Correlations in sequences of generalized eigenproblems arising in Density Functional Theory

TL;DR

This paper uncovers unexpected correlations between consecutive eigenproblems in Density Functional Theory simulations, revealing patterns in eigenvector evolution and Hamiltonian matrices that could enhance computational efficiency and inspire new theoretical approaches.

Contribution

It demonstrates the existence of correlations between successive eigenproblems in DFT sequences and analyzes their numerical properties, suggesting potential improvements in simulation methods.

Findings

Eigenvectors undergo an evolution process across sequences

Hamiltonian matrices exhibit specific informational patterns

Correlations can be exploited to improve computational efficiency

Abstract

Density Functional Theory (DFT) is one of the most used ab initio theoretical frameworks in materials science. It derives the ground state properties of a multi-atomic ensemble directly from the computation of its one-particle density \nr .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 \nr. A simulation ends when \nr\ 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. By investigating the numerical properties of the sequences of generalized eigenproblems it is shown that the eigenvectors undergo an "evolution" process. At the same time it is shown that the Hamiltonian matrices exhibit a similar evolution and manifest a specific pattern in the information they carry. Correlation between eigenproblems within a sequence is of capital importance: information extracted from the simulation at one step of the sequence could be used to compute the solution at the next step. Although they are not explored in this work, the implications could be manifold: from increasing the performance of material simulations, to the development of an improved iterative solver, to modifying the mathematical foundations of the DFT computational paradigm in use, thus opening the way to the investigation of new materials.