Improved diffusion Monte Carlo propagators for bosonic systems using Itô calculus

Abstract

The construction of importance sampled diffusion Monte Carlo (DMC) schemes accurate to second order in the time step is discussed. A central aspect in obtaining efficient second order schemes is the numerical solution of the stochastic differential equation (SDE) associated with the Fokker-Plank equation responsible for the importance sampling procedure. In this work, stochastic predictor-corrector schemes solving the SDE and consistent with Itô calculus are used in DMC simulations of helium clusters. These schemes are numerically compared with alternative algorithms obtained by splitting the Fokker-Plank operator, an approach that we analyze using the analytical tools provided by Ito; calculus. The numerical results show that predictor-corrector methods are indeed accurate to second order in the time step and that they present a smaller time step bias and a better efficiency than second order split-operator derived schemes when computing ensemble averages for bosonic systems. The possible extension of the predictor-corrector methods to higher orders is also discussed.

Exploring the importance of quantum effects in nucleation: The archetypical Nen case

“…(1). In particular, it is important to assess that a particular approximation to Green's function is associated with a well-defined order m of the discretization error O(δt m ), in order to ensure robust and accurate results that can either be extrapolated a posteriori to δt → 0, 32 or "on the fly" as recently proposed. 33 For systems presenting only a partial quantum nature and sufficiently bound such as Ne clusters, the simplest version of DMC without importance sampling (noIS-DMC) (Ref.…”

Higher order diffusion Monte Carlo propagators for linear rotors as diffusion on a sphere: Development and application to O2@Hen

“…Different from what it is usually done when testing a new projector (such as new integrators for the Langevin equation 12 ) in Cartesian space, in this case we cannot rely on the fact that the sampled limiting distribution for t → ∞ should have an error proportional to some power of the time step. In fact, it is trivial to demonstrate that any form for P (u, t) that is even in u would, in the long run, generate an uniform distribution of points on the sphere.…”

“…It is thus with the goal of addressing this paradox that we therefore set out to investigate properties of O 2 @He n using atomistic quantum simulations; this is in view of the fact that Mg@He n has already been studied extensively. 1,10 In respect to the latter goal, we note that, while quantum statistical simulations on atomic clusters have progresses to the point where it is possible to simulate a few hundreds of quantum atoms in a reasonable amount of computer time, thanks to much improved algorithms employing higher order thermal density matrices, [11][12][13][14][15][16] similar improvements have been sparse and occasional in the case of systems containing rigid bodies. The latter "state of affair" should be attributed more to the complicated topology of the coordinate space needed to describe both internal and external degrees of freedom (e.g., torsions and overall orientation) than to the lack of cunning from investigators.…”

On the convergence of diffusion Monte Carlo in non-Euclidean spaces. I. Free diffusion

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“…The second is a method that accelerates the convergence of the basic algorithm with respect to the time step. 13,14 To the best of our knowledge, these two enhancements have only been combined for the first time very recently. 7 However, several questions about the general theory of diffusion in non-Euclidean manifolds and their convergence with respect to the time step remain.…”