Reconciling alternate methods for the determination of charge distributions: a probabilistic approach to high-dimensional least-squares approximations

We propose extensions and improvements of the statistical analysis of distributed multipoles (SADM) algorithm put forth by Chipot et al. (Mol Phys 94: 881–895, 1998) for the derivation of distributed atomic multipoles from the quantum-mechanical electrostatic potential. The method is mathematically extended to general least-squares problems and provides an alternative approximation method in cases where the original least-squares problem is computationally not tractable, either because of its ill-posedness or its high-dimensionality. The solution is approximated employing a Monte Carlo method that takes the average of a random variable defined as the solutions of random small least-squares problems drawn as subsystems of the original problem. The conditions that ensure convergence and consistency of the method are discussed, along with an analysis of the computational cost in specific instances.     

Phase separation of a model binary polymer solution in an external field

The phase separation of a simple binary mixture of incompatible linear polymers in solution is investigated using an extension of the sedimentation equilibrium method, whereby the osmotic pressure of the mixture is extracted from the density profiles of the inhomogeneous mixture in a gravitational field. In Monte Carlo simulations the field can be tuned to induce significant inhomogeneity, while keeping the density profiles sufficiently smooth for the macroscopic condition of hydrostatic equilibrium to remain applicable. The method is applied here for a simplified model of ideal but mutually avoiding polymers, which readily phase separate at relatively low densities. The Monte Carlo data are interpreted with the help of an approximate bulk phase diagram calculated from a simple, second-order virial coefficient theory. By derivation of effective potentials between polymer centers of mass, the binary mixture of polymers is coarse-grained to a "soft colloid" picture reminiscent of the Widom-Rowlinson model for incompatible atomic mixtures. This approach significantly speeds up the simulations and accurately reproduces the behavior of the full monomer resolved model.     

剩余函数量子Monte Carlo方法

为量子Ｍｏｎｔｅ Ｃａｒｌｏ方法提出一条新途径－剩余函数法,引入了Ｓｃｈｒｏｅｄｉｎｇｅｒ方程剩余函数的概念,利用剩余函数将一种新的有明显物理意义的试探函数应用到量子Ｍｏｎｔｅ Ｃａｒｌｏ过程中,这种试探函数是通过一种迭进式的方式确定的,它不需要在Ｍｏｎｔｅ Ｃａｒｌｏ过程中优化参数。文中我们将给出这种试探函数的具体形式,证明由这种试探函数求出的能量期望值收敛于体系真实的能量值;文中还给出这种试探     

自由基聚合反应的Monte Carlo模拟方法

本文将化学反应动力学的MonteCarlo模拟方法运用到引发剂引发的自由基聚合反应的非稳态动力学,针对自由基聚合反应动力学数值模拟所特有的"无伸缩问题",采用"偏倚抽样法"解决了MonteCarlo模拟中的"无伸缩问题",模拟结果与非稳态动力学解的结果完全一致,此算法易推广到研究更复杂的自由基聚合反应体系。     

A simple approach to treat anisotropic elastic collisions in Monte Carlo calculations of the electron energy distribution function in cold plasmas

A simplified method allowing one to treat anisotropic electron heavy species elastic scattering in MonteCarlo models of gas discharges with the proper value for collision frequency is proposed The method is applied to an electric discharge in a Ne · Xe/HCl mixture, and the results are compared with the solution of the two-term expansion of the Boltzmann equation under the same conditions. Methods for reduction of computational time in Monte Carlo codes and the use of the Monte Carlo flux method are also discussed.     

Fourth virial coefficients of asymmetric nonadditive hard-disk mixtures

The fourth virial coefficient of asymmetric nonadditive binary mixtures of hard disks is computed with a standard Monte Carlo method. Wide ranges of size ratio (0.05 ≤ q ≤ 0.95) and nonadditivity (-0.5 ≤ Δ ≤ 0.5) are covered. A comparison is made between the numerical results and those that follow from some theoretical developments. The possible use of these data in the derivation of new equations of state for these mixtures is illustrated by considering a rescaled virial expansion truncated to fourth order. The numerical results obtained using this equation of state are compared with Monte Carlo simulation data in the case of a size ratio q = 0.7 and two nonadditivities Δ = ±0.2.     

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.     

LiF-KCl-KBr系和KF-NaCl-NaBr系熔盐相图研究

用目测变温法和X-射线相分析研究了LiF-KCl-KBr系和KF-NaCl-NaBr系熔盐相图.并在此基础上估算了熔盐溶液的正、负偏差情况,结合熔盐溶液的MonteCarlo法研究结果进行了讨论.     

Gamma distribution model to provide a direct assessment of the overall quality of quantum Monte Carlo-generated electron distributions

Our objective is to assess the accuracy of simulated quantum Monte Carlo electron distributions of atoms and molecules. Our approach is first to model the exact electron distribution by a linear combination of gamma distribution functions, with parameters chosen to exactly reproduce highly accurate literature values for a number of selected moments for the system of interest. In application to the ground-state electron distributions of helium and dihydrogen, a high level of accuracy of the model was confirmed upon comparing its predicted moments, not used in the model's parametrization, to those calculated from high-level theory. Next, we generated electron-electron and electron-nucleus distributions for dihydrogen from electron positions outputted from a variety of quantum Monte Carlo algorithms. Upon juxtaposition of the simulated distributions with the putatively exact one that we derived from the model, we quantified the error in simulated distributions. The most accurate distributions were obtained from no-compromise reptation quantum Monte Carlo, a recently developed algorithm designed to ameliorate the distributions' time-step bias. Marginally less accurate distributions were generated from fixed-node diffusion Monte Carlo with descendant counting and detailed balance.     

Quantum-mechanical evaluation of the Boltzmann operator in correlation functions for large molecular systems: a multilayer multiconfiguration time-dependent Hartree approach

It is shown that the Boltzmann operator in time correlation functions for complex molecular systems can be evaluated in a numerically exact way employing the multilayer formulation of the multiconfiguration time-dependent Hartree theory in combination with Monte Carlo importance sampling techniques. The performance of the method is illustrated by selected applications to photoinduced intervalence electron transfer reactions in the condensed phase. Furthermore, the validity of approximate schemes to evaluate the Boltzmann is discussed.     

Combining reactive and configurational-bias Monte Carlo: confinement influence on the propene metathesis reaction system in various zeolites

In order to efficiently calculate chemical equilibria of large molecules in a confined environment the reactive Monte Carlo technique is combined with the configurational-bias Monte Carlo approach. To prove that detailed balance is fulfilled the acceptance rule for this combination of particular Monte Carlo techniques is derived in detail. Notably, by using this derivation all other acceptance rules of any Monte Carlo trial moves usually carried out in combination with the configurational-bias Monte Carlo approach can be deduced from it. As an application of the combination of reactive and configurational-bias Monte Carlo the influence of different zeolitic confinements (MFI, TON, LTL, and FER) on the reaction equilibrium and the selectivity of the propene metathesis reaction system was investigated. Compared to the bulk phase the conversion is increased significantly. The authors study this reaction system in the temperature range between 300 and 600 K, and the pressure range from 1 to 7 bars. In contrast to the bulk phase, pressure and temperature have a strong influence on the composition of the reaction mixture in confinement. At low pressures and temperatures both conversion and selectivity are highest. Furthermore, the equilibrium composition is strongly dependent on the type of zeolite. This demonstrates the important role of the host structure in catalytic systems.     

A variational approach to the stochastic aspects of cellular signal transduction

Cellular signaling networks have evolved to cope with intrinsic fluctuations, coming from the small numbers of constituents, and the environmental noise. Stochastic chemical kinetics equations govern the way biochemical networks process noisy signals. The essential difficulty associated with the master equation approach to solving the stochastic chemical kinetics problem is the enormous number of ordinary differential equations involved. In this work, we show how to achieve tremendous reduction in the dimensionality of specific reaction cascade dynamics by solving variationally an equivalent quantum field theoretic formulation of stochastic chemical kinetics. The present formulation avoids cumbersome commutator computations in the derivation of evolution equations, making the physical significance of the variational method more transparent. We propose novel time-dependent basis functions which work well over a wide range of rate parameters. We apply the new basis functions to describe stochastic signaling in several enzymatic cascades and compare the results so obtained with those from alternative solution techniques. The variational Ansatz gives probability distributions that agree well with the exact ones, even when fluctuations are large and discreteness and nonlinearity are important. A numerical implementation of our technique is many orders of magnitude more efficient computationally compared with the traditional Monte Carlo simulation algorithms or the Langevin simulations.     

Comparing three stochastic search algorithms for computational protein design: Monte Carlo,replica exchange Monte Carlo,and a multistart,steepest‐descent heuristic

Computational protein design depends on an energy function and an algorithm to search the sequence/conformation space. We compare three stochastic search algorithms: a heuristic, Monte Carlo (MC), and a Replica Exchange Monte Carlo method (REMC). The heuristic performs a steepest‐descent minimization starting from thousands of random starting points. The methods are applied to nine test proteins from three structural families, with a fixed backbone structure, a molecular mechanics energy function, and with 1, 5, 10, 20, 30, or all amino acids allowed to mutate. Results are compared to an exact, “Cost Function Network” method that identifies the global minimum energy conformation (GMEC) in favorable cases. The designed sequences accurately reproduce experimental sequences in the hydrophobic core. The heuristic and REMC agree closely and reproduce the GMEC when it is known, with a few exceptions. Plain MC performs well for most cases, occasionally departing from the GMEC by 3–4 kcal/mol. With REMC, the diversity of the sequences sampled agrees with exact enumeration where the latter is possible: up to 2 kcal/mol above the GMEC. Beyond, room temperature replicas sample sequences up to 10 kcal/mol above the GMEC, providing thermal averages and a solution to the inverse protein folding problem. © 2016 Wiley Periodicals, Inc.     

An equation-free probabilistic steady-state approximation: dynamic application to the stochastic simulation of biochemical reaction networks

Stochastic chemical kinetics more accurately describes the dynamics of "small" chemical systems, such as biological cells. Many real systems contain dynamical stiffness, which causes the exact stochastic simulation algorithm or other kinetic Monte Carlo methods to spend the majority of their time executing frequently occurring reaction events. Previous methods have successfully applied a type of probabilistic steady-state approximation by deriving an evolution equation, such as the chemical master equation, for the relaxed fast dynamics and using the solution of that equation to determine the slow dynamics. However, because the solution of the chemical master equation is limited to small, carefully selected, or linear reaction networks, an alternate equation-free method would be highly useful. We present a probabilistic steady-state approximation that separates the time scales of an arbitrary reaction network, detects the convergence of a marginal distribution to a quasi-steady-state, directly samples the underlying distribution, and uses those samples to accurately predict the state of the system, including the effects of the slow dynamics, at future times. The numerical method produces an accurate solution of both the fast and slow reaction dynamics while, for stiff systems, reducing the computational time by orders of magnitude. The developed theory makes no approximations on the shape or form of the underlying steady-state distribution and only assumes that it is ergodic. We demonstrate the accuracy and efficiency of the method using multiple interesting examples, including a highly nonlinear protein-protein interaction network. The developed theory may be applied to any type of kinetic Monte Carlo simulation to more efficiently simulate dynamically stiff systems, including existing exact, approximate, or hybrid stochastic simulation techniques.     

Tests of Monte Carlo perturbation theory for the free energy of liquid copper

Monte Carlo perturbation theory, in which terms in the thermodynamic perturbation series are evaluated by Monte Carlo averaging, has potentially large advantages in efficiency for calculating free energies of liquids from ab initio potential surfaces. In order to test the accuracy of perturbation theory for liquid metals, a series of calculations has been done on liquid copper, modeled by an embedded atom potential. A simple 1/r(12) pair potential is used as the reference system. The free energy is calculated to third order in perturbation theory, and the results are compared to an exact formula. It is found that for optimal reference potential parameters, second order perturbation theory is essentially exact. Second and third order theories give accurate results for significantly nonoptimal reference parameters. The relation between perturbation theory and reweighting is discussed, and an approximate formula is derived that shows an exponential dependence of the efficiency of reweighting on the second order free energy correction. Finally, techniques for application to ab initio potentials are discussed. It is shown that with samples of 100 configurations, it is possible to obtain accuracy and precision at the level of approximately 1 meV/atom.     

Thermodynamic properties of van der Waals fluids from Monte Carlo simulations and perturbative Monte Carlo theory

Computer simulations have been performed for fluids with van der Waals potential, that is, hard spheres with attractive inverse power tails, to determine the equation of state and the excess energy. On the other hand, the first- and second-order perturbative contributions to the energy and the zero- and first-order perturbative contributions to the compressibility factor have been determined too from Monte Carlo simulations performed on the reference hard-sphere system. The aim was to test the reliability of this "exact" perturbation theory. It has been found that the results obtained from the Monte Carlo perturbation theory for these two thermodynamic properties agree well with the direct Monte Carlo simulations. Moreover, it has been found that results from the Barker-Henderson [J. Chem. Phys. 47, 2856 (1967)] perturbation theory are in good agreement with those from the exact perturbation theory.     

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.     

Thermodynamics and partitioning of homopolymers into a slit-A grand canonical Monte Carlo simulation study

Grand canonical ensemble Monte Carlo simulation (GCMC) combined with the histogram reweighting technique was used to study the thermodynamic equilibrium of a homopolymer solution between a bulk and a slit pore. GCMC gives the partition coefficients that agree with those from canonical ensemble Monte Carlo simulations in a twin box, and it also gives results that are not accessible through the regular canonical ensemble simulation such as the osmotic pressure of the solution. In a bulk polymer solution, the calculated osmotic pressure agrees very well with the scaling theory predictions both for the athermal polymer solution and the theta solution. However, one cannot obtain the osmotic pressure of the confined solution in the same way since the osmotic pressure of the confined solution is anisotropic. The chemical potentials in GCMC simulations were found to differ by a translational term from the chemical potentials obtained from canonical ensemble Monte Carlo simulations with the chain insertion method. This confirms the equilibrium condition of a polymer solution partition between the bulk and a slit pore: the chemical potentials of the polymer chain including the translational term are equal at equilibrium. The histogram reweighting method enables us to obtain the partition coefficients in the whole range of concentrations based on a limited set of simulations. Those predicted bulk-pore partition coefficient data enable us to perform further theoretical analysis. Scaling predictions of the partition coefficient at different regimes were given and were confirmed by the simulation data.     

Forward--backward semiclassical dynamics with information-guided noise reduction for a molecule in solution

The forward--backward semiclassical dynamics (FBSD) methodology is used to obtain expressions for time correlation functions of a system (atom or molecule) in solution. We use information-guided noise reduction (IGNoR) [Makri, N. Chem. Phys. Lett. 2004, 400, 446] to minimize the statistical error associated with the Monte Carlo integration of oscillatory functions. This is possible by reformulating the correlation function in terms of an oscillatory solvent-dependent contribution whose integral can be obtained analytically and a slowly varying function obtained via a grid-based iterative evaluation of solute properties. Knowledge of the exact integral of the oscillatory function, combined with correlated statistics, leads to partial cancellation of the Monte Carlo error. Application on a one-dimensional solute-solvent model shows a substantial improvement of convergence in the IGNoR-enhanced FBSD correlation function for a fixed number of Monte Carlo samples. The reduction of statistical error achieved by using the IGNoR methodology becomes more significant as the number of solvent particles increases.     

An analytical approximation for the orientation-dependent excluded volume of tangent hard sphere chains of arbitrary chain length and flexibility

Onsager-like theories are commonly used to describe the phase behavior of nematic (only orientationally ordered) liquid crystals. A key ingredient in such theories is the orientation-dependent excluded volume of two molecules. Although for hard convex molecular models this is generally known in analytical form, for more realistic molecular models that incorporate intramolecular flexibility, one has to rely on approximations or on computationally expensive Monte Carlo techniques. In this work, we provide a general correlation for the excluded volume of tangent hard-sphere chains of arbitrary chain length and flexibility. The flexibility is introduced by means of the rod-coil model. The resulting correlation is of simple analytical form and accurately covers a wide range of pure component excluded volume data obtained from Monte Carlo simulations of two-chain molecules. The extension to mixtures follows naturally by applying simple combining rules for the parameters involved. The results for mixtures are also in good agreement with data from Monte Carlo simulations. We have expressed the excluded volume as a second order power series in sin?(γ), where γ is the angle between the molecular axes. Such a representation is appealing since the solution of the Onsager Helmholtz energy functional usually involves an expansion of the excluded volume in Legendre coefficients. Both for pure components and mixtures, the correlation reduces to an exact expression in the limit of completely linear chains. The expression for mixtures, as derived in this work, is thereby an exact extension of the pure component result of Williamson and Jackson [Mol. Phys. 86, 819-836 (1995)].