Overview of Some Non –Neural Network Methods for Fitting Ab Initio Potential-Energy Databases

In this chapter, we describe results obtained by five methods that have been employed to fit ab initio potential-energy. These methods are (i) moving or modified Shepard interpolation (MSI), (ii) interpolative moving least squares (IMLS), (iii) invariant polynomials (IP), (iv) reproducing kernel Hilbert space (RKHS), and (v) a hybrid method that combines MSI and IMLS methods. The MSI and IMLS methods are described in some detail in the following. The IP and RKHS procedures are significantly more complex, and the reader is referred to the original papers for a more complete discussion of the details by which these methods are executed. The moving or modified Shepard interpolation (MSI) method was developed primarily by Collins and co-workers. The method employs electronic structure calculations to obtain the molecular potential energy at configuration points generated by an automated procedure. These data are then employed in a Shepard interpolation procedure to obtain the potential energies of the system at points other than those in the database. This procedure involves expressing the local potential about each configuration point in a Taylor series expansion. The term “moving” in the title derives from the fact that the set of internal coordinates employed in the interpolation varies from point-to-point in the database. Like all fitting methods, the MSI procedure requires the potential energy at a set of configuration points in the (3N-6) dimensional internal space of the system under investigation. These energies are generally obtained using ab initio electronic structure methods at some level of accuracy. In addition to the potential energies at each configuration point, the method also requires at least the first and second derivatives of the potential with respect to the coordinates being employed at each configuration point. These derivatives are needed to allow the local potential about a given configuration point in the database to be expressed in terms of a Taylor series expansion about that point. In principle, the MSI method may be extended to include third or fourth derivatives, but in most applications, the expansions are truncated after the quadratic terms.