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. This not only provides a detailed description of anharmonic effects but also an accurate inclusion of the coupling between the investigated vibrational states on a quantum mechanical level. All results obtained for the constructed normal modes are benchmarked against their analytical counterparts obtained from the diagonalization of the total Hessian of the entire system. Three increasingly complex systems treated at quantum chemical level of theory have been considered, namely the symmetric and asymmetric stretch vibrations of an isolated water molecule, hydroxyl groups bound to the surface of GeO2 (001), α-quartz(001) and Rutil (001) as well as crystalline Li2 NH serving as an example for a bulk material. While the data obtained for the individual systems verify the applicability of the proposed methodology, comparison to experimental data demonstrates the accuracy of this methodology despite the restriction to limit this methodology to a few selected vibrational modes. The possibility to investigate vibrational phenomena of localized normal modes without the requirement of executing costly harmonic frequency calculations of the entire system enables the application of this method to cases in which the determination of normal modes is prohibitively expensive or not available for a particular level of theory. © 2018 Wiley Periodicals, Inc.

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. The performance of the adapted method is demonstrated using exemplary harmonic and Morse problems in one and two dimensions. Furthermore, the vibrational frequencies of molecular hydrogen and water are calculated, inherently considering the influence of anharmonicity, mode-mode coupling and nuclear quantum effects. The estimation of the tunneling splitting in malonaldehyde serves as an example for a two-dimensional problem.