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 ρ near the nuclei. In particular we derive a representationwith an explicit function F, only depending on the nuclear charges and the positions of the nuclei, such that μ ∈ C 1,1 (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,α (R 3 ) for all α ∈ (0, 1). Placing one nucleus at the origin we study ρ in polar coordinates x = rω and investigate ∂ ∂r ρ(r, ω) and ∂ 2 ∂r 2 ρ(r, ω) for fixed ω as r tends to zero. We prove non-isotropic cusp conditions of first and second order, which generalize Kato's classical result.

Third Derivative of the One-Electron Density at the Nucleus

“…A general (optimal) structure-result was obtained recently [2]. For more detailed information, two possible approaches are to study limits when approaching a nucleus under a fixed angle ω ∈ S 2 , as was done in [2], and to study the spherical average of ρ (here denoted ρ ), which is mostly interesting for atoms. The existence of ρ (0), the first derivative of ρ at the nucleus, and the identity ρ (0) = −Z ρ(0) (see (1.12) below) follow immediately from Kato's classical result [12] on the 'Cusp Condition' for the associated eigenfunction (see also [10,15]).…”

“…That ρ itself is not analytic at the positions of the nuclei is already clear for the groundstate of 'Hydrogenic atoms' (N = 1); in this case, ∇ρ is not even continuous at x = 0. However, as was proved in [2], e Z|x| ρ ∈ C 1,1 (R 3 ).…”

A discontinuous Galerkin scheme for full-potential electronic structure calculations

“…We also assume throughout this paper that the potential V equals to −Z/|r| in the neighborhood of 0, and belongs to C ∞ loc (R 3 \ R) ∩ L 2 # (Ω). It was shown in [23,24,25] that the exact electron densities are analytic away from the nuclei and satisfy certain cusp conditions at the nuclei. The plane wave approximations can not have as good convergence rate as (2.1) due to the cusps at the nuclear positions.…”

“…For eigenvalue problems with singular potentials in full-potential calculations, plane waves are inefficient bases for describing the cusps at the nuclei positions [23,24,25,28]. In contrast, it is observed that a significant part of the rapid oscillations can be captured by atomic orbitals such as Gaussians and Slater-type orbitals [27,34], which have been widely used in quantum chemistry (we refer to [7,18] for their numerical analysis).…”

Exponential Convergence of Simplicial h p-FEM for H 1-Functions with Isotropic Singularities

“…We mention only nonlinear Schrödinger equations with selffocusing, density functional models in electron structure calculations (eg. [8,2,4] and the references there), nonlinear parabolic PDEs with critical growth (eg. [15] and the references there, or continuum models of crystalline solids with isolated point defects (eg.…”