Potential energy surfaces for high-energy N + O2 collisions

Dual-Level Parametrically Managed Neural Network Method for Learning a Potential Energy Surface for Efficient Dynamics

A general difficulty with machine-learned potential energy surfaces is their unreliability in regions with little or no training data. The goal of the present work is to remedy this by a low-cost method for incorporating well understood features of potential energy surfaces into an efficient data-driven machine learning algorithm. Our focus is on regions where conventional surface fitting does not need large amounts of accurate data, in particular, geometries with large separations of subsystems-where it is well recognized that the potential should reach its asymptotic form-and geometries with very close atoms-where the potential should be repulsive enough to prevent trajectories from reaching classically inaccessible regions but need not be highly quantitative. The new method involves a neural network (NN) with a parametrically managed activation function (PMAF) and two levels of electronic structure, a higher level (HL) and a lower level (LL). The resulting NN is called a dual-level parametrically managed neural network (DL-PMNN). For the present example, the HL is an accurate density functional method (CF22D/may-cc-pVTZ), and the LL is an inexpensive density functional method (MPW1K/MIDIY). We use the LL to ensure correct behavior of the potential at large and small distances; the goal is to reach HL accuracy for dynamics without making HL calculations in regions where the LL can guide the fit. To illustrate the new method, we fit the potential energy surface for dissociation of the S-H bond of ortho-fluorothiophenol in the ground electronic state, and we show that the method yields a good fit and efficient trajectory calculations without crashes.

MOLPIPx: An end-to-end differentiable package for permutationally invariant polynomials in Python and Rust

In this work, we present MOLPIPx, a versatile library designed to seamlessly integrate permutationally invariant polynomials with modern machine learning frameworks, enabling the efficient development of linear models, neural networks, and Gaussian process models. These methodologies are widely employed for parameterizing potential energy surfaces across diverse molecular systems. MOLPIPx leverages two powerful automatic differentiation engines-JAX and EnzymeAD-Rust-to facilitate the efficient computation of energy gradients and higher-order derivatives, which are essential for tasks such as force field development and dynamic simulations. MOLPIPx is available at https://github.com/ChemAI-Lab/molpipx.

Quasi-Classical Trajectory Calculation of Rate Constants Using an Ab Initio Trained Machine Learning Model (aML-MD) with Multifidelity Data

Machine learning (ML) provides a great opportunity for the construction of models with improved accuracy in classical molecular dynamics (MD). However, the accuracy of a ML trained model is limited by the quality and quantity of the training data. Generating large sets of accurate ab initio training data can require significant computational resources. Furthermore, inconsistent or incompatible data with different accuracies obtained using different methods may lead to biased or unreliable ML models that do not accurately represent the underlying physics. Recently, transfer learning showed its potential for avoiding these problems as well as for improving the accuracy, efficiency, and generalization of ML models using multifidelity data. In this work, ab initio trained ML-based MD (aML-MD) models are developed through transfer learning using DFT and multireference data from multiple sources with varying accuracy within the Deep Potential MD framework. The accuracy of the force field is demonstrated by calculating rate constants for the H + HO2 → H2 + 3O2 reaction using quasi-classical trajectories. We show that the aML-MD model with transfer learning can accurately predict the rate constants while reducing the computational cost by more than five times compared to the use of more expensive quantum chemistry training data sets. Hence, the aML-MD model with transfer learning shows great potential in using multifidelity data to reduce the computational cost involved in generating the training set for these potentials.

Parametrically Managed Activation Functions for Improved Global Potential Energy Surfaces for Six Coupled 5A' States and Fourteen Coupled 3A' States of O + O2

We report new potential energy surfaces for six coupled 5A' states and 14 coupled 3A' states of O3. The new surfaces are created by parametrically managed diabatization by deep neural network (PM-DDNN). The PM-DDNN method uses calculated adiabatic potential energy surfaces to discover and fit an underlying adiabatic-equivalent set of diabatic surfaces and their couplings and obtains the fit to the adiabatic surfaces by diagonalization of the diabatic potential energy matrix (DPEM). The procedure yields the adiabatic surfaces and their gradients, as well as the DPEM and its gradient. If desired one can also compute the nonadiabatic coupling due to the transformation. The present work improves on previous work by using a new coordinate to guide the decay of the neural network contribution to the many-body fit to the whole DPEM. The main objective was to obtain smoother potentials than the previous ones with better suitability for dynamics calculations, and this was achieved. Furthermore, we obtained suitably small deviations from the input reference data. For the six coupled 5A' surfaces, the 60,366 data below 10 eV are fit with a mean unsigned error (MUE) of 49 meV, and for the 14 coupled 3A' surfaces, the 76,733 data below 10 eV are fit with an MUE of 28 meV. The data below 5 eV fit even more accurately with MUEs of 37 meV (5A') and 20 meV (3A').

Theoretical investigation on the mechanism and kinetics of the OH•‒initiated atmospheric degradation of p-chloroaniline: Addition of ∑g-3O2 and isomerization of peroxy radicals

Atmospheric oxidation of the p-chloroaniline-OH• adduct [C6H4ClNH2-OH]• (AD-C2) by ∑g-3O2 and internal isomerization processes of peroxy radical [C6H4ClNH2-OH]•-O2 are theoretically investigated at the M06-2X/aug-cc-pVTZ and CBS-QB3//M06-2X/aug-cc-pVTZ level of theories. Potential energy surfaces (PESs) for the most efficient pathways indicated that the oxidation process begins via the complexation of individual reactants in syn mode forming PRCy-iOO-syn (y = 2,5) in an exothermic and endogenic step. The syn mode addition is favored over the anti one due to the formation of internal hydrogen bond between the hydroxyl and peroxy groups. Formation of new C5-OO bond in PRCy-iOO-syn complex is an unimolecular process which is exothermic and exoergic. This pathway is predominated over other internal conversions due to the presence of stronger intramolecular hydrogen bond. Cyclization of the produced [C6H4ClNH2-OH]•-O2 peroxy radical AD-C2-5OO-syn into the bicyclic peroxy radical AD-C2-5,6OO-syn is the last step which is strongly endothermic and endogenic. The rate coefficients are calculated by means of the RRKM theory over the temperature range 250-350 K and at a pressure range of 0.1 bar to the high-pressure limit. The RRKM rate coefficients at the M06-2X/aug-cc-pVTZ level for the first bimolecular and last unimolecular steps are in order of 10-16 cm3 molecule-1 s-1 and 10-7 s-1, respectively, while the obtained rate coefficients at the CBS-QB3//M06-2X/aug-cc-pVTZ are overestimated about two order of magnitude.

Dissociation cross sections and rates in O2 + N collisions: molecular dynamics simulations combined with machine learning

The collision-induced dissociation reaction of O2 (v, j) + N, a fundamental process in nonequilibrium air flows around reentry vehicles, has been studied systematically by applying molecular dynamics simulations on the 2A', 4A' and 6A' potential energy surfaces of NO2 in a wide temperature range. In particular, we have directly investigated the role of the 6A' surface in this process and discussed the applicability of the simplified approximate rate models proposed by Esposito et al. and Andrienko et al. based on the lowest two surfaces. The present work indicates that the state-selected dissociation of O2 + N is dominated by the 6A' surface for all except for the low-lying O2 states. Furthermore, a complete database of rovibrationally detailed cross sections and rate coefficients is a prerequisite for modeling the relevant nonequilibrium air flows in spacecraft reentry. Here, the combination of the quasi-classical trajectory (QCT) and the neural network (NN) has been proposed to predict all state-selected dissociation cross sections and further construct dissociation parameter sets. All NN-based models established in this work accurately reproduce the results calculated from QCT simulations over a wide range of rovibrational quantum numbers with R2 > 0.99. Compared with the explicit QCT simulations, the computational requirement for predicting cross sections and rates based on the NN models significantly reduces. Finally, thermal equilibrium rate coefficients computed from NN models match remarkably well the available theoretical and experimental results in the whole temperature range explored.

Experimental and Theoretical Studies of Hyperthermal N + O2 Collisions

The dynamics of hyperthermal N(4S) + O2 collisions were investigated both experimentally and theoretically. Crossed molecular beams experiments were performed at an average center-of-mass (c.m.) collision energy of ⟨Ecoll⟩ = 77.5 kcal mol-1, with velocity- and angle-resolved product detection by a rotatable mass spectrometer detector. Nonreactive (N + O2) and reactive (NO + O) product channels were identified. In the c.m. reference frame, the nonreactively scattered N atoms and reactively scattered NO molecules were both directed into the forward direction with respect to the initial direction of the reagent N atoms. On average, more than 90% of the available energy (⟨Eavl⟩ = 77.5 kcal mol-1) was retained in translation of the nonreactive products (N + O2), whereas a much smaller fraction of the available energy for the reactive pathway (⟨Eavl⟩ = 109.5 kcal mol-1) went into translation of the NO + O products, and the distribution of translational energies for this channel was broad, indicating extensive internal excitation in the nascent NO molecules. The experimentally derived c.m. translational energy and angular distributions of the reactive products suggested at least two dynamical pathways to the formation of NO + O. Quasiclassical trajectory (QCT) calculations were performed with a collision energy of Ecoll = 77 kcal mol-1 using two sets of potential energy surfaces, denoted as PES-I and PES-II, and these theoretical results were compared to each other and to the experimental results. PES-I is a reproducing kernel Hilbert space (RKHS) representation of multireference configurational interaction (MRCI) energies, while PES-II is a many-body permutation invariant polynomial (MB-PIP) fit of complete active space second order perturbation (CASPT2) points. The theoretical investigations were both consistent with the experimental suggestion of two dynamical pathways to produce NO + O, where reactive collisions may proceed on the doublet (12A') and quartet (14A') surfaces. When analyzed with this theoretical insight, the experimental c.m. translational energy and angular distributions were in reasonably good agreement with those predicted by the QCT calculations, although minor differences were observed which are discussed. Theoretical translational energy and angular distributions for the nonreactive N + O2 products matched the experimental translational energy and angular distributions almost quantitatively. Finally, relative yields for the nonreactive and reactive scattering channels were determined from the experiment and from both theoretical methods, and all results are in reasonable agreement.

Quasi-classical trajectory analysis of three-body collision induced recombination in neutral nitrogen and oxygen

We present theory and a simulation framework to model three-body collisions and gas phase recombination in dilute atom/diatom mixtures of pure oxygen (O/O2) and nitrogen (N/N2) using the Quasi-Classical Trajectory method. We formulate a three-body collision rate constant based on the lifetimes of binary collisions and initialize three-body collisions by sampling the arrival time of a third body within the lifetimes of pre-simulated binary collisions. We use this method to calculate distributions of recombined product energies, probabilities of recombination, and recombination rate constants through different collision pathways. Long-lived binary atom-diatom collisions are observed, but are too rare to play a dominant role in the recombination process for shock-heated air near the equilibrium conditions studied. The resulting recombination rate constants are within an order of magnitude of the predictions of detailed balance. Notably, the recombination simulation framework does not appeal to the principle of detailed balance and could be useful for studying conditions far from equilibrium.

Formation of N(2D) from Hyperthermal Collisions between O(3P) and NO(X2Π)

Hyperthermal collisions between O(3P) and NO(X2Π) could lead to the formation of the first electronically excited atomic nitrogen (N(2D)), which plays a key role in plasma formation in shock-heated air. This process is facilitated mainly by four doublet states, and to a much lesser extent by two quartet states. In this work, we report quasi-classical trajectory studies of this reactive process using the four analytical adiabatic potential energy surfaces for the doublet states developed previously from fitting high-level ab initio data. The reactions were found to be strongly enhanced by vibrational excitation of the NO reactant, consistent with the existence of potential energy barriers in the exit channel. Despite the large endothermicity of the reaction, the rate coefficient is significant at high temperatures, suggesting a possible role of this reaction in the hyperthermal kinetics in the shock layer of a hypersonic (re)entry vehicle.

Parametrically Managed Activation Function for Fitting a Neural Network Potential with Physical Behavior Enforced by a Low-Dimensional Potential

Machine-learned representations of potential energy surfaces generated in the output layer of a feedforward neural network are becoming increasingly popular. One difficulty with neural network output is that it is often unreliable in regions where training data is missing or sparse. Human-designed potentials often build in proper extrapolation behavior by choice of functional form. Because machine learning is very efficient, it is desirable to learn how to add human intelligence to machine-learned potentials in a convenient way. One example is the well-understood feature of interaction potentials that they vanish when subsystems are too far separated to interact. In this article, we present a way to add a new kind of activation function to a neural network to enforce low-dimensional constraints. In particular, the activation function depends parametrically on all of the input variables. We illustrate the use of this step by showing how it can force an interaction potential to go to zero at large subsystem separations without either inputting a specific functional form for the potential or adding data to the training set in the asymptotic region of geometries where the subsystems are separated. In the process of illustrating this, we present an improved set of potential energy surfaces for the 14 lowest ³A' states of O₃. The method is more general than this example, and it may be used to add other low-dimensional knowledge or lower-level knowledge to machine-learned potentials. In addition to the O₃ example, we present a greater-generality method called parametrically managed diabatization by deep neural network (PM-DDNN) that is an improvement on our previously presented permutationally restrained diabatization by deep neural network (PR-DDNN).

The role of the sextet potential energy surface in O2 + N inelastic collision processes

We have performed molecular dynamics simulations of inelastic collisions between molecular oxygen and atomic nitrogen, employing the quasi-classical trajectory method on the new doublet, quartet, and sextet analytical potential energy surfaces of NO₂. A complete database of vibrationally detailed rate coefficients is constructed in a wide temperature range for high vibrational states up to ν = 25. In particular, the present work shows that the sextet potential energy surface plays a crucial role in the rovibrational relaxation process of O₂ + N collisions. The state-to-state rate coefficients increase by a factor of 2 to 6 when we consider the contribution of this sextet potential energy surface according to the corresponding weight factor, especially for vibrational energy transfer processes in single quantum jumps and/or high-temperature regimes. Furthermore, we also provide Arrhenius-type accurate fits for the vibrational state-specific rate coefficients of this collision system to achieve the flexible application of rate coefficients in numerical codes concerning air kinetics. Our results have implications for understanding the relaxation mechanism of the collision system with degenerate electronic states.

PESPIP: Software to fit complex molecular and many-body potential energy surfaces with permutationally invariant polynomials

We wish to describe a potential energy surface by using a basis of permutationally invariant polynomials whose coefficients will be determined by numerical regression so as to smoothly fit a dataset of electronic energies as well as, perhaps, gradients. The polynomials will be powers of transformed internuclear distances, usually either Morse variables, exp(-r_i,j/λ), where λ is a constant range hyperparameter, or reciprocals of the distances, 1/r_i,j. The question we address is how to create the most efficient basis, including (a) which polynomials to keep or discard, (b) how many polynomials will be needed, (c) how to make sure the polynomials correctly reproduce the zero interaction at a large distance, (d) how to ensure special symmetries, and (e) how to calculate gradients efficiently. This article discusses how these questions can be answered by using a set of programs to choose and manipulate the polynomials as well as to write efficient Fortran programs for the calculation of energies and gradients. A user-friendly interface for access to monomial symmetrization approach results is also described. The software for these programs is now publicly available.

Reactive and Nonreactive Collisions between NO(X2Π) and O(3P) under Hyperthermal Conditions

Quasiclassical trajectory calculations are performed for hyperthermal collisions between NO(X²Π) and O(³P) on recently developed potential energy surfaces for the lowest doublet and quartet states of the NO₂ system. Three product channels are investigated, and their branching fractions are in reasonably good agreement with the recent crossed molecular beam study at 84 kcal/mol of collision energy. The dominant inelastic channel has a strong forward scattering bias and a high translational energy distribution with limited internal excitation in the scattered NO. The exchange channel has significantly higher NO internal excitation and is also forward biased. The abstraction channel producing internally excited O₂ has the smallest branching fraction and a broader angular distribution also with a forward peak. The angular and translational energy distributions in the three channels are consistent with experiment, but the agreement is not always quantitative. The sources of the differences are discussed. Finally, the impact of NO vibrational excitation on the reactive channels and the corresponding rate coefficients are reported.

Dynamics of Inelastic and Reactive Collisions of 16O(3P) with 15N18O

The dynamics of O(³P) + NO collisions were investigated at a collision energy of ⟨E_coll⟩ = 84.0 kcal mol^-1 with the use of a crossed molecular beams apparatus employing a rotatable mass spectrometer detector. This experiment was performed with beams of ¹⁶O atoms and isotopically labeled ¹⁵N¹⁸O molecules to enable the products of reactive and inelastic scattering to be distinguished. Three scattering pathways were observed: inelastic scattering (¹⁶O + ¹⁵N¹⁸O), O-atom exchange (¹⁸O + ¹⁵N¹⁶O), and O-atom abstraction (¹⁸O¹⁶O + ¹⁵N). All product channels exhibited a preponderance of forward scattering, but scattering over a broad angular range was also observed for all products. For inelastic scattering, an average of 90% of the collision energy is retained in the translation of ¹⁶O and ¹⁵N¹⁸O. On the other hand, for O-atom exchange (which also leads to O + NO products), the collision energy is partitioned roughly evenly between the translation of ¹⁸O + ¹⁵N¹⁶O and the internal excitation of ¹⁵N¹⁶O. The available energy for O-atom abstraction is significantly lower than the collision energy because of the endoergicity of this reaction, but the available energy is again roughly evenly partitioned between the translation of ¹⁸O¹⁶O + ¹⁵N and the internal excitation of the molecular (O₂) product. The relative yields of the three scattering pathways were determined to be 0.751 for inelastic scattering, 0.220 for O-atom exchange, and 0.029 for O-atom abstraction.

Potential energy surface for high-energy N + N2 collisions

Potential energy surface calculations yield physical insight into the structure of intermediates and the dynamics of molecular collisions, and they are the first step toward molecular simulations that provide physical insight into energy transfer, reaction, and dissociation probabilities. The potential energy surface for high-energy collisions of N₂ with N can be used for modeling chemical dynamics and energy transfer in atmospheric shock waves. Here we present an analytic ground-state. (⁴A'') potential energy surface for N₃ that governs electronically adiabatic collisions of N₂(¹Σ+g) with N(⁴S). The fitted surface consists of a pairwise potential based on an accurate diatomic potential energy curve plus a connected permutationally invariant polynomial (PIP) in mixed-exponential-Gaussian bond order variables (MEGs) for the three-body part. The three-body fit is based on multireference complete active space second order perturbation theory (CASPT2) calculations. The quality of the quartet N₃ fit is comparable to that for a previous fit of the NO₂ potential. We characterize two local minima of N₃, two tight transition structures, two van der Waals geometries, and the noncollinear reaction path for the symmetric exchange reaction. The nonreactive approach of an N atom to N₂ along the perpendicular bisector is more repulsive than the collinear reproach, but plots of the force on the bond versus the potential energy at the distance of closest approach allow us to infer that vibrational energy transfer should occur much more readily in high-energy collinear collisions than in high-energy perpendicular-bisector collisions.

State-to-State Transition Study of the Exchange Reaction for N(4S) and O2(X3Σg-) Collision by Quasi-Classical Trajectory

Based on the new ²A' and ⁴A' potential energy surfaces of NO₂ fitted by Varga et al., we conducted a quasi-classical trajectory study on the N(⁴S) +O₂(X³Σ_g^- ) → NO(²Π) + O(³P) reaction, focusing on the high vibrational state up to ν = 25. For different rovibrational states of O₂, within the relative translational energy (E_c) range of 0.1-30 eV, the total exchange cross section (ECS) is calculated, and it is found that the initial relative translational energy and vibration excitation have a significant effect on ECSs, while rotational excitation has little influence; the rate coefficient of the high rovibrational state of O₂ molecules at high temperatures is studied, and it is found that when the vibrational level ν of O₂ is in the range of 0-15, the value of log₁₀ k(T, ν, j) with the vibrational level ν is almost linear, while when ν is greater than 15, it becomes gentle with the increase in ν. Finally, the state-to-state rate coefficients are calculated; our results supply the advantageous state-to-state process data in the NO₂ system, and they are useful for further studying the related hypersonic gas flow at very high temperature.

Reaction of the N Atom with Electronically Excited O2 Revisited: A Theoretical Study

The kinetics of the reaction of N with electronically excited O₂ (singlet a¹Δ_g and b¹Σ_g⁺ states), potentially relevant for NO_x formation in nonthermal air plasma, is theoretically studied using the multireference second-order perturbation theory. The corresponding thermodynamically and kinetically favored reaction pathways together with possible intersystem crossings are identified. It has been revealed that the energy barrier for the N + O₂(a¹Δ_g) → NO + O reaction is approximately twice the barrier height for the counterpart process with O₂(X³Σ_g^-). The molecular oxygen in the b¹Σ_g⁺ state, in turn, proved to be even less reactive to atomic nitrogen than O₂(a¹Δ_g). Appropriate thermal rate constants for specified reaction channels are calculated by the variational transition-state theory incorporating corrections for the tunneling effect, nonadiabatic transitions, and anharmonicity of vibrations for transition states and reactants. The corresponding three-parameter Arrhenius expressions for the broad temperature range (T = 300-4000 K) are reported. At last, post-transition-state molecular dynamics simulations indicate that the N + O₂(a¹Δ_g) reaction produces vibrationally much colder NO molecules than the N + O₂(X³Σ_g^-) process.