Locality of the Thomas–Fermi–von Weizsäcker Equations

Abstract

We establish a pointwise stability estimate for the Thomas-Fermi-von Weizsäcker (TFW) model, which demonstrates that a local perturbation of a nuclear arrangement results also in a local response in the electron density and electrostatic potential. The proof adapts the arguments for existence and uniqueness of solutions to the TFW equations in the thermodynamic limit by Catto et al. (The mathematical theory of thermodynamic limits: Thomas-Fermi type models. Oxford mathematical monographs. The Clarendon Press, Oxford University Press, New York, 1998). To demonstrate the utility of this combined locality and stability result we derive several consequences, including an exponential convergence rate for the thermodynamic limit, partition of total energy into exponentially localised site energies (and consequently, exponential locality of forces), and generalised and strengthened results on the charge neutrality of local defects.

Geometry equilibration of crystalline defects in quantum and atomistic descriptions

“…It is shown in [28] and [27, Proposition 6.3, Theorem 5.11] that this site energy is well-defined and that it satisfies the same locality and homogeneity estimates (4.14) and (4.15) as the tight binding model. Thus, all assumptions (S) are again satisfied (with (S.H) holding for arbitrary s > 0) by the TFW site energy (4.18).…”

Moment Tensor Potentials: A Class of Systematically Improvable Interatomic Potentials

“…where V is called the interatomic potential. This assumption is true in most systems with shortrange interactions (as opposed to, e.g., Coulomb interaction in charged or polarized systems), refer to recent works [11,12,23] for rigorous proofs of this statement for simple QM models. Mathematically, since Dx k can be a tuple of any size (in practice limited by the maximal density of atoms), V can be understood as a family of functions each having a different number of arguments.…”

Thermodynamic Limit of Crystal Defects with Finite Temperature Tight Binding

“…Considering the simultaneous relaxation of nuclei positions is a case of great physical and mathematical interest. First steps in this direction have been taken in [34] for the Thomas-Fermivon Weizsäcker model and in [13] for a tight binding model under the simplifying assumption of a "fixed Fermi level".…”