Correlation energy of two electrons in a ball

Abstract

We study the ground-state correlation energy $E_{\mathrm{c}}$ of two electrons of opposite spin confined within a $D$-dimensional ball ($D\ge 2$) of radius $R$. In the high-density regime, we report accurate results for the exact and restricted Hartree-Fock energy, using a Hylleraas-type expansion for the former and a simple polynomial basis set for the latter. By investigating the exact limiting correlation energy $E_{\rm c}^{(0)} = \lim_{R \to 0} \Ec$ for various values of $D$, we test our recent conjecture [J. Chem. Phys. {\bf 131} (2009) 241101] that, in the large-$D$ limit, $E_{\mathrm{c}}^{(0)}\sim -{\delta }^2/8$ for any spherically-symmetric confining external potential, where $\delta =1/(D-1)$.

6 pages, 2 figures

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• Correlation energy of two electrons in the high-density limit
• Hooke's law correlation in two-electron systems
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• Electron correlation energy in confined two-electron systems
• Invariance of the correlation energy at high density and large dimension in two-electron systems
• A Tale of Two Electrons: Correlation at High Density
• Two-electron atom with a screened interaction
• Nodal surfaces and interdimensional degeneracies
• Correlation energy of anisotropic quantum dots
• Symmetry-broken local-density approximation for one-dimensional systems
• Distribution of $r_{12}\cdot p_{12}$ in quantum systems
• Contactium: A strongly correlated model system