Non-isotropic cusp conditions and regularity of the electron density of molecules at the nuclei

Abstract

We investigate regularity properties of molecular one-electron densities rho near the nuclei. In particular we derive a representation rho(x)=mu(x)*(e^F(x)) with an explicit function F, only depending on the nuclear charges and the positions of the nuclei, such that mu belongs to C^{1,1}(R^3), i.e., mu 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 mu is even C^{2,α}(R^3) for all alpha in (0,1). Placing one nucleus at the origin we study rho in polar coordinates x=r*omega and investigate rho'(r,omega) and rho''(r,omega) for fixed omega as r tends to zero. We prove non-isotropic cusp conditions of first and second order, which generalize Kato's classical result.

19 pages

References in corpus (2)

• Sharp regularity results for many-electron wave functions
• The electron density is smooth away from the nuclei

Cited by in corpus (6)

• Wavelet Scattering Regression of Quantum Chemical Energies
• Analytic structure of many-body Coulombic wave functions
• A new proof of the analyticity of the electronic density of molecules
• Analytic structure of solutions to multiconfiguration equations
• A discontinuous Galerkin scheme for full-potential electronic structure calculations
• Upper bounds of local electronic densities in molecules