Showing papers on "Quantum Monte Carlo published in 1990"

Reducing Quasi-Ergodic Behavior in Monte Carlo Simulations by J-Walking: Applications to Atomic Clusters

Abstract: A method is introduced that is easy to implement and greatly reduces the systematic error resulting from quasi‐ergodicity, or incomplete sampling of configuration space, in Monte Carlo simulations of systems containing large potential energy barriers. The method makes possible the jumping over these barriers by coupling the usual Metropolis sampling to the Boltzmann distribution generated by another random walker at a higher temperature. The basic techniques are illustrated on some simple classical systems, beginning for heuristic purposes with a simple one‐dimensional double well potential based on a quartic polynomial. The method’s suitability for typical multidimensional Monte Carlo systems is demonstrated by extending the double well potential to several dimensions, and then by applying the method to a multiparticle cluster system consisting of argon atoms bound by pairwise Lennard‐Jones potentials. Remarkable improvements are demonstrated in the convergence rate for the cluster configuration energy, ...

Correlated Monte Carlo wave functions for the atoms He through Ne

Abstract: We apply the variational Monte Carlo method to the atoms He through Ne. Our trial wave function is of the form introduced by Boys and Handy. We use the Monte Carlo method to calculate the first and second derivatives of an unreweighted variance and apply Newton’s method to minimize this variance. We motivate the form of the correlation function using the local current conservation arguments of Feynman and Cohen. Using a self‐consistent field wave function multiplied by a Boys and Handy correlation function, we recover a large fraction of the correlation energy of these atoms. We give the value of all variational parameters necessary to reproduce our wave functions. The method can be extended easily to other atoms and to molecules.

Maximum-entropy method for analytic continuation of quantum Monte Carlo data

Abstract: An outstanding problem in the simulation of condensed-matter phenomena is how to obtain dynamical information. We consider the numerical analytic continuation of imaginary-time quantum Monte Carlo data to obtain real-frequency spectral functions. This is an extremely ill-posed problem similar to the inversion of a Laplace transform. We suggest an image-reconstruction approach, which has been widely applied to data analysis in experimental research. Specifically, we apply the maximum-entropy method (ME) to the analytic continuation of quantum Monte Carlo data. We report encouraging preliminary results for the Fano-Anderson model of an impurity state in a continuum. The incorporation of additional prior information, such as sum rules and asymptotic behavior, can be expected to significantly improve results. We compare (ME) to alternative methods. We also discuss statistical error propagation for the analytic continuation problem via the likelihood function, which is independent of the choice of image-reconstruction method. This includes the sensitivity of the data to structure in the spectral function, the optimization of Monte Carlo simulations, and how to incorporate covariance in the statistical errors of the Monte Carlo method.

Quantum critical phenomena in one-dimensional Bose systems.

Abstract: We use quantum Monte Carlo techniques to study the critical properties of an interacting-boson model in one dimension. The phase diagram consists of a series of (Mott-) insulating phases at commensurate fillings and of a superfluid phase. From the critical behavior of the superfluid density and the compressibility, we measure the exponents \ensuremath{ u} and z, which agree with predictions based on a scaling analysis.

Variational quantum Monte Carlo nonlocal pseudopotential approach to solids: Formulation and application to diamond, graphite, and silicon

Abstract: A new method of calculating total energies of solids and atoms using nonlocal pseudopotentials in conjunction with the variational quantum Monte Carlo approach is presented in detail. The many-electron wave function is of the form of a Jastrow exponential factor multiplying a Slater determinant. By using pseudopotentials, the large fluctuations of the energies in the core region of the atoms which occur in quantum Monte Carlo all-electron calculations are avoided. The method is applied to calculate the binding energy and structural properties of diamond, graphite, and silicon. The results are in excellent agreement with experiment. Excellent results are also obtained for the electron affinities and ionization potentials of the carbon and silicon atoms.

Quantum Monte Carlo for the Electronic Structure of Atoms and Molecules

Abstract: Since the late 1960s molecular orbital (MO) theory has dominated the development of ab initio quantum chemistry. Specifically, the Hartree­ Fock-Roothaan (HFR) (1-3) formalism of the self-consistent field (SCF) method, which was greatly facilitated by the development of gaussian basis sets as suggested by Boys (4, 5), and the post-Hartree-Fock methods of configuration interaction (CI) (6-8), multiconfiguration SCF (MCSCF) (9, 10), many-body perturbation theory (MBPT) and coupled-cluster methods (CCM) (11), and Moller-Plesset perturbation theory (MPPT) (12). With the advent of high speed vector supercomputers the field has virtually exploded; larger molecules or very high accuracy for small molecules are now possible. In addition to such capability has come more accurate knowledge of the size of I-particle and N-particle basis sets required for high accuracy (13-16). Not only does the use of large basis sets require large memory and file-storage capability, but HFR and post-HFR algorithms contain significant bottlenecks to vectorization and parallelization. Although many new and efficient matrix methods have been developed

Critical slowing down

Abstract: The problem of critical slowing down in Monte Carlo simulations and some methods to alleviate or overcome it are reviewed: overrelaxation, multigrid and cluster algorithms.

Solving the sign problem in quantum Monte Carlo dynamics.

Abstract: A method of solving the sign problem in the Monte Carlo path-integral simulations of quantum dynamics is presented Our method is based on the distortion of integration contours in conjunction with a stationary-phase-filtering method Using this importance sampling, correlation functions for the spin-boson model have been computed for real times much longer than \ensuremath{\beta}\ensuremath{\Elzxh}

Comment on Reverse Monte Carlo Simulation

Abstract: In a recent paper McGreevy and Pusztai [1] have described an interesting simulation technique for determining the structure of disordered systems (liquids and glasses) that uses as input the experimentally measured radial distribution function g E(r 12), or equivalently, the structure factor a E(k). The essence of their procedure is to generate, via Monte Carlo, a set of particle configurations that yield a g(r 12), where r 12 [tbnd] |r 2—r 1| is the distance between particles, that is consistent with g E(r 12). From these configurations further information about the structure can be extracted, including higher-order correlation functions such as the 3-body function ρ(3) (r 1, r 2, r 3) or bond-angle distributions. Although the idea of reverse Monte Carlo (RMC) is not new (see the reference in [1]), McGreevy and co-workers have shown convincingly that it is computationally feasible and have produced results for a variety of liquids, including multicomponent systems. The RMC procedure is certainly...

The calculation of excited states with quantum Monte Carlo. II. Vibrational excited states

Abstract: Using a new Monte Carlo method for computing properties of excited quantum states, correlation function quantum Monte Carlo, we calculate the lowest ten vibrational excited state energies of H2O and H2CO in the Born–Oppenheimer approximation. The statistical errors for H2O are 0.1 cm−1 for the ground state and 15 cm−1 for the tenth excited state while for H2CO they are 2 cm−1 for the ground state and 30 cm−1 for the eighth excited state. The algorithm presented here is easily extensible to larger systems.

Potential energy surface and energy levels of (HF)2 and its D isotopomers

Abstract: A new six-dimensional analytical potential energy surface for the hydrogen bonded dimer (HF)2 is presented. It is based on the ab initio study by Kofranek et al., and uses 1070 points with energies up to 500 kJ mol-1 as well as recent correlated dispersion coefficients from a perturbation treatment by Rijks and Wormer, and the experimental Morse parameters for the HF monomer. The fit to the ab initio points contains 29 free and several constrained parameters and has a weighted standard deviation of 29·5 cm-1. A brief description of some properties of the new surface is given. Preliminary results of a diffusion quantum Monte Carlo (QMC) study for vibrational energy levels on the new surface as well as on a surface published by Bunker et al. are given. An interesting anharmonic isotope effect in the v 6 fundamental is discussed and explained.

Gaussian functions in Hylleraas‐configuration interaction calculations. V. An accurate ab initio H+3 potential‐energy surface

Abstract: The near‐equilibrium potential‐energy surface of the 1A’1 ground state of H+3 has been calculated at 69 different points with the Hylleraas‐configuration interaction method using 13s3p1d Cartesian Gaussian basis functions. This new surface is found to be substantially lower in absolute energy than all previous surface calculations. The equilibrium energy of the H+3 molecule has also been calculated with a larger 13s5p3d basis set. The minimum energy was found to be E=−1.343 827 9 hartrees at an internuclear distance of R=1.6500 bohrs in the equilateral triangle configuration. This energy is significantly (>70 cm−1) lower than the previous best published variational calculation and is outside and below the error bars of the latest quantum Monte Carlo calculation. In addition, a medium‐sized basis set of 13s4p2d orbitals was used to predict that the equilibrium separation is R=1.6499 bohrs.

Single-Cluster Monte Carlo Dynamics for the Ising Model

Abstract: We present an extensive study of a new Monte Carlo acceleration algorithm introduced by Wolff for the Ising model. It differs from the Swendsen-Wang algorithm by growing and flipping single clusters at a random seed. In general, it is more efficient than Swendsen-Wang dynamics ford>2, giving zero critical slowing down in the upper critical dimension. Monte Carlo simulations give dynamical critical exponentszw=0.33±0.05 and 0.44+0.10 ind=2 and 3, respectively, and numbers consistent withzw=0 ind=4 and mean-field theory. We present scaling arguments which indicate that the Wolff mechanism for decorrelation differs substantially from Swendsen-Wang despite the apparent similarities of the two methods.

Optimized trial functions for quantum Monte Carlo

Abstract: An algorithm to optimize trial functions for fixed‐node quantum Monte Carlo calculations has been developed based on variational random walks. The approach is applied to wave functions that are products of a simple Slater determinant and correlation factor explicitly dependent on interelectronic distance, and is found to provide improved ground‐state total energies. A modification of the method for ground‐states that makes use of a projection operator technique is shown to make possible the calculation of more accurate excited‐state energies. In this optimization method the Young tableaux of the permutation group is used to facilitate the treatment of fermion properties and multiplets. Application to ground states of H2, Li2, H3, H+3, and to the first‐excited singlets of H2, H3, and H4 are presented and discussed.

Nonlocal Monte Carlo algorithm for self-avoiding walks with fixed endpoints

Abstract: We study a new Monte Carlo algorithm for generating self-avoiding walks with variable length (controlled by a fugacityβ) and fixed endpoints. The algorithm is a hybrid of local (BFACF) and nonlocal (cut-and-paste) moves. We find that the critical slowing-down, measured in units of computer time, is reduced compared to the pure BFACF algorithm:τ CPU ∼ 〈N〉≈2.3 versus 〈N〉≈3.0. We also prove some rigorous bounds on the autocorrelation time for these and related Monte Carlo algorithms.

Slave-boson mean field versus quantum Monte Carlo results for the Hubbard model.

Abstract: The one-band Hubbard model is considered in the slave-boson formulation first introduced by Kotliar and Ruckenstein. It is shown that a mean-field approximation, where broken-symmetry states appropriate for a bipartite lattice are allowed, leads to a quantitative agreement with quantum Monte Carlo results for local observables over a wide range of interactions (0U\ensuremath{\le}20t) and doping (0n〉\ensuremath{\le}1). Thus, our saddle-point solutions constitute an excellent starting point for a systematic treatment of fluctuations.

The theory of hybrid stochastic algorithms

Abstract: These lectures introduce the family of Hybrid Stochastic Algorithms for performing Monte Carlo calculations in Quantum Field Theory. After explaining the basic concepts of Monte Carlo integration we discuss the properties of Markov processes and one particularly useful example of them: the Metropolis algorithm. Building upon this framework we consider the Hybrid and Langevin algorithms from the viewpoint that they are approximate versions of the Hybrid Monte Carlo method; and thus we are led to consider Molecular Dynamics using the Leapfrog algorithm. The lectures conclude by reviewing recent progress in these areas, explaining higher-order integration schemes, the asymptotic large-volume behaviour of the various algorithms, and some simple exact results obtained by applying them to free field theory. It is attempted throughout to give simple yet correct proofs of the various results encountered.

Computing parametric yield accurately and efficiently

Abstract: An algorithm for computing parametric yield is presented. The algorithm uses statistical modeling techniques and takes advantage of incremental knowledge of the problem to reduce significantly the number of simulations needed. Polynomial regression is used to construct simple equations mapping parameters to measurements. These simple polynomial equations can then replace circuit simulations in the Monte Carlo algorithm for computing parametric yield. The algorithm differs from previous statistical modeling algorithms using polynomial regression for three major reasons: first, the random error that is postulated in polynomial regression equations is taken into account when computing parametric yield; second, the variance of the yield is computed; and third, the algorithm is fully automated. Therefore a direct comparison with Monte Carlo methods can be made. Examples indicate that significant speed-ups can be attained over Monte Carlo methods for a large class of problems. >

Wave function optimization with a fixed sample in quantum Monte Carlo

Abstract: The optimization of trial functions consisting of a product of a single determinant and simple correlation functions is studied. The method involves minimizing the variance of the local energy over a finite number of points (sample). The role of optimization parameters, e.g., sample characteristics, initial trial function parameters, and reference energy, is examined for H2, Li2, and H2O. The extent to which cusp conditions are satisfied is also discussed. The resulting variational Monte Carlo energies 〈ΨT‖H‖ΨT〉 recover 46%–95% of the correlation energy for the simple trial function forms studied. When used as importance functions for quantum Monte Carlo calculations, these optimized trial functions recover 90%–100% of the correlation energy. Time‐step bias of the computed quantum Monte Carlo energies is found to be small.

Infinitesimal differential diffusion quantum Monte Carlo: Diatomic molecular properties

Abstract: We show how to estimate, for a given molecule, the first and higher derivatives of the expected value of an operator with respect to one or more physical parameters. This is done with high accuracy achieved by sampling to within a certain approximation from the exact electron distribution, compatible with the Hellmann–Feynman theorem. Finite difference approximations are avoided. The required derivatives of the unknown exact wave function are determined by averaging expressions involving only the total serial correlation of known quantities. The operator is not restricted to the case of the molecular Hamiltonian. This allows for computation of virtually all ground‐state properties of a molecule by a single, relatively trivial computer program. Our formulas are presented and applied in the context of a diatomic molecule (LiH), but they can be readily extended to polyatomics.

Gap states in dilute-magnetic-alloy superconductors: A quantum Monte Carlo study

Abstract: Using a combination of quantum Monte Carlo simulation, perturbation theory, and maximum-entropy analytic continuation, we calculate the density of states for a dilute-magnetic-alloy superconductor. We find that the peak location of the gap states moves more quickly with increasing Kondo temperature than predicted by Zittartz {ital et} {ital al}., and that the integrated intensity of the gap states is in extreme qualitative disagreement with that prediction.

Monte Carlo simulation of an ion-dipole mixture

Abstract: We demonstrate on a simple model of an ion-dipole mixture that the convergence of a Monte Carlo simulation at low ionic concentration may be quite slow. For a one molar concentration 105 trial moves per particle are needed for a 864 particle system to obtain a precise estimate of both the ion-dipole and ion-ion energies and the static pair correlation functions. Our results show that recently published Monte Carlo results (1989, Molec. Phys., 66, 299) on a similar system are far from being converged and question the conclusion drawn in this publication on the failure of the reference hypernetted chain equation.

Quantum Monte Carlo study of the one-dimensional symmetric Anderson lattice.

Abstract: Using a recently developed quantum Monte Carlo algorithm, we study the one-dimensional symmetric Anderson lattice. Through the use of boundary-condition averaging and imaginary-time increment extrapolation, we obtain controlled estimates for infinite-lattice properties, accurate to within a few percent, for a variety of parameters. Our results span high-temperature to, in some cases, ground-state properties. For U/${E}_{g}$\ensuremath{\lesssim}5, where U refers to the on-site Coulomb repulsion and ${E}_{g}$ is the U=0 gap, we find spin correlations which decay quickly with distance in the ground state and a saturation temperature, at which quantities saturate to their ground-state values, which is independent of U. For U/${E}_{g}$\ensuremath{\gtrsim}5, we find that nearby spin correlations decay more slowly with distance and saturation temperatures decrease with increasing U. For U/${E}_{g}$\ensuremath{\gtrsim}8, we find that the nearby-spin-correlation development above the saturation temperature is well described by an Ruderman-Kittel-Kasuya-Yosida (RKKY) lattice effective Hamiltonian with perturbatively calculated parameters. We interpret this general behavior as a smooth crossover at U/${E}_{g}$\ensuremath{\sim}5 from a basically noninteracting picture into a Kondo-lattice regime, parameterizable by RKKY interactions and a reduced effective gap. Next, using an exact canonical transformation, we point out what our results imply for superconductivity in an attractive-U Anderson lattice. Lastly, we discuss the relevance of our results to the magnetic behavior of heavy-fermion systems, which results underscore in particular the important role of RKKY interactions.

Inelastic electronic collision cross sections for Monte Carlo calculations

Abstract: For very thin absorbers, the energy losses of fast charged particles vary widely because of the stochastic nature of the interactions. In order to visualize and understand this process, a simulation with Monte Carlo calculations can be used. The distribution functions needed in these calculations, describing the electronic energy losses in single collisions of fast charged particles traversing silicon, have been found to depend only slightly on particle speed. This simplifies such calculations. A similar behaviour is expected to exist for other materials.