Correct microcanonical ensemble in molecular dynamics

Computational and Machine Learning Methods for CO2 Capture Using Metal-Organic Frameworks

Machine learning (ML) using data sets of atomic and molecular force fields (FFs) has made significant progress and provided benefits in the fields of chemistry and material science. This work examines the interactions between chemistry and materials computational science at the atomic and molecular scales for metal-organic framework (MOF) adsorbent development toward carbon dioxide (CO2) capture. Herein, a connection will be drawn between atomic forces predicted by ML algorithms and the structures of MOFs for CO2 adsorption. Our study also takes into account the successes of atomic computational screening in the field of materials science, especially quantum ML, and its relationship to ML algorithms that clarify advancements in the area of CO2 adsorption by MOFs. Additionally, we reviewed the processes for supplying data to ML algorithms for algorithm training, including text mining from scientific articles, and MOF's formula processing linked to the chemical properties of MOFs. To create ML algorithms for future research, we recommend that the digitization of scientific records can help efficiently synthesize advanced MOFs. Finally, a future vision for developing pioneer MOF synthesis routes for CO2 capture is presented in this review article.

Born-Oppenheimer Molecular Dynamics with a Linear Combination of Atomic Orbitals and Hybrid Functionals for Condensed Matter Simulations Made Possible. Theory and Performance for the Microcanonical and Canonical Ensembles

The implementation of an original Born-Oppenheimer molecular dynamics module is presented, which is able to perform simulations of large and complex condensed phase systems for sufficiently long time scales at the level of density functional theory with hybrid functionals, in the microcanonical (NVE) and canonical (NVT) ensembles. The algorithm is fully integrated in the Crystal code, a program for quantum mechanical simulations of materials, whose peculiarity stems from the use of atom-centered basis functions within a linear combination of atomic orbitals to describe the wave function. The corresponding efficiency in the evaluation of the exact Fock exchange series has led to the implementation of a rich variety of hybrid density functionals at a low computational cost. In addition, the molecular dynamics implementation benefits also from the effective MPI parallelization of the code, suited to exploit high-performance computing resources available on current generation supercomputer architectures. Furthermore, the information contained in the trajectory of the dynamics is extracted through a series of postprocessing algorithms that provide the radial distribution function, the diffusion coefficient and the vibrational density of states. In this work, we present a detailed description of the theoretical framework and the algorithmic implementation, followed by a critical evaluation of the accuracy and parallel performance (e.g., strong and weak scaling) of this approach, when ice and liquid water simulations are performed in the microcanonical and canonical ensembles.

Revisiting the pseudo-supercritical path method: An improved formulation for the alchemical calculation of solid-liquid coexistence

Alchemical free energy calculations via molecular dynamics have been applied to obtain thermodynamic properties related to solid-liquid equilibrium conditions, such as melting points. In recent years, the pseudo-supercritical path (PSCP) method has proved to be an important approach to melting point prediction due to its flexibility and applicability. In the present work, we propose improvements to the PSCP alchemical cycle to make it more compact and efficient through a concerted evaluation of different potential energies. The multistate Bennett acceptance ratio (MBAR) estimator was applied at all stages of the new cycle to provide greater accuracy and uniformity, which is essential concerning uncertainty calculations. In particular, for the multistate expansion stage from solid to liquid, we employed the MBAR estimator with a reduced energy function that allows affine transformations of coordinates. Free energy and mean derivative profiles were calculated at different cycle stages for argon, triazole, propenal, and the ionic liquid 1-ethyl-3-methyl-imidazolium hexafluorophosphate. Comparisons showed a better performance of the proposed method than the original PSCP cycle for systems with higher complexity, especially the ionic liquid. A detailed study of the expansion stage revealed that remapping the centers of mass of the molecules or ions is preferable to remapping the coordinates of each atom, yielding better overlap between adjacent states and improving the accuracy of the methodology.

Effects of structural inhomogeneity on equilibration processes in Langevin dynamics

In recent decades, computer experiments have led to an accurate and fundamental understanding of atomic and molecular mechanisms in fluids, such as different kinds of relaxation processes toward steady physical states. In this paper, we investigate how exactly the configuration of initial states in a molecular-dynamics simulation can affect the rates of decay toward equilibrium for the widely known Langevin canonical ensemble. For this purpose, we derive an original expression relating the system relaxation time τ_{sys} and the radial distribution function g(r) in the near-zero and high-density limit. We found that, for an initial state which is slightly marginally inhomogeneous in the number density of atoms, the system relaxation time τ_{sys} is much longer than that for the homogeneous case and an increasing function of the Langevin coupling constant, γ. We also found, during structural equilibration, g(r) at large distances approaches 1 from above for the inhomogeneous case and from below for the macroscopically homogeneous one.

Entropy determination for mixtures in the adiabatic grand-isobaric ensemble

The entropy change that occurs upon mixing two fluids has remained an intriguing topic since the dawn of statistical mechanics. In this work, we generalize the grand-isobaric ensemble to mixtures and develop a Monte Carlo algorithm for the rapid determination of entropy in these systems. A key advantage of adiabatic ensembles is the direct connection they provide with entropy. Here, we show how the entropy of a binary mixture A–B can be readily obtained in the adiabatic grand-isobaric ( μA, μB, P, R) ensemble, in which μA and μB denote the chemical potential of components A and B, respectively, P is the pressure, and R is the heat (Ray) function, that corresponds to the total energy of the system. This, in turn, allows for the evaluation of the entropy of mixing and the Gibbs free energy of mixing. We also demonstrate that our approach performs very well both on systems modeled with simple potentials and with complex many-body force fields. Finally, this approach provides a direct route to the determination of the thermodynamic properties of mixing and allows for the efficient detection of departures from ideal behavior in mixtures.

Systematic formulation of thermodynamic properties in the NpT ensemble

Molecular expressions for thermodynamic properties and derivatives of the Gibbs energy up to third order in the isobaric-isothermal (NpT) ensemble 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 Gibbs energy with respect to temperature and pressure. Additionally, expressions for the phase-space functions for temperature-dependent potentials are provided, which, for example, are required when quantum corrections, e.g., Feynman-Hibbs corrections, are applied in classical simulations. The derived expressions are validated by Monte Carlo simulations for the simple Lennard-Jones model fluid at three selected state points. A unique result is that the phase-space functions contain only ensemble averages of combinations of powers of enthalpy and volume. Thus, the calculation of thermodynamic properties in the NpT 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 empirical force fields or very accurate ab initio pair and nonadditive three-body potentials.

Velocity and energy distributions in microcanonical ensembles of hard spheres

In a microcanonical ensemble (constant NVE, hard reflecting walls) and in a molecular dynamics ensemble (constant NVEPG, periodic boundary conditions) with a number N of smooth elastic hard spheres in a d-dimensional volume V having a total energy E, a total momentum P, and an overall center of mass position G, the individual velocity components, velocity moduli, and energies have transformed beta distributions with different arguments and shape parameters depending on d, N, E, the boundary conditions, and possible symmetries in the initial conditions. This can be shown marginalizing the joint distribution of individual energies, which is a symmetric Dirichlet distribution. In the thermodynamic limit the beta distributions converge to gamma distributions with different arguments and shape or scale parameters, corresponding respectively to the Gaussian, i.e., Maxwell-Boltzmann, Maxwell, and Boltzmann or Boltzmann-Gibbs distribution. These analytical results agree with molecular dynamics and Monte Carlo simulations with different numbers of hard disks or spheres and hard reflecting walls or periodic boundary conditions. The agreement is perfect with our Monte Carlo algorithm, which acts only on velocities independently of positions with the collision versor sampled uniformly on a unit half sphere in d dimensions, while slight deviations appear with our molecular dynamics simulations for the smallest values of N.

Sigma method for the microcanonical entropy or density of states

We introduce a simple improvement on the method to calculate equilibrium entropy differences between classical energy levels proposed by Davis [S. Davis, Phys. Rev. E 84, 050101 (2011)]. We demonstrate that the modification is superior to the original whenever the energy levels are sufficiently closely spaced or whenever the microcanonical averaging needed in the method is carried out by importance sampling Monte Carlo. We also point out the necessary adjustments if Davis's method (improved or not) is to be used with molecular dynamics simulations.

Perturbation method to calculate the density of states

Monte Carlo switching moves ("perturbations") are defined between two or more classical Hamiltonians sharing a common ground-state energy. The ratio of the density of states (DOS) of one system to that of another is related to the ensemble averages of the microcanonical acceptance probabilities of switching between these Hamiltonians, analogously to the case of Bennett's acceptance ratio method for the canonical ensemble [C. H. Bennett, J. Comput. Phys. 22, 245 (1976)]. Thus, if the DOS of one of the systems is known, one obtains those of the others and, hence, the partition functions. As a simple test case, the vapor pressure of an anharmonic Einstein crystal is computed, using the harmonic Einstein crystal as the reference system in one dimension; an auxiliary calculation is also performed in three dimensions. As a further example of the algorithm, the energy dependence of the ratio of the DOS of the square-well and hard-sphere tetradecamers is determined, from which the temperature dependence of the constant-volume heat capacity of the square-well system is calculated and compared with canonical Metropolis Monte Carlo estimates. For these cases and reference systems, the perturbation calculations exhibit a higher degree of convergence per Monte Carlo cycle than Wang-Landau (WL) sampling, although for the one-dimensional oscillator the WL sampling is ultimately more efficient for long runs. Last, we calculate the vapor pressure of liquid gold using an empirical Sutton-Chen many-body potential and the ideal gas as the reference state. Although this proves the general applicability of the method, by its inherent perturbation approach the algorithm is suitable for those particular cases where the properties of a related system are well known.

Periodic boundary condition induced breakdown of the equipartition principle and other kinetic effects of finite sample size in classical hard-sphere molecular dynamics simulation

We examine consequences of the non-Boltzmann nature of probability distributions for one-particle kinetic energy, momentum, and velocity for finite systems of classical hard spheres with constant total energy and nonidentical masses. By comparing two cases, reflecting walls (NVE or microcanonical ensemble) and periodic boundaries (NVEPG or molecular dynamics ensemble), we describe three consequences of the center-of-mass constraint in periodic boundary conditions: the equipartition theorem no longer holds for unequal masses, the ratio of the average relative velocity to the average velocity is increased by a factor of [N/(N-1)]1/2, and the ratio of average collision energy to average kinetic energy is increased by a factor of N/(N-1). Simulations in one, two, and three dimensions confirm the analytic results for arbitrary dimension.

Pressure derivatives in the classical molecular-dynamics ensemble

The calculation of thermodynamic state variables, particularly derivatives of the pressure with respect to density and temperature, in conventional molecular-dynamics simulations is considered in the frame of the comprehensive treatment of the molecular-dynamics ensemble by Lustig [J. Chem. Phys. 100, 3048 (1994)]. This paper improves the work of Lustig in two aspects. In the first place, a general expression for the basic phase-space functions in the molecular-dynamics ensemble is derived, which takes into account that a mechanical quantity G is, in addition to the number of particles, the volume, the energy, and the total momentum of the system, a constant of motion. G is related to the initial position of the center of mass of the system. Secondly, the correct general expression for volume derivatives of the potential energy is derived. This latter result solves a problem reported by Lustig [J. Chem. Phys. 109, 8816 (1998)] and Meier [Computer Simulation and Interpretation of the Transport Coefficients of the Lennard-Jones Model Fluid (Shaker, Aachen, 2002)] and enables the correct calculation of the isentropic and isothermal compressibilities, the speed of sound, and, in principle, all higher pressure derivatives. The derived equations are verified by calculations of several state variables and pressure derivatives up to second order by molecular-dynamics simulations with 256 particles at two state points of the Lennard-Jones fluid in the gas and liquid regions. It is also found that it is impossible for systems of this size to calculate third- and higher-order pressure derivatives due to the limited accuracy of the algorithm employed to integrate the equations of motion.

Transport coefficients of the Lennard-Jones model fluid. II Self-diffusion

In an extensive computer simulation study, the transport coefficients of the Lennard-Jones model fluid were determined with high accuracy from equilibrium molecular-dynamics simulations. In the frame of time-correlation function theory, the generalized Einstein relations were employed to evaluate the transport coefficients. This second of a series of four papers presents the results for the self-diffusion coefficient, and discusses and interprets the behavior of this transport coefficient in the fluid region of the phase diagram. The uncertainty of the self-diffusion data is estimated to be 1% in the gas region and 0.5% at high-density liquid states. With the very accurate data, even fine details in the shape of the self-diffusion isotherms are resolved, and the previously little-investigated behavior of the self-diffusion coefficient at low-density gaseous states is analyzed in detail. Finally, aspects of the mass transport mechanisms on the molecular scale are explored by an analysis of the velocity autocorrelation functions.

Transport coefficients of the Lennard-Jones model fluid. I. Viscosity

In an extensive computer simulation study, the transport coefficients of the Lennard-Jones model fluid were determined with high accuracy from equilibrium molecular-dynamics simulations. In the frame of time-correlation function theory, the generalized Einstein relations were employed to evaluate the transport coefficients. This first of a series of four papers presents the results for the viscosity, and discusses and interprets the behavior of this transport coefficient in the fluid region of the phase diagram. Moreover, the kinetic-kinetic, kinetic-potential, and potential-potential viscosity contributions are resolved over the whole range of fluid states, and their characteristic dependence on temperature and density is described. Finally, an additional analysis of the shear-stress correlation functions reveals aspects of the momentum-transport mechanisms on the molecular scale.

Adiabatic invariance with first integrals of motion

The construction of a microthermodynamic formalism for isolated systems based on the concept of adiabatic invariance is an old but seldom appreciated effort in the literature, dating back at least to P. Hertz [Ann. Phys. (Leipzig) 33, 225 (1910)]. An apparently independent extension of such formalism for systems bearing additional first integrals of motion was recently proposed by Hans H. Rugh [Phys. Rev. E 64, 055101 (2001)], establishing the concept of adiabatic invariance even in such singular cases. After some remarks in connection with the formalism pioneered by Hertz, it will be suggested that such an extension can incidentally explain the success of a dynamical method for computing the entropy of classical interacting fluids, at least in some potential applications where the presence of additional first integrals cannot be ignored.

Molecular dynamics ensemble, equation of state, and ergodicity

The variant of the NVE ensemble known as the molecular dynamics ensemble was recently redefined by Ray and Zhang [Phys. Rev. E 59, 4781 (1999)] to include the specification of a time invariant G (a function of phase and, explicitly, the time) in addition to the total linear momentum M. We reformulate this ensemble slightly as the NVEMR ensemble, in which R/N is the center-of-mass position, and consider the equation of state of the hard-sphere system in this ensemble through both the virial function and the Boltzmann entropy. We test the quasiergodic hypothesis by a comparison of old molecular dynamics and Monte Carlo results for the compressibility factor of the 12-particle, hard-disk systems. The virial approach, which had previously been found to support the hypothesis in the NVEM ensemble, remains unchanged in the NVEMR ensemble. The entropy S approach depends on whether S is defined through the phase integral over the energy sphere or the energy shell, the parameter straight theta being 0 or 1, respectively. The ergodic hypothesis is found to be supported for straight theta=0 but not for straight theta=1.

Theory of stress fluctuations

The current status of the theory of stress fluctuations is marked by two circumstances: no currently available formulas are valid for a metallic system, and a series of contradictory formulas remains unresolved. Here we derive formulas for shear- and isotropic-stress energy fluctuations, in the primary statistical mechanics ensembles. These formulas are valid for a classical monatomic system representing a metal or nonmetal, in cubic crystal, amorphous solid, or liquid phases. Current contradictions in fluctuation formulas are resolved through the following observations. First, the expansion of a dynamical variable A in terms of the fluctuations explicit in a given ensemble distribution, for example deltaA=adeltaN+bdeltaH in the grand canonical ensemble, is correct if and only if deltaA is a function only of deltaN and deltaH. The common use of this expansion has produced incorrect fluctuation formulas. Second, the thermodynamic fluctuations of Landau and Lifshitz do not correspond to statistical mechanics fluctuations, and the two types of fluctuations have essentially different values.

Dynamical temperature for spin systems

A transcription of Rugh's geometrical approach to temperature is given for classical Heisenberg spin systems. For the simple case of a paramagnet with small and large numbers of spins we verify the approach. A numerical check for long spin chains using spin dynamics shows its practicality.