A hybrid Schrödinger/Gaussian beam solver for quantum barriers andsurface hopping

Shi Jin; Peng Qi

doi:10.3934/krm.2011.4.1097

Article Contents

2011, Volume 4, Issue 4: 1097-1120. Doi: 10.3934/krm.2011.4.1097

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A hybrid Schrödinger/Gaussian beam solver for quantum barriers and surface hopping

Shi Jin^1, and
Peng Qi^2,

1.
Department of Mathematics, University of Wisconsin-Madison, 480 Lincoln Drive, Madison, WI 53706
2.
Department of Mathematics, University of Wisconsin, 480 Lincoln Drive, Madison, WI 53706

Received: April 30, 2011

Revised: September 30, 2011

Published: November 2011

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Abstract

In this paper, we propose a hybrid method coupling a Schrödinger solver and the Gaussian beam method for the numerical simulation of quantum tunneling through potential barriers or surface hopping across electronic potential energy surfaces. The idea is to use a Schrödinger solver near potential barriers or zones where potential energy surfaces cross, and the Gaussian beam method--which is much more efficient than a direct Schrödinger solver--elsewhere. Buffer zones are used to convert data between the Schrödinger solver and the Gaussian beam solver. Numerical examples show that this method indeed captures quantum tunneling and surface hopping accurately, with a computational cost much lower than a direct quantum solver in the entire domain.

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Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C35.

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