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.     

Improved density of states Monte Carlo method based on recycling of rejected states

In this paper a new algorithm is presented that improves the efficiency of Wang and Landau algorithm or density of states (DOS) Monte Carlo simulations by employing rejected states. The algorithm is shown to have a performance superior to that of the original Wang-Landau [F. Wang and D. P. Landau, Phys. Rev. Lett. 86, 2050 (2001)] algorithm and the more recent configurational temperature DOS algorithm. The performance of the method is illustrated in the context of results for the Lennard-Jones fluid.     

Maximum probability domains from Quantum Monte Carlo calculations

Although it would be tempting to associate the Lewis structures to the maxima of the squared wave function |Psi|2, we prefer in this paper the use of domains of the three-dimensional space, which maximize the probability of containing opposite-spin electron pairs. We find for simple systems (CH4, H2O, Ne, N2, C2H2) domains comparable to those obtained with the electron localization function (ELF) or by localizing molecular orbitals. The different domains we define can overlap, and this gives an interesting physical picture of the floppiness of CH5+ and of the symmetric hydrogen bond in FHF-. The presence of multiple solutions has an analogy with resonant structures, as shown in the trans-bent structure of Si2H2. Correlated wave functions were used (MCSCF or Slater-Jastrow) in the Variational Quantum Monte Carlo framework.     

精确固定节面量子Monte Carlo差值法

提出了精确固定节面量子Monte Carlo差值法, 这个新算法能够在精确固定节面量子Monte Carlo方法的基础上直接计算两个体系之间的能量差, 且使计算结果的统计误差达到10-5 hartree 数量级, 获得电子相关能90%以上. 我们把这个新算法应用于分子势能面的研究中, 使用一个“刚性移动”模型, 利用Jacobi变换使分子两个几何构型的能量计算具有很好的正相关性, 因而能得到准确的能量差值, 由此就可以得到精确的分子势能面.     

Rydberg states with quantum Monte Carlo

Calculations on Rydberg states are performed using quantum Monte Carlo methods. Excitation energies and singlet-triplet splittings are calculated for two model systems, the carbon atom (3P and 1P) and carbon monoxide ((1Sigma and 3Sigma). Kohn-Sham wave functions constructed from open-shell localized Hartree-Fock orbitals are used as trial and guide functions. The fixed-node diffusion quantum Monte Carlo (FN-DMC) method depends strongly on the wave function's nodal hypersurface. Nodal artefacts are investigated for the ground state of the carbon atom. Their effect on the FN-DMC results can be analyzed quantitatively. FN-DMC leads to accurate excitation energies but to less accurate singlet-triplet splittings. Variational Monte Carlo calculations are able to reproduce the experimental results for both the excitation energies and the singlet-triplet splittings.     

A new Monte Carlo method for getting the density of states of atomic cluster systems

A novel Monte Carlo flat histogram algorithm is proposed to get the classical density of states in terms of the potential energy, g(E(p)), for systems with continuous variables such as atomic clusters. It aims at avoiding the long iterative process of the Wang-Landau method and controlling carefully the convergence, but keeping the ability to overcome energy barriers. Our algorithm is based on a preliminary mapping in a series of points (called a σ-mapping), obtained by a two-parameter local probing of g(E(p)), and it converges in only two subsequent reweighting iterations on large intervals. The method is illustrated on the model system of a 432 atom cluster bound by a Rydberg type potential. Convergence properties are first examined in detail, particularly in the phase transition zone. We get g(E(p)) varying by a factor 10(3700) over the energy range [0.01 < E(p) < 6000 eV], covered by only eight overlapping intervals. Canonical quantities are derived, such as the internal energy U(T) and the heat capacity C(V)(T). This reveals the solid to liquid phase transition, lying in our conditions at the triple point. This phase transition is further studied by computing a Lindemann-Berry index, the atomic cluster density n(r), and the pressure, demonstrating the progressive surface melting at this triple point. Some limited results are also given for 1224 and 4044 atom clusters.     

Monte Carlo modelling of the polymer glass transition

We are proposing a lattice model with chemical input for the computer modelling of the polymer glass transition. The chemical input information is obtained by a coarse graining procedure applied to a microscopic model with full chemical detail. We use this information on Bisphenol-A-Polycarbonate to predict it's Vogel-Fulcher temperature out of a dynamic Monte Carlo Simulation. The microscopic structure of the lattice model is that of a genuine amorphous material, and the structural relaxation obeys the time temperature superposition.     

Melting of palladium clusters – density of states determination by Monte Carlo simulation

The density of states (DOS) has been calculated for the metal clusters Pd₁₃, Pd₅₅ and Pd₁₄₇ using the recently proposed reference system equilibration (RSE) method. The interaction within the clusters was described by a many-body alloy potential. Using this DOS, the caloric curve of Pd₁₃ has been calculated and excellent agreement with canonical Monte Carlo simulations is obtained. For Pd₅₅ and Pd₁₄₇, the solid and one molten isomers have been isolated in order to calculate the DOS for the isomers separately. The melting of the clusters occurs when the DOS for the solid and the molten isomers are equal. Comparison with previous microcanonical Monte Carlo simulations shows that the number of statistically equivalent molten isomers are 1.1×10¹⁸ for Pd₅₅ and 4.1×10⁴¹ for Pd₁₄₇.     

Equilibrium states of self-assembly systems: Monte Carlo simulations

We investigated the equilibrium states of the self-assembly of amphiphilic molecules in water. The amphiphiles are represented by chains of the type H1T4, where H is the hydrophilic part of the molecule and T is its hydrophobic portion formed by four monomers. We have performed Monte Carlo simulations on a two-dimensional lattice, in which each water molecule occupies a single site, and the amphiphiles occupy five sites of the lattice. We have determined the aggregate distribution curves for the system at low concentration and fixed temperature. We have shown that the criterion to determine the equilibrium states of the system, based on the stabilization of energy curves as a function of the simulation time, is not reliable. The best way to ensure that the equilibrium state was reached was to follow the route to equilibrium of all aggregate sizes of the system.     

Spectroscopic data for the LiH molecule from pseudopotential quantum Monte Carlo calculations

Quantum Monte Carlo and quantum chemistry techniques are used to investigate pseudopotential models of the lithium hydride (LiH) molecule. Interatomic potentials are calculated and tested by comparing with the experimental spectroscopic constants and well depth. Two recently developed pseudopotentials are tested, and the effects of introducing a Li core polarization potential are investigated. The calculations are sufficiently accurate to isolate the errors from the pseudopotentials and core polarization potential. Core-valence correlation and core relaxation are found to be important in determining the interatomic potential.     

Quantum Monte Carlo calculations for ground and excited states

A brief overview of the diffusion quantum Monte Carlo method is given. We illustrate the application to ground‐state calculations by a study of the relative stability of carbon clusters near the crossover to fullerene stability, thereby determining the smallest stable fullerene. The application to excited states is illustrated via a study of excitonic states in small hydrogenated silicon clusters. © 2001 John Wiley & Sons, Inc. Int J Quantum Chem, 2001     

Monte Carlo estimation of kinetic parameters in polymerization reactions

A general method for estimating kinetic parameters in polymerization reactions using Monte Carlo simulation to represent the models of the reactions is developed. From a statistical point of view, the procedure is a Bayesian one in which a posterior probability density surface (PPDS) is calculated for points on a grid in the parameter space. A smoothing function is fitted to the PPDS, then a posterior probability region, which is similar to a confidence region, is calculated for the parameters. An application to a relatively trivial example, the Mayo–Lewis copolymerization model is shown in detail. Many other potential applications are suggested.     

On the calculation of absolute free energies from molecular-dynamics or Monte Carlo data

We propose a method for calculating absolute free energies from Monte Carlo or molecular-dynamics data. The method is based on the identity that expresses the partition function Q as a Boltzmann average: 1Q=w(p,x)exp[betaH(p,x)], where w(p,x) is an arbitrary weight function such that its integral over the phase space is equal to 1. In practice, to minimize statistical errors the weight function is chosen such that the regions of the phase space where sampling statistics are poor are excluded from the average. The "ideal" weight function would be the equilibrium phase-space density exp[-betaH(p,x)]Q itself. We consider two methods for constructing the weight function based on different estimates of the equilibrium phase-space density from simulation data. In the first method, it is chosen to be a Gaussian function, whose parameters are obtained from the covariance matrix of the atomic coordinates. In the second, a clustering algorithm is used to attempt partitioning the data into clusters corresponding to different basins of attraction visited by the system. The weight function is then constructed as a superposition of Gaussians calculated for each cluster separately. We show that these strategies can be used to improve upon previous methods of estimating absolute entropies from covariance matrices.     

Isotropic-nematic phase transition in the Lebwohl-Lasher model from density of states simulations

Density of states Monte Carlo simulations have been performed to study the isotropic-nematic (IN) transition of the Lebwohl-Lasher model for liquid crystals. The IN transition temperature was calculated as a function of system size using expanded ensemble density of states simulations with histogram reweighting. The IN temperature for infinite system size was obtained by extrapolation of three independent measures. A subsequent analysis of the kinetics in the model showed that the transition occurs via spinodal decomposition through aggregation of clusters of liquid crystal molecules.     

Accurate and unbiased estimation of power-law exponents from single-emitter blinking data

Single emitter blinking with a power-law distribution for the on and off times has been observed on a variety of systems including semiconductor nanocrystals, conjugated polymers, fluorescent proteins, and organic fluorophores. The origin of this behavior is still under debate. Reliable estimation of power exponents from experimental data is crucial in validating the various models under consideration. We derive a maximum likelihood estimator for power-law distributed data and analyze its accuracy as a function of data set size and power exponent both analytically and numerically. Results are compared to least-squares fitting of the double logarithmically transformed probability density. We demonstrate that least-squares fitting introduces a severe bias in the estimation result and that the maximum likelihood procedure is superior in retrieving the correct exponent and reducing the statistical error. For a data set as small as 50 data points, the error margins of the maximum likelihood estimator are already below 7%, giving the possibility to quantify blinking behavior when data set size is limited, e.g., due to photobleaching.     

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.     

Monte Carlo study of the coil-to-globule transition of a model polymeric system

Monte Carlo computer simulations of single, flexible, self-avoiding chains on a cubic lattice have been performed upon conditions of increasing segment–segment cohesive energy (deteriorating solvent quality). The simulations spanned a wide range of chain lengths (20–10,000, i.e., up to molecular weights of a few millions) and cohesive energies (0.0–0.45k_BT, i.e., from athermal to very poor solvents). The chain length dependence of the chain size in poor solvents was characterized by a wide plateau of almost null growth for intermediate chain lengths. This plateau was linked to the development of the incipient constant density core, while genuine power law dependence (1/3) was not reached even for the longest chains and poorest solvents simulated here. The mere appearance of a core required substantial chain lengths (higher than 1000; molecular weights of a few hundred thousands), while short chains underwent a gradual densification devoid of any qualitative changes in the density distribution. Sufficiently long chains became more but not quite spherical and underwent a reasonably sharp second order phase transition. The findings were generally in agreement with predictions of mean-field theory and with the use of the standard scaling variables, despite slight inconsistencies. Nevertheless, the results stress the fact that short chains never form a constant density core and that core-dominance on the globule's properties (“volume approximation”) is only valid for extraordinarily long chains [molecular weight of O(10⁹)], an effect linked to the relatively diffuse nature of the surface layer and originating from chain connectivity in conjunction with spherical geometry. © 2006 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 44: 3651–3666, 2006     

A Monte Carlo method for finding important ligand fragments from receptor data

A simulated annealing method for finding important ligand fragments is described. At a given temperature, ligand fragments are randomly selected and randomly placed within the given receptor cavity, often replacing or forming bonds with existing ligand fragments. For each new ligand fragment combination, the bonded, nonbonded, polarization and solvation energies of the new ligand–receptor system are compared to the previous configuration. Acceptance or rejection of the new system is decided using the Boltzmann distribution , where E is the energy difference between the old and new systems, k is the Boltzmann constant and T is the temperature. Thus, energetically unfavorable fragment switches are sometimes accepted, sacrificing immediate energy gains in the interest of finding a system with minimum energy. By lowering the temperature, the rate of unfavorable switches decreases and energetically favorable combinations become more difficult to change. The process is terminated when the frequency of switches becomes too small. As a test, the method predicted positions and types of important ligand fragments for neuraminidase that were in accord with the known ligand, sialic acid.