Correlation method for variance reduction of Monte Carlo integration in RS-HDMR

The High Dimensional Model Representation (HDMR) technique is a procedure for efficiently representing high-dimensional functions. A practical form of the technique, RS-HDMR, is based on randomly sampling the overall function and utilizing orthonormal polynomial expansions. The determination of expansion coefficients employs Monte Carlo integration, which controls the accuracy of RS-HDMR expansions. In this article, a correlation method is used to reduce the Monte Carlo integration error. The determination of the expansion coefficients becomes an iteration procedure, and the resultant RS-HDMR expansion has much better accuracy than that achieved by direct Monte Carlo integration. For an illustration in four dimensions a few hundred random samples are sufficient to construct an RS-HDMR expansion by the correlation method with an accuracy comparable to that obtained by direct Monte Carlo integration with thousands of samples.     

Quantum Monte Carlo study of first-row atoms using transcorrelated variational Monte Carlo trial functions

The effect of using the transcorrelated variational Monte Carlo (TC-VMC) approach to construct a trial function for fixed node diffusion Monte Carlo (DMC) energy calculations has been investigated for the first-row atoms, Li to Ne. The computed energies are compared with fixed node DMC energies obtained using trial functions constructed from Hartree-Fock and density functional levels of theory. Despite major VMC energy improvement with TC-VMC trial functions, no improvement in DMC energy was observed using these trial functions for the first-row atoms studied. The implications of these results on the nodes of the trial wave functions are discussed.     

Monte carlo integration with oscillatory integrands: implications for feynman path integration in real time

A new method is described for the Monte Carlo evaluation of integrals of the form

exp[iS(x)] that occur in the Feynman path integral representation of the time evolution operator, exp(−iHt/h). The method is general, strictly Monte Carlo based (and thus applicable to high dimensionality), and has the desirable feature that the stationary phase (i.e. semiclassical) approximation to the integral is obtained in its worst limit. Application to a non-trivial test case (the Airy integral) illustrates these features.     

Quantum Monte Carlo with Jastrow-valence-bond wave functions

We consider the use in quantum Monte Carlo calculations of two types of valence bond wave functions based on strictly localized active orbitals, namely valence bond self-consistent-field and breathing-orbital valence bond wave functions. Complemented by a Jastrow factor, these Jastrow-valence-bond wave functions are tested by computing the equilibrium well depths of the four diatomic molecules C(2), N(2), O(2), and F(2) in both variational Monte Carlo and diffusion Monte Carlo. We show that it is possible to design compact wave functions based on chemical grounds that are capable of describing both static and dynamic electron correlations. These wave functions can be systematically improved by inclusion of valence bond structures corresponding to additional bonding patterns.     

Monte Carlo study of electron correlation functions for small molecules

A study is made of electron-electron correlation functions for use in trial wave functions for small molecules. New forms are proposed that have only a few variational parameters, and these parameters have physical meanings that are easily discerned. Total energies for H₂, LiH and Li₂ computed using these correlation functions are presented, and comparison is made with previous forms, including the Jastrow-Pade form often used in Monte Carlo studies. We further treat the possibility that correlation depends not only on the separation of a pair of electrons but also on the location of the electron pair relative to the nuclei — indicative of a density-dependent or many body correlation effect. Our results indicate that such a many-body correlation effect is weakly present.This work was supported by the Director, Office of Energy Research, Office of Basic Energy Sciences, Chemical Sciences Division of the U.S. Department of Energy under Contract No. DE-AC03-76SF00098     

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.     

Monte Carlo calculation of angular transmission functions for small angle scattering

We define an angular transmission function η in the center of mass system. The convolution of the differential cross section σ with η yields the signal in the laboratory system. For the case of elastic small angle scattering by spherically symmetric potentials we calculate η by a Monte Carlo method. Random positions are taken in the beam defining collimators, resulting in a trajectory with a deflection angle at the scattering centre. These deflection angles are transformed to the c.m. system with the small angle tranformation formulae. From the distribution we calculate η as a histogram and the central moments of η. The function η depends on the velocity ratio and on the mass ratio of the scattering partners. We store the results in such a way that the central moments can be calculated afterwards for all mass and velocity ratios. By using the central moments the convolution integral can be reduced to a simple weighted sum of σ-values at different scattering angles. The r.m.s. deviations of the central moments scale with N^{$\text{1}\text{2}$}, with N the number of Monte Carlo trajectories. A typical deviation is 1% in the second order moment for N = 2 × 10⁴, increasing slightly with increasing order of the moments. This method of calculation gives a large degree of freedom for optimisation of the collimation geometry. The use of an angular transmission function defined in the center of mass system gives a good insight in the experimental reflection of the physical events. As an example we apply the method to the case of small angle scattering of Ar as a primary beam by Kr as a secondary beam and the inverse configuration of Kr by Ar.     

Optimization of quantum Monte Carlo wave functions by energy minimization

We study three wave function optimization methods based on energy minimization in a variational Monte Carlo framework: the Newton, linear, and perturbative methods. In the Newton method, the parameter variations are calculated from the energy gradient and Hessian, using a reduced variance statistical estimator for the latter. In the linear method, the parameter variations are found by diagonalizing a nonsymmetric estimator of the Hamiltonian matrix in the space spanned by the wave function and its derivatives with respect to the parameters, making use of a strong zero-variance principle. In the less computationally expensive perturbative method, the parameter variations are calculated by approximately solving the generalized eigenvalue equation of the linear method by a nonorthogonal perturbation theory. These general methods are illustrated here by the optimization of wave functions consisting of a Jastrow factor multiplied by an expansion in configuration state functions (CSFs) for the C2 molecule, including both valence and core electrons in the calculation. The Newton and linear methods are very efficient for the optimization of the Jastrow, CSF, and orbital parameters. The perturbative method is a good alternative for the optimization of just the CSF and orbital parameters. Although the optimization is performed at the variational Monte Carlo level, we observe for the C2 molecule studied here, and for other systems we have studied, that as more parameters in the trial wave functions are optimized, the diffusion Monte Carlo total energy improves monotonically, implying that the nodal hypersurface also improves monotonically.     

Rheology of polymer brush under oscillatory shear flow studied by nonequilibrium Monte Carlo simulation

The rheological behaviors of polymer brush under oscillatory shear flow were investigated by nonequilibrium Monte Carlo simulation. The grafted chain under oscillatory shear flow exhibited a waggling behavior like a flower, and the segments were found to have different oscillatory phases along the chain contour. Stress tensor was further obtained based on the statistics of sampled configuration distribution functions. The simulation reproduced the abrupt increase of the first normal stress difference N(1) with the flow velocity over a critical value, as observed in the experiment of Klein et al. [Nature (London) 352, 143 (1991)]. However, our simulation did not reproduce the brush thickening with shear velocity increased, which was suggested to be responsible for the abrupt increase of N(1) in the above-mentioned paper. This simulation demonstrates that the increase of normal stress might be an inherent behavior of polymer brush due to chain deformation under flow.     

Nonequilibrium Monte Carlo simulation of lattice block copolymer chains subject to oscillatory shear flow

This paper has extended nonequilibrium Monte Carlo (MC) approach to simulate oscillatory shear flow in a lattice block copolymer system. Phase transition and associated rheological behaviors of multiple self-avoiding chains have been investigated. Stress tensor has been obtained based upon sampled configuration distribution functions. At low temperatures, micellar structures have been observed and the underlying frequency-dependent rheological properties exhibit different initial slopes. The simulation outputs are consistent with the experimental observations in literature. Chain deformation during oscillatory shear flow has also been revealed. Although MC simulation cannot account for hydrodynamic interaction, the highlight of our simulation approach is that it can, at small computing cost, investigate polymer chains simultaneously at different spatial scales, i.e., macroscopic rheological behaviors, mesoscopic self-assembled structures, and microscopic chain configurations.     

Monte Carlo simulation and thermodynamic integration applied to protein charge transfer

We introduce a combination of Monte Carlo simulation and thermodynamic integration methods to address a model problem in free energy computations, electron transfer in proteins. The feasibility of this approach is tested using the ferredoxin protein from Clostridium acidurici. The results are compared to numerical solutions of the Poisson-Boltzmann equation and data from recent molecular dynamics simulations on charge transfer in a protein complex, the NrfHA nitrite reductase of Desulfovibrio vulgaris. Despite the conceptual and computational simplicity of the Monte Carlo approach, the data agree well with those obtained by other methods. A link to experiments is established via the cytochrome subunit of the bacterial photosynthetic reaction center of Rhodopseudomonas viridis.     

Linear expansions of correlated functions: Variational Monte Carlo case study

The relative performance of trial wave functions expressed as linear combination of correlated exponentials has been tested on a variety of systems. The results are compared against other correlated functions commonly used in the literature to assess the capabilities of the proposed ansatz. A possible departure from the simple exponential functional form used in previous works is discussed, along with its advantages and drawbacks. We also discuss how to implement an efficient optimization procedure for this correlated basis set. ©1999 John Wiley & Sons, Inc. Int J Quant Chem 74: 23–33, 1999     

A new variational Monte Carlo trial wavefunction with directional Jastrow functions

The addition of a directional term to standard Jastrow functions was investigated in variational quantum Monte Carlo calculations for the LiH molecule using wavefunctions composed of Slater determinants combined with Padé and with Schmidt–Moskowitz functions. The recovery of correlation energy was improved significantly by the added term: from 52% to 71% for the Padé function and from 61% to 78% for the Schmidt–Moskowitz function.     

A simple methodology for analyzing association effects on response functions via Monte Carlo simulations

A simple methodology was developed to analyze association effects on the thermodynamic response functions for a pure self-associated fluid via Monte Carlo simulations. The procedure essentially involves expressing the residual energy and volume of the fluid in terms of these properties for two hypothetical fluids consisting of monomers and associated molecules, respectively. This allows the thermodynamic response functions to be expressed in a perturbative form as a combination of the values for the property in the monomeric fluid and the contribution of association (the perturbative term). The proposed methodology was used to determine both contributions to the isobaric heat capacity and to the temperature and pressure derivatives of the volume for OPLS methanol along the 50 MPa isobar from 220 to 1500 K. Based on the results, both terms exert a substantial influence on the isobaric heat capacity; by contrast, the association term for the volumetric properties is negligible. These results are consistent with those of a previous work involving simulations with the same model under identical thermodynamic conditions but a different approach. They are also compared with others previously reported in context. Moreover, a comprehensive study of the different types of clusters present in the fluid was performed and the results were related to thermodynamic properties. A strong correlation between the heat capacity of the monomeric fluid and this structural analysis was found.     

Density functional theory guided Monte Carlo simulations: application to melting of Na13

We present a density functional theory (DFT) based Monte Carlo simulation method in which a simple energy function gets fitted on-the-fly to DFT energies and gradients. The fitness of the energy function gets tested periodically using the classical importance function technique [R. Iftimie, D. Salahub, D. Wei, and J. Schofield, J. Chem. Phys. 113, 4852 (2000)]. The function is updated to fit the DFT energies and gradients of the most recent structures visited whenever it fails to achieve a preset accuracy. In this way, we effectively break down the problem of fitting the entire potential energy surface (PES) into many easier problems, which are to fit small local regions of the PES. We used the scaled Morse potential empirical function to guide a DFT Monte Carlo simulation of Na(13) at various temperatures. The use of empirical function guide produced a computational speed-up of about 7 in our test system without affecting the quality of the results.     

Monte Carlo simulation of liquid hydroxylamine using ab initio intermolecular potential functions

A potential model for intermolecular interactions between hydroxylamine (NH₂OH) molecules based on ab initio quantum mechanical calculations is reviewed and critically assessed by analyzing results from a Monte Carlo simulation of liquid hydroxylamine. The liquid structure is studied in detail using radial, energy, and angular distribution functions, coordination numbers, and their distribution. Results indicate a large first solvation shell (5.3 Å), which contains 13 molecules, out of which only 4 are truly bonded by nonlinear, low-energy hydrogen bonds. These are of either the OH…O or the OH…N type, as NH…O and NH…N linear bonds are considerably suppressed, and no cyclic dimers are found. The dependence of the structural and physical properties on the simulation characteristics has also been investigated.     

Monte Carlo simulations of pure liquid substituted benzenes with OPLS potential functions

Intermolecular potential functions have been developed for use in computer simulations of substituted benzenes. Previously reported optimized potentials for liquid simulations (OPLS) for benzene and organic functional groups were merged and tested in Monte Carlo statistical mechanics simulations for the pure liquids of toluene, m-cresol, anisole, aniline, and benzonitrile at 25°C at 1 atm. The merged potential functions yielded acceptable thermodynamic results for the liquids except in the case of aniline, for which the error in the heat of vaporization was 12%. This was remedied by enhancing the polarity of the model to be more consistent with the observed dipole moment of aniline. Overall, the average errors in computed heats of vaporization and densities were then 2 and 1%, respectively. The structures of the liquids were characterized through energy and radial distribution functions. For m-cresol and aniline, the molecules participate in averages of 1.6 and 1.4 hydrogen bonds, respectively. Condensed phase effects on the torsional energies for anisole, m-cresol, and aniline were found to be small; m-cresol has a slightly enhanced tendency to be nonplanar in the liquid than in the gas phase, while anisole shows the opposite pattern. © 1993 John Wiley & Sons, Inc.     

Monte Carlo correction programme for PC

A Monte Carlo correction program for quantitative microanalysis on PC computer is introduced in this paper. The elastic scattering is described by the screened Rutherford cross section. Instead of computing the energy loss according to the actual path between two scatterings we have defined the Bethe inelastic cross section determined by the Bethe-slowing-down approximation. It is assumed that it causes no angular departure of the scattered electron. In the second model we took into account the angular dependence of inelastic scattering assuming that the primary electron interacts with quasi-free atom electrons. On the basis of these two models analytical Monte Carlo programmes were developed and experimentally tested on some oxide glass. Our results are fully comparable to those obtained by ten world microprobe laboratories using classical ZAF correction or Bence-Albee methods. We have found that introducing angular part of the inelastic cross section analytical results did not significantly change. All of our results were carried out for bulk specimens but extending it to layers is under the development.