UC Irvine

This work revisits a general problem in numerical analysis; the efficient sampling of any d-dimensional distribution function. Monte Carlo methods are the standard approach when addressing a general distribution function, as they are able to produce a set of points in configuration space following any distribution. However, there are apparent flaws with this approach: namely, the inevitable phenomenon commonly referred to as “gaps and islands” where some regions in configuration space are oversampled while other regions are under- sampled. The well-known “curse of dimensionality” and its exponential scaling, combined with inefficient sampling, results in these methods quickly becoming unfeasible for meaningful applications. To this end we introduce a new sampling method, Quasi-Regular Grids, which results in an optimally distributed set of points that maintain local uniformity while simultaneously sampling any desired distribution. These new grids are then applied to the challenging problem of computing the quantum vibrational spectra for both model and molecular systems. We quantitatively establish the scaling properties of our method using an analytic model, the Morse potential, contrasting our results to the current best practices in the literature. We then tackle a chemical system of interest, formaldehyde, and again show the method to be superior to recent work published on the same system. Finally we present preliminary results for the spectral calculations of water. This research establishes a completely general method with numerous applications expanding far beyond spectra calculations, and will surely be an active area of further research in the future.