Capturing the electron–electron cusp with the coupling-constant averaged exchange–correlation hole: A case study for Hooke’s atoms

Hou, Lin; Irons, Tom J. P.; Wang, Yanyong; Furness, James W.; Wibowo-Teale, Andrew M.; Sun, Jianwei

Abstract

In density-functional theory, the exchange–correlation (XC) energy can be defined exactly through the coupling-constant (λ) averaged XC hole n¯xc(r, r′), representing the probability depletion of finding an electron at r′ due to an electron at r. Accurate knowledge of n¯xc(r, r′) has been crucial for developing XC energy density-functional approximations and understanding their performance for molecules and materials. However, there are very few systems for which accurate XC holes have been calculated since this requires evaluating the one- and two-particle reduced density matrices for a reference wave function over a range of λ while the electron density remains fixed at the physical (λ = 1) density. Although the coupled-cluster singles and doubles (CCSD) method can yield exact results for a two electron system in the complete basis set limit, it cannot capture the electron–electron cusp using finite basis sets. Focusing on Hooke’s atom as a two-electron model system for which certain analytic solutions are known, we examine the effect of this cusp error on the XC hole calculated using CCSD. The Lieb functional is calculated at a range of coupling constants to determine the λ-integrated XC hole. Our results indicate that, for Hooke’s atoms, the error introduced by the description of the electron–electron cusp using Gaussian basis sets at the CCSD level is negligible compared to the basis set incompleteness error. The system-, angle-, and coupling-constant averaged XC holes are also calculated and provide a benchmark against which the Perdew–Burke–Ernzerhof and local density approximation XC hole models are assessed.

Citation

Hou, L., Irons, T. J. P., Wang, Y., Furness, J. W., Wibowo-Teale, A. M., & Sun, J. (2024). Capturing the electron–electron cusp with the coupling-constant averaged exchange–correlation hole: A case study for Hooke’s atoms. Journal of Chemical Physics, 160(1), Article 014103. https://doi.org/10.1063/5.0173370

• Journal Article Type: Article
• Acceptance Date: Nov 15, 2023
• Online Publication Date: Jan 5, 2024
• Publication Date: Jan 7, 2024
• Deposit Date: Jan 23, 2024
• Publicly Available Date: Jan 24, 2024
• Journal: The Journal of Chemical Physics
• Print ISSN: 0021-9606
• Electronic ISSN: 1089-7690
• Publisher: American Institute of Physics
• Peer Reviewed: Peer Reviewed
• Volume: 160
• Issue: 1
• Article Number: 014103
• DOI: https://doi.org/10.1063/5.0173370
• Keywords: Coupled-cluster methods, Density functional theory, Exchange interactions, Local density approximations, Correlation energy, Basis sets, Coupling constants, Density-matrix
• Public URL: https://nottingham-repository.worktribe.com/output/29272344
• Publisher URL: https://pubs.aip.org/aip/jcp/article/160/1/014103/2932466/Capturing-the-electron-electron-cusp-with-the