Algorithms for the electronic and vibrational properties of nanocrystals

Abstract

Solving the electronic structure problem for nanoscale systems remains a computationallychallenging problem. The numerous degrees of freedom, both electronic and nuclear, makethe problem impossible to solve without some effective approximations. Here we illustratesome advances in algorithm developments to solve the Kohn–Sham eigenvalue problem, i.e.we solve the electronic structure problem within density functional theory usingpseudopotentials expressed in real space. Our algorithms are based on a nonlinearChebyshev filtered subspace iteration method, which avoids computing explicit eigenvectorsexcept at the first self-consistent-field iteration. Our method may be viewed as an approachto solve the original nonlinear Kohn–Sham equation by a nonlinear subspace iterationtechnique, without emphasizing the intermediate linearized Kohn–Sham eigenvalueproblems. Replacing the standard iterative diagonalization at each self-consistent-fielditeration by a Chebyshev subspace filtering step results in a significant speed-up, often anorder of magnitude or more, over methods based on standard diagonalization. We illustratethis method by predicting the electronic and vibrational states for silicon nanocrystals.