Hyperspherical Potential Curves Research Articles

Study of the van der Waals Rare Gas Trimers $$\hbox {Ne}_3$$ Ne 3 , $$\hbox {Ar}_3$$ Ar 3 , $$\hbox {Kr}_3$$ Kr 3 , and $$\hbox {Xe}_3$$ Xe 3 Using Hyperspherical Coordinates

We study the bosonic van der Waals rare gas trimers $$\hbox {Ne}_3$$ , $$\hbox {Ar}_3$$ , $$\hbox {Kr}_3$$ , and $$\hbox {Xe}_3$$ in zero total angular momentum, $$J=0$$ , states. The three-body Schrodinger equation in hyperspherical coordinates is solved using the slow variable discretization approach. We calculate the $$J=0$$ trimer bound state energy levels as well as their average root-mean-square radii. The adiabatic hyperspherical potential curves converging near the three-body dissociation threshold are also studied.

Universality and chaoticity in ultracold K+KRb chemical reactions

A fundamental question in the study of chemical reactions is how reactions proceed at a collision energy close to absolute zero. This question is no longer hypothetical: quantum degenerate gases of atoms and molecules can now be created at temperatures lower than a few tens of nanokelvin. Here we consider the benchmark ultracold reaction between, the most-celebrated ultracold molecule, KRb and K. We map out an accurate ab initio ground-state potential energy surface of the K2Rb complex in full dimensionality and report numerically-exact quantum-mechanical reaction dynamics. The distribution of rotationally resolved rates is shown to be Poissonian. An analysis of the hyperspherical adiabatic potential curves explains this statistical character revealing a chaotic distribution for the short-range collision complex that plays a key role in governing the reaction outcome.

Calculations of the binding energies of weakly bound He-He-Ba molecules

The three-body Schrdinger equation is approximately solved in the hyperspherical coordinates and the binding energies of the three-body weakly bound systems are calculated with the purpose to find if He-He-Ba trimers could exist. Using the special feature of the B-spline function like the flexible and highly localized properties, hypersphercial potentials are obtained by modifying the knots distribution of the B-spline basis of different weakly bound three-atom systems. Employing the best empirical interaction potentials between each pair of particles, we obtain that in the ground state binding energies of the weakly bound typical three-atom systems, the bindings of the molecules, 4He-4He-138Ba, 4He-3He-138Ba and 3He-3He-138Ba are possible. The binding energies of these systems are shown in the order of 1 Kelvin, each system could support only one bound state. These weakly bound molecules can exist only in a very cold environment. To get insight into the geometry of the molecules, the features of the channel functions associated with the hyperspherical potential curves of each system are investigated.

Weakly bound states of the He-He-Ca triatomic system

We investigate for the existence of the weakly bound He-He-Ca molecules. By employing the best empirical interaction between each pair of particles, the Schr\"odinger equation for the triatomic system is solved using hyperspherical coordinates in the adiabatic approximation. The bound states are found for each of the ${}^{3}$He-${}^{3}$He-${}^{40}$Ca, ${}^{3}$He-${}^{4}$He-${}^{40}$Ca, and ${}^{4}$He-${}^{4}$He-${}^{40}$Ca trimers, respectively. The binding energies of such molecules are estimated. We discuss the features of the channel functions associated with the hyperspherical potential curves of each system to obtain insight into the geometry of the molecules.

S-wave resonances in the positron–helium scattering

We have performed a three-body calculation for investigation of S-wave resonances in the positron–helium scattering in the hyperspherical coordinates. B-splines are used to obtain accurate hyperspherical potential curves and channel functions. An adiabatical hyperspherical approach is used to predict semi-quantitatively the position of the resonances. A multichannel calculation with the stabilization method is also used to extract accurately the positions and widths of resonances. We find four resonances associated with the He+–Ps(n = 2) channel. The resonance parameters for the lowest two resonances are in reasonable agreement with the calculations of Kar and Ho (2004 J. Phys. B: At. Mol. Opt. Phys. 37 3177), and two more new resonances below the Ps(n = 2) threshold are identified.

Doubly excited P, D and F unnatural parity states of hydrogen negative ion using correlated wavefunctions

We have investigated the doubly excited 1,3Pe, 1,3D° and 1,3Fe states of the hydrogen negative ion using highly accurate correlated wavefunctions. The doubly excited 2p2 3Pe metastable bound state (so-called second bound state) energy of H− is obtained using the Ritz variational principle. The upper bound of the energy of the 2p2 3Pe bound states is −0.125 355 451 242 au, which is comparable with the available best results −0.125 355 451 24 (Bylicki M and Bednarz E 2003 Phys. Rev. A 67 022503). We employ the complex-coordinate rotation method to extract resonance parameters. The resonance parameters (both resonance position and width) of H− for 1,3Pe, 1,3D° below the n = 3, 4, 5, and 6 thresholds and 1,3Fe states below the n = 4, 5, and 6 thresholds are reported. We have also employed the stabilization method to extract resonance parameters for the 1,3Pe states. In addition to Feshbach resonances lying below the n = 3, 4, 5, 6 hydrogen thresholds, we have calculated one 3D° shape resonances lying above the H (n = 3) threshold. Resonance parameters obtained from present work are comparable with the reported results using CI-type basis functions and hyperspherical potential curves. The 1Pe resonance state, and some of 1,3Pe, 1,3D°, 1,3Fe resonance states are reported for the first time using the explicitly correlated exponential wavefunctions.

Hyperspherical analysis of radial correlations in four-electron atoms

Hyperspherical coordinates are used to study electron correlations in the radial motion of a four-electron atom within the model of s{sup 4} configurations. We identify groups of hyperspherical adiabatic potential curves that support singly, doubly, triply, and quadruply excited states. By examining the nodal structures of the channel functions, we show that it is possible to relate the number of nodal surfaces in the hyperangles and the principal quantum numbers used in the independent particle picture.

Protonium formation in thep¯−Hcollision at low energies by a diabatic approach

We present a diabatization technique in combination with the recently developed hyperspherical close coupling (HSCC) method. In contrast to the strict diabatization, our simple diabatization procedure transforms only sharp avoided crossings in the adiabatic hyperspherical potential curves into real crossings. With this approach, the weak collision channels can be removed from the close-coupling calculations. This method is used to study the antiproton-hydrogen collision at low energies. In the case of a scaled down (anti)proton mass, we show that a 10-channel calculation is enough to obtain converged cross sections at low energies. The results also indicate that protonium formation occurs mostly to the lowest states of the different excited protonium manifolds.

Analytical functions for the calculation of hyperspherical potential curves of atomic systems

We present angular basis functions for the Schr\"odinger equation of two-electron systems in hyperspherical coordinates. By using the hyperspherical adiabatic approach, the wave functions of two-electron systems are expanded in analytical functions, which generalizes the Jacobi polynomials. We show that these functions, obtained by selecting the diagonal terms of the angular equation, allow efficient diagonalization of the Hamiltonian for all values of the hyperspherical radius. The method is applied to the determination of the ${}^{1}{S}^{e}$ energy levels of the ${\mathrm{Li}}^{+}$ and we show that the precision can be improved in a systematic and controllable way.

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.

Full Ambiguity-Free Quantum Treatment ofD++H(1s)Charge Transfer Reactions at Low Energies

Cross sections for the nonresonant charge transfer process D{sup +}+H (1s){yields}D(1s)+H {sup +} at low energies are calculated in hyperspherical coordinates. The method is free from all the inherent ambiguities associated with the conventional Born-Oppenheimer approach, such as the incorrect asymptotic energies and spurious couplings. However, like the Born-Oppenheimer approach, we show that hyperspherical potential curves and coupling terms have to be calculated only once to obtain results for all partial waves. Feshbach and shape resonances of HD{sup +} near the H(1s) threshold are also calculated. (c) 1999 The American Physical Society.

Low-Energy Recombination of Identical Bosons by Three-Body Collisions

The threshold recombination coefficient for three bosons whose two-body scattering length $a$ is positive and large compared with the nominal range of the potentials is shown to proceed via a crossing of hyperspherical adiabatic potential curves at $R\ensuremath{\approx}2.6a$. The hidden crossing theory gives a simple expression for the rate in terms of the scattering length and a phase $\ensuremath{\Delta}$. The phase depends upon the details of the interaction, and varies from species to species, however an upper limit to the rate which depends only upon the scattering length emerges from the hidden crossing theory.

Hyperspherical analysis of doubly and triply excited states of Li

A computational method for calculating the hyperspherical adiabatic potential curves for three-electron atomic systems is presented. This method allows us to obtain accurate potential curves for any symmetries more efficiently. The potential curves for the $\mathrm{Li}{(}^{2}{S}^{e})$ symmetry are analyzed. For the ground state, the energy calculated using the single channel adiabatic approximation is in good agreement with experiment. For doubly excited states, in the region of small and medium hyperradius the potential curves are similar to those for the doubly excited states of two-electron atoms and these curves can be classified using the same set of $K$, $T$, and $A$ quantum numbers. For triply excited states, the potential curves are used to show the different Rydberg series that converge to the doubly excited states of ${\mathrm{Li}}^{+}$. We also illustrate the rotor structure in the energies of triply excited states of Li.

Nonexistence of Resonances inH2−

The controversy of the existence of resonances in ${\mathrm{H}}^{2\ensuremath{-}}$ ions is revisited. We calculate the hyperspherical adiabatic potential curves for ${\mathrm{H}}^{2\ensuremath{-}}$. For $^{4}S^{0}$ symmetry all the potential curves are repulsive, and thus no bound or resonance states exist. This conclusion contradicts the recent calculations of Sommerfeld et al. [Phys. Rev. Lett. 77, 470 (1996); Phys. Rev. A 55, 1903 (1997)], where a $^{4}S^{0}$ resonance has been predicted. We also calculate the potential curves for several values of the nuclear charge $Z$ ranging between 1 and 2 and show that no resonance or bound state can exist for $Z$ less than about 1.6. Based on the general ordering of energies of intrashell states, we show the conclusion of Sommerfeld et al. is in contradiction with experiment.

Hyperspherical hierarchy of three-electron radial excitations

We investigate hierarchical representation of three-electron radial excitations, relying on the replacement of a sequence of ordered individual electron radial coordinates by a hierarchical sequence of hyperspherical radial variables. In the spirit of the hyperspherical method we argue that the latter variables allow adiabatic quasiseparability of the three-electron wave function. We examine this by approximately evaluating hyperspherical adiabatic potential curves and channel functions for the Li${(s}^{3})$ configuration. The hierarchical representation permits one to display manifold aspects of the triply excited states with a pair of radial quantum numbers.

Hyperspherical close-coupling calculation of D-wave positronium formation and excitation cross sections in positron–hydrogen scattering

The hyperspherical close-coupling method is used to calculate the elastic, positronium-formation, and excitation cross sections for positron collisions with atomic hydrogen at energies below the H (n = 4) threshold for the J = 2 partial wave. The resonances below each inelastic threshold were also analyzed. The adiabatic hyperspherical potential curves are used to identify the nature of these resonances.

Hyperspherical approach to three-electron atomic systems.

The hyperspherical coordinates approach to three-electron atomic systems is further refined by taking into account the symmetry of the electrons analytically. Spins are treated explicitly for both doublet and quartet cases to reduce the three-electron Schr\odinger equation to a set of coupled two-dimensional partial differential equations in a compact, symmetric form ready for numerical calculations. It is shown that the resulting equations can be accurately solved using B splines. By adopting the adiabatic approximation, the hyperspherical potential curves for singly, doubly, and triply excited states for the $^{2}$${\mathit{P}}^{\mathit{o}}$ symmetry of Li are obtained. \textcopyright{} 1996 The American Physical Society.

Comparative studies of excitations and resonances in H-, Ps-, and e++H systems.

The hyperspherical close coupling method for general Coulomb three-body systems is used to examine {ital e}{sup {minus}}+H, {ital e}{sup +}+Ps, and {ital e}{sup +}+H collisions to investigate how the excitation cross sections and resonances depend on the masses, the charges, and the quantum statistics of the system. Hyperspherical potential curves are used to help unravel the nature of Feshbach resonances, shape resonances, and overlapping resonances in these systems.

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.

Hyperspherical approach to Coulombic three-body systems with different masses

The authors adopted mass-weighted hyperspherical coordinates to study the properties of Coulombic three-body systems where all three particles are different. Using an adiabatic approximation, they applied the finite-element method to the two-dimensional eigenvalue problems at fixed hyperradius. The authors have calculated the adiabatic hyperspherical potential curves, and examined the wavefunctions (in terms of density plots) and the non-adiabatic coupling terms for a number of three-body systems. By fixing the masses of two of the particles, they examined how these properties vary with the mass of the third particle. The existence of stable bound states versus the masses of the systems is also investigated.