Hyperspherical Coordinates Research Articles

Statistical adiabatic channel model for termolecular reactions.

In this work, we present a statistical adiabatic channel model for termolecular reactions, A + B + C → Products. Our approach relies on hyperspherical coordinates, where the adiabatic channels are readily defined in the hyper-radius after averaging the hyperangular degrees of freedom. In this way, we find a general expression for termolecular rate constants. We focus on ion-neutral association reactions to test our approach's accuracy and predictive power, finding a good agreement between theory and experiment, especially in those reactions' temperature dependence.

Ortho-Para Conversion for H+ + H2 Collision in Low Temperature: A Fully Close-Coupled Time-Dependent Wave Packet Study.

The collision-induced rate coefficients of ortho-para conversion for the H+ + H2 reaction provide accurate information to probe the lifetime of cold environments in interstellar media. Rotationally resolved reaction probabilities are calculated at the low collision energy regime (0 < Ecol ≤ 0.3 eV) by employing the coupled three-dimensional (3D) time-dependent wave packet (TDWP) formalism in hyperspherical coordinates on a recently constructed ab initio ground adiabatic potential energy surface of H3+ [J. Chem. Phys. 2014, 141, 204306] for the process H+ + H2 (v = 0, j = 0-5) → H+ + H2 (v' = 0, j'). Cross-sections are then computed from the converged reaction probabilities as a function of total angular momentum (J) over the same energy regime and subsequently employed to obtain the rate constants for the ortho-to-para (O-P) and para-to-ortho (P-O) conversions and their ratio. The ratio of ortho-para conversion shows (a) appropriate convergence with the inclusion of a higher number of initial rotational states as well as a reasonable agreement with the results from a quantum statistical method and (b) a peak at lower temperature that could be due to the available collision energy for transitions involving lower rotational states (j = 0 → j' = 1).

The Interaction-Asymptotic Region Decomposition Method in Jacobi Coordinates: Triatomic Reactive Scatterings.

The interaction-asymptotic region decomposition (IARD) technique has been proven to be a good solution to the long-standing coordinate problem in reactive scattering calculations. In this work, the IARD technique was further developed using Jacobi coordinates for the interaction region, instead of the previously used hyperspherical coordinates. Although the Jacobi coordinate may not be as optimal as the hyperspherical coordinates for describing the interaction region in reactive scattering processes, it has simpler kinetic operators and provides a more physically intuitive picture. By developing an intermediate interpolation method, which could efficiently transform the overlapped wave functions from the asymptotic regions to the interaction region, the new implementation of the IARD technique for triatomic reactive scatterings is similarly efficient and accurate. The differential cross sections of the H+H2, and product state-resolved reaction probabilities of the F+HD and16O+36O2 reactions, which involve products of extremely low translational energy and are challenging for a single coordinate-based method, were calculated as numerical examples to show the ability of the new method.

Sulfur recombination: A direct approach.

This work presents a direct three-body recombination approach of the sulfur recombination reaction, S + S + M → S2 + M, at temperatures between 100 and 500K. Our calculations for M = Ar, based on a classical trajectory approach in hyperspherical coordinates, show excellent agreement with the experimental measurement at T = 298K of Fair and Thrush [Trans. Faraday Soc. 65, 1208 (1969)]. Similarly, we find that the production of S2 strongly depends on the SAr product, the other possible reaction channel. Finally, using the classical threshold law, we check sulfur recombination with another third body, M = H2S, and find no significant change in the rate.

Product state-resolved reactive scattering calculations using stair shaped grids in hyperspherical coordinates for the quantum wave packet method

Product state-resolved reactive scattering calculations using stair shaped grids in hyperspherical coordinates for the quantum wave packet method

Development of the Time-Independent Methods for the Cl + H2/F + HD Reaction Using Hyper-Spherical Coordinates Including (Full) Spin-Orbit Characteristics.

Recently, a combined study of high-resolution molecular crossed beam experiment and accurate full-dimensional time-dependent theory, including full spin-orbit characteristics on the effect of electronic spin and orbital angular momenta in the F + HD reaction, was reported by some of us, focusing on the partial wave resonance phenomenon (Science 2021, 371, 936-940). It revealed that the time-dependent theory could explain all of the details observed in the high-resolution experiment. Here, we develop two time-independent close-coupling methods using hyperspherical coordinates, including the two-state model, where only a part of the spin-orbit characteristics is considered, and the six-state model, where the full spin-orbit characteristics is considered. With these two newly developed theoretical models and the adiabatic theoretical model, the detailed reaction dynamics of the F + HD (v = 0, j = 0) reaction and the Cl + H2 (v = 0, j = 0) reaction are investigated and compared. Some of the results are compared with the time-dependent quantum wave packet theory and the experimental observations, and good agreements have been obtained, which suggests the validity of the pure-procession approximation in the six-state model using different theoretical methods. This work demonstrates the ability of the reactive scattering theory including full spin-orbit characteristics for describing the reactions of a halogen atom plus hydrogen molecule and its isotopologues.

Multivariate Gaussian process surrogates for predicting basic structural parameters of refractory non-dilute random alloys

Refractory non-dilute random alloys consist of two or more principal refractory metals with complex interactions that modify their basic structural properties such as lattice parameters and elastic constants. Atomistic simulations (ASs) are an effective method to compute such basic structural parameters. However, accurate predictions from ASs are computationally expensive due to the size and number of atomistic structures required. To reduce the computational burden, multivariate Gaussian process regression (MVGPR) is proposed as a surrogate model that only requires computing a small number of configurations for training. The elemental atom percentage in the hyper-spherical coordinates is demonstrated to be an effective feature for surrogate modeling. An additive approximation of the full MVGPR model is also proposed to further reduce computations. To improve surrogate accuracy, active learning is used to select a small number of alloys to simulate. Numerical studies based on AS data show the accuracy of the surrogate methodology and the additive approximation, as well as the effectiveness and robustness of the active learning for selecting new alloy designs to simulate.

Py3BR: A software for computing atomic three-body recombination rates.

The three-body recombination reaction, or ternary association, is a termolecular reaction leading to a molecule after a three-body encounter that plays a vital role in many relevant scenarios in chemical physics. Here, we introduce the Python 3-Body Recombination program, which is dedicated to the computation of atomic three-body recombination rate coefficients. The software is based on a classical trajectory approach in hyperspherical coordinates after mapping the three-body problem as a single particle in a higher-dimensional space. This theoretical approach is fully general and applicable to any ion-atom-atom or atom-atom-atom three-body process. The predictive power of the methodology has been tested in several different experimental scenarios, reaching a good description of every system. The code structure is presented alongside examples and tests to ensure the software's capacity. In addition, the performance of the software after parallelization is shown.

Bayesian estimation in generalized linear models for longitudinal data with hyperspherical coordinates

Under the framework of generalized linear models (GLM), the generalized estimating equation (GEE) method is typically applied for longitudinal data analysis. However, there are a series of problems due to the misspecification of the within-subject correlation structure, especially in Bayesian estimation. To handle these difficulties, in this paper, we construct a class of generalized estimating equations for longitudinal data with hyperspherical coordinates (HPC) and propose a Bayesian approach established through empirical likelihood (EL). Additionally, an efficient Markov chain Monte Carlo (MCMC) procedure is developed for the required computation of the posterior distribution. As proved by the simulation studies and an application to a real longitudinal data set, our method not only performs better than traditional empirical likelihood estimation and Bayesian estimation with partial autocorrelations (PAC) but also is suitable for non-Gaussian data.

Theoretical Development of the Interaction-Asymptotic Region Decomposition Method for Tetratomic Reactive Scattering.

An accurate and efficient time-dependent wave packet method is proposed for solving the product state-resolved reaction probabilities of the tetratomic reactive system. In this method, the entire scattering process is divided into the interaction region and multiple asymptotic regions, sharing the same spirit as the interaction-asymptotic region decomposition (IARD) approach in a triatomic reactive scattering process. The hyperspherical coordinate is adopted in the interaction region, while the corresponding Jacobi coordinate is employed in each asymptotic region. Therefore, in this IARD method, the "coordinate problem", the difficulty of expressing the wave function in the entire region using a single coordinate system, can be effectively avoided, and only a very small number of the grid points (or the basis functions) are required. For the numerical illustration, the typical tetratomic reaction H2 + OH with zero total angular momentum is calculated, and compared with other quantum wave packet methods. Our proposed IARD method for the tetratomic reactive system is much more efficient and accurate.

Energy spectrum and applications of Eckart plus Hellmann potential in hyperspherical coordinates

We investigate the approximate bound-state solutions of the N-dimensional Schrödinger equation with a combination of Eckart and Hellmann potentials via the asymptotic iteration approach. By the use of the Greene-Aldrich approximation, which is valid for small values of the screening parameter, we establish the N-dimensional energy spectrum and the N-dimensional radial wave function in approximate analytic form. In hyperspherical coordinates, the normalized radial wave function is expressed in terms of hypergeometric and Jacobi polynomials. To double-check the energy spectrum in hyperspherical coordinates, we utilize polynomial solutions in view of the asymptotic iteration approach. The expressions for the expectation values of inverse position, square of inverse position, kinetic energy, and square of momentum are derived in hyperspherical coordinates by using the Hellmann-Feynmann theorem. We also present the analytically calculated energy eigenvalues for Eckart plus Hellmann potential. We deduce special forms of the relevant potential, such as Eckart, Hellmann, and Hulthén potentials.

A time-dependent quantum wavepacket method on stair-shaped grids for reactive scattering using the hyperspherical coordinates

Development of efficient numerical methods for quantum reactive scattering process by using time-dependent quantum wavepacket method is always necessary in the field of quantum chemistry, due to the complexity of the system, and the large computational efforts required. To perform quantum reactive scattering calculation in hyperspherical coordinates, as the hyper-radial (ρ) increases, the potential well along the angular degree of the freedom (χ) becomes narrower and narrower, which thus requires denser and denser grid points for well describing the diatomic vibrational motion. This makes the time dependent wavepacket packet calculations prohibitively difficult using hyperspherical coordinate when the reaction involves long range interaction potential or requires long absorption potential. In this study, we developed a stair-shaped grid based quantum wavepacket method to overcome this difficulty. In this method, the hyper-radial is partitioned into multidomains in the form of a stair-shape. In each domain, different grid point numbers could be applied for the angular degree of the freedom (χ). In this way, the computational effort could be saved. For numerical example, the stair-shaped grid based quantum wavepacket method is applied to compute the total reaction probability of the F+HCl (v0,j0)→ HF+Cl reaction , which involves long range interaction potential. The results are compared with those calculated with the reactant coordinate-based (RCB) Jacobi coordinate and Interaction Asymptotic Region Decomposition (IARD) method.

A single resonance Regge pole dominates the forward-angle scattering of the state-to-state F + H2 → FH + H reaction at Etrans = 62.09 meV.

The aim of the present paper is to bring clarity, through simplicity, to the important and long-standing problem: does a resonance contribute to the forward-angle scattering of the F + H2 reaction? We reduce the problem to its essentials and present a well-defined, yet rigorous and unambiguous, investigation of structure in the differential cross sections (DCSs) of the following three state-to-state reactions at a translational energy of 62.09 meV: F + H2(vi = 0, ji = 0, mi = 0) → FH(vf = 3, jf = 0, 1, 2, mf = 0) + H, where vi, ji, mi and vf, jf, mf are the initial and final vibrational, rotational and helicity quantum numbers respectively. Firstly, we carry out quantum-scattering calculations for the Fu-Xu-Zhang potential energy surface, obtaining accurate numerical scattering matrix elements for indistinguishable H2. The calculations use a time-independent method, with hyperspherical coordinates and an enhanced Numerov method. Secondly, the following theoretical techniques are employed to analyse structures in the DCSs: (a) full and Nearside-Farside (NF) partial wave series (PWS) and local angular momentum theory, including resummations of the full PWS up to second order. (b) The recently introduced "CoroGlo" test, which lets us distinguish between glory and corona scattering at forward angles for a Legendre PWS. (c) Six asymptotic (semiclassical) forward-angle glory theories and three asymptotic farside rainbow theories, valid for rainbows at sideward-scattering angles. (d) Complex angular momentum (CAM) theories of forward and backward scattering, with the Regge pole positions and residues computed by Thiele rational interpolation. Thirdly, our conclusions for the three PWS DCSs are: (a) the forward-angle peaks arise from glory scattering. (b) A broad (hidden) farside rainbow is present at sideward angles. (c) A single Regge pole contributes to the DCS across the whole angular range, being most prominent at forward angles. This proves that a resonance contributes to the DCSs for the three transitions. (d) The diffraction oscillations in the DCSs arise from NF interference, in particular, interference between the Regge pole and direct subamplitudes.

Lifetimes and decay mechanisms of isotopically substituted ozone above the dissociation threshold: matching quantum and classical dynamics.

Energies and lifetimes of vibrational resonances were computed for 18O-enriched isotopologue 50O3 = {16O16O18O and 16O18O16O} of the ozone molecule using hyperspherical coordinates and the method of complex absorbing potential. Various types of scattering resonances were identified, including roaming OO-O rotational states, the series corresponding to continuation of bound vibrational resonances of highly excited bending or symmetric stretching vibrational modes. Such a series become metastable above the dissociation limit. The coupling between the vibrationally excited O2 fragment and rotational roaming gives rise to Feshbach type resonances in ozone. Different paths for the formation and decay of symmetric 16O18O16O and asymmetric species 16O16O18O were also identified. The symmetry properties of the total rovibronic wave functions of the 18O-enriched isotopologues are discussed in the context of allowed dissociation channels.

Analysis of quantum systems subject to inverse trigonometry Scarf plus Coulomb potential in N-dimensions

We present approximate solutions of the N-dimensional Klein–Gordon equation in the presence of a superposition of inverse trigonometry Scarf potential and Coulomb potential through asymptotic iteration approach. With the purpose of dealing with the -state, we use Greene-Aldrich approximation which is valid for the small values of the screening parameter. The analytical expressions for the relativistic energy eigenvalues and the corresponding eigenfunctions are established in hyperspherical coordinates. The normalized hyper-radial wavefunction is expressed in terms of hypergeometric and Jacobi polynomials. We analyze the dependences of the relativistic energy levels on the dimensions. We also discuss the effect of the screening parameter on the relativistic energy spectrum. The expectation values of inverse position the square of inverse position kinetic energy and square of momentum in N-dimensional space are reported using the Hellmann-Feynmann theorem. Discussing the special cases of the relevant potential, we show the accuracy of our results.

High-Energy Quantum Dynamics of the 15N + o-14N14N Rovibrational Activation and Isotope Exchange Processes.

We report full quantum reaction probabilities, computed within the framework of time-independent quantum mechanics using hyperspherical coordinates, for the 15N + 14N14N inelastic and reactive collision processes, restricted to total angular momentum J = 0, for kinetic energies up to 4.5 eV. We take advantage of the nonzero (i = 1) nuclear spin of 14N, leading to the existence of two nuclear spin isomers of 14N14N, namely, ortho- and para-14N14N, to restrict the study to the ortho molecular nitrogen species, with even rotational quantum number j = 0, 2, ... states. Specifically, we start with diatomic reagents ortho-14N14N in the initial rotational state j = 0. A comparison with similar works previously published by other groups using time-dependent wave packet and quasi-classical trajectory methods for the 14N + 14N14N fully symmetric collision is given. We find that reactive processes 15N + 14N14N involving atom exchange do not happen for collision energies less than 2.2 eV. Collisions at energies of around 2.0 eV are most effective for populating reactants' rovibrational states, that is, for inelastic scattering, whereas those at energies close to 5.0 eV yield a newly formed 14N15N isotopologue in a wide variety of excited vibrational levels.

Coupled three-dimensional quantum mechanical wave packet study of proton transfer in H2+ + He collisions on accurate abinitio two-state diabatic potential energy surfaces.

We have carried out fully close-coupled three dimensional quantum mechanical wave packet dynamical calculations for the reaction He+H2+→HeH++H on the ground electronic adiabatic potential energy surface and on the lowest two electronic states of newly constructed abinitio calculated diabatic potential energy surfaces for the system [Naskar et al., J. Phys. Chem. A 127, 3832 (2023)]. With the reactant diatom (H2+) in its roto-vibrational ground state (v = 0, j = 0), the calculations have been carried out in hyperspherical coordinates to obtain the reaction attributes. Convergence profiles of the reaction probability with respect to the total angular momentum quantum number at different collision energies are presented for the title reaction. State-to-state as well as initial state selected integral reaction cross sections are calculated from the fully converged reaction probabilities over a range of collision energies. The integral cross section values computed using the two-state diabatic potential energy surfaces are significantly lower than those obtained using the ground electronic state adiabatic potential energy surface and are in much better agreement with the available experimental results than the latter for total energy greater than 1.1eV. Therefore, it becomes clear that it is important to include the nonadiabatic coupling terms for a quantitative prediction of the dynamical observables.

Beyond Born-Oppenheimer Constructed Diabatic Potential Energy Surfaces for HeH2.

First-principles based beyond Born-Oppenheimer theory has been employed to construct multistate global Potential-Energy Surfaces (PESs) for the HeH2+ system by explicitly incorporating the Nonadiabatic Coupling Terms (NACTs). Adiabatic PESs and NACTs for the lowest four electronic states (12A', 22A', 32A' and 42A') are evaluated as functions of hyperangles for a grid of fixed values of the hyperradius in hyperspherical coordinates. Conical intersection between different states are validated by integrating the NACTs along appropriately chosen contours. Subsequently, adiabatic-to-diabatic (ADT) transformation angles are determined by solving the ADT equations to construct the diabatic potential matrix for the HeH2+ system which are smooth, single-valued, continuous, and symmetric and are suitable for performing accurate scattering calculations for the titled system.

Bayesian estimation for longitudinal data in a joint model with HPCs

In longitudinal data analysis, linear models are typically utilized. However, deriving the Bayesian estimation with respect to the misspecification of the correlation structure is a challenging task. In this article, we construct a joint mean–covariance model with angles or hyperspherical coordinates (HPCs) for which we then present a Bayesian framework. Based on the connection with the semipartial correlations (SPCs), we focus on the selection (sparsity) priors on these angles. An efficient Markov chain Monte Carlo (MCMC) algorithm is developed for the proposed model, and the positive definiteness of the correlation matrix in posterior computation is automatically guaranteed by our method. Ultimately, we compare the performance of our joint model with some recent methods focusing only on the correlation matrix by using simulations and clinical trial data on smoking.

Low-lying bound and resonance states in Λ61Ca and ΛΛ62Ca

In this paper, the energy, and wavefunctions of the low-lying bound and resonance states of [Formula: see text]Ca and [Formula: see text]Ca are studied. For [Formula: see text]Ca we adopted [Formula: see text]Ca two-body cluster model while for [Formula: see text]Ca we adopted [Formula: see text]Ca three-body cluster model. Two models of potentials, namely SH: Hulthén type and WS: Woods–Saxon type potentials, each having a spin-orbit coupling term is chosen for the [Formula: see text]-nucleus pair while three-range Gaussian (TRG) Nijmegen hardcore and softcore potential models are chosen for [Formula: see text] pair. The resulting few-body (two- and three-body) Schrödinger equations are solved numerically to get the bound state energies and wavefunctions. For the three-body calculation, we used hyperspherical coordinates. The formalism of supersymmetric quantum mechanics is applied to construct a one-parameter family of isospectral potential to investigate the resonance states of the [Formula: see text]Ca ([Formula: see text]Ca) core-[Formula: see text] two-body system. To study the effect of replacement of the outer core valence nucleon(s) of [Formula: see text]Ca by [Formula: see text] hyperon(s), two- and three-body cluster model calculations for the bound states of [Formula: see text]Ca and [Formula: see text]Ca are also carried out using the same set of two-body potentials, as used for [Formula: see text]Ca and [Formula: see text]Ca respectively. And a clear enhancement in the one- and two-[Formula: see text] separation energies and a shrinking of the overall size of the nucleus is indicated. The computed results are compared with some of those found in the literature.