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Tailored grids for numerical simulation of quantum molecular dynamics
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9.To see that the Fourier transform of an analytic function (on the real axis) decays exponentially (or faster) in the outer portion of Fourier space, consider evaluating the Fourier transform by the method of steepest descent. Poles are necessarily off the real axis, and therefore determine exponentially small asymptotic forms.
10.In DVR, the potential energy is diagonal. A separable kinetic energy (as in the case of Cartesian coordinates) is a sum of matrices with each term a direct product of an one-dimensional kinetic energy matrix and identity matrices for the other degrees of freedom.
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20.We use the method of Cullum and Willoughby, as implemented by Matthew Bramley, to eliminate spurious eigenvalues and compute eigenvectors by inverse iteration.
21.Conversion of eigenvectors from Lanczos representation back to DVR requires either that the Lanczos vectors be saved, or the Lanczos iteration be repeated as the desired set of DVR eigenvectors is constructed. We used the latter method which provided much simpler memory management with only a factor of 2 overhead, i.e., the Lanczos iteration is performed twice.
22.Because of the factor in the function used to construct the log-contour plots, all contours (no matter which are selected) are drawn next to nodal lines whatever the value of in the region.
23.Note that the DVR wave function data are interpolated via Eq. (2) so the wave functions defined by different grids can be compared at the same points.
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26.We performed computations for T-shaped with ℏ set below 1, and with ℏ set above 1, to see if there is a regime between the quantum and classical limits where the envelope of the density of states is not well described by the Weyl formula. Such a regime was not found.
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