Permutationally Invariant Potential Energy Surfaces

Abstract

Over the past decade, about 50 potential energy surfaces (PESs) for polyatomics with 4-11 atoms and for clusters have been calculated using the permutationally invariant polynomial method. This is a general, mainly linear least-squares method for precise mathematical fitting of tens of thousands of electronic energies for reactive and nonreactive systems. A brief tutorial of the methodology is given, including several recent improvements. Recent applications to the formic acid dimer (the current record holder in size for a reactive system), the H-HO complex, and four protonated water clusters [H(HO)] are given. The last application also illustrates extension to large clusters using the many-body representation.

A critical comparison of neural network potentials for molecular reaction dynamics with exact permutation symmetry

“…ij , where p ij = exp(ÀaR ij ), and Ŝ is the symmetrization operator, which consists of all the permutation operations between atoms of the same element in the system. 10 In practice, all PIPs up to a specific maximum order are included in the input layer. As discussed above, many PIPs are redundant.…”

“…While the accuracy of empirical potentials relying on physical approximations is often limited, numerous approaches have been developed to represent PESs based on very flexible and purely mathematical functional forms, which can be grouped into two categories: (1) interpolation methods, which provide error-free energies for the available reference data but interpolate in between, such as cubic splines, reproducing kernel Hilbert space (RKHS), 1 interpolating moving least square (IMLS) 2 and modified Shepard interpolation (MSI) methods; 3 and (2) fitting methods relying on specific functional forms such as polynomials within the many body expansion regime, 4,5 sum-of-product forms 6,7 and permutation invariant polynomials (PIPs). [8][9][10] In spite of all these methods, the accurate description of global PESs has remained a formidable challenge even for comparably small polyatomic systems, because they often exhibit a complex topology with several reactants and products, saddle points and intermediates, and in general it is impossible to derive suitable functional forms based on physical considerations. A new promising class of PESs relies on machine learning (ML) techniques.…”

“…36 In addition, Lu et al employed the HD-NNP approach to fit the PES of the H 2 + SH reaction based on ab initio calculated points, and the quantum dynamical results of the HD-NNP and PIP-NN PESs have been found to be essentially the same. 37 Building on the extensive work of Bowman and coworkers on molecular PESs based on permutation invariant polynomials (PIPs), 10,29 in 2013, Guo and co-workers proposed to use PIPs in the input layer of a NN to obtain another type of NN potential with exact permutation invariance. 24,25 In practice, all PIPs of Bowman and co-workers, 8,9 truncated at a specified total degree, are used in the input layer of the NN.…”

“…Each NN i corresponds to an equation like eqn (1) with the input variables {G i,t }. The exchange of the two identical atoms 1 and 2 does not affect {G 3,t } and {G 4,t } at all, as shown in eqn(10) and(11). Therefore, NN 3 and NN 4 , which describe the environments of different elements, do not use the same NN: both the architectures and the fitting parameters of NN 3 and NN 4 are not restricted.…”

Regularised atomic body-ordered permutation-invariant polynomials for the construction of interatomic potentials

“…Similar effects can be seen in a related field, the fitting of the potential energy surfaces of molecules to high level, wave function based quantum chemistry calculations. This endeavour has a rich history [14][15][16][17][18][19], which also includes high-dimensional nonparametric fits that are very accurate for small systems (a handful of atoms), yet it is recognised that once the dimensionality reaches a few tens, the fitting task becomes extremely difficult.…”