Hyperspherical Coordinate Method Research Articles

Ion-atom-atom three-body recombination: From the cold to the thermal regime.

We present a study on ion-atom-atom reaction A + A + B+ in a wide range of systems and collision energies ranging from 100 μK to 105 K, analyzing two possible products: molecules and molecular ions. The dynamics is performed via a direct three-body formalism based on a classical trajectory method in hyperspherical coordinates developed in Pérez-Ríos et al. [J. Chem. Phys. 140, 044307 (2014)]. Our chief finding is that the dissociation energy of the molecular ion product acts as a threshold energy, separating the low- and high-energy regimes. In the low-energy regime, the long-range tail of the three-body potential dictates the fate of the reaction and the main reaction product. On the contrary, in the high-energy regime, the short-range of atom-atom and atom-ion interaction potential dominate the dynamics, enhancing molecular formation.

Hyperradial Distribution Function of Few-Body Problems: A New Arena for Extreme Value Theory

This work explores classical capture models for few-body systems via a Monte Carlo method in hyperspherical coordinates. In particular, we focus on van der Waals and charged-induced dipole interactions. As a result, we notice that, independently of the number of particles and interparticle interaction, the capture hyperradial distribution function follows a Fr\'echet distribution, a special type of the generalized extreme value distribution. Besides, we elaborate on the fundamentals of such universal feature using the general extreme value theory, thus, establishing a connection between extreme value theory and few-body physics.

Classical threshold law for the formation of van der Waals molecules.

We study the role of pairwise long-range interactions in the formation of van der Waals molecules through direct three-body recombination processes A + B + B → AB + B, based on a classical trajectory method in hyperspherical coordinates developed in our earlier works [J. Pérez-Ríos et al., J. Chem. Phys. 140, 044307 (2014); M. Mirahmadi and J. Pérez-Ríos, J. Chem. Phys. 154, 034305 (2021)]. In particular, we find the effective long-range potential in hyperspherical coordinates with an exact expression in terms of dispersion coefficients of pairwise potentials. Exploiting this relation, we derive a classical threshold law for the total cross section and the three-body recombination rate yielding an analytical expression for the three-body recombination rate as a function of the pairwise long-range coefficients of the involved partners.

On the formation of van der Waals complexes through three-body recombination.

In this work, we show that van der Waals molecules X-RG (where RG is the rare gas atom) may be created through direct three-body recombination collisions, i.e., X + RG + RG → X-RG + RG. In particular, the three-body recombination rate at temperatures relevant for buffer gas cell experiments is calculated via a classical trajectory method in hyperspherical coordinates [Pérez-Ríos et al., J. Chem. Phys. 140, 044307 (2014)]. As a result, it is found that the formation of van der Waals molecules in buffer gas cells (1 K ≲ T ≲ 10 K) is dominated by the long-range tail (distances larger than the LeRoy radius) of the X-RG interaction. For higher temperatures, the short-range region of the potential becomes more significant. Moreover, we notice that the rate of formation of van der Walls molecules is of the same order of the magnitude independent of the chemical properties of X. As a consequence, almost any X-RG molecule may be created and observed in a buffer gas cell under proper conditions.

Two-nucleon emitters within a pseudostate method: The case of Be6 and Be16

Background: Since the first experimental observation, two-nucleon radioactivity has gained renewed attention over the past fifteen years. The $^6$Be system is the lightest two-proton ground-state emitter, while $^{16}$Be has been recently proposed to be the first two-neutron ground-state emitter ever observed. A proper understanding of their properties and decay modes requires a reasonable description of the three-body continuum. Purpose: Study the ground-state properties of $^6$Be and $^{16}$Be within a general three-body model and investigate their nucleon-nucleon correlations in the continuum. Method: The pseudostate (PS) method in hyperspherical coordinates, using the analytical transformed harmonic oscillator (THO) basis for three-body systems, is used to construct the $^6$Be and $^{16}$Be ground-state wave functions. These resonances are approximated as a stable PS around the known two-nucleon separation energy. Effective core-$N$ potentials, constrained by the available experimental information on the binary subsystems $^5$Li and $^{15}$Be, are employed in the calculations. Results: The ground state of $^{16}$Be is found to present a strong dineutron configuration, with the valence neutrons occupying mostly an $l=2$ state relative to the core. The results are consistent with previous $R$-matrix calculations for the actual continuum. The case of $^6$Be shows a clear symmetry with respect to its mirror partner, the two-neutron halo $^6$He: The diproton configuration is dominant, and the valence protons occupy an $l=1$ orbit. Conclusions: The PS method is found to be a suitable tool in describing the properties of unbound $\text{core}+N+N$ ground states. For both $^{16}$Be and $^6$Be, the results are consistent with previous theoretical studies and confirm the dominant dinucleon configuration. This favors the picture of a correlated two-nucleon emission.

The H2+ + He proton transfer reaction: quantum reactive differential cross sections to be linked with future velocity mapping experiments

We construct the velocity map images of the proton transfer reaction between helium and molecular hydrogen ion . We perform simulations of imaging experiments at one representative total collision energy taking into account the inherent aberrations of the velocity mapping in order to explore the feasibility of direct comparisons between theory and future experiments planned in our laboratory. The asymptotic angular distributions of the fragments in a 3D velocity space is determined from the quantum state-to-state differential reactive cross sections and reaction probabilities which are computed by using the time-independent coupled channel hyperspherical coordinate method. The calculations employ an earlier ab initio potential energy surface computed at the FCI/cc-pVQZ level of theory. The present simulations indicate that the planned experiments would be selective enough to differentiate between product distributions resulting from different initial internal states of the reactants.

A consistent four-body CDCC model of low-energy reactions: Application to9Be+208Pb

We investigate the $^9$Be + $^{208}$Pb elastic scattering, breakup and fusion at energies around the Coulomb barrier. The three processes are described simultaneously, with identical conditions of calculations. The $^{9}$Be nucleus is defined in an $\alpha + \alpha$ + n three-body model, using the hyperspherical coordinate method. We first analyze spectroscopic properties of $^9$Be, and show that the model provides a fairly good description of the low-lying states. The scattering with $^{208}$Pb is then studied with the Continuum Discretized Coupled Channel (CDCC) method, where the $\alpha+\alpha$ + n continuum is approximated by a discrete number of pseudostates. Optical potentials for the $\alpha$+ $^{208}$Pb and n+ $^{208}$Pb systems are taken from the literature. We present elastic-scattering and fusion cross sections at different energies.

Theoretical study of very low-energy collision in Coulombic three-body systems

SynopsisWe present a systematic study for Coulombic three-body systems in very low-energy regime with mass ratio varying over several orders of magnitudes. Accurate phase shifts and scattering lengths are obtained by the Hyperspherical elliptic coordinate method. We identify the mass ratio dependence of the scattering lengths.

Four-body effects on 9Be+208Pb scattering and fusion around the Coulomb barrier

We investigate the 9Be+208Pb elastic scattering and fusion at energies around the Coulomb barrier. The 9Be nucleus is described in α + α + n three-body model, using the hyperspherical coordinate method. The scattering with 208Pb is then studied with the Continuum Discretized Coupled Channel (CDCC) method, where the α + α + n continuum is approximated by a discrete number of pseudostates. Optical potentials for the α+208Pb and n+208Pb systems are taken from the literature. We present elastic-scattering and fusion cross sections at different energies, and investigate the convergence with respect to the truncation of the α + α + n continuum. A good agreement with experiment is obtained, considering that there is no parameter fitting. We show that continuum effects increase at low energies.

Ab initiostudy of high-lying doubly excited states of helium in static electric fields: Complex-scaling generalized pseudospectral method in hyperspherical coordinates

We present a complex-scaling (CS) generalized pseudospectral (GPS) method in hyperspherical coordinates (HSC) for $\mathit{ab}$ initio and accurate treatment of the resonance energies and autoionization widths of two-electron atomic systems in the presence of a strong dc electric field. The GPS method allows nonuniform and optimal spatial discretization of the two-electron Hamiltonian in HSC with the use of only a modest number of grid points. The procedure is applied for the first precision calculation of the energies and autoionization widths for the high-lying ${}^{1}\phantom{\rule{-0.16em}{0ex}}{S}^{e}$, ${}^{1}\phantom{\rule{-0.16em}{0ex}}{P}^{o}$, ${}^{1}\phantom{\rule{-0.16em}{0ex}}{D}^{e}$, and ${}^{1}\phantom{\rule{-0.16em}{0ex}}{F}^{o}$ ($n=10$--20) doubly excited resonance states of He atoms. In addition, we present a theoretical prediction of the energies and widths of high-lying doubly excited resonance states of ${}^{1}\phantom{\rule{-0.16em}{0ex}}{P}^{o}$ ($n=$ 8--15) in external dc electric field strengths of 3.915--10.44 kV/cm. The calculated dc-field perturbed high-lying resonance energies are in good agreement with the latest experimental data.

An inverse hyper-spherical harmonics-based formulation for reconstructing 3D volumetric lung deformations

A method to estimate the deformation operator for the 3D volumetric lung dynamics of human subjects is described in this paper. For known values of air flow and volumetric displacement, the deformation operator and subsequently the elastic properties of the lung are estimated in terms of a Green's function. A Hyper-Spherical Harmonic (HSH) transformation is employed to compute the deformation operator. The hyper-spherical coordinate transformation method discussed in this paper facilitates accounting for the heterogeneity of the deformation operator using a finite number of frequency coefficients. Spirometry measurements are used to provide values for the airflow inside the lung. Using a 3D optical flow-based method, the 3D volumetric displacement of the left and right lungs, which represents the local anatomy and deformation of a human subject, was estimated from 4D-CT dataset. Results from an implementation of the method show the estimation of the deformation operator for the left and right lungs of a human subject with non-small cell lung cancer. Validation of the proposed method shows that we can estimate the Young's modulus of each voxel within a 2% error level.

FAST LiH DESTRUCTION IN REACTION WITH H: QUANTUM CALCULATIONS AND ASTROPHYSICAL CONSEQUENCES

We present a quantum-mechanical study of the exothermic 7LiH reaction with H. Accurate reactive probabilities and rate coefficients are obtained by solving the Schrödinger equation for the motion of the three nuclei on a single Born–Oppenheimer potential energy surface and using a coupled-channel hyperspherical coordinate method. Our new rates indeed confirm earlier, qualitative predictions and some previous theoretical calculations, as discussed in the main text. In the astrophysical domain, we find that the depletion process largely dominates for redshift (z) between 400 and 100, a range significant for early universe models. This new result from first-principle calculations leads us to definitively surmise that LiH should be already destroyed when the survival processes become important. Because of this very rapid depletion reaction, the fractional abundance of LiH is found to be drastically reduced, so that it should be very difficult to manage to observe it as an imprinted species in the cosmic background radiation. The present findings appear to settle the question of LiH observability in the early universe. We further report several state-to-state computed reaction rates in the same range of temperatures of interest for the present problem.

A NEW HYPERCENTERAL POTENTIAL TO STUDY A NONRELATIVISTIC N-BODY SYSTEM

The Hyperspherical coordinates method (HCM), allows one to obtain the spectrum of the system and develop the classification of excited states. We have solved the radial Schrödinger wave equation exactly in three dimensions for an N -body problem. Using the obtained results, we have studied the N family baryon masses.

A theoretical investigation on the spectrum of the Ar trimer for high rotational excitations

A detailed study of the rovibrational spectrum of the Ar trimer is performed by means of an exact hyperspherical coordinate (HC) method and a variational approach based on distributed Gaussian functions (DGFs) to describe the interparticle distances. The good agreement observed between the energy levels obtained with both procedures for high values of the total angular momentum (J=15 and 20) reveals the quality of the DGF method to describe the rotation of the title system. Rotational constants for the lowest bound states, obtained as averages for each vibrational state, have been obtained and compared to previous results. A detailed analysis of density probability functions obtained by means of the HC approach for rovibrational states at J=0 and 20 shows close similitudes thus supporting the vibration-rotation separation adopted within the DGF scheme for the Ar(3) system.

Hyperspherical approach to description of electron correlation in negative ion of exciton

The binding energy of excitonium negative ion for the ground1,3S-states in bulk semiconductors GaAs and ZnO in the hyperspherical coordinate method was found. Angular and radial correlations between electrons in gerade and ungerade states were taken into account by channel functions, that are the eigenfunctions of Hamiltonian on the surface of the sphere in the three-dimensional configuration space. Energy values were calculated using the adiabatic and Born-Oppenheimer approximations. The obtained energy values are in agreement with those obtained using variational method.

Quantum stereodynamics of the F+H2→HF+H reaction by the stereodirected S-matrix approach

Reaction stereodynamics can be studied in quantum mechanics using alternative representations of the S matrix. In this paper we employ the equations for the orthogonal transformations (expressed in terms of Wigner 3j symbols) that convert the S matrix from the body fixed (|jΩ〉) representation into the stereodirected one (|νΩ〉). This representation is characterized by the introduction of the steric quantum number ν, which in the vector model of quantum mechanics is put into correspondence with given precession cones of attack of the incoming atom on the diatomic molecule for the reactants' channels, and of cones of escape for the departing atom away from the diatomic molecule for the products' channels. The angles of aperture of such cones are determined from the uncertainty principle. As the ν quantum number increases (semiclassical limit), the grid of discrete values of the precession cones more finely scans the angle between the Jacobi vectors. Using a time-independent hyperspherical coordinate method we have generated the full S matrix including all open reactive and inelastic channels for two potential energy surfaces corresponding to the F + H2 → HF + H reaction and they have been used to calculate, via |jΩ〉→|νΩ〉 matrix transformations, the attack and exit cumulative reaction probabilities. During the calculations, we have distinguished between ortho-H2 and para-H2. Clear stereodynamical effects have being identified, in particular, regarding the reaction entrance channel, that F-atom attacks are preferred at the transition state (bent) geometry, while for the exit channel the H-atom departs in a collinear geometry by the H-end side of HF.

Time-dependent quantum dynamical study of C(1D) + H2 ? CH + H and S(1D) + H2 ? SH + H reactive collisions

A time-dependent method is used to calculate reaction probabilities for J = 0 for C(1D) + H2 (v = 0, j = 0) → CH(v′, j′) + H and S(1D) + H2 (v = 0, j = 0) → SH (v′, j′) + H indirect reactive collisions. The state-to-state probabilities are dominated by a strong resonance structure associated with the formation of a long-lived triatomic complex during the collision. We compare our results with those obtained using a time-independent hyperspherical coordinates method. The agreement is very good: in particular, the narrow resonances obtained with the wave packet method correspond exactly to those of the time-independent results.

Two-channel Siegert pseudostate approach to a three-body Coulomb problem:S-wave resonances ofPs−(N=2,3)

The recently developed two-channel Siegert pseudostate method of Sitnikov and Tolstikhin [Phys. Rev. A 67, 032714 (2003)] is here applied to the case of ${\mathrm{Ps}}^{\ensuremath{-}}$ ${(e}^{+}ee)$ exploiting the hyperspherical elliptic coordinates method. S-wave resonances and antiresonances are calculated and classified according to their locations in the complex plane of the uniformization parameter. The resonance positions and widths are compared to other accurate calculations.

Highlights and Prospects of Hyperspherical Approach in Few-Body Quantum-Mechanical Systems

The binding energy of excitonium negative ion for ground 1S-state in bulk conductors: Ge, Si, CdSe and for green and yellow lines of Cu2O in hyperspherical coordinate method are found. Angular and radial correlations between electrons are taken into account by the channel functions, which are the eigen-functions of operator on the surface of sphere in six-dimensional sphere. The calculation of energies have been done using the adiabatic and Born-Oppenheimer approximations. In Born-Oppenheimer approximation is enough to give a binding energy with only 1.2% error, in adiabatic approximation this error drops to 0.7 %.

Description of electrons correlation in 1,3 S-states of negative ion of positronium

The energy of ground1,3S-states of negative positronium ion in the hyperspherical coordinates method has been calculated in Born-Oppenheimer and adiabatic approximations accounting the contribution of angle and radial correlation. Quasi-crossing points for autodetaching adiabatic potentials were revealed, which forms a structure in nonadiabatic potential. The received energy values have been compared with results of calculations performed using other methods. It was shown that in order to receive energy values with accuracy of 10−4 a.u. it is necessary to include more than 70 elements into the basis for determination of adiabatic potentials or to use Pade approximation on the basis dimension.