Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

The Journal of Chemical Physics

Binding energy and geometry of e+A (A=Li,Na) by the hyperspherical approach

Hui-Li Han, Yong Li, Xianzhou Zhang, Ting-Yun Shi

Published Jun 27, 2008• 16 citations
Publisher

Abstract

We calculate the binding energy and geometry of the weakly bound e(+)Li and e(+)Na systems within the framework of hyperspherical coordinates. The Schrodinger equation in hyperangular coordinates is solved at a series of fixed hyper-radii using B-splines and the resulting coupled hyper-radial equation is solved using the slow variable discretization method developed by Tolstikhin et al. [J. Phys. B 29, L389 (1996)]. Great efforts are made in optimizing the distribution of B-splines to overcome the slow convergence of the binding energy and geometrical quantities. This approach allows us to obtain the results with improved convergence that are in good agreement with the best values reported to date. In addition, an analysis of the structure of the two systems is also made and the e(+)Na system is seen to exhibit quantum halo features.

DOI
10.1063/1.2944271

Related papers

Calculation of loosely bound levels for three-body quantum systems using hyperspherical coordinates with a mapping procedure

Viatcheslav Kokoouline, Françoise Masnou-Seeuws

In view of modelization of experiments involving cold atoms and molecules, we develop a method that allows us to calculate weakly bound levels of triatomic molecules. The method combines (1) the hyperspherical coordinates to describe interparticle motion in the three-body system, (2) the solution of the Schr\"odinger equation in two steps: determination of adiabatic states for a fixed hyper-radius and then solution of a set of coupled hyper-radial equations using the slow variable representation of Tolstikhin et al. [J. Phys. B: At. Mol. Opt. Phys. 29, L389 (1996)], (3) and a mapping procedure that reduces considerably the number of basis functions needed to represent wave functions of weakly bound levels. We apply the method to the three different systems: the helium trimer $^{4}\mathrm{He}_{3}$, isotopomers of the $\mathrm{H}_{3}{}^{+}$ ion, and finally a model three-body problem involving three nucleons. For all these systems, we show that the suggested method provides accurate results.

Physical Review A•Jan 10, 2006•38 citations

Hyperspherical Calculations on Electron Affinity and Geometry for Li − and Na −

Han Hui-Li, Xianzhou Zhang, Ting-Yun Shi

Using a model potential to describe the interaction between the core and the valence electron, we perform hyperspherical calculations for electron affinity and geometry of the weakly bound Li − and Na − systems. In our calculation, channel functions are expanded in terms of B-splines. Using the special properties of B-splines, we make the knot distributions more precisely, characterizing the behaviour of channel functions. This improves the convergence greatly. Our results are in good agreement with the other theoretical and experimental values.

Chinese Physics Letters•Dec 1, 2007•2 citations

A theoretical study of Ne3 using hyperspherical coordinates and a slow variable discretization approach

Hiroya Suno

We study theoretically the ground and excited bound states of the bosonic rare gas van der Waals trimer Ne(3). A slow variable discretization approach is adopted to solve the nuclear Schrödinger equation, in which the Schrödinger equation in hyperangular coordinates is solved using basis splines at a series of fixed finite-element methods discrete variable representation hyper-radii. We consider not only zero total nuclear orbital angular momentum, J = 0, states but also J > 0 states. By using the best empirical neon dimer interaction potentials, all the bound state energy levels of Ne(3) will be calculated for total angular momenta up to J = 6, as well as their average root-mean-square radii. We also analyze the wave functions in hyperspherical coordinates for several selected bound states.

The Journal of Chemical Physics•Oct 6, 2011•17 citations

Geometrical structure of helium triatomic systems: comparison with the neon trimer

Hiroya Suno

Abstract The structural properties of helium triatomic systems are studied using hyperspherical coordinates. A slow variable discretization approach is adopted to solve the three-body Schrödinger equation, in which the Schrödinger equation in hyperangular coordinates is solved using basis splines at a series of fixed FEM-DVR (finite-element methods–discrete variable representation) hyperradii. We focus on studying the geometrical structure of the 4 He 3 and 3 He 4 He 2 triatomic systems. Using the bound state wave functions obtained, we calculate and analyze the one-dimensional pair distribution and angle distribution functions as well as the two-dimensional angle–angle distributions. All these bound states are found to exhibit such a floppy nature that classifying them into particular geometrical shapes does not appear to be sensible. A comparison will be made with some bound states of the neon trimer, which are expected to be more tightly bound.

Journal of Physics B Atomic Molecular and Optical Physics•Nov 24, 2015•11 citations

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

Gou Qing-Dong, Yong Li

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.

Acta Physica Sinica•Jan 1, 2015•0 citations

Hyperspherical calculations of ultralow-energy collisions in Coulomb three-body systems

Yueming Zhou, Shinichi Watanabe, Oleg I. Tolstikhin +1 more

Quantum mechanical calculations of ultralow-energy collision of Coulomb three-body systems in the hyperspherical elliptic coordinates are presented. The nonadiabatic coupling between the hyperradius and hyperangular variables is treated with the slow-variable discretization method in combination with the $R$-matrix propagation technique. For scattering state calculations, the two-dimensional matching procedure using Gailitis's method [M. Gailitis, J. Phys. B 9, 843 (1976); C. Noble and R. Nesbet, Comput. Phys. Commun. 33, 399 (1984)] is implemented to determine the boundary conditions between the internal and the asymptotic wave functions. This method is proved to be very efficient and gives very accurate results. Taking advantage of this method, we accurately calculate the scattering phase shifts and the scattering lengths of Coulomb three-body systems with mass ratio varying over several orders of magnitudes. We observed jumps of the scattering length from $\ensuremath{-}\ensuremath{\infty}$ to $\ensuremath{\infty}$ at certain mass ratios and monotonic decreases between two jumps. These are closely related to the binding energy of the highest bound state through Levinson's theorem [N. Levinson, K. Dan. Vidensk. Selsk. Mat. Fys. Medd. 25, 9 (1949)]. Our calculations provide a comprehensive perspective to the scattering length from the variation of mass ratio of Coulomb three-body systems.

Physical Review A•Sep 28, 2015•6 citations
We calculate the binding energy and geometry of the weakly bound e(+)Li and e(+)Na systems within the framework of hyperspherical coordinates. The Schrodinger equation in hyperangular coordinates is solved at a series of fixed hyper-radii using B-splines and the resulting coupled hyper-radial equation is solved using the slow variable discretization method developed by Tolstikhin et al. [J. Phys. B 29, L389 (1996)]. Great efforts are made in optimizing the distribution of B-splines to overcome the slow convergence of the binding energy and geometrical quantities. This approach allows us to obtain the results with improved convergence that are in good agreement with the best values reported to date. In addition, an analysis of the structure of the two systems is also made and the e(+)Na system is seen to exhibit quantum halo features.
In view of modelization of experiments involving cold atoms and molecules, we develop a method that allows us to calculate weakly bound levels of triatomic molecules. The method combines (1) the hyperspherical coordinates to describe interparticle motion in the three-body system, (2) the solution of the Schr\"odinger equation in two steps: determination of adiabatic states for a fixed hyper-radius and then solution of a set of coupled hyper-radial equations using the slow variable representation of Tolstikhin et al. [J. Phys. B: At. Mol. Opt. Phys. 29, L389 (1996)], (3) and a mapping procedure that reduces considerably the number of basis functions needed to represent wave functions of weakly bound levels. We apply the method to the three different systems: the helium trimer 4He3^{4}\mathrm{He}_{3}4He3​, isotopomers of the H3+\mathrm{H}_{3}{}^{+}H3​+ ion, and finally a model three-body problem involving three nucleons. For all these systems, we show that the suggested method provides accurate results.
Using a model potential to describe the interaction between the core and the valence electron, we perform hyperspherical calculations for electron affinity and geometry of the weakly bound Li − and Na − systems. In our calculation, channel functions are expanded in terms of B-splines. Using the special properties of B-splines, we make the knot distributions more precisely, characterizing the behaviour of channel functions. This improves the convergence greatly. Our results are in good agreement with the other theoretical and experimental values.
We study theoretically the ground and excited bound states of the bosonic rare gas van der Waals trimer Ne(3). A slow variable discretization approach is adopted to solve the nuclear Schrödinger equation, in which the Schrödinger equation in hyperangular coordinates is solved using basis splines at a series of fixed finite-element methods discrete variable representation hyper-radii. We consider not only zero total nuclear orbital angular momentum, J = 0, states but also J > 0 states. By using the best empirical neon dimer interaction potentials, all the bound state energy levels of Ne(3) will be calculated for total angular momenta up to J = 6, as well as their average root-mean-square radii. We also analyze the wave functions in hyperspherical coordinates for several selected bound states.
Abstract The structural properties of helium triatomic systems are studied using hyperspherical coordinates. A slow variable discretization approach is adopted to solve the three-body Schrödinger equation, in which the Schrödinger equation in hyperangular coordinates is solved using basis splines at a series of fixed FEM-DVR (finite-element methods–discrete variable representation) hyperradii. We focus on studying the geometrical structure of the 4 He 3 and 3 He 4 He 2 triatomic systems. Using the bound state wave functions obtained, we calculate and analyze the one-dimensional pair distribution and angle distribution functions as well as the two-dimensional angle–angle distributions. All these bound states are found to exhibit such a floppy nature that classifying them into particular geometrical shapes does not appear to be sensible. A comparison will be made with some bound states of the neon trimer, which are expected to be more tightly bound.
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.
Quantum mechanical calculations of ultralow-energy collision of Coulomb three-body systems in the hyperspherical elliptic coordinates are presented. The nonadiabatic coupling between the hyperradius and hyperangular variables is treated with the slow-variable discretization method in combination with the RRR-matrix propagation technique. For scattering state calculations, the two-dimensional matching procedure using Gailitis's method [M. Gailitis, J. Phys. B 9, 843 (1976); C. Noble and R. Nesbet, Comput. Phys. Commun. 33, 399 (1984)] is implemented to determine the boundary conditions between the internal and the asymptotic wave functions. This method is proved to be very efficient and gives very accurate results. Taking advantage of this method, we accurately calculate the scattering phase shifts and the scattering lengths of Coulomb three-body systems with mass ratio varying over several orders of magnitudes. We observed jumps of the scattering length from $\ensuremath{-}\ensuremath{\infty}$ to $\ensuremath{\infty}$ at certain mass ratios and monotonic decreases between two jumps. These are closely related to the binding energy of the highest bound state through Levinson's theorem [N. Levinson, K. Dan. Vidensk. Selsk. Mat. Fys. Medd. 25, 9 (1949)]. Our calculations provide a comprehensive perspective to the scattering length from the variation of mass ratio of Coulomb three-body systems.