Correlations in sequences of generalized eigenproblems arising in Density Functional Theory

Density Functional Theory (DFT) is one of the most used ab initio theoretical frameworks in materials science. It derives the ground state properties of multi-atomic ensembles directly from the computation of their one-particle density n(r). In DFT-based simulations the solution is calculated through a chain of successive self-consistent cycles; in each cycle a series of coupled equations (Kohn-Sham) translates to a large number of generalized eigenvalue problems whose eigenpairs are the principal means for expressing n(r). A simulation ends when n(r) has converged to the solution within the required numerical accuracy. This usually happens after several cycles, resulting in a process calling for the solution of many sequences of eigenproblems. In this paper, the authors report evidence showing unexpected correlations between adjacent eigenproblems within each sequence and suggest the investigation of an alternative computational approach: information extracted from the simulation at one step of the sequence is used to compute the solution at the next step. The implications are multiple: from increasing the performance of material simulations, to the development of a mixed direct-iterative solver, to modifying the mathematical foundations of the DFT computational paradigm in use, thus opening the way to the investigation of new materials.