Path-integral virial estimator for reaction-rate calculation based on the quantum instanton approximation

The quantum instanton approximation is a type of quantum transition-state theory that calculates the chemical reaction rate using the reactive flux correlation function and its low-order derivatives at time zero. Here we present several path-integral estimators for the latter quantities, which characterize the initial decay profile of the flux correlation function. As with the internal energy or heat-capacity calculation, different estimators yield different variances (and therefore different convergence properties) in a Monte Carlo calculation. Here we obtain a virial (-type) estimator by using a coordinate scaling procedure rather than integration by parts, which allows more computational benefits. We also consider two different methods for treating the flux operator, i.e., local-path and global-path approaches, in which the latter achieves a smaller variance at the cost of using second-order potential derivatives. Numerical tests are performed for a one-dimensional Eckart barrier and a model proton transfer reaction in a polar solvent, which illustrates the reduced variance of the virial estimator over the corresponding thermodynamic estimator.     

Fuzzy robust estimation of central location

The problem of a new robust algorithm for estimation of central location has been described in a mathematically simpler way using the fuzzy sets theory. It was compared with ordinary mean estimator and other robust estimators - median, 5% trimmed mean and Huber-, Tukey-, Hampel-, and Andrews-type M-estimators. The performance of Fuzzy 1-means algorithm (FM) proposed is demonstrated by applying it to different data sets from published literature and has been shown to exceed that of conventional ordinary mean estimator and equals or often exceeds that of the most robust estimators.     

Low variance energy estimators for systems of quantum Drude oscillators: treating harmonic path integrals with large separations of time scales

In the effort to develop atomistic models capable of accurately describing nanoscale systems with complex interfaces, it has become clear that simple treatments with rigid charge distributions and dispersion coefficients selected to generate bulk properties are insufficient to predict important physical properties. The quantum Drude oscillator model, a system of one-electron pseudoatoms whose "pseudoelectrons" are harmonically bound to their respective "pseudonuclei," is capable of treating many-body polarization and dispersion interactions in molecular systems on an equal footing due to the ability of the pseudoatoms to mimic the long-range interactions that characterize real materials. Using imaginary time path integration, the Drude oscillator model can, in principle, be solved in computer operation counts that scale linearly with the number of atoms in the system. In practice, however, standard expressions for the energy and pressure, including the commonly used virial estimator, have extremely large variances that require untenably long simulation times to generate converged averages. In this paper, low-variance estimators for the internal energy are derived, in which the large zero-point energy of the oscillators does not contribute to the variance. The new estimators are applicable to any system of harmonic oscillators coupled to one another (or to the environment) via an arbitrary set of anharmonic interactions. The variance of the new estimators is found to be much smaller than standard estimators in three example problems, a one-dimensional anharmonic oscillator and quantum Drude models of the xenon dimer and solid (fcc) xenon, respectively, yielding 2-3 orders of magnitude improvement in computational efficiency.     

Classification for high-throughput data with an optimal subset of principal components

High-throughput data have been widely used in biological and medical studies to discover gene and protein functions. Due to the high dimensionality, principal component analysis (PCA) is often involved for data dimension reduction. However, when a few principal components (PCs) are selected for dimension reduction or considered for dimension determination, they are typically ranked by their variances, eigenvalues. However, this approach is not always effective in subsequent multivariate analysis, particularly classification. To maximize information from data with a subset of the components, we apply a different ranking criterion, canonical variate criterion, which considers within- and between-group variance rather than total variance in the classical criterion. Four prevalent classification methods are considered and compared using leave-one-out cross-validation. These methods are illustrated with three real high-throughput data sets, two microarray data sets and a nuclear magnetic resonance spectra data set.     

Measuring surface topography by scanning electron microscopy. II. Analysis of three estimators of surface roughness in second dimension and third dimension.

Scanning electron microscopy (SEM) is widely used in surface studies and continuous efforts are carried out in the search of estimators of different surface characteristics. By using the variogram, we developed two of these estimators that were used to characterize the surface roughness from the SEM image texture. One of the estimators is related to the crossover between fractal region at low scale and the periodic region at high scale, whereas the other estimator characterizes the periodic region. In this work, a full study of these estimators and the fractal dimension in two dimensions (2D) and three dimensions (3D) was carried out for emery papers. We show that the obtained fractal dimension with only one image is good enough to characterize the roughness surface because its behavior is similar to those obtained with 3D height data. We show also that the estimator that indicates the crossover is related to the minimum cell size in 2D and to the average particle size in 3D. The other estimator has different values for the three studied emery papers in 2D but it does not have a clear meaning, and these values are similar for those studied samples in 3D. Nevertheless, it indicates the formation tendency of compound cells. The fractal dimension values from the variogram and from an area versus step log-log graph were studied with 3D data. Both methods yield different values corresponding to different information from the samples.     

Estimating equilibrium ensemble averages using multiple time slices from driven nonequilibrium processes: theory and application to free energies, moments, and thermodynamic length in single-molecule pulling experiments

Recently discovered identities in statistical mechanics have enabled the calculation of equilibrium ensemble averages from realizations of driven nonequilibrium processes, including single-molecule pulling experiments and analogous computer simulations. Challenges in collecting large data sets motivate the pursuit of efficient statistical estimators that maximize use of available information. Along these lines, Hummer and Szabo developed an estimator that combines data from multiple time slices along a driven nonequilibrium process to compute the potential of mean force. Here, we generalize their approach, pooling information from multiple time slices to estimate arbitrary equilibrium expectations. Our expression may be combined with estimators of path-ensemble averages, including existing optimal estimators that use data collected by unidirectional and bidirectional protocols. We demonstrate the estimator by calculating free energies, moments of the polymer extension, the thermodynamic metric tensor, and the thermodynamic length in a model single-molecule pulling experiment. Compared to estimators that only use individual time slices, our multiple time-slice estimators yield substantially smoother estimates and achieve lower variance for higher-order moments.     

Ratio control variate method for efficiently determining high-dimensional model representations

The High-Dimensional Model Representation (HDMR) technique is a family of approaches to efficiently interpolate high-dimensional functions. RS(Random Sampling)-HDMR is a practical form of HDMR based on randomly sampling the overall function, and utilizing orthonormal polynomial expansions to approximate the RS-HDMR component functions. The determination of the expansion coefficients for the component functions employs Monte Carlo integration, which controls the accuracy of the RS-HDMR interpolation. The control variate method is an established approach to improve the accuracy of Monte Carlo integration. However, this method is often not practical for an arbitrary function f(x) because there is no general way to find the analytical control variate function h(x), which needs to be very similar to f(x). In this article, we show that truncated RS-HDMR expansions can be used as h(x) for arbitrary f(x), and a more stable iterative ratio control variate method was developed for the determination of the expansion coefficients for the RS-HDMR component functions. As the asymptotic error (standard deviation) of the estimator given by the ratio control variate method is proportional to 1/N(sample size), it is more efficient than the direct Monte Carlo integration, whose error is proportional to 1/square root(N). In an illustration of a four-dimensional atmospheric model a few hundred random samples are sufficient to construct an RS-HDMR expansion by the ratio control variate method with an accuracy comparable to that obtained by direct Monte Carlo integration with thousands of samples.     

Path-integral virial estimator based on the scaling of fluctuation coordinates: application to quantum clusters with fourth-order propagators

We first show that a simple scaling of fluctuation coordinates defined in terms of a given reference point gives the conventional virial estimator in discretized path integral, where different choices of the reference point lead to different forms of the estimator (e.g., centroid virial). The merit of this procedure is that it allows a finite-difference evaluation of the virial estimator with respect to temperature, which totally avoids the need of higher-order potential derivatives. We apply this procedure to energy and heat-capacity calculations of the (H(2))(22) and Ne(13) clusters at low temperature using the fourth-order Takahashi-Imada [J. Phys. Soc. Jpn. 53, 3765 (1984)] and Suzuki [Phys. Lett. A 201, 425 (1995)] propagators. This type of calculation requires up to third-order potential derivatives if analytical virial estimators are used, but in practice only first-order derivatives suffice by virtue of the finite-difference scheme above. From the application to quantum clusters, we find that the fourth-order propagators do improve upon the primitive approximation, and that the choice of the reference point plays a vital role in reducing the variance of the virial estimator.     

Dynamic information for cardiotoxin protein desorption from a methyl-terminated self-assembled monolayer using steered molecular dynamics simulation

Dynamic information, such as force, structural change, interaction energy, and potential of mean force (PMF), about the desorption of a single cardiotoxin (CTX) protein from a methyl-terminated self-assembled monolayer (SAM) surface was investigated by means of steered molecular dynamics (SMD) simulations. The simulation results indicated that Loop I is the first loop to depart from the SAM surface, which is in good agreement with the results of the nuclear magnetic resonance spectroscopy experiment. The free energy landscape and the thermodynamic force of the CTX desorption process was represented by the PMF and by the derivative of PMF with respect to distance, respectively. By applying Jarzynski's equality, the PMF can be reconstructed from the SMD simulation. The PMFs, calculated by different estimators based upon Jarzynski's equality, were compared with the conventional umbrella sampling method. The best estimation was obtained by using the fluctuation-dissipation estimator with a pulling velocity of v = 0.25 nm/ns for the present study.     

Density-dependent analysis of nonequilibrium paths improves free energy estimates II. A Feynman-Kac formalism

The nonequilibrium fluctuation theorems have paved the way for estimating equilibrium thermodynamic properties, such as free energy differences, using trajectories from driven nonequilibrium processes. While many statistical estimators may be derived from these identities, some are more efficient than others. It has recently been suggested that trajectories sampled using a particular time-dependent protocol for perturbing the Hamiltonian may be analyzed with another one. Choosing an analysis protocol based on the nonequilibrium density was empirically demonstrated to reduce the variance and bias of free energy estimates. Here, we present an alternate mathematical formalism for protocol postprocessing based on the Feynmac-Kac theorem. The estimator that results from this formalism is demonstrated on a few low-dimensional model systems. It is found to have reduced bias compared to both the standard form of Jarzynski's equality and the previous protocol postprocessing formalism.     

Evaluation of configurational entropy methods from peptide folding-unfolding simulation

A 4-micros molecular dynamics simulation of the second beta-hairpin of the B1 domain of streptococcal protein G is used to characterize the free energy surface and to evaluate different configurational entropy estimators. From the equilibrium folding-unfolding trajectory, 200 000 conformers are clustered according to their root-mean-square deviation (RMSD). The height of the free energy barrier between pairs of clusters is found to be significantly correlated with their pairwise RMSD. Relative free energies and relative configurational entropies of the clusters are determined by explicit evaluation of the partition functions of the different clusters. These entropies are used to evaluate different entropy estimators for the largest 20 clusters as well as a subensemble comprising exclusively extended conformers. It is found that the quasi-harmonic entropy estimator operating in dihedral angle space performs better than the one using Cartesian coordinates. A recent generalization of the quasi-harmonic approach that computes Shannon entropies of probability distributions obtained by projecting the conformers along the eigenvectors of the covariance matrix performs similarly well. For the best entropy estimators, a linear correlation coefficient between 0.92 and 0.97 is found. Unexpectedly, when correlations between dihedral angles are neglected, the agreement with the reference entropies improved.     

Structure-activity relationships of pyrethroid insecticides. Part 2. The use of molecular dynamics for conformation searching and average parameter calculation

Summary Molecular dynamics simulations have been performed on a number of conformationally flexible pyrethroid insecticides. The results indicate that molecular dynamics is a suitable tool for conformational searching of small molecules given suitable simulation parameters. The structures derived from the simulations are compared with the static conformation used in a previous study. Various physicochemical parameters have been calculated for a set of conformations selected from the simulations using multivariate analysis. The averaged values of the parameters over the selected set (and the factors derived from them) are compared with the single conformation values used in the previous study.     

Cu-Ni合金结构的分子动力学模拟

采用EAM作用势对Cu-Ni合金的结构特性进行了MD模拟研究.通过FZ结构因子可发现,Cu含量的变化对结构因子的波动影响很小,键取向序参数和键对也表现出相似的变化规律,这表明液态Cu-Ni合金对成份变化不敏感,体系中的化学序较弱.将Cu70Ni30合金熔体的FZ结构因子与Waseda的实验结果进行对比,发现二者吻合得较好,表明EAM势可以很好地描绘Cu-Ni合金的结构特性.在快速冷却过程中,除了Cu20Ni80合金外,其他合金成份的双体分布函数的第二峰都发生了劈裂,标志着体系最终形成了非晶结构,而Cu20Ni80合金的双体分布函数却表现出晶体峰的特征.通过对键取向序参数、键型指数以及铜镍原子的有效扩散系数的分析表明,在快速冷却过程中,Cu20Ni80合金最终形成了hcp晶体结构.     

Zero-variance zero-bias quantum Monte Carlo estimators for the electron density at a nucleus

We derive new quantum Monte Carlo (QMC) estimators for the electronic density at the position of a point nucleus using the zero-variance and zero-bias principles. The resulting estimators are highly efficient, and are significantly simpler to implement and use than alternative methods, as they contain no adjustable parameters. In addition, they can be used in both variational and diffusion QMC calculations. Our best estimator is used to calculate the most accurate available estimates of the total electron density at the nucleus for the first-row atoms Li-Ne, the Ar atom, and the diatomic molecules B(2), N(2), and F(2).     

A novel fusion approach based on induced ordered weighted averaging operators for chemometric data analysis

This paper proposes a novel approach for the estimation of spectroscopic data by combining the predictions of an ensemble of estimators using the induced ordered weighted averaging (IOWA) fusion operators. For ensemble generation, we use Gaussian process regression (GPR) and extreme learning machine (ELM) estimators associated with different kernels. To render the model selection issue of ELM as efficiently as in the GPR Bayesian estimation method, we develop an automatic solution based on the powerful differential evolution (DE) algorithm. During the fusion process, the IOWA operator needs two things: (1) an order‐inducing value; and (2) a way to determine its weights. For the order‐inducing value, we propose to use the residual of each estimated output value. Because we cannot compute the true residual, we explore the idea of estimating the residuals themselves by associating to each estimator of the ensemble a second estimator of the same kind called a residual estimator. To learn the weights associated with these nonlinear operators, the proposed method relies on the concept of prioritized aggregation, where we generate the weights directly from the estimated residuals. Experimental results obtained on three real spectroscopic datasets confirm the interesting capabilities of the proposed IOWA fusion method. Copyright © 2013 John Wiley & Sons, Ltd.     

Comparison of biased and unbiased estimators of variances of qualitative and semi-quantitative results of testing

Unbiased estimators of within-laboratory and between-laboratory (or within reference material unit and between-unit) variances of results of qualitative and semi-quantitative testing are formulated and discussed. Qualitative and semi-quantitative test results were treated as binary nominal and ordinal values, respectively, in framework of the newly developed ordinal analysis of variance (ORDANOVA). It is shown that the difference of the unbiased and the biased estimators of a within-laboratory variance does not exceed 5 %, when the number of replicate tests in a laboratory is larger than 20. Such a difference is increasing when the replicate number is decreasing, not depending on the number of laboratories and the between-laboratory variation, since both the unbiased and the biased estimators are based on the averaged within-laboratory variances. The difference of the unbiased and the biased estimators of the between-laboratory variance depends not only on the number of replicates, but also on the number of laboratories and on the ratio of the contributions to the total variance (the between-laboratory variance and the averaged within-laboratory variance). This difference does not exceed 5 %, when the number of replicates and the number of laboratories are larger than 20 and the ratio of the between-laboratory to the averaged within-laboratory variances does not yield 1. For a limited size of experiment (smaller numbers of replicates and laboratories), the difference is increasing with the size decreasing and can be significant.     

BAR-based optimum adaptive steered MD for configurational sampling

The equilibrium and nonequilibrium adaptive alchemical free energy simulation methods optimum Bennett's acceptance ratio and optimum crooks' equation (OCE), based on the statistically optimal bidirectional reweighting estimator named Bennett's Acceptance Ratio or Crooks' equation, perform initial sampling in the staging alchemical transformation and then determine the importance rank of different states via the time-derivative of the variance. The method is proven to give speedups compared with the equal time rule. In the current work, we extend the time derivative of variance guided adaptive sampling method to the configurational space, falling in the term of steered MD (SMD). The SMD approach biasing physically meaningful collective variable (CV) such as one dihedral or one distance to pulling the system from one conformational state to another. By minimizing the variance of the free energy differences along the pathway in an optimized way, a new type of adaptive SMD (ASMD) is introduced. As exhibits in the alchemical case, this adaptive sampling method outperforms the traditional equal-time SMD in nonequilibrium stratification. Also, the method gives much more efficient calculation of potential of mean force than the selection criterion-based ASMD scheme, which is proven to be more efficient than traditional SMD. The OCE workflow is periodicity-of-CV dependent while ASMD is not. The performance is demonstrated in a dihedral flipping case and two distance pulling cases, accounting for periodic and nonperiodic CVs, respectively. © 2019 Wiley Periodicals, Inc.     

Monte Carlo Simulation in Electron Probe Microanalysis. Comparison of Different Simulation Algorithms

Practical aspects of Monte Carlo simulation of EPMA experiments are considered. Simulations are performed using the general-purpose Monte Carlo code system Penelope, which is briefly described. This code includes geometry tools and variance reduction methods that allow the practical simulation of x-ray spectra from samples with complex geometries in moderate computing times. The reliability of simple interaction models and approximations, which have been frequently used in EPMA studies, is analyzed by studying their effects on the simulated x-ray spectra.     

铝锌合金的微观结构及抗腐蚀行为研究

Electro-chemical experiment was carded out to test the corrosion rates of aluminium-zinc hot-dip coating. It is shown that 5.3% aluminium-zinc alloy （weight ratio） has superior anti-corrosion property. The determined microstructure has displayed amorphous structure composed of nanometer sized particle of the system. The analysis indicated large negative change of Gibbs energy of 5.3% aluminium-zinc system. Molecular dynamics simulation showed that 5.3% aluminium-zinc system has very different behavior from other systems. A phase transition of this particular system was observed from simulation. The transition temperature was determined around 400 K. The simulation indicated that 5.3% aluminium-zinc system is amorphous over temperature range from 300 to 900 K, supporting the inference from experiments that amorphous solid of aluminium-zinc alloy has special anti-corrosion character.