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Chemical Physics Letters

Improved exponential split operator method for solving the time-dependent Schrödinger equation

André D. Bandrauk, Hai Shen

Published Jan 1, 1991• 170 citations
Publisher
DOI
10.1016/0009-2614(91)90232-x

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The time-dependent Schrödinger Equation (TDSE) is a parabolic partial differential equation (PDE) comparable to a diffusion equation but with imaginary time. Due to its first order time derivative, exponential integrators or propagators are natural methods to describe evolution in time of the TDSE, both for time-independent and time-dependent potentials. Two splitting methods based on Fer and/or Magnus expansions allow for developing unitary factorizations of exponentials with different accuracies in the time step △t. The unitary factorization of exponentials to high order accuracy depends on commutators of kinetic energy operators with potentials. Fourth-order accuracy propagators can involve negative or complex time steps, or real time steps only but with gradients of potentials, i.e. forces. Extending the propagators of TDSE's to imaginary time allows to also apply these methods to classical many-body dynamics, and quantum statistical mechanics of molecular systems.
The authors introduce a family of explicit and unconditionally stable algorithms for solving linear differential equations which contain a time-dependent Hermitian operator. Rigorous upper bounds are derived for two different 'time-ordered' approximation schemes and for errors resulting from approximating a time-ordered exponential by an ordinary exponential operator. The properties and the usefulness of the product formula algorithms are examined by applying them to the problem of Zener tunnelling. The most efficient algorithm is employed to solve the time-dependent Schrodinger equation for a wavepacket incident on a time-modulated rectangular barrier.