Path integral molecular dynamics methods: Application to neon

Feynman's path integral formulation of quantum statistical mechanics, which has commonly been applied be Monte Carlo methods, is now also implemented by traditional molecular dynamics simulations of the microcanonical ensemble and in the Nosé-Hoover method simulating the isothermal-isobaric ensemble. In this article these two methods are applied to solid and liquid neon, in which quantum effects are not negligible. The validity of the procedure is shown by comparison with Monte Carlo and Brownian Dynamics computer simulations and with experiment. © 1995 by John Wiley & Sons, Inc.     

Quantum monte carlo methods for constrained systems

The torsional ground state for ethane, the torsional, rotational, and mixed torsional and rotational ground state of propane are computed with a version of diffusion Monte Carlo adapted to handle the geometric complexity of curved spaces such as the Ramachandra space. The quantum NVT ensemble average for the mixed torsional and rotational degrees of freedom of propane is computed, using a version of Monte Carlo path integral, also adapted to handle curved spaces. These three problems are selected to demonstrate the generality and the applicability of the approaches described. The spaces of coordinates can be best constructed from the parameters of continuous Lie groups, and alternative methods based on vector spaces, where extended Lagrangian terms would be too cumbersome to implement. We note that the geometric coupling between the torsions and the rotations of propane produces a substantial effect on the ground state energy of propane, and that the quantum effects on the energy of propane are quite large even well above room temperature. © 2014 Wiley Periodicals, Inc.     

Hybrid quantum/classical path integral approach for simulation of hydrogen transfer reactions in enzymes

A hybrid quantum/classical path integral Monte Carlo (QC-PIMC) method for calculating the quantum free energy barrier for hydrogen transfer reactions in condensed phases is presented. In this approach, the classical potential of mean force along a collective reaction coordinate is calculated using umbrella sampling techniques in conjunction with molecular dynamics trajectories propagated according to a mapping potential. The quantum contribution is determined for each configuration along the classical trajectory with path integral Monte Carlo calculations in which the beads move according to an effective mapping potential. This type of path integral calculation does not utilize the centroid constraint and can lead to more efficient sampling of the relevant region of conformational space than free-particle path integral sampling. The QC-PIMC method is computationally practical for large systems because the path integral sampling for the quantum nuclei is performed separately from the classical molecular dynamics sampling of the entire system. The utility of the QC-PIMC method is illustrated by an application to hydride transfer in the enzyme dihydrofolate reductase. A comparison of this method to the quantized classical path and grid-based methods for this system is presented.     

Iterative Monte Carlo formulation of real-time correlation functions

We present an iterative Monte Carlo path integral methodology for evaluating thermally averaged real-time correlation functions. Standard path integral Monte Carlo methods are used to sample paths along the imaginary time contour. Propagation of the density matrix is performed iteratively on a grid composed of the end points of the sampled paths. Minimally oscillatory propagators are constructed using energy filtering techniques. A single propagation yields the values of the correlation function at all intermediate time points. Model calculations suggest that the method yields accurate results over several oscillation periods and the statistical error grows slowly with increasing propagation time.     

Gibbs free-energy estimates from direct path-sampling computations

We have implemented a path-sampling scheme enabling a direct estimation of Gibbs free energy. This scheme consists of a Monte Carlo sampling of constant-pressure Langevin paths, followed by an ensemble averaging carried out over the Markov chain of paths. In practice, we sample an umbrella path ensemble, which requires to rigorously define a statistical weight for the paths, equivalent of the Boltzmann weight. This statistical weight is a function of an effective work related to the path. The umbrella ensemble is chosen so that its work histogram overlaps with the histograms corresponding to the ensembles of forward and backward paths. We have finally investigated the relations between numerical efficiency and overlapping properties of the various work histograms. This analysis yields a built-in criterion for diagnosing the convergence during a single-run simulation.     

Path ensembles and path sampling in nonequilibrium stochastic systems

Markovian models based on the stochastic master equation are often encountered in single molecule dynamics, reaction networks, and nonequilibrium problems in chemistry, physics, and biology. An efficient and convenient method to simulate these systems is the kinetic Monte Carlo algorithm which generates continuous-time stochastic trajectories. We discuss an alternative simulation method based on sampling of stochastic paths. Utilizing known probabilities of stochastic paths, it is possible to apply Metropolis Monte Carlo in path space to generate a desired ensemble of stochastic paths. The method is a generalization of the path sampling idea to stochastic dynamics, and is especially suited for the analysis of rare paths which are not often produced in the standard kinetic Monte Carlo procedure. Two generic examples are presented to illustrate the methodology.     

Utilization of Geant4 Monte Carlo simulation method for studying attenuation of photons in normal and heavy concretes at high energy values

The mass attenuation coefficients, μ/ρ, the linear attenuation coefficients, μ and the mean free path, MFP of normal and heavy concretes; ordinary, hematite-serpentine, ilmenite-limonite, basalt-magnetite, ilmenite, steel-scrap and steel-magnetite have been investigated using Geant4 Monte Carlo method at photon energies 1.5, 2, 3, 4, 5 and 6 MeV. The mass attenuation coefficients, the linear attenuation coefficient and the mean free path for the concretes were found dependent upon chemical compositions, density and the photon energy. The linear attenuation coefficient values for the selected concretes increase with the density and decrease with the energy. The mean free path thickness for the all the concretes decrease with increase in iron content and increase with increase the photon energy. The Geant4 Monte Carlo simulation method results have been compared with experimental and theoretical XCOM data, and showed a very good agreement. Our investigation validates the utilization of the Geant4 Monte Carlo simulation method for high energy photon interactions.     

Quantum Monte Carlo simulations of selected ammonia clusters (n = 2-5): isotope effects on the ground state of typical hydrogen bonded systems

Variational Monte Carlo, diffusion Monte Carlo, and stereographic projection path integral simulations are performed on eight selected species from the (NH(3))(n), (ND(3))(n), (NH(2)D)(n), and (NH(3))(n-1)(ND(3)) clusters. Each monomer is treated as a rigid body with the rotation spaces mapped by the stereographic projection coordinates. We compare the energy obtained from path integral simulations at several low temperatures with those obtained by diffusion Monte Carlo, for two dimers, and we find that at 4 K, the fully deuterated dimer energy is in excellent agreement with the ground state energy of the same. The ground state wavefunction for the (NH(3))(2-5) clusters is predominantly localized in the global minimum of the potential energy. In all simulations of mixed isotopic substitutions, we find that the heavier isotope is almost exclusively the participant in the hydrogen bond.     

Monte Carlo simulation of Li+ motion in polyethylene based on polarization energy calculations and informed by data compression analysis

We present an n-fold way kinetic Monte Carlo simulation of the hopping motion of Li+ ions in polyethylene on a grid of mesh 0.36 A superimposed on the voids of the rigid polymer. The structure of the polymer is derived from a higher-order simulation, and the energy of the ion at each site is derived by the self-consistent polarization field method. The ion motion evolves in time from free flight through anomalous diffusion to normal diffusion, with the average energy tending to decrease with increasing temperature through thermal annealing. We compare the results with those of hopping models with probabilistic energy distributions of increasing complexity by analyzing the mean-square displacement and the average energy of an ensemble of ions. The Gumbel distribution describes the ion energy statistics in this system better than the usual Gaussian distribution does; including energy correlation greatly affects the ion dynamics. The analysis uses the standard data compression program GZIP, which proves to be a powerful tool for data analysis by giving a measure of recurrences in the ion path.     

Extrapolation of the time-step bias in diffusion quantum Monte Carlo by a differential sampling technique

A new method of eliminating the finite-time-step error inherent in diffusion quantum Monte Carlo is presented, utilizing an improved version of the existing differential techniques. An implementation is described and results of several small but representative calculations are discussed. The pertinent computation requirements on these systems were reduced by up to a factor of five by the new algorithm. It is speculated that this method may be easily applied to other quantum Monte Carlo and discretized path integral Monte Carlo techniques having related finite step-size errors with a possibility of obtaining similar good results.     

Hybrid Monte Carlo implementation of the Fourier path integral algorithm

This paper formulates a hybrid Monte Carlo implementation of the Fourier path integral (FPI-HMC) approach with partial averaging. Such a hybrid Monte Carlo approach allows one to generate collective moves through configuration space using molecular dynamics while retaining the computational advantages associated with the Fourier path integral Monte Carlo method. In comparison with the earlier Metropolis Monte Carlo implementations of the FPI algorithm, the present HMC method is shown to be significantly more efficient for quantum Lennard-Jones solids and suggests that such algorithms may prove useful for efficient simulations of a range of atomic and molecular systems.     

Hydration structure around a non-electrolyte molecule as revealed by monte carlo calculation: have you seen the “iceberg”?

Monte Carlo calculations have been carried out for pure water and an infinitely dilute aqueous solution of methanol at 298.15 K at ordinary density by the Metropolis method in NTV ensemble. The hydration structure around methanol revealed by a graphic display technique indicates a structure-forming effect near the hydrophobic group.     

On the computational calculation of surface free energies for the disordered semihexagonal reconstructed Au(100) surface

Previously we developed a general method for calculating the free energy of any surface constrained to a distinct surface excess number/density. In this paper we show how to combine a range of such surfaces, whose free energies have been calculated, to produce an ad hoc semigrand canonical ensemble of surfaces from which ensemble surface properties can be calculated, including the ensemble surface free energy. We construct such an ensemble for the disordered Au(100) semihexagonal reconstructed surface using a Glue model potential at 1000 K and calculate the ensemble surface free energy to be 0.088 18 eVA(2). The ensemble average surface lateral density was found to be 1.375 (with respect to the bulk), which is in agreement with previous grand canonical Monte Carlo studies.     

The partial averaging Fourier path Integral approach based on the harmonic reference path

The Monte Carlo Fourier path integral approach has proved to be quite useful in calculating equilibrium thermodynamic properties. One of its advantages is that it can be formulated in such a way as to include higher order terms using the partial averaging technique, which includes the contribution from higher terms usually neglected by the discretized path integral approach. In the original approach, the Feynman path integral is evaluated via a free-particle reference state. Here, using a new expression for the Feynman paths expanded around a harmonic reference path, we derive an alternative formulation for the density matrix element and its corresponding partial averaging expression. Received: 16 September 1998 / Accepted: 30 October 1998 / Published online: 1 February 1999     

Rigid quantum Monte Carlo simulations of condensed molecular matter: water clusters in the n=2-->8 range

The numerical advantage of quantum Monte Carlo simulations of rigid bodies relative to the flexible simulations is investigated for some simple systems. The results show that if high frequency modes in molecular condensed matter are predominantly in the ground state, the convergence of path integral simulations becomes nonuniform. Rigid body quantum parallel tempering simulations are necessary to accurately capture thermodynamic phenomena in the temperature range where the dynamics are influenced by intermolecular degrees of freedom; the stereographic projection path integral adapted for quantum simulations of asymmetric tops is a significantly more efficient strategy compared with Cartesian coordinate simulations for molecular condensed matter under these conditions. The reweighted random series approach for stereographic path integral Monte Carlo is refined and implemented for the quantum simulation of water clusters treated as an assembly of rigid asymmetric tops.     

A path integral influence functional for excess electron in fluids: Density-functional formulation

In this paper, we propose a path integral influence functional from a solvent to determine a self-correlation function of a quantum particle in classical simple fluid. It is shown that the influence functional is related to a grand potential functional of the pure solvent under a three-dimensional external field arising from a classical isomorphic polymer, on which the quantum particle is mapped. The influence functional can be calculated from the self-correlation function, the solute-solvent and the solvent-solvent pair correlation function. The obtained equation of the self-correlation function is applied to an excess electron problem in fluid helium. The Fourier path-integral Monte Carlo method is employed to perform the path integral of the electron. The solute-solvent pair correlation function is estimated from a reference interaction site model integral equation. These results obtained form our proposed influence functional and from that proposed by Chandler, Singh, and Richardson are compared with those provided by a path integral Monte Carlo simulation with the explicit helium solvent.     

Path integral formulation for many-electron system

The Feynman path integral method is applied to the many-electron problem of quantum chemistry. We begin with constructing new closure relations in terms of the linear combination of atomic orbital (LCAO) coefficients and investigate the transition amplitude and the partition function of the system in question; then a “classical path of electrons,” which is described by the time-dependent Hartree-Fock-Roothaan equation, is obtained by minimizing the action integral of the system with respect to the “electron coordinate.” The next order approximation is obtained by evaluating the deviation from this classical path, which is approximately written by a Gaussian integral. The result is expected to be the random-phase approximation. © 1996 John Wiley & Sons, Inc.     

Modified Monte Carlo simulation of adsorption in a spatially inhomogeneous porous system

A modified Monte Carlo method in conjunction with the canonical and grand canonical ensembles is proposed for simulating adsorption in spatially inhomogeneous porous systems. Unlike the traditional Monte Carlo simulation in terms of the grand canonical ensemble, the simulation for the regions of pore space having no direct communication with the bulk phase is performed in local conditions of the canonical ensemble.     

Path integral ground state study of finite-size systems: application to small (parahydrogen)N (N=2-20) clusters

We use the path integral ground state method to study the energetic and structural properties of small para-H2 clusters of sizes ranging from 2 to 20 molecules. A fourth order formula is used to approximate the short imaginary-time propagator and two interaction potentials are considered. Our results are compared to those of exact basis set calculations and other quantum Monte Carlo methods when available. We find that for all cluster sizes considered, our results show a lower ground state energy than literature values obtained by diffusion Monte Carlo and variational Monte Carlo. For the dimer and trimer, ground state energies are in good agreement with exact results obtained using the discrete variable representation. Structural properties are found to be insensitive to the choice of interaction potential. We explore the use of Pekeris coordinates to analyze the importance of linear arrangement in trimers and for trimers within clusters of larger size.     

Monte Carlo method for computing density of states and quench probability of potential energy and enthalpy landscapes

The thermodynamics and kinetics of a many-body system can be described in terms of a potential energy landscape in multidimensional configuration space. The partition function of such a landscape can be written in terms of a density of states, which can be computed using a variety of Monte Carlo techniques. In this paper, a new self-consistent Monte Carlo method for computing density of states is described that uses importance sampling and a multiplicative update factor to achieve rapid convergence. The technique is then applied to compute the equilibrium quench probability of the various inherent structures (minima) in the landscape. The quench probability depends on both the potential energy of the inherent structure and the volume of its corresponding basin in configuration space. Finally, the methodology is extended to the isothermal-isobaric ensemble in order to compute inherent structure quench probabilities in an enthalpy landscape.