Adiabatic Hyperspherical Research Articles

Variational description of continuum states in terms of integral relations

Two integral relations derived from the Kohn Variational Principle (KVP) are used for describing scattering states. In usual applications the KVP requires the explicit form of the asymptotic behavior of the scattering wave function. This is not the case when the integral relations are applied since, due to their short range nature, the only condition for the scattering wave function $\Psi$ is that it be the solution of $(H-E)\Psi=0$ in the internal region. Several examples are analyzed for the computation of phase-shifts from bound state type wave functions or, in the case of the scattering of charged particles, it is possible to obtain phase-shifts using free asymptotic conditions. As a final example we discuss the use of the integral relations in the case of the Hyperspherical Adiabatic method.

Integral Relations for Three-Body Continuum States with the Adiabatic Expansion

The application of the hyperspherical adiabatic expansion to describe three-body scattering states suffers from the problem of very slow convergence. Contrary to what happens for bound states, a huge number of hyper-radial equations has to be solved, and even if done, the extraction of the scattering amplitude is problematic. In this Letter we show how to obtain accurate scattering phase shifts using the hyperspherical adiabatic expansion. To this aim two integral relations, derived from the Kohn variational principle, are used. The convergence of this procedure is as fast as for bound states.

Scattering States of Three-Body Systems with the Hyperspherical Adiabatic Method

In this paper, we investigate the feasibility of employing the Hyperspherical Adiabatic (HA) basis set to describe continuum states of the Helium trimer molecule.

Continuum three-body states using the hyperspherical adiabatic basis set

In this paper we investigate the feasibility of employing the hyperspherical adiabatic (HA) basis set to describe continuum three-nucleon states. In particular, the HA expansion is compared with the simpler expansion on hyperspherical harmonics (HH). A practical numerical application is presented using the MT-III potential.

Three-Nucleon Continuum by Means of the Hyperspherical Adiabatic Method

This paper investigates the possible use of the Hyperspherical Adiabatic basis in the description of scattering states of a three-body system. In particular, we analyze a 1+2 collision process below the three-body breakup. The convergence patterns for the observables of interest are analyzed by comparison to a unitary equivalent Hyperspherical Harmonic expansion. Furthermore, we compare and discuss two different possible choices for describing the asymptotic configurations of the system, related to the use of Jacobi or hyperspherical coordinates. In order to illustrate the difficulties and advantages of the approach two simple numerical applications are shown in the case of neutron-deuteron scattering at low energies using s-wave interactions. We found that the optimization driven by the Hyperspherical Adiabatic basis is not as efficient for scattering states as in bound state applications.

The Negative and Positive Charged Two-dimensional Three-body Systems in Adiabatic Hyperspherical Approach

Binding energies of ground 1S-states for two-dimensional three-body systems (e−e+e−, e−p+e−, He and Li+), which were received using the hyperspherical adiabatic approach in Born-Oppenheimer approximation, are presented. For description of electron correlation we use K-harmonics, which contain ground angular momentum equal to 0, 2, 4 and 6. We used expansion of potential energy operator in series on multipoles including up to forty terms for description of angular correlations. The values of binding energies are four times bigger than in three-dimensional cases with accuracy of about one-two percents. The convergence of energy values according to expansion parameters for channel function is shown. Avoided-crossing regions (R ≈ 0.2−5 a.u.) between adiabatic potential U2(R) and U3(R) were also found.

Symmetric elastic and spin-flip low-energy collisions of the hydrogen-isotope mesic atoms in the adiabatic hyperspherical approach

The reduced adiabatic hyperspherical (RAHS) basis suggested previously is used to calculate elastic and spin-flip cross sections in the processes (aμ)F+a → (aμ)F′+a, a=(p, d, t), for collision energies 10−3≤ɛ≤102 eV. The rapid convergence of the method is demonstrated: to achieve an accuracy of ⋍1% in the calculated cross sections, it is sufficient to use N≤10 of the basis RAHS functions. The comparison of the obtained results with the previous ones is presented.

Spin-Flip and Elastic Processes in Slow Collisions of Mesic Atoms

The reduced adiabatic hyperspherical (RAHS) basis suggested previously is used for the calculation of elastic and spin-flip cross sections in the processes (aμ)F + a → (aμ)F′ + a, a = (p, d, t), for collision energies 10−3 ≤ e ≤ 102 eV. The rapid convergence of the method is demonstrated. A comparison of the obtained results with previous ones is presented.

Adiabatic Hyperspherical Approach to the Problems of Muon Catalyzed Fusion

The adiabatic hyperspherical approach (AHSA) is applied for the numerical investigation of the scattering processes and resonances in Coulomb three-body mesic atomic systems. The results of the calculations of elastic and inelastic cross sections in low-energy collisions aμ + b (a, b = p, d, t), energies, lifetimes and local characteristics of resonant states of mesic molecular ions nHeaμ+ (n = 3, 4) are presented.

A New Method for Solving an Eigenvalue Problem for a System of Three Coulomb Particles within the Hyperspherical Adiabatic Representation

The quantum mechanical three-body problem with Coulomb interaction is formulated within the adiabatic representation method using the hyperspherical coordinates. The Kantorovich method of reducing the multidimensional problem to the one-dimensional one is used. A new method for computing variable coefficients (potential matrix elements of radial coupling) of a resulting system of ordinary second-order differential equations is proposed. It allows the calculation of the coefficients with the same precision as the adiabatic functions obtained as solutions of an auxiliary parametric eigenvalue problem. In the method proposed, a new boundary parametric problem with respect to unknown derivatives of eigensolutions in the adiabatic variable (hyperradius) is formulated. An efficient, fast, and stable algorithm for solving the boundary problem with the same accuracy for the adiabatic eigenfunctions and their derivatives is proposed. The method developed is tested on a parametric eigenvalue problem for a hydrogen atom on a three-dimensional sphere that has an analytical solution. The accuracy, efficiency, and robustness of the algorithm are studied in detail. The method is also applied to the computation of the ground-state energy of the helium atom and negative hydrogen ion.

Calculation of resonant states of the (dµ3He)J = 0 ion in the adiabatic hyperspherical approach

The resonant state of the Coulomb three-body system d + (µ3He) (J = 0) below the (µ d)1s + 3He threshold is investigated using the adiabatic hyperspherical (AHS) approach. The effective numerical method for the scattering problem in the AHS basis is developed. This allows one to study in detail the energy dependence of the three-body wavefunction, its local characteristics (G- and -factors) and the phase shift in the resonance range. The resonance position E0 = 70.82 eV and the width = 3.8 ×10-4 eV are obtained as the parameters of the Breit-Wigner formula which reproduces the numerical results for the regularized normalization constant N2(E). The numerically calculated G-factor depends strongly on E within the resonance width. Its E dependence is reproduced with high accuracy by a three-parameter formula. In contrast to the G-factor, the numerically obtained -factors do not depend on E noticeably in the resonance range.

ASYMPT: a program for calculating asymptotics of hyperspherical potential curves and adiabatic potentials

A FORTRAN program is presented which calculates asymptotics of potential curves and adiabatic potentials with an accuracy of O( ρ −2 ) in the framework of the hyperspherical adiabatic (HSA) approach. Here, ρ is the hyperradius. It is shown that matrix elements of the equivalent operator corresponding to the perturbation ρ −2 have a simple form in the basis of the Coulomb parabolic functions in the body-fixed frame and can be easily computed for high values of total orbital momentum and threshold number. The second-order corrections to the adiabatic curves are obtained as the eigenvalues of the corresponding secular equation. The eigenvectors computed are used to calculate the relevant corrections to matrix elements of potential coupling. The asymptotic potentials obtained can be used for the calculation of the energy levels and radial wave functions of two-electron systems in the adiabatic and coupled-channel approximations of the HSA approach and also in scattering calculations.

Exact and avoided crossings of adiabatic hyperspherical potential curves

Exact crossing of potential curves related to the adiabatic hyperspherical (AH) approach to a three-body Coulomb system is studied. Analytic structure of the AH potential curves, harmonics, and coupling matrix elements near crossing points in the complex plane of hyperradius is investigated. Results are applied to derive some basic features of avoided crossings of the AH potential curves at real hyperradii.

Quantum three-body scattering problem in the adiabatic hyperspherical representation

The adiabatic hyperspherical (AH) approach to the quantum three-body problem is considered. It is proven that the AH harmonics are complete and differentiable with respect to the hyperradius for a large class of two-body potentials. For the case of short-range potentials, the scattering theory is studied in the framework of the AH approach. The asymptotics of the AH potential curves, harmonics, and coupling matrix elements are derived, as well as the structure of the three-body wave operators and, the S-matrix in the AH representation is investigated.

Adiabatic hyperspherical representation in barycentric coordinates for helium-like systems

The hyperspherical adiabatic (HSA) representation is introduced in barycentric coordinates for helium-like systems in which isotopic and correlation effects are taken into account exactly. Different methods of forming the HSA basis are considered. The general classification of the HSA basis states is suggested, and examples of correlation diagrams are presented. The complete classification of doubly excited states of two-electron systems is given. Different classification schemes are compared. The convergence of the HSA expansion is investigated numerically. The ground-state energy value of the negative hydrogen ion H- is calculated in the six-channel approximation within an accuracy of 10-5 au.

Adiabatic hyperspherical approach to the Coulomb three-body problem: Theory and numerical method

The adiabatic hyperspherical (AH) approach to the three-body Coulomb bound-state problems is considered. The variational method of computation of the AH harmonics potential curves and coupling matrix elements is developed. The method takes into account the asymptotic behaviour of the AH harmonics at large and small values of the hyperradius ρ. The developed method allows to perform calculations with high accuracy and stability for any hyperradius ρ∈(0,∞) with only a few AH harmonics. The efficiency of the method and its convergence is illustrated by calculations of energy levels of the mesic moleculesddμ anddtμ.

Bi- and unimolecular resonances of the collinear reaction F+DBr⇄FDBr⇄FD+Br

The reaction probability function P(ν′←ν‖E) of the bimolecular collinear reaction F+DBr(ν)→FD(ν′)+Br shows sharp resonance peaks at certain collision energies Ebi. On the other hand, the unimolecular dissociation FDBr*(ν1ν3)→F+DBr(ν), FD+Br(ν) exhibits long-lived behavior for particular resonance wave functions Ψν1ν3 at energies Euni. Good numerical agreement is found in a comparison of bi- and unimolecular resonance energies and lifetimes by two complementary methods: (i) S-matrix propagation in hyperspherical coordinates for the bimolecular reaction; (ii) the diagonally corrected vibrational adiabatic hyperspherical (DIVAH) method to determine unimolecular resonances, coupled with the fast Hankel transform (FHT) method to propagate them in time. The agreement found suggests a correlation between bi- and unimolecular resonances by the reaction mechanism F+DBr(ν)⇄FDBr*(ν1ν3)⇄FD(ν′)+Br, where the unimolecular resonance states Ψν1ν3 serve as traps or doorway states, blocking or catalyzing the bimolecular reaction.

Prediction and interpretation of collinear reactive scattering resonances by the diagonal corrected vibrational adiabatic hyperspherical model

One-dimensional single-channel scattering calculations on an effective potential are used to discuss resonances in collinear reactions and to predict their energetic location almost quantitatively. The effective potential includes the vibrationally adiabatic potential and the adiabatic diagonal correction in Delves' mass-weighted polar coordinates (hyperspherical coordinates). This diagonal corrected vibrational adiabatic hyperspherical (DIVAH) model is tested for a variety of reactions (H + H2, H + MuH, I + HI, F + H2) and a detailed discussion of the effective potential and its physical implications is given.