Chemical Applications of Multi-Dimensional Moving Least Squares Fitting Methods for Hypersurfaces
A. Kawano, Y. Guo, and D. L. Thompson,*
University of Missouri
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.