A fast configuration space method for solving local Kohn–Sham equations

Abstract

This paper describes algorithmic components of a program package for solving the Kohn–Sham equations in coordinate space. The primary application is envisioned to be in the area of quantum dots and metallic clusters, where basis function expansions are difficult to apply. A number of techniques are introduced for solving different aspects of the problem: the eigenvalue problem is solved by using a fourth order factorization of the evolution operator e − ϵH , which is significantly more efficient than the conventional second order factorization. The wave functions are orthogonalized using a subspace orthogonalization procedure generating a noticeably faster convergence than the conventional Gram–Schmidt orthogonalization method. The self-consistency problem arising from the non-linearity of the Kohn–Sham equations is solved with a Newton–Raphson procedure, which is shown to be equivalent to linear response theory. A collective approximation for the static density–density response function, rooted in Feynman’s theory of excitations in quantum fluids, leads to a version of the algorithm that maintains the convergence properties and only needs the computation of occupied states of the system. This approximation makes the implementation of the algorithm in three dimensions practical. Converged solutions are obtained from very rough guesses within a view iterations. The method is applied to two-dimensional quantum dots with random impurities and to metal clusters. We report statistical properties of ionization and excitation energies of spin-polarized quantum dots and optimized structural properties of small Na clusters.