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BIRS Workshop Lecture Videos

Using time–dependent Gaussian basis sets in quantum dynamics simulations Worth, Graham

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Propagating a multi–dimensional wavepacket using the time–dependent Schr ̈odinger Equation is a computationally hard problem that scales exponentially with the number of degrees of freedom in the system. In contrast to the traditional grid–based approach, one way to ease the scaling is to describe the evolving wavepacket by a superposition of Gaussian functions, often referred to as Gaussian Wavepackets (GWPs). There are a variety of algorithms for the propagation of the GWPs that provide the time– dependent basis set. All have advantages and disadvantages, but most use classical trajectories which leads to good scaling, but poor convergence and problems in dealing with quantum phenomena such as tunneling. There is also the problem in selecting the initial functions. All suffer from numerical problems due to the non–orthonormality of the basis set, which can lead to linear dependence and singularities in the equations of motion. One method that promises to overcome the convergence and initial selection problem is based on the MCTDH wavepacket propagation method [1,2]. This method will be presented to show its potential. It variationally couples the evolving basis functions as well as the expansion coefficients, and as the result the functions follow “quantum trajectories”. In addition, it is straightforward to combine the GWP basis with grid–based methods for better convergence when some modes are strongly quantum mechanical [3]. [1] Burghardt, Meyer and Cederbaum. J. Chem. Phys. 111, 2927 (1999) [2] Worth, Robb and Burghardt. Farad. Discuss. 127, 307 (2004) [3] Burghardt, Giri and Worth. J. Chem. Phys. 129, 174104 (2008)

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