Adiabatic Hyperspherical Approach Research Articles

Adiabatic hyperspherical potentials with localized correlated Gaussians

The adiabatic hyperspherical approach is a natural extension of the well-known three-dimensional polar coordinate method to solving a Schr\odinger equation of a few-body system. To evaluate the matrix element of an adiabatic Hamiltonian at a fixed hyper-radius is crucially important in that approach, but due to the difficulty of its calculation real applications have been limited mostly up to four-body systems. To resolve this limitation I introduce a localized hyper-radial function and show that the matrix element needed for $N$-body system can be obtained using correlated Gaussians with arbitrary angular momentum. I demonstrate its feasibility in the systems of $N=3\text{--}6\phantom{\rule{4pt}{0ex}}\ensuremath{\alpha}$ particles. It is pointed out that an extension into correlated Gaussians with double global vectors is desirable for further realistic descriptions of the hyperangular motion.

Resonant States of the H3– Molecule and Its Isotopologues D2H– and H2D–

Using the ground potential energy surface [Ayouz, M.; etal. J. Chem. Phys. 2010, 132, 194309] of the H3– molecule, we have determined the energies and widths of the complex resonant levels of H3– located up to 4000 cm–1 above the dissociation limit H– + H2(νd = 0,jd = 0). Bound and resonant levels of the H2D– and D2H– isotopologues have been also characterized within the same energy range. The method combines the hyperspherical adiabatic approach, slow variable discretization method, and complex absorbing potential. These results represent the first step for modeling the dynamics of the associated diatom–negative ion collision at low energy involving rotational quenching of the diatom and reactive nucleus exchange via the weak tunneling effect through the potential barrier of the potential energy surface.

Analytic Expression for Three-Body Recombination Rates into Deep Dimers

We investigate three-body recombination rates into deep dimers in cold atomic gases with large scattering length within hyper-spherical adiabatic zero-range approach. We derive closed analytic expressions for the rates for one- and two-species gases. Although the deep dimers are beyond the zero-range theory the latter can still describe the recombination into deep dimers by use of one additional short-range absorption parameter. The recombination rate, as function of the scattering length, retains the known universal behavior—the fourth power trend with characteristic log-periodic peaks—however increasing the short-range absorption broadens the peaks until they are eventually completely smeared out. Increasing the heavy-to-light mass ratio in a two-species system with two identical bosons and a third particle decreases the distance between the peaks and increases the overall scale of the recombination rate.

Adiabatic hyperspherical approach to large-scale nuclear dynamics

We formulate a fully microscopic approach to large-scale nuclear dynamics using a hyperradius as a collective coordinate. An adiabatic potential is defined by taking account of all possible configurations at a fixed hyperradius, and its hyperradius dependence plays a key role in governing the global nuclear motion. In order to go to larger systems beyond few-body systems, we suggest basis functions of a microscopic multicluster model, propose a method for calculating matrix elements of an adiabatic Hamiltonian with use of Fourier transforms, and test its effectiveness.

KANTBP 3.0: New version of a program for computing energy levels, reflection and transmission matrices, and corresponding wave functions in the coupled-channel adiabatic approach

A FORTRAN program for calculating energy values, reflection and transmission matrices, and corresponding wave functions in a coupled-channel approximation of the adiabatic approach is presented. In this approach, a multidimensional Schrodinger equation is reduced to a system of the coupled second-order ordinary differential equations on a finite interval with the homogeneous boundary conditions of the third type at the left- and right-boundary points for continuous spectrum problem. The resulting system of these equations containing the potential matrix elements and first-derivative coupling terms is solved using high-order accuracy approximations of the finite element method. As a test desk, the program is applied to the calculation of the reflection and transmission matrices and corresponding wave functions for the two-dimensional problem with different barrier potentials.

POTHEA: A program for computing eigenvalues and eigenfunctions and their first derivatives with respect to the parameter of the parametric self-adjoined 2D elliptic partial differential equation

POTHEA: A program for computing eigenvalues and eigenfunctions and their first derivatives with respect to the parameter of the parametric self-adjoined 2D elliptic partial differential equation

Resonant States of the H3– Molecule and Its Isotopologues D2H– and H2D–

Using the ground potential energy surface[M. Ayouz \textit{et\, al}. J. Chem Phys \textbf{132} 194309 (2010)] of the H$_3^-$ molecule, we have determined the energies and widths of the complex resonant levels of H$_3^-$ located up to 4000 cm$^{-1}$ above the dissociation limit H$^-$+H$_2(v_d=0,j_d=0)$. Bound and resonant levels of the H$_2$D$^-$ and D$_2$H$^-$ isotopologues have been also characterized within the same energy range. The method combines the hyperspherical adiabatic approach, slow variable discretization method, and complex absorbing potential. These results represent the first step for modeling the dynamics of the associated diatom-negative ion collision at low energy involving rotational quenching of the diatom and reactive nucleus exchange via weak tunneling effect through the potential barrier of the potential energy surface.

Binding energies of lower odd states of negative hydrogen ion in a onedimensional space

The binding energy values for ground and lower autoionizing states of the negative ion of hydrogen atom were found using an elliptic coordinate system in the framework of hyperspherical adiabatic approach. Adiabatic potentials were determined by diagonalization of the system’s Hamiltonian on the ellipse using eigenfunctions of the generalized angular momentum operator at different dimensions of the basis. Radial equation in the Born-Oppenheimer approximation is solved numerically. The obtained binding energy values are in satisfactory agreement with the existing experimental results and theoretical calculations performed in other approaches.

KANTBP 2.0: New version of a program for computing energy levels, reaction matrix and radial wave functions in the coupled-channel hyperspherical adiabatic approach

KANTBP 2.0: New version of a program for computing energy levels, reaction matrix and radial wave functions in the coupled-channel hyperspherical adiabatic approach

Three-body recombination in one dimension

We study the three-body problem in one dimension for both zero and finite range interactions using the adiabatic hyperspherical approach. Particular emphasis is placed on the threshold laws for recombination, which are derived for all combinations of the parity and exchange symmetries. For bosons, we provide a numerical demonstration of several universal features that appear in the three-body system, and discuss how certain universal features in three dimensions are different in one dimension. We show that the probability for inelastic processes vanishes as the range of the pair-wise interaction is taken to zero and demonstrate numerically that the recombination threshold law manifests itself for large scattering length.

A Study of Confined Helium Atom

The helium atom confined by a spherical parabolic potential well isstudied employing the adiabatic hyperspherical approach method. Total energiesof the ground and three low-excited states are obtained as a function of theconfined potential radii. We find that the energies of a spherical parabolicpotential well are in good agreement with those of an impenetrable sphericalbox for the larger confined potential radius. We find also that theconfinement may cause accidental degeneracies between levels with differentlow-excited states and the inversion of the energy values. The results forthe three-dimensional spherical potential well and the two-dimensionaldisc-like potential well are compared with each other. We find that theenergy difference between states in a two-dimensional parabolic potential isalso obviously larger than the corresponding levels for a sphericalparabolic potential.

DETERMINATION OF BINDING ENERGY OF THE GROUND STATE FOR ONE-DIMENSIONAL HELIUM ATOM IN HYPERSPHERICAL APPROACH

Ground state binding energy for one-dimensional helium atom has been obtained in hyperspherical adiabatic approach. Logarithmic divergences in matrix elements of the potential energy operator were removed using the theory of residue, which led to the appearance of their complex values. An additional region of attraction was found for small values of radial variable. The width of this region depends on the dimension of basis, on which the adiabatic potentials are determined. The present results are compared with the calculations obtained in other approaches.

A Study of a Confined Hydrogen Negative Ion

The ground and three low-excited states of the hydrogen negative ionconfined by a spherical harmonic oscillator potential are studied employingthe adiabatic hyperspherical approach method. Total energies are obtained asa function of the confined potential radii. We find that the confinement maycause accidental degeneracies between levels with different low-excitedstates and the inversion of the energy values.

KANTBP: A program for computing energy levels, reaction matrix and radial wave functions in the coupled-channel hyperspherical adiabatic approach

A FORTRAN 77 program is presented which calculates energy values, reaction matrix and corresponding radial wave functions in a coupled-channel approximation of the hyperspherical adiabatic approach. In this approach, a multi-dimensional Schrödinger equation is reduced to a system of the coupled second-order ordinary differential equations on the finite interval with homogeneous boundary conditions of the third type. The resulting system of radial equations which contains the potential matrix elements and first-derivative coupling terms is solved using high-order accuracy approximations of the finite-element method. As a test desk, the program is applied to the calculation of the energy values and reaction matrix for an exactly solvable 2D-model of three identical particles on a line with pair zero-range potentials. Program title: KANTBP Catalogue identifier: ADZH_v1_0 Program summary URL: http://cpc.cs.qub.ac.uk/summaries/ADZH_v1_0.html Program obtainable from: CPC Program Library, Queen's University, Belfast, N. Ireland Licensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.html No. of lines in distributed program, including test data, etc.: 4224 No. of bytes in distributed program, including test data, etc.: 31 232 Distribution format: tar.gz Programming language: FORTRAN 77 Computer: Intel Xeon EM64T, Alpha 21264A, AMD Athlon MP, Pentium IV Xeon, Opteron 248, Intel Pentium IV Operating system: OC Linux, Unix AIX 5.3, SunOS 5.8, Solaris, Windows XP RAM: depends on (a) the number of differential equations; (b) the number and order of finite-elements; (c) the number of hyperradial points; and (d) the number of eigensolutions required. Test run requires 30 MB Classification: 2.1, 2.4 External routines: GAULEG and GAUSSJ [W.H. Press, B.F. Flanery, S.A. Teukolsky, W.T. Vetterley, Numerical Recipes: The Art of Scientific Computing, Cambridge University Press, Cambridge, 1986] Nature of problem: In the hyperspherical adiabatic approach [J. Macek, J. Phys. B 1 (1968) 831–843; U. Fano, Rep. Progr. Phys. 46 (1983) 97–165; C.D. Lin, Adv. Atom. Mol. Phys. 22 (1986) 77–142], a multi-dimensional Schrödinger equation for a two-electron system [A.G. Abrashkevich, D.G. Abrashkevich, M. Shapiro, Comput. Phys. Comm. 90 (1995) 311–339] or a hydrogen atom in magnetic field [M.G. Dimova, M.S. Kaschiev, S.I. Vinitsky, J. Phys. B 38 (2005) 2337–2352] is reduced by separating the radial coordinate ρ from the angular variables to a system of second-order ordinary differential equations which contain potential matrix elements and first-derivative coupling terms. The purpose of this paper is to present the finite-element method procedure based on the use of high-order accuracy approximations for calculating approximate eigensolutions for such systems of coupled differential equations. Solution method: The boundary problems for coupled differential equations are solved by the finite-element method using high-order accuracy approximations [A.G. Abrashkevich, D.G. Abrashkevich, M.S. Kaschiev, I.V. Puzynin, Comput. Phys. Comm. 85 (1995) 40–64]. The generalized algebraic eigenvalue problem AF=EBF with respect to pair unknowns (E,F) arising after the replacement of the differential problem by the finite-element approximation is solved by the subspace iteration method using the SSPACE program [K.J. Bathe, Finite Element Procedures in Engineering Analysis, Englewood Cliffs, Prentice–Hall, New York, 1982]. The generalized algebraic eigenvalue problem (A−EB)F=λDF with respect to pair unknowns (λ,F) arising after the corresponding replacement of the scattering boundary problem in open channels at fixed energy value, E, is solved by the LDLT factorization of symmetric matrix and back-substitution methods using the DECOMP and REDBAK programs, respectively [K.J. Bathe, Finite Element Procedures in Engineering Analysis, Englewood Cliffs, Prentice–Hall, New York, 1982]. As a test desk, the program is applied to the calculation of the energy values and reaction matrix for an exactly solvable 2D-model of three identical particles on a line with pair zero-range potentials described in [Yu. A. Kuperin, P.B. Kurasov, Yu.B. Melnikov, S.P. Merkuriev, Ann. Phys. 205 (1991) 330–361; O. Chuluunbaatar, A.A. Gusev, S.Y. Larsen, S.I. Vinitsky, J. Phys. A 35 (2002) L513–L525; N.P. Mehta, J.R. Shepard, Phys. Rev. A 72 (2005) 032728-1-11; O. Chuluunbaatar, A.A. Gusev, M.S. Kaschiev, V.A. Kaschieva, A. Amaya-Tapia, S.Y. Larsen, S.I. Vinitsky, J. Phys. B 39 (2006) 243–269]. For this benchmark model the needed analytical expressions for the potential matrix elements and first-derivative coupling terms, their asymptotics and asymptotics of radial solutions of the boundary problems for coupled differential equations have been produced with help of a MAPLE computer algebra system. Restrictions: The computer memory requirements depend on: (a) the number of differential equations; (b) the number and order of finite-elements; (c) the total number of hyperradial points; and (d) the number of eigensolutions required. Running time: The running time depends critically upon: (a) the number of differential equations; (b) the number and order of finite-elements; (c) the total number of hyperradial points on interval [0,ρmax]; and (d) the number of eigensolutions required.

Calculation of three-body resonances using slow-variable discretization coupled with a complex absorbing potential

We developed a method to calculate positions and widths of three-body resonances. The method combines the hyperspherical adiabatic approach, slow variable discretization method (Tolstikhin et al., J. Phys. B: At. Mol. Opt. Phys. 29, L389 (1996)), and a complex absorbing potential. The method can be used to obtain resonances having short-range or long-range wave functions. In particular, we applied the method to obtain very shallow three-body Efimov resonances for a model system (Nielsen et al., Phys. Rev. A 66, 012705 (2002)).

Feature of a Confined Positronium Negative Ion by aSpherical Parabolic Potential

The ground and three low-excited states of the positroniumnegative ion confined by a spherical harmonic oscillator potentialare studied employing the adiabatic hyperspherical approach method.Total energies are obtained as a function of the confined potentialradii. We find that the confinement may cause accidental degeneraciesbetween levels with different low-excited states and the inversion ofthe energy values.

Calculation of Energies of the Ground and Low Excited Statesof a Confined Helium Atom in a Spherical Parabolic Well

Making use of the adiabatic hyperspherical approach, we report acalculation for the energy spectrum of the ground and low-excited statesof the confined helium atom in a spherical parabolic well. We find thatthe energies of a spherical parabolic well are in good agreement withthose of an impenetrable spherical box for the larger confined potentialradius. However, the energy values of a spherical parabolic well aremuch lower than those of an impenetrable spherical box for small valuesof rc. We also find that the confinement may cause accidentaldegeneracies between levels with different low-excited states and theinversion of the energy values.

Calculation of resonances in the Coulomb three-body system with two disintegration channels in the adiabatic hyperspherical approach

The method of calculation of the resonance characteristics is developed for the metastable states of the Coulomb three-body system with two disintegration channels. It is based on the numerical solution of the scattering problem in the framework of the adiabatic hyperspherical approach. The energy dependence of K-matrix in the resonance region is calculated with the use of the stabilization method. The resonance position E0 and the partial widths Γ1, Γ2 are obtained by fitting of the numerically calculated Kij(E) with the help of the general Breit–Wigner formula which takes into account the non-zero background inelastic scattering. The method developed is applied to the calculation of the parameters of the three lowest metastable states of the mesic molecular ion dt μ.

Energy Spectrum of Helium Confined to a Two-Dimensional Space**Theproject supported by National Natural ScienceFoundation of China under Grant Nos. 10475021 and 10275014

Making use of the adiabatic hyperspherical approach, we report a calculationfor the energy spectrum of the ground and low-excited states of atwo-dimensional helium in a magnetic field. The results show that the groundand low-excited states of helium in low-dimensional space are more stablethan those in three-dimensional space and there may exist more bound states.

Bound states of the barium atom by the hyperspherical approach

We present a nonadiabatic hyperspherical calculation of the highly excited and low lying doubly excited states of the barium atom using effective potentials for the two optically active electrons' interactions. Within the hyperspherical adiabatic approach the investigation of the spectra is performed with potential curves and nonadiabatic couplings of a unique radial variable, which allows clear identification of the states. The convergence of energy is obtained within well established bound limits, and the precision is comparable to accurate configuration interaction calculations. A very good agreement with experimental results is obtained with only few nonadiabatic couplings.