Shepard interpolation method

Energy derivatives are essential for the computation of dynamics properties. There are several dynamics-related methods available in gamess. The intrinsic reaction coordinate (IRC) or minimum energy path (MEP) follows the infinitely damped path from a first-order saddle point (transition state) to the minima connected to that transition state. In addition to providing an analysis of the process by which a chemical reaction occurs (e.g. evolution of geometric structure and wavefunction), the IRC is a common starting point for the study of dynamics. Example are variational transition state theory (VTST [55]) and the modified Shepard interpolation method developed by Collins and co-workers... 

In the Shepard interpolation method, the potential is written in the form... 

Crespos C, CoUins MA, Pijper E, Kroes GJ (2004) Application of the modified Shepard interpolation method to the determination of the potential energy surface for a molecule-surface reaction H2-b Pt(lll). J Chem Phys 120 2392... 

Dfaz C, Olsen RA, Busnengo HF, Kroes GJ (2010) Dynamics on six-dimensional potential energy surfaces for Hz/Cu(lll) Corrugation reducing procedure versus modified Shepard interpolation method and PW91 versus RPBE. J Phys Chem C 114 11192... 

Shepard Interpolation Method for Generating Potential Energy Surfaces for Dynamics Calculations. 

Gradient methods discussed above use a quadratic function (energy, gradient and approximate Hessian) to model the energy surface near the transition state. Distance-weighted interpolants provide a more flexible functional form that can interpolate arbitrarily spaced points with a smooth differentiable function. For a gradient-based optimization, the Shepard interpolation functions seem appropriate... 

Gordon, W.J., Wixom, J.A. 1978. Shepard s Method of Metric Interpolation to Bivariate and Multivariate Interpolation, in Mathematics and Computation 32 (141) 253-264. 

Using a suitable quantum mechanical electronic structure method, the energy, gradient, and Hessian can be calculated at an arbitrary geometry, q, which is called an interpolation point or a Shepard point. Near q Vii(q), V(q), and V22(q) may be expanded as a Taylor series, yielding... 

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