Interpolating moving least-squares methods for fitting potential energy surfaces: Detailed analysis of one-dimensional applications

Abstract

We present the basic formal and numerical aspects of higher degree interpolated moving least-squares ͑IMLS͒ methods. For simplicity, applications of these methods are restricted to two one-dimensional ͑1D͒ test cases: a Morse oscillator and a 1D slice of the HN 2 →HϩN 2 potential energy surface. For these two test cases, we systematically examine the effect of parameters in the weight function ͑intrinsic to IMLS methods͒, the degree of the IMLS fit, and the number and placement of potential energy points. From this systematic study, we discover compact and accurate representations of potentials and their derivatives for first-degree and higher-degree ͑up to nine degree͒ IMLS fits. We show how the number of ab initio points needed to achieve a given accuracy declines with the degree of the IMLS. We outline automatic procedures for ab initio point selection that can optimize this decline.

On representing chemical environments

“…Fitting potential energy surfaces (PESs) and electrostatic multipole moment surfaces of small molecules to data generated by first principles electronic structure calculations has been a mainstay of computational chemistry for decades 7,[9][10][11][12][13][14][15][16][17][18][19][20][21][22][23][24][25][26][27] . Typically, when modelling the PES of a small group of atoms, the list of pairwise distances is used or, equivalently, some transformed version of the interatomic distances, e.g.…”

Direct dynamics simulations using Hessian-based predictor-corrector integration algorithms

“…28,30,31,[34][35][36][37] However, an essential component of the current interpolation is that it gives the correct trajectory for the specified initial condition. In future work it would be of interest to compare the computational efficiency of our interpolation scheme with the other proposed schemes, given the requirement they give the correct trajectory.…”

“…In addition, the possibility of using ab initio points, from previous integration steps of the current trajectory and those from previous trajectories in the ensemble, to interpolate a locally accurate PES for the immediate integration step, without additional ab initio calculations, should also be investigated. 66 The interpolation schemes proposed in earlier work 28,[34][35][36][37] should be considered, but as discussed above it is important for the interpolation scheme to give the correct trajectory for the specified initial condition.…”

“…[25][26][27] A number of different algorithms have been presented for developing analytic PESs by interpolating ab initio data points. [28][29][30][31][32][33][34][35][36][37] Ischtwan and Collins have proposed 28 and used extensively 29 a Shepherd interpolation scheme. In recent work 30 this method has been modified by using Bayesian analysis to define confidence regions for the interpolation.…”

“…Eckert and Werner 36 have proposed a least-square interpolation scheme, related to the method of Collins and co-workers, but different in that it retains strict locality in a rectangular grid. Maisuradze et al 37 have presented an interpolating moving least-squares method, which removes a possible discontinuity in the method of Ischtwan and Collins. There is considerable interest in determining the chemical dynamics, whether classical, semiclassical, or quantum, predicted by a specific electronic structure theory. In the same way, as one has studied how stationary point and reaction path properties depend on the level of the electronic structure theory, one can investigate the relationship between the predicted chemical dynamics and the electronic structure theory method.…”