Analysis of the divide-and-conquer method for electronic structure calculations

TL;DR

This paper analyzes the accuracy of the divide-and-conquer method in electronic structure calculations, demonstrating exponential decay of density differences under certain conditions and highlighting the importance of the gap assumption.

Contribution

It provides a rigorous proof of the exponential decay of density differences and emphasizes the critical role of the gap assumption for the method's accuracy.

Findings

Density difference decays exponentially with distance from boundary

Gap assumption is essential for accuracy

Numerical examples show loss of accuracy when gap assumption fails

Abstract

We study the accuracy of the divide-and-conquer method for electronic structure calculations. The analysis is conducted for a prototypical subdomain problem in the method. We prove that the pointwise difference between electron densities of the global system and the subsystem decays exponentially as a function of the distance away from the boundary of the subsystem, under the gap assumption of both the global system and the subsystem. We show that gap assumption is crucial for the accuracy of the divide-and-conquer method by numerical examples. In particular, we show examples with the loss of accuracy when the gap assumption of the subsystem is invalid.