Abstract.
We investigate regularity properties of molecular one-electron densities ρ near the nuclei. In particular we derive a representation
with an explicit function \({\mathcal{F}}\), only depending on the nuclear charges and the positions of the nuclei, such that \(\mu \in C^{1,1}({{\mathbb{R}}}^{3})\), i.e., μ has locally essentially bounded second derivatives. An example constructed using Hydrogenic eigenfunctions shows that this regularity result is sharp. For atomic eigenfunctions which are either even or odd with respect to inversion in the origin, we prove that μ is even \(C^{2,\alpha}({{\mathbb{R}}}^{3})\) for all \(\alpha \in (0,1)\). Placing one nucleus at the origin we study ρ in polar coordinates x = rw and investigate \(\frac{\partial}{\partial r}\rho(r, w)\) and \(\frac{\partial^{2}}{\partial r^{2}}\rho(r,w)\) for fixed w as r tends to zero. We prove non-isotropic cusp conditions of first and second order, which generalize Kato’s classical result.
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Communicated by Claude Alain Pillet.
Submitted: March 13, 2006. Accepted: October 5, 2006.
© 2007 by the authors. This article may be reproduced in its entirety for non-commercial purposes.
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Fournais, S., Sørensen, T.Ø., Hoffmann-Ostenhof, M. et al. Non-Isotropic Cusp Conditions and Regularity of the Electron Density of Molecules at the Nuclei. Ann. Henri Poincaré 8, 731–748 (2007). https://doi.org/10.1007/s00023-006-0320-1
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DOI: https://doi.org/10.1007/s00023-006-0320-1