Permutation invariant polynomial neural network approach to fitting potential energy surfaces

A rigorous, general, and simple method to fit global and permutation invariant potential energy surfaces (PESs) using neural networks (NNs) is discussed. This so-called permutation invariant polynomial neural network (PIP-NN) method imposes permutation symmetry by using in its input a set of symmetry functions based on PIPs. For systems with more than three atoms, it is shown that the number of symmetry functions in the input vector needs to be larger than the number of internal coordinates in order to include both the primary and secondary invariant polynomials. This PIP-NN method is successfully demonstrated in three atom-triatomic reactive systems, resulting in full-dimensional global PESs with average errors on the order of meV. These PESs are used in full-dimensional quantum dynamical calculations.

The permutation invariant polynomial-neural network (PIP-NN) method for constructing highly accurate potential energy surfaces (PESs) for gas phase molecules is extended to molecule-surface interaction PESs. The symmetry adaptation in the NN fitting of a PES is achieved by employing as the input symmetry functions that fulfill both the translational symmetry of the surface and permutation symmetry of the molecule. These symmetry functions are low-order PIPs of the primitive symmetry functions containing the surface periodic symmetry. It is stressed that permutationally invariant cross terms are needed to avoid oversymmetrization. The accuracy and efficiency are demonstrated in fitting both a model PES for the H2 + Cu(111) system and density functional theory points for the H2 + Ag(111) system.

An accurate full-dimensional global potential energy surface (PES) is developed for the title reaction. While the long-range interactions in the reactant asymptote are represented by an analytical expression, the interaction region of the PES is fit to more than 81,000 of ab initio points at the UCCSD(T)-F12b/AVTZ level using the permutation invariant polynomial neural network approach. Fully symmetric with respect to permutation of all four hydrogen atoms, the PES provides a faithful representation of the ab initio points, with a root mean square error of 1.8 meV or 15 cm(-1). The reaction path for this exoergic reaction features an attractive and barrierless entrance channel, a submerged saddle point, a shallow H4O(+) well, and a barrierless exit channel. The rate coefficients for the title reaction and kinetic isotope effect have been determined on this PES using quasi-classical trajectories, and they are in good agreement with available experimental data. It is further shown that the H2O(+) rotational enhancement of reactivity observed experimentally can be traced to the submerged saddle point. Using our recently proposed Sudden Vector Projection model, we demonstrate that a rotational degree of freedom of the H2O(+) reactant is strongly coupled with the reaction coordinate at this saddle point, thus unraveling the origin of the pronounced mode specificity in this reaction.

As the most abundant molecule in the universe, collisions involving H2 have important implications in astrochemistry. Collisions between hydrogen molecules also represent a prototype for assessing various dynamic methods for understanding fundamental few-body processes. In this work, we develop a new and highly accurate full-dimensional potential energy surface (PES) covering all reactive channels of the H2 + H2 system, which extends our previously reported H2 + H2 nonreactive PES [J. Chem. Theory Comput., 2021, 17, 6747] by adding 39,538 additional ab initio points calculated at the MRCI/AV5Z level in the reactive channels. The global PES is represented with high fidelity (RMSE = 0.6 meV for a total of 79,000 points) by a permutation invariant polynomial neural network (PIP-NN) and is suitable for studying collision-induced dissociation, single-exchange, as well as four-center exchange reactions. Preliminary quasi-classical trajectory studies on the new PIP-NN PES reveal strong vibrational enhancement of all reaction channels.

The study of K + H2 collision has a long experimental history, but there have been few theoretical studies due to lack of a global potential energy surface (PES). In this study, a new global PES for the ground state of KH2 system was constructed based on numerous ab initio points, using the permutation invariant polynomial neural network method. The root mean square error (RMSE) of PES is very small (5.64 meV). On the new PES, time-dependent quantum wave packet (TDWP) and quasiclassical trajectory (QCT) calculations were carried out to study the dynamics of the K(2S) + H2(X1Σ+g) → H(2S) + KH(X1Σ+) reaction. Dynamics results show that (i) the K + H2(v = 0) → H + KH reaction scarcely occurred, (ii) the K + H2(v = 1) → H + KH reaction took place in small quantities, and (iii) the K + H2(v = 2) → H + KH reaction occurred in large quantities. This indicates that vibrational energy of the reactant is significantly more effective at promoting the reaction than the translational energy. This characteristic stems from a major physical model in reactive collisions: the vibrationally excited H2 molecule and K atom collide first in a T-geometric configuration and the vibrational motion of the H2 molecule helps separate the two H atoms a large distance after the collision. At a large H-H distance, a broad well exists on the PES, so the heavy K atom could pull back the light H atom to initiate the reaction. Similar to the reactive channel, vibrational excitation of the reactant also has a significant effect on the collision-induced dissociation channel.

The N+(3P) + H2(X1Σ) → NH+(X2Π) + H(2S) reaction is important for initiating the chain reaction of ammonia synthesis in the universe. To study the dynamics of this reaction, a global accurate potential energy surface (PES) of the ground state NH was constructed by combining numerous high-level ab initio energy points with the permutation invariant polynomial neural network method. Utilizing this newly constructed PES, time-dependent wave packet calculations were performed on the state-to-state reactions of N+(3P0) + H2 (v = 0, j = 0) and N+(3P0) + D2 (v = 0, j = 0) in order to study the microscopic reaction mechanisms and dynamical isotope effects. Isotope effects have a significant influence on the rovibrational state distributions and state resolved angle distributions of the product. The total differential cross sections (DCSs) present in the aforementioned reactions are dominated by both forward and backward scattering, as expected from the observable deep well along the reaction path. Meanwhile, the rovibrational state-resolved DCSs show that both reactions are not entirely statistical at the state-to-state level.

The H + CH3OH reaction, which plays an important role in combustion and the interstellar medium, presents a prototypical system with multi channels and tight transition states. However, no globally reliable potential energy surface (PES) has been available to date. Here we develop global analytical PESs for this system using the permutation-invariant polynomial neural network (PIP-NN) and the high-dimensional neural network (HD-NN) methods based on a large number of data points calculated at the level of the explicitly correlated unrestricted coupled cluster single, double, and perturbative triple level with the augmented correlation corrected valence triple-ζ basis set (UCCSD(T)-F12a/AVTZ). We demonstrate that both machine learning PESs are able to accurately describe all dynamically relevant reaction channels. At a collision energy of 20 kcal/mol, quasi-classical trajectory calculations reveal that the dominant channel is the hydrogen abstraction from the methyl site, yielding H2 + CH2OH. The reaction of this major channel takes place mainly via the direct rebound mechanism. Both the vibrational and rotational states of the H2 product are relatively cold, and large portions of the available energy are converted into the product translational motion.

The reactive collision between Ca and H2 molecules has attracted great interest experimentally due to the key role of the product CaH molecule in the field of astrophysics and cold molecules. However, quantum dynamics calculations for this system have not been reported due to the lack of a global potential energy surface (PES). Herein, a globally accurate PES of the ground-state CaH2 is developed by combining 11365 high-level ab initio points and permutation invariant polynomial neural network method. Based on the newly constructed PES, the state-to-state quantum dynamics calculations for the Ca(1S) + H2 (v 0 = 0, j 0 = 0) → CaH + H reaction are carried out using the time-dependent wave packet method. The dynamic results reveal that the reaction follows the complex-forming mechanism near the reactive threshold, whereas both the indirect insertion mechanism and direct abstraction mechanism have effects at higher collision energies. The newly constructed PES can be used to further study the influence of isotope substitution, rovibrational excitation, and spatial orientation of reactant molecules on reaction dynamics.

The permutation invariant polynomial-neural network (PIP-NN) approach is extended to fit intermolecular potential energy surfaces (PESs). Specifically, three PESs were constructed for the Ne-C2H2 system. PES1 is a full nine-dimensional PIP-NN PES directly fitted to ∼42 000 ab initio points calculated at the level of CCSD(T)-F12a/cc-pCVTZ-F12, while the other two consist of the six-dimensional PES for C2H2 [H. Han, A. Li, and H. Guo, J. Chem. Phys. 141, 244312 (2014)] and an intermolecular PES represented in either the PIP (PES2) or PIP-NN (PES3) form. The comparison of fitting errors and their distributions, one-dimensional cuts and two-dimensional contour plots of the PESs, as well as classical trajectory collisional energy transfer dynamics calculations shows that the three PESs are very similar. We conclude that full-dimensional PESs for non-covalent interacting molecular systems can be constructed efficiently and accurately by the PIP-NN approach for both the constituent molecules and intermolecular parts.

The initial state specific quantum wave packet dynamics studies of the H + HBr (v0 = 0, j0 = 0-2) reaction were performed using a new global potential energy surface (PES) of the ground state of the BrH2 system for the collision energy ranging from 0.01 to 2.0 eV. The PES was constructed using the permutation invariant polynomial neural network method based on approximately 63 000 ab initio points, which were calculated by the multireference configuration interaction method with AVTZ and AVQZ basis sets. To improve the accuracy of the PES, Davidson's correction and spin-orbit coupling effects were considered in the ab initio calculation and the basis set was extrapolated to complete basis set limit. The new PES was compared with the previous ones and also the available experimental data, which suggests that the new PES is more accurate. The state-to-state quantum wave packet dynamics was carried out using the reactant-coordinate based approach. The reaction probabilities, integral and differential cross sections, rovibrational state distributions of product and rate constants, etc., were compared with the available theoretical and experimental studies. In general, the present work is in better agreement with the available experimental data. The quantum dynamics studies suggest that the rotational excitation of HBr has little effect on the reaction.

A new global potential energy surface for the BH2 molecule and dynamics studies of the B(2p2P) + H2 reaction

A new global potential energy surface of X2A′ state of Li2H system and quantum dynamics calculation of the H + Li2 → Li + LiH reaction

The ro-vibrational spectrum of the simplest Criegee intermediate (CH2OO) has been determined quantum mechanically based on nine-dimensional potential energy and dipole surfaces for its ground electronic state. The potential energy surface is fitted to more than 50 000 high-level ab initio points with a root-mean-square error of 25 cm(-1), using a recently proposed permutation invariant polynomial neural network method. The calculated rotational constants, vibrational frequencies, and spectral intensities of CH2OO are in excellent agreement with experiment. The potential energy surface provides a valuable platform for studying highly excited vibrational and unimolecular reaction dynamics of this important molecule.

A machine learning method is proposed for representing the elements of diabatic potential energy matrices (PEMs) with high fidelity. This is an extension of the so-called permutation invariant polynomial-neural network (PIP-NN) method for representing adiabatic potential energy surfaces. While for one-dimensional irreducible representations the diagonal elements of a diabatic PEM are invariant under exchange of identical nuclei in a molecular system, the off-diagonal elements require special symmetry consideration, particularly in the presence of a conical intersection. A multiplicative factor is introduced to take into consideration the particular symmetry properties while maintaining the PIP-NN framework. We demonstrate here that the extended PIP-NN approach is accurate in representing diabatic PEMs, as evidenced by small fitting errors and by the reproduction of absorption spectra and product branching ratios in both H2O( ) and NH3( ) non-adiabatic photodissociation.

A new and more accurate full-dimensional global potential energy surface (PES) for the ground electronic state of the ClH2O system is developed by fitting 15 777 points obtained using an explicitly correlated unrestricted coupled-cluster method with single, double, and perturbative triple excitations (UCCSD(T)-F12b). The fitting is carried out using the permutation invariant polynomial-neural network (PIP-NN) method and has an error of 6.9 meV. The new PES has a slightly lower barrier for the atmospherically important HCl + OH → Cl + H2O reaction than the previous PES based on multi-reference configuration interaction (MRCI) calculations. As a result, it should provide a better characterization of the kinetics. Quantum dynamical calculations of reaction probabilities for both the forward and reverse reactions are performed on this new PES and compared with those on the MRCI PES. They reveal notable differences, resulting apparently from subtle differences in the PESs.