Numerical integration of intracule pair densities with optimized multicenter grids
Abstract
Intracule pair densities provide insight into electron correlation, but their routine analysis in extended systems is limited by the cost of numerical integration and the restriction of analytical methods to Gaussian basis functions. We present an efficient multicenter integration scheme for intracule densities and their moments. By adapting the topological fuzzy Voronoi cell formalism, we partition intracule space around interatomic displacement vectors, concentrating quadrature points near secondary density maxima that standard single-center grids poorly resolve. Benchmarks on linear alkanes show improved convergence with increasing molecular size, achieving relative errors below 0.02% in electron-electron repulsion energies with fewer quadrature points than single-center methods. The partitioning also enables the decomposition of global two-electron properties into spatial contributions associated with short- and long-range electron-pair separations. Coupling the multicenter grid with kernel density estimation reconstructs smooth radial intracule distributions at large interelectronic distances without the dense angular grids required by conventional surface integration. Validation on the S22 dataset supports the reliability of the method across diverse molecular systems. These developments facilitate intracule analysis in polyatomic systems and the investigation of electron correlation and dispersion interactions.
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BibTeX
@misc{ylla2026numerical,
title = {{Numerical integration of intracule pair densities with optimized multicenter grids}},
author = {Markel Ylla and Jesus M. Ugalde and Eduard Matito and Eloy Ramos-Cordoba},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.01598},
eprint = {2610.01598},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.01598},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Numerical integration of intracule pair densities with optimized multicenter grids
AU - Markel Ylla
AU - Jesus M. Ugalde
AU - Eduard Matito
AU - Eloy Ramos-Cordoba
PY - 2026
DA - 2026/10/01
PB - arXiv
DO - 10.48550/arxiv.2610.01598
UR - https://arxiv.org/abs/2610.01598
AB - Intracule pair densities provide insight into electron correlation, but their routine analysis in extended systems is limited by the cost of numerical integration and the restriction of analytical methods to Gaussian basis functions. We present an efficient multicenter integration scheme for intracule densities and their moments. By adapting the topological fuzzy Voronoi cell formalism, we partition intracule space around interatomic displacement vectors, concentrating quadrature points near secondary density maxima that standard single-center grids poorly resolve. Benchmarks on linear alkanes show improved convergence with increasing molecular size, achieving relative errors below 0.02% in electron-electron repulsion energies with fewer quadrature points than single-center methods. The partitioning also enables the decomposition of global two-electron properties into spatial contributions associated with short- and long-range electron-pair separations. Coupling the multicenter grid with kernel density estimation reconstructs smooth radial intracule distributions at large interelectronic distances without the dense angular grids required by conventional surface integration. Validation on the S22 dataset supports the reliability of the method across diverse molecular systems. These developments facilitate intracule analysis in polyatomic systems and the investigation of electron correlation and dispersion interactions.
ER -