Chemical Applications of Multi-Dimensional Moving Least Squares Fitting Methods for Hypersurfaces

A. Kawano, Y. Guo, and D. L. Thompson,*
University of Missouri
      and
A. F. Wagner and M. Minkoff
Argonne National Laboratory

Interpolating moving least-squares (IMLS) methods are an accurate and reliable way for generating hypersurfaces from sparse calculations of points on the surface. IMLS generalizes the mathematical underpinnings of the popular Modified Shepard method used in chemistry but, unlike that method, does not require computed derivatives or Hessians of the surface for an accurate fit. In past years, joint work at our two institutions showed how IMLS could be used to automatically fit potential energy surfaces (PES) in chemistry by directing electronic structure methods to performed calculations at IMLS-selected geometries. More recently, we have completed three projects. First we have for the first time carried out classical trajectories and computed kinetics properties (e.g., the energy-specific dissociation rate) on higher degree IMLS surfaces. With the proper choice of the fitting basis set, we demonstrate convergence of the computed dissociation rate of HOOH with as little as a few hundred ab initio points for the full 6D PES. Second, we dramatically improved the efficiency of IMLS by developing a flexible method to exclude remote points from the least squares fitting procedure IMLS requires for each surface evaluation. The number of included points varies according to the density of points in the local vicinity of the evaluation point. In the case of the 6D HOOH PES, on average more than 90% of the ab initio points can be excluded, resulting in a full order of magnitude reduction in time-to-solution. The third project is the installation of a general, “black box” version of the IMLS method on our website.** The user must supply the fitting basis functions, the criteria for excluding remote points, and the ab initio data set. The code will then supply value and, if requested, derivatives at any desired evaluation point. Future projects will be directed at improving efficiency by improved basis sets and exclusion methods, further reducing the density of the ab initio data set by incorporating ab initio derivatives, and by large scale PES applications in an automatic generation mode.

* Principal Investigator.
** http://www.chem.missouri.edu/thompson/SciDAC.html.