M. P. Nightingale

A diffusion Monte Carlo algorithm with very small time-step errors

We propose modifications to the simple diffusion Monte Carlo algorithm that greatly reduce the time‐step error. The improved algorithm has a time‐step error smaller by a factor of 70 to 300 in the energy of Be, Li2 and Ne. For other observables the improvement is yet larger. The effective time step possible with the improved algorithm is typically a factor of a few hundred larger than the time step used in domain Green function Monte Carlo. We also present an optimized 109 parameter trial wave function for Be which, used in combination with our algorithm, yields an exceedingly accurate ground state energy. A simple solution to the population control bias in diffusion Monte Carlo is also discussed.

Quantum Monte Carlo methods in physics and chemistry

Preface. 1. Basics, Quantum Monte Carlo and Statistical Mechanics M.P. Nightingale. 2. Stochastic Diagonalization H. de Raedt, et al. 3. World-Line Quantum Monte Carlo R.T. Scalettar. 4. Variational Monte Carlo in Solids S. Fahy. 5. Variational Monte Carlo Basics and Applications to Atoms and Molecules C.J. Umrigar. 6. Calculations of Exchange Frequencies with Path Integral Monte Carlo: Solid 3He Adsorbed on Graphite B. Bernu, D. Ceperley. 7. Static Response of Homogeneous Quantum Fluids by Diffusion Monte Carlo G. Senatore, et al. 8. Equilibrium and Dynamical Path Integral Methods: An Introduction J.D. Doll, et al. 9. Diffusion Monte Carlo L. Mitas. 10. Fermion Monte Carlo M.H. Kalos, F. Pederiva. 11. Quantum Monte Carlo in Nuclear Physics J. Carlson. 12. Reputation Quantum Monte Carlo: A Round-Trip Tour from Classical Diffusion to Quantum Mechanics S. Baroni, S. Moroni. 13. Quantum Monte Carlo for Lattice Fermions A. Muramatsu. 14. Phase Separation in the 2D Hubbard Model: A Challenging Application of Fixed-Node QMC G.B. Bachelet, A.C. Cosentini. 15. Constrained Path Monte Carlo for Fermions Shiwei Zhang. 16. Serial and Parallel Random Number Generation M. Mascagni. 17. Fixed-Node DMC for Fermions on a Lattice: Application to Doped Fullerides E. Koch, et al. 18. Index.

Optimization of ground- and excited-state wave functions and van der Waals clusters.

A quantum Monte Carlo method is introduced to optimize excited-state trial wave functions. The method is applied in a correlation function Monte Carlo calculation to compute ground- and excited-state energies of bosonic van der Waals clusters of up to seven particles. The calculations are performed using trial wave functions with general three-body correlations.

Accuracy of electronic wave functions in quantum Monte Carlo: The effect of high-order correlations

Compact and accurate wave functions can be constructed by quantum Monte Carlo methods. Typically, these wave functions consist of a sum of a small number of Slater determinants multiplied by a Jastrow factor. In this paper we study the importance of including high-order, nucleus-three-electron correlations in the Jastrow factor. An efficient algorithm based on the theory of invariants is used to compute the high-body correlations. We observe significant improvements in the variational Monte Carlo energy and in the fluctuations of the local energies but not in the fixed-node diffusion Monte Carlo energies. Improvements for the ground states of physical, fermionic atoms are found to be smaller than those for the ground states of fictitious, bosonic atoms, indicating that errors in the nodal surfaces of the fermionic wave functions are a limiting factor.

MANY-BODY TRIAL WAVE FUNCTIONS FOR ATOMIC SYSTEMS AND GROUND STATES OF SMALL NOBLE GAS CLUSTERS

Clusters of sizes ranging from two to five are studied by variational quantum Monte Carlo techniques. The clusters consist of Ar, Ne, and hypothetical lighter (‘‘1/2‐Ne’’) atoms. A general form of trial function is developed for which the variational bias is considerably smaller than the statistical error of currently available diffusion Monte Carlo estimates. The trial functions are designed by a careful analysis of long‐ and short‐range behavior as a function of inter‐atomic distance; at intermediate distances, on the order of the average nearest neighbor distance, the trial functions are constructed to have considerable variational freedom. A systematic study of the relative importance of n‐body contributions to the quality of the optimized trial wave function is made with 2≤n≤5. Algebraic invariants are employed to deal efficiently with the many‐body interactions.

Monte Carlo computation of correlation times of independent relaxation modes at criticality

We investigate aspects of universality of Glauber critical dynamics in two dimensions. We compute the critical exponent

Universality in two-dimensional Ising models

z

Chiral exponents of the square-lattice frustrated XY model: A Monte Carlo transfer-matrix calculation

and numerically corroborate its universality for three different models in the static Ising universality class and for five independent relaxation modes. We also present evidence for universality of amplitude ratios, which shows that, as far as dynamic behavior is concerned, each model in a given universality class is characterized by a single non-universal metric factor which determines the overall time scale. This paper also discusses in detail the variational and projection methods that are used to compute relaxation times with high accuracy.

Conformal Anomaly and Critical Exponents of the XY Ising Model

We have used finite-size scaling and transfer matrix techniques to calculate accurately the critical exponents of three two-dimensional Ising-like models for which no exact solution is available: two spin-12 models with crossing bonds, and a spin-1 model. The results for the temperature and magnetic exponents are very close to the exact results for exactly solvable models which are assumed to be in the same universality class. Differences are between 10-5 and a few times 10-4, and within the apparent numerical uncertainties. We also present an estimate of the critical point of the spin-1 model, and some preliminary results concerning universal properties of critical amplitudes.

Surface and Bulk Transitions in Three-Dimensional O(n) Models

Thermal and chiral critical exponents of the fully frustrated [ital XY] model on a square lattice are obtained from a finite-size scaling analysis of the free energy of chiral domain walls. Data were obtained by extensive Monte Carlo transfer-matrix computations for infinite strips of widths up to 14 lattice spacings. Two transfer matrices were implemented, one for each of two principal lattice directions. The results of both are consistent, but the critical exponents differ significantly from the pure Ising values. This is in agreement with other recent Monte Carlo simulations. Our results also support the identification of the critical behavior of this model with that along the line of transitions of simultaneous ordering or becoming critical of Ising and planar rotor degrees of freedom in the [ital XY]-Ising model studied recently.