Efficient iterative method for solving the Dirac–Kohn–Sham density functional theory

We present for the first time an efficient iterative method to directly solve the four-component Dirac–Kohn–Sham (DKS) density functional theory. Due to the existence of the negative energy continuum in the DKS operator, the existing iterative techniques for solving the Kohn–Sham systems cannot be efficiently applied to solve the DKS systems. The key component of our method is a novel filtering step (F) which acts as a preconditioner in the framework of the locally optimal block preconditioned conjugate gradient (LOBPCG) method. The resulting method, dubbed the LOBPCG-F method, is able to compute the desired eigenvalues and eigenvectors in the positive energy band without computing any state in the negative energy band. The LOBPCG-F method introduces mild extra cost compared to the standard LOBPCG method and can be easily implemented. We demonstrate our method in the pseudopotential framework with a planewave basis set which naturally satisfies the kinetic balance prescription. Numerical results for Pt2, Au2, TlF, and Bi2Se3 indicate that the LOBPCG-F method is a robust and efficient method for investigating the relativistic effect in systems containing heavy elements.