Interpolating moving least-squares methods for fitting potential energy surfaces: Improving efficiency via local approximants

Abstract

The local interpolating moving least-squares (IMLS) method for constructing potential energy surfaces is investigated. The method retains the advantageous features of the IMLS approach in that the ab initio derivatives are not required and high degree polynomials can be used to provide accurate fits, while at the same time it is much more efficient than the standard IMLS approach because the least-squares solutions need to be calculated only once at the data points. Issues related to the implementation of the local IMLS method are investigated and the accuracy is assessed using HOOH as a test case. It is shown that the local IMLS method is at the same level of accuracy as the standard IMLS method. In addition, the scaling of the method is found to be a power law as a function of number of data points N, N(-q). The results suggest that when fitting only to the energy values for a d-dimensional system by using a Qth degree polynomial the power law exponent q approximately Qd when the energy range fitted is large (e.g., E<100 kcalmol for HOOH), and q>Qd when the energy range fitted is smaller (E<30 kcalmol) and the density of data points is higher. This study demonstrates that the local IMLS method provides an efficient and accurate means for constructing potential energy surfaces.

Haptic quantum chemistry

“…22 There exists also a variant of the IMLS method that does not require to solve the least-squares equations for every evaluation point via local approximants. 27 The local interpolation moving least-squares method (L-IMLS) requires to calculate at every ab initio data point, the set of coefficients {a i }, which results in an N 3 m matrix. At every evaluation point, it is then simply necessary to calculate a weighted average over the coefficients of the data points near the evaluation point.…”

Potential energy surface of the 12A′ Li2 + Li doublet ground state

“…The weights w i (Z) dictate the effective range at which a given ab initio point will contribute to the global fit and the effective contribution to the fit. We use Guo et al's [37] form of the weight function, which introduces a cutoff function S(χ) to the unnormalized weight function v i ( Z − i ) so as to smoothly go to zero at a given cutoff radius R cut . The cutoff function is given [37] by…”

“…We use Guo et al's [37] form of the weight function, which introduces a cutoff function S(χ) to the unnormalized weight function v i ( Z − i ) so as to smoothly go to zero at a given cutoff radius R cut . The cutoff function is given [37] by…”

Potential energy surface fitting by a statistically localized, permutationally invariant, local interpolating moving least squares method for the many-body potential: Method and application to N4

“…The process we have described is the local interpolating moving least squares (L-IMLS) approach. The term L-IMLS and its application to PES fitting in physical chemistry are due to Dawes et al 7 and Guo et al, 11 who developed the method for systems of higher dimensionality. Their research makes use of weight functions similar to those in Eqs.…”

“…6 In the present paper, we describe an alternate strategy, which we pursued concurrently, for fitting a six-dimensional surface to the tetranitrogen dataset. Our approach is a new version of the local interpolating moving least squares (L-IMLS) method, as investigated by Dawes et al [7][8][9][10] and Guo et al, 11 with three improvements. (1) Our method treats pairwise interactions separately from many-body interactions.…”