Towards a Direct Numerical Solution of Schr&amp;ouml;dinger’s Equation for (e, 2e) Reactions

Jones, S.T.; Stelbovics, A. T.

doi:10.1071/ph98109

Cited by 5 publications

(6 citation statements)

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“…where the sparse matrices, A, B and C, and the − → χ vector are determined from the coefficients of a two-dimensional 3-point Numerov finite difference formula, modified to allow for arbitrary changes in grid spacing, including the change from real to complex coordinates at R 0 . This propagation technique is based on that used by Poet (1980) for e-H scattering and more recently by Jones and Stelbovics (1999) and Jones and Stelbovics (2002) for their benchmark calculations for the Temkin-Poet model for e-H ionization. We have previously given details for the l = 0 modified Numerov formula (Bartlett and Stelbovics 2004), which uses the eight nearest-neighbour grid points in its calculations.…”

Section: L70mentioning

confidence: 99%

Iteratively-coupled propagating exterior complex scaling method for electron–hydrogen collisions

Bartlett¹,

Stelbovics²,

Bray³

2004

J. Phys. B: At. Mol. Opt. Phys.

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A newly-derived iterative coupling procedure for the propagating exterior complex scaling (PECS) method is used to efficiently calculate the electronimpact wavefunctions for atomic hydrogen. An overview of this method is given along with methods for extracting scattering cross sections. Differential scattering cross sections at 30 eV are presented for the electron-impact excitation to the n = 1, 2, 3 and 4 final states, for both PECS and convergent close coupling (CCC), which are in excellent agreement with each other and with experiment. PECS results are presented at 27.2 eV and 30 eV for symmetric and asymmetric energy-sharing triple differential cross sections, which are in excellent agreement with CCC and exterior complex scaling calculations, and with experimental data. At these intermediate energies, the efficiency of the PECS method with iterative coupling has allowed highly accurate partial-wave solutions of the full Schrödinger equation, for L 50 and a large number of coupled angular momentum states, to be obtained with minimal computing resources.The method for evaluating the scattered wavefunction for the electron-impact of atomic hydrogen using exterior complex scaling (ECS) was pioneered by Rescigno et al (1999), and using integral methods to obtain ionization cross sections (Baertschy et al 2001a), an excellent agreement with experimental results was obtained. The absence of any systematic approximations in the ECS method, and its universal nature (by directly calculating the underlying Schrödinger equation of the collision processes), gives tremendous potential for its application to collisions more complex than electron-hydrogen. Existing implementations of the ECS method for electron-hydrogen, however, are very computationally intensive. So, in order to apply the method to larger systems, it is necessary to find strategies to reduce its computational complexity.

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Section: L70mentioning

confidence: 99%

Iteratively-coupled propagating exterior complex scaling method for electron–hydrogen collisions

Bartlett¹,

Stelbovics²,

Bray³

2004

J. Phys. B: At. Mol. Opt. Phys.

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“…To begin the propagation at the first column (i = 1) we note that A (1) is a null vector and D (0) and − → E (0) are not required to be known, and in fact are also null. We can therefore reduce (51) and (52) to…”

Section: Propagation Methodsmentioning

confidence: 99%

“…The Numerov formula has been used successfully for e-H model problems for scattering [30] and ionization [52] and the full e-H scattering problem [54], but to our knowledge has not previously been adapted for variable grids. Jones and Stelbovics [55] used a grid-doubling method that allowed their grid spacing to be increased by integer multiples (whilst continuing to use evenly spaced points for the Numerov formula).…”

Section: Numerov Formulaementioning

confidence: 99%

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A complete numerical approach to electron–hydrogen collisions

Bartlett

2006

J. Phys. B: At. Mol. Opt. Phys.

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This tutorial presents an extensive computational study of electron-impact scattering and ionization of atomic hydrogen and hydrogenic ions, through the solution of the non-relativistic Schrödinger equation in coordinate space using propagating exterior complex scaling (PECS). It details the complete numerical and computational development of the PECS method, which enables highly computationally-efficient solution of these collision systems. Benchmark results are presented for a complete range of electron-hydrogen collisions, including discrete elastic and inelastic scattering both below and above the ionization threshold energy, very low-energy ionizing collisions through to moderately high-energy ionizing collisions, ground-state and excited-state targets and charged hydrogenic targets with Z 4. Total ionization cross sections through to fully differential cross sections, both in-plane and out-ofplane, are given and are found to be in excellent accord with other state-of-theart methods and measurements, where available. We also review our recent confirmation (Bartlett and Stelbovics 2004 Phys. Rev. Lett. 93 233201) of the Wannier and related threshold laws for e-H collisions.

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“…Nevertheless, a solution can be found by the standard method of minimizing the sum of the squares of the residuals [the differences between the left-and right-hand sides of equations (11)]. Previously we found [5] that the least-squares method is generally stabler than keeping any subset of just N equations (11).…”

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Complete Numerical Solution of Electron-Hydrogen Model Collision Problem above the Ionization Threshold

Jones

Stelbovics

2000

Phys. Rev. Lett.

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Benchmark results are presented for electrons colliding with hydrogen atoms in the S-wave (TemkinPoet) model collision problem, which neglects angular momentum. Complete results (elastic, inelastic, and ionization), accurate to 1%, are obtained by numerically integrating Schrödinger's equation subject to correct asymptotic boundary conditions. This marks the first time direct matching to asymptotic boundary conditions has been shown to yield convergent ionization amplitudes for a Coulomb three-body problem. Results are presented for impact energies of 54.4 and 40.8 eV, where comparison with other theories is available.PACS numbers: 34.80. Dp, 31.15.Fx, 34.10. + x, 34.80.Bm The Temkin-Poet [1,2] model of electron-hydrogen scattering is now widely regarded as an ideal testing ground for the development of general methods intended for the full three-body Coulomb problem. Although only s states are included for both the projectile and the atomic electrons, this model problem still contains most of the features that make the real scattering problem hard to solve. Indeed, even in this simplified model, converged energy distributions for ionization cannot generally be obtained via the close-coupling formalism [3]. Any general method that cannot obtain complete, converged results for this model problem will face similar difficulties when applied to the full electron-hydrogen system. For this reason, we believe it is essential to develop a general numerical method capable of solving the Temkin-Poet model completely before angular momentum is included. Here we report such a method. Complete, precision results for e 2

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Towards a Direct Numerical Solution of Schrödinger’s Equation for (e, 2e) Reactions

Cited by 5 publications

References 1 publication

Iteratively-coupled propagating exterior complex scaling method for electron–hydrogen collisions

Iteratively-coupled propagating exterior complex scaling method for electron–hydrogen collisions

A complete numerical approach to electron–hydrogen collisions

Complete Numerical Solution of Electron-Hydrogen Model Collision Problem above the Ionization Threshold

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