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arXiv:physics/0206010

Matrix Inverse Free Method for Solving Quantum Electron Dynamics on Unstructured Grid

Katsuhiro Watanabe, Akihito Kikuchi

Published Jun 4, 2002• 0 citations
arXivHTMLPDF

Abstract

A matrix inverse free method to solve time-dependent Schrodinger equation is presented. The method is not subject to form of Hamiltonian and adopting real space grid system such as structured and unstructured grid, and achieves the order N algorithm even if we adopt unstructured grid systems to adjust complex regions. We have demonstrated some error evaluation problems that have exact solution and application in which it has non-Schrodinger type Hamiltonian, and we have successfully simulated the time-evolution of wave function for all of the case.

DOI
10.48550/arxiv.physics/0206010

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A matrix inverse free method to solve time-dependent Schrodinger equation is presented. The method is not subject to form of Hamiltonian and adopting real space grid system such as structured and unstructured grid, and achieves the order N algorithm even if we adopt unstructured grid systems to adjust complex regions. We have demonstrated some error evaluation problems that have exact solution and application in which it has non-Schrodinger type Hamiltonian, and we have successfully simulated the time-evolution of wave function for all of the case.
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A fast and stable method is formulated to compute the time evolution of a wave function by numerically solving the time-dependent Schrodinger equation. This method is a real-space-real-time evolution method implemented by several computational techniques such as Suzuki's exponential product, Cayley's form, the finite differential method, and an operator named adhesive operator. This method conserves the norm of the wave function, manages periodic conditions and adaptive mesh refinement technique, and is suitable for vector- and parallel-type supercomputers. Applying this method to some simple electron dynamics, we confirmed the efficiency and accuracy of the method for simulating fast time-dependent quantum phenomena.
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Dynamical electronic- and vibrational-structure theories have received a growing interest in the past few years due to their ability to simulate spectra recorded with ultrafast experimental techniques. The exact time evolution of a molecular system can, in principle, be obtained from the time-dependent version of full configuration interaction. Such an approach is, however, limited to few-atom systems due to the exponential increase of its cost with the system dimension. In the present work, we overcome this unfavorable scaling by employing the time-dependent density matrix renormalization group (TD-DMRG) which parametrizes the time-dependent wave function as a matrix product state. The time-dependent Schrödinger equation is then integrated with a sweep-based algorithm, as in standard time-independent DMRG. Unlike other TD-DMRG approaches, the one presented here leads to a set of coupled equations that can be integrated exactly. The resulting theory enables us to study real- and imaginary-time evolutions of Hamiltonians comprising more than 20 degrees of freedom that are challenging for current state-of-the-art quantum dynamics algorithms. We apply our algorithm to the simulation of quantum dynamics of models of increasing complexity, ranging from simple excitonic Hamiltonians to more complex ab initio vibronic ones.
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