Abstract
We establish a pointwise stability estimate for the Thomas–Fermi–von Weiz-säcker (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.
Highlights
Locality properties of electronic structure models are a key premise in certain state of the art numerical algorithms
A stronger notion is the locality of the mechanical response, which is a fundamental premise underpinning the construction of interatomic potentials and of multi-scale algorithms such as hybrid quantum mechanics/molecular mechanics (QM/MM) schemes [17]
The two main results of this work, Theorems 3.1 and 3.2, are stability and exponential locality estimates for the Thomas–Fermi–von Weizsäcker (TFW) model, which apply to general condensed nuclear configurations
Summary
Locality properties of electronic structure models are a key premise in certain state of the art numerical algorithms. A stronger notion is the locality of the mechanical response, which is a fundamental premise underpinning the construction of interatomic potentials and of multi-scale algorithms such as hybrid QM/MM schemes [17] (here, it is termed “strong locality”). This latter category of locality is less well studied, the only result in this direction being the locality of non-selfconsistent tight binding models [14]. Our main technical result to achieve this is the following pointwise stability estimate for the TFW equations, which establishes the locality of the electron response to changes in the nuclear configuration. Under different conditions, are given in Theorems 3.1 and 3.2
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