Numerical time-dependent Schrödinger description of charge-exchange collisions

Riley, Merle E.; Ritchie, Burke

doi:10.1103/physreva.59.3544

Cited by 40 publications

(26 citation statements)

References 19 publications

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“…The time-dependent Schroedinger's equation [(12)] is solved in the time and three Cartesian coordinates for each electron of H 2 using an algorithm described previously [11]. (Preliminary He-atom calculations were presented previously [12].)…”

Section: Calculations and Resultsmentioning

confidence: 99%

“…In the nonrelativistic regime of electron velocity, (2a)-(9) may be evaluated using Schroedinger's equation, which follows on exactly eliminating (2b) in favor of (2a) and dropping contributions in the resulting equation for the large component, ( ⃗ , ), of order −2 . The current is evaluated in the nonrelativistic limit using ( ⃗ , ) = ( ⃗ , ) and ( ⃗ , ) given by (11) in the regime ( − ) ≪ 2 , where ( ⃗ , ) obeys the time-dependent Schroedinger's equation as follows:…”

Section: Quantum Trajectories In the Nonrelativistic Limitmentioning

confidence: 99%

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Quantum-Dynamical Theory of Electron Exchange Correlation

Ritchie

Weatherford

2013

Advances in Physical Chemistry

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The relationship between the spin of an individual electron and Fermi-Dirac statistics (FDS), which is obeyed by electrons in the aggregate, is elucidated. The relationship depends on the use of spin-dependent quantum trajectories (SDQT) to evaluate Coulomb's law between any two electrons as an instantaneous interaction in space and time rather than as a quantum-mean interaction in the form of screening and exchange potentials. Hence FDS depends in an ab initio sense on the inference of SDQT from Dirac's equation, which provides for relativistic Lorentz invariance and a permanent magnetic moment (or spin) in the electron's equation of motion. Schroedinger's time-dependent equation can be used to evaluate the SDQT in the nonrelativistic regime of electron velocity. Remarkably FDS is a relativistic property of an ensemble of electron, even though it is of order 0 in the nonrelativistic limit, in agreement with experimental observation. Finally it is shown that covalent versus separated-atoms limits can be characterized by the SDQT. As an example of the use of SDQT in a canonical structure problem, the energies of the 1 Σ and 3 Σ states of H 2 are calculated and compared with the accurate variational energies of Kolos and Wolniewitz.

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Section: Calculations and Resultsmentioning

confidence: 99%

Section: Quantum Trajectories In the Nonrelativistic Limitmentioning

confidence: 99%

Quantum-Dynamical Theory of Electron Exchange Correlation

Ritchie

Weatherford

2013

Advances in Physical Chemistry

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“…These include e.g. alternating-direction-implicit (ADI) methods [5][6][7], splitoperator and fast Fourier transform (FFT) methods [8,9], mixed finite-difference/harmonic expansion methods [10][11][12][13] and B-spline methods [7,14].…”

Section: Introductionmentioning

confidence: 99%

Radiation boundary conditions for the numerical solution of the three-dimensional time-dependent Schrödinger equation with a localized interaction

Heinen

Kull

2009

Phys. Rev. E

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Exact radiation boundary conditions on the surface of a sphere are presented for the single-particle time-dependent Schrödinger equation with a localized interaction. With these boundary conditions, numerical computations of spatially unbounded outgoing wave solutions can be restricted to the finite volume of a sphere. The boundary conditions are expressed in terms of the free-particle Greenfunction for the outside region. The Green-function is analytically calculated by an expansion in spherical harmonics and by the method of Laplace transformation. For each harmonic number a discrete boundary condition between the function values at adjacent radial grid points is obtained. The numerical method is applied to quantum tunneling through a spherically symmetric potential barrier with different angular momentum quantum numbers l. Calculations for l = 0 are compared with exact theoretical results.

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“…On the grid coordinates the splitting method [10,11] results in the difference scheme of the second order accuracy. Here, the equation (1) is solved iteratively in time with the fast complex Fourier transform [11] on a spatial grid with a plane of symmetry (the collision plane).…”

Section: The Solving Of a Time-dependent Schrödinger Equationmentioning

confidence: 99%