Quantum monte carlo methods for constrained systems

Abstract

The torsional ground state for ethane, the torsional, rotational, and mixed torsional and rotational ground state of propane are computed with a version of diffusion Monte Carlo adapted to handle the geometric complexity of curved spaces such as the Ramachandra space. The quantum NVT ensemble average for the mixed torsional and rotational degrees of freedom of propane is computed, using a version of Monte Carlo path integral, also adapted to handle curved spaces. These three problems are selected to demonstrate the generality and the applicability of the approaches described. The spaces of coordinates can be best constructed from the parameters of continuous Lie groups, and alternative methods based on vector spaces, where extended Lagrangian terms would be too cumbersome to implement. We note that the geometric coupling between the torsions and the rotations of propane produces a substantial effect on the ground state energy of propane, and that the quantum effects on the energy of propane are quite large even well above room temperature.

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

“…We have noted that, occasionally and for reasons we do not yet fully understand, the inclusion of a second order branching strategy improves the convergence of DMC compared to the expected linear behaviour. 5 For example, we have observed that the 2-sphere mapped by stereographic projections displays a nonlinear convergence behavior when second order branching is used, and we observe no degradation of this property in the overall walk over the simulation time. 12 Brownian motion and the related stochastic calculus in Riemann non-Euclidean manifolds have been investigated in detail in a number of recent works.…”

“…(20) requires a number of derivatives of the metric tensor, though a derivative free version of the simplified Ito-Taylor method has been proposed as well. 17 The latter may prove useful for future applications of DMC in molecular physics, such as, e.g., for simulations in Ramachandra space, 5 where the expressions for the metric tensor can become quite convoluted, and the evaluations of derivative can perhaps become expensive. The analytical derivative version is quite useful for our purposes, since it is conducive to the convergence analysis for curved manifolds that we perform here.…”

“…[21][22][23] This definition of a manifold is too general for our purposes and, in addition, we require that the points in M (or each U i ) be reached from a Euclidean space of higher dimension R d ′ , through a variety of constructions, including the application of holonomic constraints or by formulating a manifold from the parameter space of continuous Lie groups. 5,20 Let Φ be composed of a set of functions that map the homomorphism χ µ i in U i to Cartesian coordinates x µ in an open set of R d ′ . Φ must have a unique inverse, be continuous, infinitely differentiable, and integrable bijection at every point in U i except for a subset of points of zero measure.…”

“…[5][6][7][8][9] Most recently, we have used it to study the ground state of ethane and propane in an external field, 5 to perform simulations of the hydrogen molecule on ammonia clusters, 6 and to study oxygen in helium clusters. 7 These demanding applications of DMC have required a number of enhancements to the basic algorithm.…”

Assessment of the Effects of Anisotropic Interactions among Hydrogen Molecules and Their Isotopologues: A Diffusion Monte Carlo Investigation of Gas Phase and Adsorbed Clusters

“…For the task we set out to accomplish, we have deemed diffusion Monte Carlo (DMC) , a suitable technique given its ability to accurately describe anharmonicity in all degrees of freedom. Indeed, DMC has become a particularly convenient technique thanks to its recently improved accuracy and efficiency in treating the quantum ground state of systems such as He droplets and other molecular clusters. − Specifically, DMC is a ground state technique that allows one to obtain both energetics (i.e., the ground state energy E 0 ) and structural details at a limited computational cost, as the version commonly employed when treating molecular species scales linearly with the number of configurations sampled during the simulation and with the cost associated with the PES evaluation. The particular implementation of DMC employed in this study has previously been described in detail , and compared with alternative schemes. ,, Thus, we shall discuss only the features that are relevant for this work for the sake of brevity.…”