Permutation Invariant Polynomial Research Articles

A Neural Network Potential Energy Surface and Quantum Dynamics Study of Ca(1S) + H2(v 0 = 0, j 0 = 0) → CaH + H Reaction.

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.

New Full-Dimensional Reactive Potential Energy Surface for the H4 System.

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.

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 (PES) for the ground state of the BH2 system was constructed using permutation invariant polynomial neural network. In ab initio calculation, the AVQZ basis set and multi-reference configuration interaction method was used. The topographic features of the new PES were discussed in detail. To further clarify the accuracy of PES, the dynamics studies of the B(2p2P) + H2 reaction were carried out using time-dependent wave packet method. The dynamics properties such as reaction probabilities, integral cross sections, differential cross sections (DCSs) and rate constants were reported. The DCSs indicated that the “complex-forming” reaction mechanism plays a dominant role at low collision energies and the forward abstraction mechanism dominates the reaction at high collision energies.

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.

Reaction dynamics for the Cl(2P) + XCl → XCl + Cl(2P) (X = H, D, Mu) reaction on a high-fidelity ground state potential energy surface.

In this work, the dynamics of a prototypical heavy-light-heavy abstract reaction, Cl(2P) + HCl → HCl + Cl(2P), is investigated both by constructing a new potential energy surface (PES) and by rate coefficient calculations. Both the permutation invariant polynomial neural network method and the embedded atom neural network (EANN) method, based on abinitio MRCI-F12+Q/AVTZ level points, are used for obtaining globally accurate full-dimensional ground state PES, with the corresponding total root mean square error being only 0.043 and 0.056 kcal/mol, respectively. In addition, this is also the first application of the EANN in a gas-phase bimolecular reaction. The saddle point of this reaction system is confirmed to be nonlinear. In comparison with both the energetics and rate coefficients obtained on both PESs, we find that the EANN is reliable in dynamic calculations. A full-dimensional approximate quantum mechanical method, ring-polymer molecular dynamics with a Cayley propagator, is employed to obtain the thermal rate coefficients and kinetic isotopic effects of the title reaction Cl(2P) + XCl→ XCl + Cl(2P) (H, D, Mu) on both new PESs, and the kinetic isotope effect (KIE) is also obtained. The rate coefficients reproduce the experimental results at high temperatures perfectly but with moderate accuracy at lower temperatures, but the KIE is with high accuracy. The similar kinetic behavior is supported by quantum dynamics using wave packet calculations as well.

Hidden symmetries and large N factorisation for permutation invariant matrix observables

Permutation invariant polynomial functions of matrices have previously been studied as the observables in matrix models invariant under SN, the symmetric group of all permutations of N objects. In this paper, the permutation invariant matrix observables (PIMOs) of degree k are shown to be in one-to-one correspondence with equivalence classes of elements in the diagrammatic partition algebra Pk (N). On a 4-dimensional subspace of the 13-parameter space of SN invariant Gaussian models, there is an enhanced O(N) symmetry. At a special point in this subspace, is the simplest O(N) invariant action. This is used to define an inner product on the PIMOs which is expressible as a trace of a product of elements in the partition algebra. The diagram algebra Pk (N) is used to prove the large N factorisation property for this inner product, which generalizes a familiar large N factorisation for inner products of matrix traces invariant under continuous symmetries.

Permutation invariant polynomial neural network based diabatic ansatz for the (E + A) × (e + a) Jahn-Teller and Pseudo-Jahn-Teller systems.

In this work, the permutation invariant polynomial neural network (PIP-NN) approach is employed to construct a quasi-diabatic Hamiltonian for system with non-Abelian symmetries. It provides a flexible and compact NN-based diabatic ansatz from the related approach of Williams, Eisfeld, and co-workers. The example of H3 + is studied, which is an (E + A) × (e + a) Jahn-Teller and Pseudo-Jahn-Teller system. The PIP-NN diabatic ansatz is based on the symmetric polynomial expansion of Viel and Eisfeld, the coefficients of which are expressed with neural network functions that take permutation-invariant polynomials as input. This PIP-NN-based diabatic ansatz not only preserves the correct symmetry but also provides functional flexibility to accurately reproduce ab initio electronic structure data, thus resulting in excellent fits. The adiabatic energies, energy gradients, and derivative couplings are well reproduced. A good description of the local topology of the conical intersection seam is also achieved. Therefore, this diabatic ansatz completes the PIP-NN based representation of DPEM with correct symmetries and will enable us to diabatize even more complicated systems with complex symmetries.

The global potential energy surface of the RbH2 system and dynamics studies of the H + RbH → Rb + H2 reaction

A new global potential energy surface of the ground state of the RbH2system is constructed by using the permutation invariant polynomial neural network method.In the ab initiocalculations, the aug-cc-pVQZ basis set for H, Def2-QZVPP basis set for Rb, and multi-reference configuration interaction method with Davidson correction are used. The topographical features of the new PES are discussed in detail, and no wells and barriers in the minimum energy path are found. Thereafter, specific initial-state dynamics calculations of the H + RbH (v = 0 ∼ 2,j = 0) → Rb + H2reaction are carried out based on the new PES by means of the time-dependent quantum wave packet method with a second-order split operator. The dynamics results indicate that the vibrational excitation of RbH molecules reduces and increases the reaction probability when the collision energy is below 0.4 eV and above 0.4 eV, respectively.

Gaussianity and typicality in matrix distributional semantics

Constructions in type-driven compositional distributional semantics associate large collections of matrices of size $D$ to linguistic corpora. We develop the proposal of analysing the statistical characteristics of this data in the framework of permutation invariant matrix models. The observables in this framework are permutation invariant polynomial functions of the matrix entries, which correspond to directed graphs. Using the general 13-parameter permutation invariant Gaussian matrix models recently solved, we find, using a dataset of matrices constructed via standard techniques in distributional semantics, that the expectation values of a large class of cubic and quartic observables show high Gaussianity at levels between 90 to 99 percent. Beyond expectation values, which are averages over words, the dataset allows the computation of standard deviations for each observable, which can be viewed as a measure of typicality for each observable. There is a wide range of magnitudes in the measures of typicality. The permutation invariant matrix models, considered as functions of random couplings, give a very good prediction of the magnitude of the typicality for different observables. We find evidence that observables with similar matrix model characteristics of Gaussianity and typicality also have high degrees of correlation between the ranked lists of words associated to these observables.

Permutation invariant Gaussian two-matrix models

We construct the general permutation invariant Gaussian two-matrix model for matrices of arbitrary size D. The parameters of the model are given in terms of variables defined using the representation theory of the symmetric group S D . A correspondence is established between the permutation invariant polynomial functions of the matrix variables (the observables of the model) and directed colored graphs, which sheds light on stability properties in the large D counting of these invariants. A refined counting of the graphs is given in terms of double cosets involving permutation groups defined by the local structure of the graphs. Linear and quadratic observables are transformed to an S D representation theoretic basis and are used to define the convergent Gaussian measure. The perturbative rules for the computation of expectation values of graph-basis observables of any degree are given in terms of the representation theoretic parameters. Explicit results for a number of observables of degree up to four are given along with a Sage programme that computes general expectation values.

Dynamics of C<sup>+</sup> + H<sub>2</sub> reaction based on a new potential energy surface

The multi-reference interaction method is explicitly dependent on the electron-electron distance, and ACVQZ basis set is used in the <i>ab initio</i> calculation. The potential energy surface (PES) is fitted by using the permutation invariant polynomial neural network method based on 18222 <i>ab initio</i> points. In addition, the topographical features of the PES are compared with available theoretical and experimental data. The results indicate that the present PES is more accurate and can be applied to any type of dynamic study. In order to validate the PES, the dynamic study of the C<sup>+</sup> + H<sub>2</sub> → H + CH<sup>+</sup> reaction is carried out by using the quasi-classical trajectory method in a collision energy range of 0.4–1.0 eV. The integral cross sections and differential cross sections are calculated and compared with previous theoretical studies. For the integral cross section, the present results are, in general, in good agreement with previous theoretical studies, both of which increase with collision energy increasing. The forward and backward symmetric differential cross sections indicate that the “complex-forming” mechanism plays a dominant role in the reaction.

Adiabatic potential energy surfaces and photodissociation mechanisms for highly excited states of H2O

Full-dimensional adiabatic potential energy surfaces of the electronic ground state X̃ and nine excited states Ã, Ĩ, B̃, C̃, D̃, D̃′, D̃″, Ẽ′ and F̃ of H2O molecule are developed at the level of internally contracted multireference configuration interaction with the Davidson correction. The potential energy surfaces are fitted by using Gaussian process regression combining permutation invariant polynomials. With a large selected active space and extra diffuse basis set to describe these Rydberg states, the calculated vertical excited energies and equilibrium geometries are in good agreement with the previous theoretical and experimental values. Compared with the well-investigated photodissociation of the first three low-lying states, both theoretical and experimental studies on higher states are still limited. In this work, we focus on all the three channels of the highly excited state, which are directly involved in the vacuum ultraviolet photodissociation of water. In particular, some conical intersections of D̃–Ẽ′, Ẽ′-F̃, ÖĨ and Ĩ–C̃ states are clearly illustrated for the first time based on the newly developed potential energy surfaces (PESs). The nonadiabatic dissociation pathways for these excited states are discussed in detail, which may shed light on the photodissociation mechanisms for these highly excited states.

A neural network potential energy surface and quantum dynamics studies for the Ca+(2S) + H2 → CaH+ + H reaction.

Reactive collisions of Ca+ ions with H2 molecules play a crucial role in ultracold chemistry, quantum information and other cutting-edge fields, and have been widely studied experimentally, but the corresponding theoretical studies have not been reported due to the lack of an applicable potential energy surface (PES). Herein, a globally accurate PES of the ground-state CaH2+ is constructed using the permutation invariant polynomial neural network method based on 27 780 ab initio points calculated at the multi-reference configuration interaction level. On the new PES, the quantum time-dependent wave packet calculations are performed to study the dynamics mechanisms of the Ca+(2S) + H2(ν0 = 0, j0 = 0) → CaH+ + H reaction. The calculated results suggest that the reaction follows a direct abstraction process when the collision energy is below 5.0 eV. The dynamics results would have a great reference significance for the experimental research of this reactive system at a finer level, and further dynamics studies, such as the effects of isotope substitution and rovibrational excitations of the reactant molecule, could be carried out on this newly constructed PES.

The Peculiar Interaction of Trifluoride Anions with Cryogenic Rare Gas Matrices.

In this contribution, we present theoretical modeling of the interaction between rare gas matrices and a trifluoride guest anion, as well as its quantitative effect on measured vibrational spectra. Using a combination of coupled-cluster electronic structure calculations and a many-body potential expansion coupled with permutation invariant polynomial fitting and anharmonic vibrational spectrum simulations, we shed light on the origin of the trifluoride matrix effects observed experimentally. The theoretical spectra are found to reproduce accurately the measured data while providing deeper insights into the effects of the guest-host interaction. The investigations reveal that neon can only stabilize trifluoride in hexagonal cavities formed by double vacancies, while argon can host the anion in a variety of cavities ranging from zero to two defects in the matrix. The origin of this structural variability can be traced back to the disparate strengths of the host-host interactions in neon and argon. The present work demonstrates the importance of theoretical modeling to complement matrix isolation experiments, which alone do not provide direct information about the structure of the matrices or about the physical origin of their interaction and of their spectroscopic signature.

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

A new potential energy surface (PES) of the ground state (X 2 A′) of Li 2 H was built using 18,403 ab initio points. The MRCI-F12 method with AVQZ basis set were used in the ab initio calculation. Furthermore, the widely used permutation invariant polynomial neural network method was adopted in the fitting process. The features of new PES were compared with previous theoretical and experimental studies. The results indicated that the spectroscopic parameters of PES are in good agreement with experimental data. In addition, the dynamics calculations of the H + Li 2 → Li + LiH reaction were carried out using time-dependent wave packet method based on the new PES. The dynamics properties such as reaction probability, integral cross section and thermal rate constant were reported and compared with previous theoretical studies.

A new global potential energy surface of the SH2+(X4A'') system and quantum calculations for the S+ + H2(v = 0-3, j = 0) reaction.

A new global potential energy surface (PES) for the ground state of the SH2+(X4A'') system is constructed using a permutation invariant polynomial neural network method. In ab initio calculations, the MRCI-F12 method with the AVTZ basis set is used. Furthermore, the dynamics calculations of the S+ + H2(v = 0-3, j = 0) → SH+ + H reaction are carried out based on the new PES. The reaction probabilities and integral cross sections are compared with available theoretical calculations. Present values are in general good agreement with the previous theoretical studies. However, some discrepancies can still be found due to different PESs used in the calculation. Furthermore, the vibrational energy of the reactant molecule can significantly enhance the reactivity compared to the translational energy. The differential cross sections indicated that the reaction mechanism is changed from the "head-on" rebound mechanism to the tripping mechanism with the increasing number of initial vibrational excitation state.

Constructing feed-forward artificial neural networks to fit potential energy surfaces for molecular simulation of high-temperature gas flows.

Kinetic rates for thermochemical nonequilibrium models are generally computed from quasiclassical trajectory (QCT) calculations on accurate ab initio potential energy surfaces (PES). In this article, we use a feed-forward artificial neural network (ANN) to fit existing single-point energies for N_{2}+N_{2} interactions [Bender et al., J. Chem. Phys. 143, 054304 (2015)JCPSA60021-960610.1063/1.4927571] to construct a PES suitable for molecular simulation of high-temperature gas flows. We then perform detailed comparisons with a widely used N_{4} PES that was built using the permutation invariant polynomials (PIP) method. Specific physical considerations in the construction of the ANN for this application are detailed. Translation, rotation, and permutation invariance are precisely satisfied by mapping the interatomic distances onto a set of permutation invariant inputs, known as fundamental invariants (FI) that generate the permutation invariant polynomial ring. The diatomic energy is imposed by decomposing the total potential energy into a sum of a two-body and a many-body energy contribution. To obtain the correct dynamical behavior with the most basic, yet computationally efficient ANN, spurious long-distance interactions must be removed to avoid incorrect physical behavior at the dissociation threshold. We use a simple apodization function to smoothly taper off to zero any residual many-body interaction at large separations. Both accuracy and performance of the FI-ANN PES are assessed. QCT calculations are used to compute dissociation probabilities and vibrational energy distributions at various equilibrium temperatures. Excellent agreement with the results obtained from the PIP PES is found. For our test case, the ANN PES is also significantly more computationally efficient than the PIP PES at comparable root-mean-square error levels.

Theoretical Investigations of Rate Coefficients for H + O3 and HO2 + O Reactions on a Full-Dimensional Potential Energy Surface.

In this work, a machine learning method is used to construct a high-fidelity multichannel global reactive potential energy surface (PES) for the HO3 system from 21452 high-level ab initio calculations at the explicitly correlated multireference configuration interaction (MRCI-F12) level of theory. The permutation invariance of the PES with respect to the three identical oxygen atoms is enforced using permutation invariant polynomials (PIPs) in the input layer of a neural network (NN). This PIP-NN representation is highly faithful to the ab initio points, with a root-mean-square error of 0.20 kcal/mol. Using this PES, the kinetics of H + O3 → OH + O2 (R1) and HO2 + O → OH + O2 (R2) reactions were investigated using a quasi-classical trajectory method over a wide temperature range (200-2000 K). It was found that the calculated thermal rate coefficients of R1 and R2, exhibiting positive and negative temperature dependences, respectively, are in reasonably good agreement with most experimental measured values. These temperature dependences can be attributed to the presence and absence of an entrance channel potential barrier.

Accurate Global Potential Energy Surfaces for the H + CH3OH Reaction by Neural Network Fitting with Permutation Invariance.

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.

A full-dimensional ab initio intermolecular potential energy surface and ro-vibrational spectra for N2-HF and N2-DF.

A full-dimensional ab initio intermolecular potential energy surface (IPES) is reported in this paper for van der Waals complex N2-HF. The calculations were performed by employing the explicitly correlated coupled cluster [CCSD (T)-F12a] method with the augmented correlation-consistent aug-cc-pVTZ basis set plus bond functions. The basis set superposition error was corrected by the full counterpoise procedure. About 55 000 ab initio points were calculated and then fitted by the permutation invariant polynomial neural network approach with a root-mean-square error of 0.433 cm-1. The potential energy surface features two equivalent linear minima with a well depth of 811.012 cm-1 separated by a barrier of 635.836 cm-1. The ro-vibrational energy levels for N2-HF and N2-DF were calculated based on the vibrationally averaged 4D IPESs with the radial discrete variable representation/angular finite basis representation method and Lanczos propagation algorithm. The calculated frequencies and the relative line intensities in the HF (DF) stretching band agree well with the available observed spectra. The theoretical band origins are all red shifted relative to the isolated HF (DF) molecule and reproduce the experimental values well. The results of ro-vibrational state calculations demonstrate the high accuracy of our new PES.