Nonspectral Methods for Solving the Schrödinger Equation for Electronic and Vibrational Problems

To compute molecular and material properties, one must solve the electronic and/or vibrational Schrödinger equation. This Perspective briefly presents nonspectral (pseudospectral and grid) methods to solve the Schrödinger equation and their recent and promising applications to vibrational and electronic problems. Pseudospectral and grid methods facilitate parallelization, which is of critical importance for modern material modeling applications. By obviating the need to choose basis functions for which exact integrals are possible, they also enable one to use smaller basis sets. For the vibrational problem, pseudospectral and grid methods, in principle, make it possible to compute a vibrational spectrum without a potential surface, simply from values of the potential at points. This is possible if optimized basis functions are used. Iterative eigensolvers are very advantageous when simpler basis functions are used because matrix–vector products can be evaluated efficiently.