Cool walking: a new Markov chain Monte Carlo sampling method

Effective relaxation processes for difficult systems like proteins or spin glasses require special simulation techniques that permit barrier crossing to ensure ergodic sampling. Numerous adaptations of the venerable Metropolis Monte Carlo (MMC) algorithm have been proposed to improve its sampling efficiency, including various hybrid Monte Carlo (HMC) schemes, and methods designed specifically for overcoming quasi-ergodicity problems such as Jump Walking (J-Walking), Smart Walking (S-Walking), Smart Darting, and Parallel Tempering. We present an alternative to these approaches that we call Cool Walking, or C-Walking. In C-Walking two Markov chains are propagated in tandem, one at a high (ergodic) temperature and the other at a low temperature. Nonlocal trial moves for the low temperature walker are generated by first sampling from the high-temperature distribution, then performing a statistical quenching process on the sampled configuration to generate a C-Walking jump move. C-Walking needs only one high-temperature walker, satisfies detailed balance, and offers the important practical advantage that the high and low-temperature walkers can be run in tandem with minimal degradation of sampling due to the presence of correlations. To make the C-Walking approach more suitable to real problems we decrease the required number of cooling steps by attempting to jump at intermediate temperatures during cooling. We further reduce the number of cooling steps by utilizing "windows" of states when jumping, which improves acceptance ratios and lowers the average number of cooling steps. We present C-Walking results with comparisons to J-Walking, S-Walking, Smart Darting, and Parallel Tempering on a one-dimensional rugged potential energy surface in which the exact normalized probability distribution is known. C-Walking shows superior sampling as judged by two ergodic measures.     

Transition network based on equilibrium sampling: a new method for extracting kinetic information from Monte Carlo simulations of protein folding

We propose a new type of transition network for modeling of protein dynamics. The nodes of the network correspond to the conformations taken from random sampling of equilibrium ensemble available, e.g., by Monte Carlo simulations. Although this approach does not provide absolute values of folding/unfolding rates, it allows identification of reaction pathways, transition state ensemble, and, eventually, intermediates. The new method is verified by a comparison with direct molecular dynamic simulations performed for a coarse-grained Gō-like model of proteins. As an illustrative example, we analyze kinetics of formation of a small β-hairpin (Trp zipper 1) in the all-atom representation.     

Excited electronic state calculations by the transcorrelated variational Monte Carlo method: application to a helium atom

We have implemented the excited electronic state calculations for a helium atom by the transcorrelated variational Monte Carlo (TC-VMC) method. In this method, Jastrow-Slater-type wave function is efficiently optimized not only for the Jastrow factor but also for the Slater determinant. Since the formalism for the TC-VMC method is based on the variance minimization, excited states as well as the ground state calculations are feasible. It is found that both the first and the second excitation energies given by TC-VMC are much closer to the experimental data than those given by the variational Monte Carlo method with using the Hartree-Fock orbitals. The successful results in the TC-VMC method are considered to be due to the nodal optimization of the wave functions.     

Efficient sampling algorithms for Monte Carlo based treatment planning

Efficient sampling algorithms are necessary for producing a fast Monte Carlo based treatment planning code. This study evaluates several aspects of a photon-based tracking scheme and the effect of optimal sampling algorithms on the efficiency of the code. Four areas were tested: pseudo-random number generation, generalized sampling of a discrete distribution, sampling from the exponential distribution, and delta scattering as applied to photon transport through a heterogeneous simulation geometry. Generalized sampling of a discrete distribution using the cutpoint method can produce speedup gains of one order of magnitude versus conventional sequential sampling. Photon transport modifications based upon the delta scattering method were implemented and compared with a conventional boundary and collision checking algorithm. The delta scattering algorithm is faster by a factor of six versus the conventional algorithm for a boundary size of 5 mm within a heterogeneous geometry. A comparison of portable pseudo-random number algorithms and exponential sampling techniques is also discussed.     

应用Monte Carlo平均值法计算STO双中心重迭积分

把Monte Carlo方法引进STO双中心重叠积分的计算中,结果表明,它不仅计算简便、快速、很容易在计算机上实现,而且具有较高的精确度,有望推广应用于更复杂的多中心分子积分中.     

Energy-consistent pseudopotentials for quantum Monte Carlo calculations

The authors present scalar-relativistic energy-consistent Hartree-Fock pseudopotentials for the main-group elements. The pseudopotentials do not exhibit a singularity at the nucleus and are therefore suitable for quantum Monte Carlo (QMC) calculations. They demonstrate their transferability through extensive benchmark calculations of atomic excitation spectra as well as molecular properties. In particular, they compute the vibrational frequencies and binding energies of 26 first- and second-row diatomic molecules using post-Hartree-Fock methods, finding excellent agreement with the corresponding all-electron values. They also show their pseudopotentials give superior accuracy than other existing pseudopotentials constructed specifically for QMC. Finally, valence basis sets of different sizes (VnZ with n=D,T,Q,5 for first and second rows, and n=D,T for third to fifth rows) optimized for our pseudopotentials are also presented.     

A rare event sampling method for diffusion Monte Carlo using smart darting

We identify a set of multidimensional potential energy surfaces sufficiently complex to cause both the classical parallel tempering and the guided or unguided diffusion Monte Carlo methods to converge too inefficiently for practical applications. The mathematical model is constructed as a linear combination of decoupled Double Wells [(DDW)(n)]. We show that the set (DDW)(n) provides a serious test for new methods aimed at addressing rare event sampling in stochastic simulations. Unlike the typical numerical tests used in these cases, the thermodynamics and the quantum dynamics for (DDW)(n) can be solved deterministically. We use the potential energy set (DDW)(n) to explore and identify methods that can enhance the diffusion Monte Carlo algorithm. We demonstrate that the smart darting method succeeds at reducing quasiergodicity for n ? 100 using just 1 × 10(6) moves in classical simulations (DDW)(n). Finally, we prove that smart darting, when incorporated into the regular or the guided diffusion Monte Carlo algorithm, drastically improves its convergence. The new method promises to significantly extend the range of systems computationally tractable by the diffusion Monte Carlo algorithm.     

An efficient sampling algorithm for variational Monte Carlo

We propose a new algorithm for sampling the N-body density mid R:Psi(R)mid R:(2)R(3N)mid R:Psimid R:(2) in the variational Monte Carlo framework. This algorithm is based upon a modified Ricci-Ciccotti discretization of the Langevin dynamics in the phase space (R,P) improved by a Metropolis-Hastings accept/reject step. We show through some representative numerical examples (lithium, fluorine, and copper atoms and phenol molecule) that this algorithm is superior to the standard sampling algorithm based on the biased random walk (importance sampling).     

A general framework for non-Boltzmann Monte Carlo sampling

Non-Boltzmann sampling (NBS) methods have been extensively employed in recent years, mainly due to their ability to enhance ergodicity in simulations of complex systems. In addition, they make possible reliable computation of equilibrium properties (ensemble averages, free-energy differences, and potentials of mean force) over continuous ranges of thermodynamic conditions. In this work, we put forward a general and systematic framework for NBS methods that allows a single set of equations and procedures to be applied to diverse systems. Moreover, we show how to exploit simulation data most effectively by obtaining continuous profiles of any mechanical properties, including structural quantities not directly related to the ensemble parameters. Finally, we demonstrate the usefulness of the developed formulation by applying it to spin systems, Lennard-Jones fluids, and a model protein molecule (both in isolation and in the proximity of a flat wall).     

Quantum Monte Carlo calculations on Be and LiH

Quantum Monte Carlo (QMC) methods, as recently developed for molecular systems, are applied to Be and LiH. The importance of the trial wavefunction, written as a product of a correlation factor and an orbital part, is emphasised. It is shown that significant improvements in the accuracy of the approach are achieved if multi-configuration wavefunctions are used in preference to self-consistent field wavefunctions. Various forms of the correlation factor are investigated.     

A new algorithm for the fixed-node quantum Monte Carlo method

A novel algorithm is proposed for the fixed-node quantum Monte Carlo (FNQMC) method.In contrast to previous procedures,its "guiding function" is not optimized prior to diffusion quantum Monte Carlo (DMC) computation but synchronistically in the diffusion process The new algorithm can not only save CPU time,but also make both of the optimization and diffusion carried out according to the same sampling fashion,reaching the goal to improve each other This new optimizing procedure converges super-linearly,and thus can accelerate the particle diffusion During the diffusion process,the node of the "guiding function" changes incessantly,which is conducible to reducing the "fixed-node error" The new algorithm has been used to calculate the total energies of states X3B1 and a1A1 of CH2 as well as π-X2B1 and λ-2A1 of NH2 The singlet-triplet energy splitting (λEsT) in CH2 and π energy splitting in NH2 obtained with this present method are (45 542±1.840) and (141.644±1.589) kJ/mol,respectively The calculated     

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     

Optimum and efficient sampling for variational quantum Monte Carlo

Quantum mechanics for many-body systems may be reduced to the evaluation of integrals in 3N dimensions using Monte Carlo, providing the Quantum Monte Carlo ab initio methods. Here we limit ourselves to expectation values for trial wave functions, that is to variational quantum Monte Carlo. Almost all previous implementations employ samples distributed as the physical probability density of the trial wave function, and assume the central limit theorem to be valid. In this paper we provide an analysis of random error in estimation and optimization that leads naturally to new sampling strategies with improved computational and statistical properties. A rigorous lower limit to the random error is derived, and an efficient sampling strategy presented that significantly increases computational efficiency. In addition the infinite variance heavy tailed random errors of optimum parameters in conventional methods are replaced with a Normal random error, strengthening the theoretical basis of optimization. The method is applied to a number of first row systems and compared with previously published results.     

Monte Carlo sampling algorithm for searching a scale-transformed energy space of polypeptides

A Monte Carlo sampling algorithm for searching a scale-transformed conformational energy space of polypeptides is presented. This algorithm is based on the assumption that energy barriers can be overcome by a uniform sampling of the logarithmically transformed energy space. This algorithm is tested with Met-enkephalin. For comparison, the entropy sampling Monte Carlo (ESMC) simulation is performed. First, the global minimum is easily found by the optimization of a scale-transformed energy space. With a new Monte Carlo sampling, energy barriers of 3000 kcal/mol are frequently overcome, and low-energy conformations are sampled more efficiently than with ESMC simulations. Several thermodynamic quantities are calculated with good accuracy.     

Conformational distribution of heptaalanine: Analysis using a new Monte Carlo chain growth method

A distribution of conformations of heptaalanine is obtained using a new Monte Carlo (MC) method that grows the chain atom by atom. Resulting configurations are energy minimized and a detailed analysis is performed of the minimum-energy conformers using a method of classification that partitions ?ψ space. The MC-generated configurations are compared with those generated from high-temperature molecular dynamics (MD) simulations. It is found that the new method generates a wide distribution of low-energy conformers at least 10 times more quickly than the MD. An analysis of the generated energy minima demonstrates that they can be divided into clusters in the space defined by the five pairs of ?—ψ angles of the inner residues. The space occupied by the minima populating each cluster is restricted. © 1992 by John Wiley & Sons, Inc.     

Monte Carlo and modified Tanford-Kirkwood results for macromolecular electrostatics calculations

The understanding of electrostatic interactions is an essential aspect of the complex correlation between structure and function of biological macromolecules. It is also important in protein engineering and design. Theoretical studies of such interactions are predominantly done within the framework of Debye-Hückel theory. A classical example is the Tanford-Kirkwood (TK) model. Besides other limitations, this model assumes an infinitesimally small macromolecule concentration. By comparison to Monte Carlo (MC) simulations, it is shown that TK predictions for the shifts in ion binding constants upon addition of salt become less reliable even at moderately macromolecular concentrations. A simple modification based on colloidal literature is suggested to the TK scheme. The modified TK models suggested here satisfactorily predict MC and experimental shifts in the calcium binding constant as a function of protein concentration for the calbindin D(9k) mutant and calmodulin.     

Temperature accelerated Monte Carlo (TAMC): a method for sampling the free energy surface of non-analytical collective variables

We introduce a new method to simulate the physics of rare events. The method, an extension of the Temperature Accelerated Molecular Dynamics, comes in use when the collective variables introduced to characterize the rare events are either non-analytical or so complex that computing their derivative is not practical. We illustrate the functioning of the method by studying the homogeneous crystallization in a sample of Lennard-Jones particles. The process is studied by introducing a new collective variable that we call Effective Nucleus Size N. We have computed the free energy barriers and the size of critical nucleus, which result in agreement with data available in the literature. We have also performed simulations in the liquid domain of the phase diagram. We found a free energy curve monotonically growing with the nucleus size, consistent with the liquid domain.