Probing vibrational coupling via a grid-based quantum approach-an efficient strategy for accurate calculations of localized normal modes in solid-state systems.

In this work an approach to investigate the properties of strongly localized vibrational modes of functional groups in bulk material and on solid-state surfaces is presented. The associated normal mode vectors are approximated solely on the basis of structural information and obtained via diagonalization of a reduced Hessian. The grid-based Numerov procedure in one and two dimensions is then applied to an adequate scan of the respective potential surface yielding the associated vibrational wave functions and energy eigenvalues.

Pushing the limit for the grid-based treatment of Schrödinger's equation: a sparse Numerov approach for one, two and three dimensional quantum problems.

The general Numerov method employed to numerically solve ordinary differential equations of second order was adapted with a special focus on solving Schrödinger's equation. By formulating a hierarchy of novel stencil expressions for the numerical treatment of the Laplace operator in one, two and three dimensions the method could not only be simplified over the standard Numerov scheme. The improved framework enables the natural use of matrix sparsity to reduce the memory demand and the associated computing time, thus enabling the application of the method to larger problems.