Coordinate systems and analytic expansions for three-body atomic wavefunctions. II. Closed form wavefunction to second order in r

Gottschalk, J E; Abbott, Paul; Maslen, E. N.

doi:10.1088/0305-4470/20/8/024

Cited by 40 publications

(21 citation statements)

References 26 publications

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“…and △ is the two-dimensional Laplacian over the hyperangles. All e nm with n ≤ 2 are known analytically plus a few additional terms with higher n [83][84][85][86]. Morgan proved that the series is convergent everywhere [87], and it has been shown that variational calculations converge faster when a single power of a logarithm is included in the basis [7] [94].…”

Section: B Three Particle Coalescencementioning

confidence: 99%

Pseudospectral calculation of the wave function of helium and the negative hydrogen ion

Grabowski

Chernoff

2010

Phys. Rev. A

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We study the numerical solution of the nonrelativistic Schrödinger equation for two-electron atoms in ground and excited S states using pseudospectral (PS) methods of calculation. The calculation achieves convergence rates for the energy, Cauchy error in the wave function, and variance in local energy that are exponentially fast for all practical purposes. The method requires three separate subdomains to handle the wave function's cusplike behavior near the two-particle coalescences. The use of three subdomains is essential to maintaining exponential convergence and is more computationally efficient than a single subdomain. A comparison of several different treatments of the cusps suggests that the simplest prescription is sufficient. We investigate two alternate methods for handling the semi-infinite domain, one which involves a sequence of truncated versions of the domain and the other which employs an algebraic mapping of the semi-infinite domain to a finite one and imposes no explicit cutoffs on the wave function. The latter prescription proves superior. For many purposes it proves unnecessary to handle the three-particle coalescence in a special way. The presence of logarithmic terms in the exact solution is expected to limit the convergence to being nonexponential but the only clear evidence of that is the rate of convergence of derivatives near the three-particle coalescence point. Higher resolution than achieved in this work will ultimately be needed to see its limiting effect on other measures of error. As developed and applied here the PS method has many virtues: no explicit assumptions need be made about the asymptotic behavior of the wave function near cusps or at large distances, the local energy (Hψ/ψ) is exactly equal to the calculated global energy at all collocation points, local errors go down everywhere with increasing resolution, the effective basis using Chebyshev polynomials is complete and simple, and the method is easily extensible to other bound states. As the number of collocation points grows, the method achieves exponential convergence up to the resolution tested. This study serves as a proof-of-principle of the method for more general two-and possibly three-electron applications.

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Section: B Three Particle Coalescencementioning

confidence: 99%

Pseudospectral calculation of the wave function of helium and the negative hydrogen ion

Grabowski

Chernoff

2010

Phys. Rev. A

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“…where σ l (1), τ l (1) and φ l (1) are defined by Eqs. (16),(36) and (55), respectively. It is easy to verify that the results (107) confirm the validity of the conditions…”

Section: The Boundary Conditions For the Afc Componentsmentioning

confidence: 99%

“…The works [10][11][12][13][14][15][16][17][18] were devoted to investigation of the properties and to calculations of the angular Fock coefficients (AFCs), ψ k,p (α, θ). It has been proven that the AFCs satisfy (see, e.g., [15] or [19]) the Fock recurrence relation (FRR)…”

Section: Introductionmentioning

confidence: 99%

Two-particle atomic coalescences: Boundary conditions for the Fock coefficient components

Liverts

2016

Phys. Rev. A

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The exact values of the presently determined components of the angular Fock coefficients at the two-particle coalescences were obtained and systematized. The Green Function approach was successfully applied to simplify the most complicated calculations. The boundary conditions for the Fock coefficient components in the hyperspherical angular coordinates, which follows from the Kato cusp conditions for the two-electron wave function in the natural interparticle coordinates, were derived. The validity of the obtained boundary conditions was verified on examples of all the presently determined components. The additional boundary conditions are not arising from the Kato cusp conditions were obtained as well. The Wolfram Mathematica was used intensively.

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“…(2) Such logarithmic terms are characteristic of 3PC behavior [22][23][24] and using a wave function based on (2) yields a continuous local energy for the helium atom. 25 The present work aims to elucidate the form of the wave function at three-electron coalescences for three electrons, in various spin states, confined to a harmonic potential in D = 2 or D = 3 dimensions.…”

Section: Introductionmentioning

confidence: 99%

Communication: Three-electron coalescence points in two and three dimensions

Loos

Bloomfield

Gill

2015

The Journal of Chemical Physics

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The form of the wave function at three-electron coalescence points is examined for several spin states using an alternative method to the usual Fock expansion. We find that, in two- and three-dimensional systems, the non-analytical nature of the wave function is characterized by the appearance of logarithmic terms, reminiscent of those that appear as both electrons approach the nucleus of the helium atom. The explicit form of these singularities is given in terms of the interelectronic distances for a doublet and two quartet states of three electrons in a harmonic well.

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Coordinate systems and analytic expansions for three-body atomic wavefunctions. II. Closed form wavefunction to second order in r

Cited by 40 publications

References 26 publications

Pseudospectral calculation of the wave function of helium and the negative hydrogen ion

Pseudospectral calculation of the wave function of helium and the negative hydrogen ion

Two-particle atomic coalescences: Boundary conditions for the Fock coefficient components

Communication: Three-electron coalescence points in two and three dimensions

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