementing such whole-molecule transfers. In EXE, such moves occur via a target molecule that undergoes transitions through a number of intermediate coupling states. To minimize the tunneling time between the fully coupled and fully decoupled states, the intermediate states could be either: (i) sampled with an optimal frequency distribution (the sampling problem) or (ii) selected with an optimal spacing distribution (staging problem). The sampling issue is addressed by determining the biasing weights that would allow generating an optimal ensemble; discretized versions of this algorithm (well suited for small number of coupling stages) are also presented. The staging problem is addressed by selecting the intermediate stages in such a way that a flat histogram is the optimized ensemble. The validity of the advocated methods is demonstrated by their application to two model problems, the solvation of large hard spheres into a fluid of small and large spheres, and the vapor-liquid equilibrium of a chain system.

Simulation of the density of states in isothermal and adiabatic ensembles

This paper provides a unified treatment of the fundamental methods used to obtain the density of states via molecular simulations with isothermal ensembles (IEs) and adiabatic ensembles (AEs). Our analysis and results show that provides a natural bridge to go back and forth between IE and AE simulation data. They also underline the difference between the density of states of potential energy macrostates and that of total energy macrostates Omega, even though both provide access to the thermodynamic properties of the system. Visited-states approaches and transition matrix methods are described and applied to the Lennard-Jones fluid to target omega and Omega as functions of energy and volume macrostates. It is shown that one can obtain omega via a generalized acceptance-ratio formula that is applicable regardless of the conditions at which the ensemble is simulated. In this way, one can obtain while performing conventional IE or AE simulations, and do it at no extra cost and with a higher accuracy than is achievable with histogram methods.

On the use of transition matrix methods with extended ensembles

Different extended ensemble schemes for non-Boltzmann sampling (NBS) of a selected reaction coordinate lambda were formulated so that they employ (i) "variable" sampling window schemes (that include the "successive umbrella sampling" method) to comprehensibly explore the lambda domain and (ii) transition matrix methods to iteratively obtain the underlying free-energy eta landscape (or "importance" weights) associated with lambda. The connection between "acceptance ratio" and transition matrix methods was first established to form the basis of the approach for estimating eta(lambda). The validity and performance of the different NBS schemes were then assessed using as lambda coordinate the configurational energy of the Lennard-Jones fluid. For the cases studied, it was found that the convergence rate in the estimation of eta is little affected by the use of data from high-order transitions, while it is noticeably improved by the use of a broader window of sampling in the variable window methods. Finally, it is shown how an "elastic" window of sampling can be used to effectively enact (nonuniform) preferential sampling over the lambda domain, and how to stitch the weights from separate one-dimensional NBS runs to produce a eta surface over a two-dimensional domain.

Freezing line of the Lennard-Jones fluid: A phase switch Monte Carlo study

We report a phase switch Monte Carlo (PSMC) method study of the freezing line of the Lennard-Jones (LJ) fluid. Our work generalizes to soft potentials the original application of the method to hard-sphere freezing and builds on a previous PSMC study of the LJ system by Errington [J. Chem. Phys. 120, 3130 (2004)]. The latter work is extended by tracing a large section of the Lennard-Jones freezing curve, the results of which we compare with a previous Gibbs-Duhem integration study. Additionally, we provide new background material regarding the statistical-mechanical basis of the PSMC method and extensive implementation details.