A basis-set-free reference for the exact exchange-energy density and functional development
Abstract
The exact local exchange-energy density, obtained from the Slater potential of the Hartree-Fock exchange hole, is a central ingredient of Becke's B05 and B13 functionals and of many other local hybrids. It is usually evaluated in a Gaussian basis, so the local quantity retains a pointwise basis-set incompleteness error. We describe here a basis-set-free numerical code that solves the Kohn-Sham problem on real-space grids for atoms, diatomics, and polyatomics, and returns the exact local exchange-energy density together with the density, its derivatives, the kinetic-energy density, and the reverse Becke-Roussel hole quantities. Reconstructed from the published record of Becke's non-public NUMOL code, AXEL reproduces numerical atomic references through krypton, scalar-relativistic uranium orbital energies, and Becke's diatomic spectroscopic constants. We then extend the original local-density numerical framework to generalized-gradient and hybrid functionals. Comparisons with the independent adaptive-multiwavelet code MRChem give sub-millihartree agreement for Hartree-Fock and PBE0. The same calculations agree with large Gaussian-basis references to a similar level of accuracy. Atom-centred eXact Electronic Limit (AXEL) is intended as both a basis-set-free numerical reference and a platform for exchange-hole-based density-functional development.
Related
- Analytic Energy Gradients for Unrestricted Mixed-Reference Spin-Flip Time-Dependent Density Functional Theory
- The Evolution of Electronic Structure in Conjugated Molecules Revealed by Mixed-Reference Spin-Flip Time-Dependent Density Functional Theory and Density-Based Analyses: II. Polyacenes
- ROSE: Deep Learning exchange-correlation functional from reduced density-matrix representations
- Machine-learned Laplacian-level density functional from exact exchange-correlation potentials and energies
- Is there a density-gradient expansion for the electronic exchange energy?
BibTeX
@misc{price2026basissetfree,
title = {{A basis-set-free reference for the exact exchange-energy density and functional development}},
author = {Alastair J. A. Price},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.19031},
eprint = {2610.19031},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.19031},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - A basis-set-free reference for the exact exchange-energy density and functional development
AU - Alastair J. A. Price
PY - 2026
DA - 2026/10/13
PB - arXiv
DO - 10.48550/arxiv.2610.19031
UR - https://arxiv.org/abs/2610.19031
AB - The exact local exchange-energy density, obtained from the Slater potential of the Hartree-Fock exchange hole, is a central ingredient of Becke's B05 and B13 functionals and of many other local hybrids. It is usually evaluated in a Gaussian basis, so the local quantity retains a pointwise basis-set incompleteness error. We describe here a basis-set-free numerical code that solves the Kohn-Sham problem on real-space grids for atoms, diatomics, and polyatomics, and returns the exact local exchange-energy density together with the density, its derivatives, the kinetic-energy density, and the reverse Becke-Roussel hole quantities. Reconstructed from the published record of Becke's non-public NUMOL code, AXEL reproduces numerical atomic references through krypton, scalar-relativistic uranium orbital energies, and Becke's diatomic spectroscopic constants. We then extend the original local-density numerical framework to generalized-gradient and hybrid functionals. Comparisons with the independent adaptive-multiwavelet code MRChem give sub-millihartree agreement for Hartree-Fock and PBE0. The same calculations agree with large Gaussian-basis references to a similar level of accuracy. Atom-centred eXact Electronic Limit (AXEL) is intended as both a basis-set-free numerical reference and a platform for exchange-hole-based density-functional development.
ER -