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arXiv · 0810.2426

A combined R-matrix eigenstate basis set and finite-differences propagation method for the time-dependent Schr\"{od}dinger equation: the one-electron case

L. A. A. Nikolopoulos·J. S. Parker·K. T. Taylor
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Abstract

In this work we present the theoretical framework for the solution of the time-dependent Schrödinger equation (TDSE) of atomic and molecular systems under strong electromagnetic fields with the configuration space of the electron's coordinates separated over two regions, that is regions $I$ and $II$. In region $I$ the solution of the TDSE is obtained by an R-matrix basis set representation of the time-dependent wavefunction. In region $II$ a grid representation of the wavefunction is considered and propagation in space and time is obtained through the finite-differences method. It appears this is the first time a combination of basis set and grid methods has been put forward for tackling multi-region time-dependent problems. In both regions, a high-order explicit scheme is employed for the time propagation. While, in a purely hydrogenic system no approximation is involved due to this separation, in multi-electron systems the validity and the usefulness of the present method relies on the basic assumption of R-matrix theory, namely that beyond a certain distance (encompassing region $I$) a single ejected electron is distinguishable from the other electrons of the multi-electron system and evolves there (region II) effectively as a one-electron system. The method is developed in detail for single active electron systems and applied to the exemplar case of the hydrogen atom in an intense laser field.

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L. A. A. Nikolopoulos, J. S. Parker, K. T. Taylor. 2008-10-14. A combined R-matrix eigenstate basis set and finite-differences propagation method for the time-dependent Schr\"{od}dinger equation: the one-electron case. https://doi.org/10.1103/physreva.78.063420

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