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In density functional theory the exchange-correlation (XC) energy functionalcan be defined exactly through the coupling-constant ($\lambda$) averaged XChole $\bar{n}_\text{xc}(\mathbf{r},\mathbf{r}')$, representing the probabilitydepletion of finding an electron at $\mathbf{r}$ due to an electron at$\mathbf{r}$. Accurate knowledge of $\bar{n}_\text{xc}(\mathbf{r},\mathbf{r}')$has been crucial for developing various XC energy density functionalapproximations and understanding their performance for real molecules andmaterials. However, there are very few systems for which accurate XC holes havebeen calculated, since this requires evaluating the one- and two-particlereduced density matrices for a reference wave function over a range of$\lambda$ whilst the electron density remains fixed at the physical($\lambda=1$) density. Although the coupled-cluster singles and doubles (CCSD)method can yield exact results for a two-electron system in the complete basisset limit, it cannot capture the electron-electron cusp with commonly usedfinite basis sets. In this study, focusing on the Hooke's atom as atwo-electron model system for which certain analytic solutions are known, weexamine the effect of this cusp error on the XC hole calculated using CCSD. TheLieb functional is calculated at a range of coupling constants to determine the$\lambda$-integrated XC hole. Our results indicate that, for the Hooke's atoms,the error introduced by the description of the electron-electron cusp usingGaussian basis sets at the CCSD level is negligible compared to the basis setincompleteness error. The system-, angle- and coupling-constant-averaged XChole is calculated using the same approach and provides a benchmark againstwhich the Perdew-Burke-Ernzerhof (PBE) and local density approximation (LDA) XChole models are assessed.

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