Hyperspherical approach to Coulombic three-body systems with different masses

Zhou, Yan; Lin, C. D.; Shertzer, J.

doi:10.1088/0953-4075/26/21/026

Cited by 24 publications

(20 citation statements)

References 24 publications

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“…For hydrogen, µ = i corresponds e + + H and µ = j corresponds to Ps(1s) + H + . We solve the coupled equations given in equation (3) for 2R 2 ε µ (R) and f µ,I (R; θ, φ), µ = i, j using the finite element method [43][44][45]. It is computationally intensive to solve a complex generalized eigenvalue problem for thousands of values of R. In order to maximize efficiency, we use Rayleigh quotient iteration [43].…”

Section: Hyperspherical Hidden Crossing Methods (Hhcm) For Three-body mentioning

confidence: 99%

Hyperspherical hidden crossing method applied to Ps(1s)-formation in low energy e⁺− H, e⁺− Li and e⁺− Na collisions

Ward

Shertzer

2012

New J. Phys.

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The hyperspherical hidden crossing method with the correction term (HHCM +cor ) has been used to calculate partial wave Ps(1s)-formation cross sections for low-energy e + − H, e + − Li and e + − Na collisions. The Stückelberg phase varies in a systematic way as a function of atomic number Z , incident positron momentum k + , and total orbital angular momentum L and provides an explanation for the small S-wave cross section for all three systems. 13 References 13 Including the correction term does not compromise the simplicity of the method; it simply provides an improved value for the wave vector K (R). However, there is one undesirable consequence of using the correction term. The correction term diverges at the branch point, and therefore we include it only on the real axis. As a result, the cross section becomes path dependent. The correction term can be included in a consistent way by following the prescription for the paths given in [6]; we refer to these calculations as HHCM +cor .The HHCM has previously been applied with success to three-body collisions, including electron excitation [1] and impact ionization [1,[7][8][9][10][11][12]. Earlier HHCM calculations for Ps(1s) formation in e + − H collisions included an asymptotic approximation to the correction term (for the P-and D-waves) and provided an explanation for the small S-wave Ps(1s)-formation cross section in terms of the Stückelberg phase [2]. The HHCM was also used to treat near-threshold positron impact ionization of hydrogen [13]. The HHCM enables one to understand why the S-wave cross section for any process for e + − H collisions (including ionization) that proceeds via the Ps(1s) formation channel is small. Although the HHCM was derived for three-body Coulomb systems, it has been successfully used to study the recombination of three identical bosons with zero range two-body potentials [14]. We previously used the HHCM to calculate partial wave Ps(1s)-formation cross sections for e + − Li collisions [15]. Despite the success of these calculations, the application of the HHCM has been somewhat limited.We have carried out a systematic study of Ps(1s) formation in positron collisions with H, Li and Na using the HHCM +cor . The HHCM +cor results for e + − H and e + − Na reported here are new; we also include the previously published HHCM +cor e + − Li results [6] for completeness. We examine the behavior of the Stückelberg phase as a function of atomic number Z , incident positron momentum k + , and total orbital angular momentum L for the three atoms with a single l = 0 valence electron. For the alkalis, we use a model potential to describe the interaction of the valence electron and the positron with the ion core.

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Section: Hyperspherical Hidden Crossing Methods (Hhcm) For Three-body mentioning

confidence: 99%

Hyperspherical hidden crossing method applied to Ps(1s)-formation in low energy e⁺− H, e⁺− Li and e⁺− Na collisions

Ward

Shertzer

2012

New J. Phys.

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“…In this section we briefly review the adiabatic hyperspherical method, 31,32,37,38 namely, we derive a set of differential equations that provide an approximate solution to the Schrödinger equation for N interacting He atoms. The numerical methods used for solving these equations are discussed in Sec.…”

Section: Adiabatic Approachmentioning

confidence: 99%

Monte Carlo hyperspherical description of helium cluster excited states

Blume

Greene

2000

The Journal of Chemical Physics

106

112

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Path integral Monte Carlo simulation of the absorption spectra of an Al atom embedded in heliumThe Jϭ0 many-body Schrödinger equation for 4 He N clusters with Nϭ3 -10 is solved numerically by combining Monte Carlo methods with the adiabatic hyperspherical approximation. We find ground state and excited state energies for these systems with an adiabatic separation scheme that reduces the problem to motion in a one-dimensional effective potential curve as a function of the hyperspherical radius R. We predict the number of Jϭ0 bound states for these clusters, and also the HeϩHe NϪ1 elastic scattering lengths up to Nϭ10. For Nϭ5 -10, these are the first such calculations reported.

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“…. , ρ n−1 ) in terms of a complete set of hyperangular channel functions ν (R; ) that depend parametrically on the hyperradius R and hyperradial weight functions F νE (R) [3,7,8,15]:…”

Section: System Hamiltonian and Hyperspherical Frameworkmentioning

confidence: 99%

“…The channel functions ν (R; ) form a complete set in the (3n − 4)-dimensional Hilbert space associated with the hyperangular degrees of freedom [3,7,8,15] …”

Section: System Hamiltonian and Hyperspherical Frameworkmentioning

confidence: 99%

“…It is a basis-set expansion-type approach, which combines elements of the aspects (i)-(iii) mentioned above. In particular, the use of hyperspherical coordinates [3][4][5][6][7][8][9][10][11][12][13][14][15] within the framework of explicitly correlated Gaussian (ECG) basis functions [16] allows us to take advantage of the machinery developed for bound-state calculations while at the same time enables us to describe the scattering continuum. Although significant progress has been made [1,3,[17][18][19][20], in general, the determination of scattering quantities is significantly more involved than that of bound-state quantities, and the four-, five-, and higher-body scattering continua are comparatively poorly understood.…”

Section: Introductionmentioning

confidence: 99%

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Hyperspherical explicitly correlated Gaussian approach for few-body systems with finite angular momentum

Rakshit

Blume

2012

Phys. Rev. A

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Within the hyperspherical framework, the solution of the time-independent Schrödinger equation for a nparticle system is divided into two steps: the solution of a Schrödinger-type equation in the hyperangular degrees of freedom and the solution of a set of coupled Schrödinger-type hyperradial equations. The solutions to the former provide effective potentials and coupling matrix elements that enter into the latter set of equations. This paper develops a theoretical framework to determine the effective potentials, as well as the associated coupling matrix elements, for few-body systems with finite angular momentum L = 1 and negative and positive parity . The hyperangular channel functions are expanded in terms of explicitly correlated Gaussian basis functions, and relatively compact expressions for the matrix elements are derived. The developed formalism is applicable to any n; however, for n 6, the computational demands are likely beyond present-day computational capabilities. A number of calculations relevant to cold-atom physics are presented, demonstrating that the developed approach provides a computationally efficient means to solving four-body bound and scattering problems with finite angular momentum on powerful desktop computers. Details regarding the implementation are discussed.

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Hyperspherical approach to Coulombic three-body systems with different masses

Cited by 24 publications

References 24 publications

Hyperspherical hidden crossing method applied to Ps(1s)-formation in low energy e⁺− H, e⁺− Li and e⁺− Na collisions

Hyperspherical hidden crossing method applied to Ps(1s)-formation in low energy e⁺− H, e⁺− Li and e⁺− Na collisions

Monte Carlo hyperspherical description of helium cluster excited states

Hyperspherical explicitly correlated Gaussian approach for few-body systems with finite angular momentum

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Hyperspherical approach to Coulombic three-body systems with different masses

Cited by 24 publications

References 24 publications

Hyperspherical hidden crossing method applied to Ps(1s)-formation in low energy e+− H, e+− Li and e+− Na collisions

Hyperspherical hidden crossing method applied to Ps(1s)-formation in low energy e+− H, e+− Li and e+− Na collisions

Monte Carlo hyperspherical description of helium cluster excited states

Hyperspherical explicitly correlated Gaussian approach for few-body systems with finite angular momentum

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Product

Resources

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Hyperspherical hidden crossing method applied to Ps(1s)-formation in low energy e⁺− H, e⁺− Li and e⁺− Na collisions

Hyperspherical hidden crossing method applied to Ps(1s)-formation in low energy e⁺− H, e⁺− Li and e⁺− Na collisions