Physical Review A

Three-body bound states and structure calculations of the He2−Ne system using the slow variable discretization method

Muhammad Saleem Shahzad, Yong Li

Published Sep 22, 2014• 1 citation

Abstract

The ${\mathrm{He}}_{2}\text{\ensuremath{-}}\mathrm{Ne}$ system with the combination of the isotopes of helium ($^{3}\mathrm{He}$ and $^{4}\mathrm{He}$) is used in structure calculations. Three-body channel functions and adiabatic potential functions have been calculated using the $B$-spline method. We use these channel functions in the slow variable discretization method to calculate the three-body bound states of the ${\mathrm{He}}_{2}\text{\ensuremath{-}}\mathrm{Ne}$ system. Using this method, two bound states for the ${}^{4}{\mathrm{He}}_{2}\text{\ensuremath{-}}{}^{20}\mathrm{Ne}$ system and one bound state for each of the $^{4}\mathrm{He}\text{\ensuremath{-}}^{3}\mathrm{He}\text{\ensuremath{-}}^{20}\mathrm{Ne}$ and $^{3}\mathrm{He}{}_{2}\text{\ensuremath{-}}^{20}\mathrm{Ne}$ systems have been found. The geometry of the trimer is also analyzed using the probability densities and expectation values.

DOI: 10.1103/physreva.90.032510

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

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Physical Review A

Three-body bound states and structure calculations of the He2−Ne system using the slow variable discretization method

Muhammad Saleem Shahzad, Yong Li

Published Sep 22, 2014• 1 citation

Publisher

Abstract

The

{\mathrm{He}}_{2}\text{\ensuremath{-}}\mathrm{Ne}

system with the combination of the isotopes of helium (

^{3}\mathrm{He}

and

^{4}\mathrm{He}

) is used in structure calculations. Three-body channel functions and adiabatic potential functions have been calculated using the

B

-spline method. We use these channel functions in the slow variable discretization method to calculate the three-body bound states of the

{\mathrm{He}}_{2}\text{\ensuremath{-}}\mathrm{Ne}

system. Using this method, two bound states for the

{}^{4}{\mathrm{He}}_{2}\text{\ensuremath{-}}{}^{20}\mathrm{Ne}

system and one bound state for each of the

^{4}\mathrm{He}\text{\ensuremath{-}}^{3}\mathrm{He}\text{\ensuremath{-}}^{20}\mathrm{Ne}

and

^{3}\mathrm{He}{}_{2}\text{\ensuremath{-}}^{20}\mathrm{Ne}

systems have been found. The geometry of the trimer is also analyzed using the probability densities and expectation values.

DOI: 10.1103/physreva.90.032510

Related papers

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.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.

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Dario Bressanini

Despite their apparent simplicity, the properties of the two helium trimers, (4)He3 and (3)He(4)He2, are still not completely understood. In particular, the existence of a bound state of the asymmetric trimer (3)He(4)He2 was established many years ago, using different theoretical approaches, and later it was experimentally detected. However its structural properties have not been thoroughly investigated so far, probably because an accurate theoretical description of this very weakly bound system is computationally quite demanding. In this work we give for the first time an accurate and complete theoretical description of the geometrical structure of this fragile system using quantum Monte Carlo techniques employing the TTY potential and compare its properties with those of (4)He2 and (4)He3. We compute average values of distances and angles, along with the angle-angle distribution function: a two-dimensional probability distribution well suited to discuss the shape of a trimer. Our analysis shows that the lighter isotope is very diffuse and can be found at large distances from the other two atoms, but also close to the center of mass of the system in nearly linear configurations. For this system the concept of "equilibrium structure" is meaningless and all kinds of three-atom configurations must be taken into account in its description.Despite their apparent simplicity, the properties of the two helium trimers, (4)He3 and (3)He(4)He2, are still not completely understood. In particular, the existence of a bound state of the asymmetric trimer (3)He(4)He2 was established many years ago, using different theoretical approaches, and later it was experimentally detected. However its structural properties have not been thoroughly investigated so far, probably because an accurate theoretical description of this very weakly bound system is computationally quite demanding. In this work we give for the first time an accurate and complete theoretical description of the geometrical structure of this fragile system using quantum Monte Carlo techniques employing the TTY potential and compare its properties with those of (4)He2 and (4)He3. We compute average values of distances and angles, along with the angle-angle distribution function: a two-dimensional probability distribution well suited to discuss the shape of a trimer. Our analysis shows that the lighter isotope is very diffuse and can be found at large distances from the other two atoms, but also close to the center of mass of the system in nearly linear configurations. For this system the concept of "equilibrium structure" is meaningless and all kinds of three-atom configurations must be taken into account in its description.

The Journal of Physical Chemistry AMay 14, 20148 citations

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.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 AJan 10, 200638 citations

Adiabatic hyperspherical study of the helium trimer

Brett D. Esry, C. D. Lin, Chris H. Greene

The adiabatic hyperspherical method is used to investigate the spectrum of the helium trimer and to explore effects on the spectrum of varying the helium masses as well as the interaction strength. When a realistic helium dimer potential is adopted, only two isotopic combinations are observed to have three-body bound states: $^{4}\mathrm{He}_{3}$ and $^{4}\mathrm{He}_{2}^{3}$ He. By artificially adjusting the dimer potential, we also determine the range of interaction strengths for which halo states can result. The existence of Efimov states for both physical and unphysical systems is also examined. \textcopyright{} 1996 American Institute of Physics.The adiabatic hyperspherical method is used to investigate the spectrum of the helium trimer and to explore effects on the spectrum of varying the helium masses as well as the interaction strength. When a realistic helium dimer potential is adopted, only two isotopic combinations are observed to have three-body bound states: $^{4}\mathrm{He}_{3}$ and $^{4}\mathrm{He}_{2}^{3}$He. By artificially adjusting the dimer potential, we also determine the range of interaction strengths for which halo states can result. The existence of Efimov states for both physical and unphysical systems is also examined. \textcopyright{} 1996 American Institute of Physics.

Physical Review AJul 1, 1996214 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.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 SinicaJan 1, 20150 citations

Adiabatic hyperspherical study of triatomic helium systems

Hiroya Suno, Brett D. Esry

The $^{4}\mathrm{He}_{3}$ system is studied using the adiabatic hyperspherical representation. We adopt the current state-of-the-art helium interaction potential including retardation and the nonadditive three-body term to calculate all low-energy properties of the triatomic $^{4}\mathrm{He}$ system. The bound state energies of the $^{4}\mathrm{He}$ trimer are computed as well as the $^{4}\mathrm{He}+^{4}\mathrm{He}_{2}$ elastic scattering cross sections, the three-body recombination, and collision induced dissociation rates at finite temperatures. We also treat the system that consists of two $^{4}\mathrm{He}$ and one $^{3}\mathrm{He}$ atoms, and compute the spectrum of the isotopic trimer $^{4}\mathrm{He}_{2}^{3}\mathrm{He}$ , the $^{3}\mathrm{He}+^{4}\mathrm{He}_{2}$ elastic scattering cross sections, the rates for three-body recombination, and the collision induced dissociation rate at finite temperatures. The effects of retardation and the nonadditive three-body term are investigated. Retardation is found to be significant in some cases, while the three-body term plays only a minor role for these systems.The $^{4}\mathrm{He}_{3}$ system is studied using the adiabatic hyperspherical representation. We adopt the current state-of-the-art helium interaction potential including retardation and the nonadditive three-body term to calculate all low-energy properties of the triatomic $^{4}\mathrm{He}$ system. The bound state energies of the $^{4}\mathrm{He}$ trimer are computed as well as the $^{4}\mathrm{He}+^{4}\mathrm{He}_{2}$ elastic scattering cross sections, the three-body recombination, and collision induced dissociation rates at finite temperatures. We also treat the system that consists of two $^{4}\mathrm{He}$ and one $^{3}\mathrm{He}$ atoms, and compute the spectrum of the isotopic trimer $^{4}\mathrm{He}_{2}^{3}\mathrm{He}$, the $^{3}\mathrm{He}+^{4}\mathrm{He}_{2}$ elastic scattering cross sections, the rates for three-body recombination, and the collision induced dissociation rate at finite temperatures. The effects of retardation and the nonadditive three-body term are investigated. Retardation is found to be significant in some cases, while the three-body term plays only a minor role for these systems.

arXiv:0809.5111Physical Review ADec 1, 200883 citations

The

{\mathrm{He}}_{2}\text{\ensuremath{-}}\mathrm{Ne}

system with the combination of the isotopes of helium (

^{3}\mathrm{He}

and

^{4}\mathrm{He}

) is used in structure calculations. Three-body channel functions and adiabatic potential functions have been calculated using the

B

-spline method. We use these channel functions in the slow variable discretization method to calculate the three-body bound states of the

{\mathrm{He}}_{2}\text{\ensuremath{-}}\mathrm{Ne}

system. Using this method, two bound states for the

{}^{4}{\mathrm{He}}_{2}\text{\ensuremath{-}}{}^{20}\mathrm{Ne}

system and one bound state for each of the

^{4}\mathrm{He}\text{\ensuremath{-}}^{3}\mathrm{He}\text{\ensuremath{-}}^{20}\mathrm{Ne}

and

^{3}\mathrm{He}{}_{2}\text{\ensuremath{-}}^{20}\mathrm{Ne}

systems have been found. The geometry of the trimer is also analyzed using the probability densities and expectation values.

^{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.

^{4}\mathrm{He}_{3}

and

^{4}\mathrm{He}_{2}^{3}

He. By artificially adjusting the dimer potential, we also determine the range of interaction strengths for which halo states can result. The existence of Efimov states for both physical and unphysical systems is also examined. \textcopyright{} 1996 American Institute of Physics.

The

^{4}\mathrm{He}_{3}

system is studied using the adiabatic hyperspherical representation. We adopt the current state-of-the-art helium interaction potential including retardation and the nonadditive three-body term to calculate all low-energy properties of the triatomic

^{4}\mathrm{He}

system. The bound state energies of the

^{4}\mathrm{He}

trimer are computed as well as the

^{4}\mathrm{He}+^{4}\mathrm{He}_{2}

elastic scattering cross sections, the three-body recombination, and collision induced dissociation rates at finite temperatures. We also treat the system that consists of two

^{4}\mathrm{He}

and one

^{3}\mathrm{He}

atoms, and compute the spectrum of the isotopic trimer

^{4}\mathrm{He}_{2}^{3}\mathrm{He}

, the

^{3}\mathrm{He}+^{4}\mathrm{He}_{2}

elastic scattering cross sections, the rates for three-body recombination, and the collision induced dissociation rate at finite temperatures. The effects of retardation and the nonadditive three-body term are investigated. Retardation is found to be significant in some cases, while the three-body term plays only a minor role for these systems.