An overlap fitted chain of spheres exchange method

TLDR

The chain‑of‑spheres (COS) algorithm in the RIJCOSX SCF procedure approximates the exchange term by analytically integrating over one electron coordinate while numerically integrating over the other coordinates. This study seeks to improve COS efficiency by reducing numerical errors. By constructing a fitting matrix Q that aligns the numerically and analytically evaluated overlap matrices, the authors test various grid sizes and compare results to fully analytic and earlier COS approximations. The fitted‑grid approach achieves chemical accuracy with about 30 % faster single‑point energies, improves total and reaction energies, and yields geometries within the inherent method error, though with slightly lower accuracy than the original COS.

Abstract

The “chain of spheres” (COS) algorithm, as part of the RIJCOSX SCF procedure, approximates the exchange term by performing analytic integration with respect to the coordinates of only one of the two electrons, whereas for the remaining coordinates, integration is carried out numerically. In the present work, we attempt to enhance the efficiency of the method by minimizing numerical errors in the COS procedure. The main idea is based on the work of Friesner and consists of finding a fitting matrix, \documentclass[12pt]{minimal}\begin{document}$\mathbf {Q}$\end{document}Q, which leads the numerical and analytically evaluated overlap matrices to coincide. Using \documentclass[12pt]{minimal}\begin{document}$\mathbf {Q}$\end{document}Q, the evaluation of exchange integrals can indeed be improved. Improved results and timings are obtained with the present default grid setup for both single point calculations and geometry optimizations. The fitting procedure results in a reduction of grid sizes necessary for achieving chemical accuracy. We demonstrate this by testing a number of grids and comparing results to the fully analytic and the earlier COS approximations. This turns out to be favourable for total and reaction energies, for which chemical accuracy can now be reached with a corresponding ∼30% speedup over the original RIJCOSX procedure for single point energies. Results are slightly less favourable for the accuracy of geometry optimizations, but the procedure is still shown to yield geometries with errors well below the method inherent errors of the employed theoretical framework.