Pseudopotentials on grids: Application to the electronic, optical, and vibrational properties of silicon nanocrystals

Solving for the quantum properties of nano-scale systems is a "grand challenge" problem. The size of a nanocrystal or nanowire can exceed thousands of atoms; however, such systems can be very different than a macroscopic one and require an explicit "quantum mechanical" solution. Until recently there were virtually no methods for describing these systems with the same accuracy that one would expect for small molecules or clusters. Here we outline a method that can be applied to systems of this size and illustrate some representative applications. Our approach capitalizes on several algorithmic and conceptual advances. The key physical ingredients include pseudopotentials and density functional theory. The pseudopotential approximation allows us to set the length and energy scales to those of the valence electron states, considerably simplifying the representation of the wave functions, while density functional theory allows us to map the all electron problem onto an equivalent "one-electron" problem, i.e., the Kohn-Sham problem. This "pseudopotential-density functional theory" combination has become the "standard" for large scale electronic structure problems. However, this powerful combination cannot be applied to nanoscale systems containing thousands of atoms without improved algorithms. We will present several algorithmic advances that are centered on a real-space or grid solution of the Kohn-Sham equation. These advances, including subspace filtering methods, allow one to handle systems with thousands of atoms. We will illustrate this approach to silicon nanocrystals by predicting the role of quantum confinement on the electronic, optical and vibrational properties for this technologically important system.