David L. Freeman

Reducing quasi-ergodic behavior in Monte carlo simulations by J-walking : applications to atomic clusters

A method is introduced that is easy to implement and greatly reduces the systematic error resulting from quasi‐ergodicity, or incomplete sampling of configuration space, in Monte Carlo simulations of systems containing large potential energy barriers. The method makes possible the jumping over these barriers by coupling the usual Metropolis sampling to the Boltzmann distribution generated by another random walker at a higher temperature. The basic techniques are illustrated on some simple classical systems, beginning for heuristic purposes with a simple one‐dimensional double well potential based on a quartic polynomial. The method’s suitability for typical multidimensional Monte Carlo systems is demonstrated by extending the double well potential to several dimensions, and then by applying the method to a multiparticle cluster system consisting of argon atoms bound by pairwise Lennard‐Jones potentials. Remarkable improvements are demonstrated in the convergence rate for the cluster configuration energy, ...

Phase changes in 38-atom Lennard-Jones clusters. I. A parallel tempering study in the canonical ensemble

The heat capacity and isomer distributions of the 38-atom Lennard-Jones cluster have been calculated in the canonical ensemble using parallel tempering Monte Carlo methods. A distinct region of temperature is identified that corresponds to equilibrium between the global minimum structure and the icosahedral basin of structures. This region of temperatures occurs below the melting peak of the heat capacity and is accompanied by a peak in the derivative of the heat capacity with temperature. Parallel tempering is shown to introduce correlations between results at different temperatures. A discussion is given that compares parallel tempering with other related approaches that ensure ergodic simulations.

Phase changes in 38-atom Lennard-Jones clusters. II. A parallel tempering study of equilibrium and dynamic properties in the molecular dynamics and microcanonical ensembles

We study the 38-atom Lennard-Jones cluster with parallel tempering Monte Carlo methods in the microcanonical and molecular dynamics ensembles. A new Monte Carlo algorithm is presented that samples rigorously the molecular dynamics ensemble for a system at constant total energy, linear and angular momenta. By combining the parallel tempering technique with molecular dynamics methods, we develop a hybrid method to overcome quasiergodicity and to extract both equilibrium and dynamical properties from Monte Carlo and molecular dynamics simulations. Several thermodynamic, structural, and dynamical properties are investigated for LJ38, including the caloric curve, the diffusion constant and the largest Lyapunov exponent. The importance of insuring ergodicity in molecular dynamics simulations is illustrated by comparing the results of ergodic simulations with earlier molecular dynamics simulations.

Partial averaging approach to Fourier coefficient path integration

The recently introduced method of partial averaging is developed into a general formalism for computing simple Cartesian path integrals. Examples of its application to both harmonic and anharmonic systems are given. For harmonic systems, where analytical results can be derived, both imaginary and complex time evolution is discussed. For two representative anharmonic systems, Monte Carlo path integral simulations of the imaginary time propagator (statistical density matrix) are presented. Connections with other Cartesian path integral techniques are stressed.

Theoretical studies of the energetics and structures of atomic clusters

Comparative calculations of the binding energy and structure of relaxed closed‐shell clusters of icosahedral and cuboctahedral point group symmetry are reported. The atoms are presumed to interact via either the Lennard‐Jones or the Aziz–Chen (HFD–C) pair potential. The IC structure is found to be lower in total energy for less than 14 shells (10 179 atoms) in the Lennard‐Jones case and for less than 13 shells (8217 atoms) in the HFD–C case. Detailed energetics are analyzed in order to elucidate the mechanism for the transition from icosahedral to cuboctahedral symmetry.

Quantum Monte Carlo study of the thermodynamic properties of argon clusters: The homogeneous nucleation of argon in argon vapor and ‘‘magic number’’ distributions in argon vapor

The thermodynamic properties of clusters of argon atoms are studied by a combination of classical and quantum mechanical Monte Carlo methods. The argon atoms are represented by Lennard‐Jones interactions and internal energies, free energies, and entropies are calculated as a function of temperature and cluster size. For the argon system quantum effects and anharmonicity corrections are found to be simultaneously important for a temperature range from 15 to 20 K. By examining local minima in the free energy of formation of argon clusters as a function of cluster size, magic numbers in the Boltzmann mass distribution are observed at n=7, 13, and 19 under some conditions of temperature and pressure. In some cases magic numbers are predicted in the quantum and not in the classical calculation. The entropy changes associated with cluster growth are found to be insensitive to cluster size. Quantum corrections are calculated to nucleation rates and found to be very important at low temperatures.

Stationary phase Monte Carlo methods: An exact formulation

Abstract We present here a general and formally exact stationary phase Monte Carlo method. This method is aimed at the generic problem of performing many-dimensional averages of highly oscillatory integrands, a central problem in the path integral treatment of quantum dynamics. The relationship between our previous developments, the work by Filinov and the recent work by Makri and Miller is clarified. We present an implementation of the stationary phase Monte Carlo approach that is viable, formally exact and that eliminates the need for derivatives beyond first order.

Quantum Monte Carlo dynamics: The stationary phase Monte Carlo path integral calculation of finite temperature time correlation functions

We present a numerically exact procedure for the calculation of an important class of finite temperature quantum mechanical time correlation functions. The present approach is based around the stationary phase Monte Carlo (SPMC) method, a general mathematical tool for the calculation of high dimensional averages of oscillatory integrands. In the present context the method makes possible the direct numerical path integral calculation of real‐time quantum dynamical quantities for times appreciably greater than the thermal time (βℏ). Illustrative applications involving finite temperature anharmonic motion are presented. Issues of importance with respect to future applications are identified and discussed.

Extending J Walking to Quantum Systems: Applications to Atomic Clusters

The J‐walking (or jump‐walking) method is extended to quantum systems by incorporating it into the Fourier path integral Monte Carlo methodology. J walking can greatly reduce systematic errors due to quasiergodicity, or the incomplete sampling of configuration space in Monte Carlo simulations. As in the classical case, quantum J walking uses a jumping scheme to overcome configurational barriers. It couples the usual Metropolis sampling to a distribution generated at a higher temperature where the sampling is sufficiently ergodic. The J‐walker distributions used in quantum J walking can be either quantum or classical, with classical distributions having the advantage of lower storage requirements, but the disadvantage of being slightly more computationally intensive and having a more limited useful temperature range. The basic techniques are illustrated first on a simple one‐dimensional double well potential based on a quartic polynomial. The suitability of J walking for typical multidimensional quantum Mo...

Toward a Monte Carlo Theory of Quantum Dynamics

We consider in the present paper an extension of numerical path integral methods for use in computing finite temperature time correlation functions. We demonstrate that coordinate rotation techniques extend appreciably the time domain over which Monte Carlo methods are of use in the construction of such correlation functions.