Calculation and interpretation of classical turning surfaces in solids

Aaron D Kaplan,John P Perdew,Kieron Burke,Stewart J Clark

doi:10.1038/s41524-020-00479-0

Aaron D Kaplan, John P Perdew + Show 2 more

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https://doi.org/10.1038/s41524-020-00479-0

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Abstract

Classical turning surfaces of Kohn–Sham potentials separate classically allowed regions (CARs) from classically forbidden regions (CFRs). They are useful for understanding many chemical properties of molecules but need not exist in solids, where the density never decays to zero. At equilibrium geometries, we find that CFRs are absent in perfect metals, rare in covalent semiconductors at equilibrium, but common in ionic and molecular crystals. In all materials, CFRs appear or grow as the internuclear distances are uniformly expanded. They can also appear at a monovacancy in a metal. Calculations with several approximate density functionals and codes confirm these behaviors. A classical picture of conduction suggests that CARs should be connected in metals, and disconnected in wide-gap insulators, and is confirmed in the limits of extreme compression and expansion. Surprisingly, many semiconductors have no CFR at equilibrium, a key finding for density functional construction. Nonetheless, a strong correlation with insulating behavior can still be inferred. Moreover, equilibrium bond lengths for all cases can be estimated from the bond type and the sum of the classical turning radii of the free atoms or ions.

Highlights

Modern Kohn–Sham (KS) density functional theory (DFT)[1] calculations produce a KS potential, vs(r), which, while not a physical observable, has proven useful in providing physical and chemical insight
No bulk metal that we studied had a classically forbidden regions (CFRs) at equilibrium, but covalent semiconductors lack CFRs at equilibrium
CFRs emerged in all narrow gap insulators studied here when expanded by 40% of the equilibrium volume

Summary

INTRODUCTION

Modern Kohn–Sham (KS) density functional theory (DFT)[1] calculations produce a KS potential, vs(r), which, while not a physical observable, has proven useful in providing physical and chemical insight. In the limit of extreme compression, solids become metallic, while, in the less-physical limit of extreme expansion, they become insulators In the former case, there are no classical turning surfaces, i.e., all space is classically allowed for the most energetic electrons, while in the latter, all atoms (or ions) are isolated spheres from which classical electrons could not escape. Pronounced structures in the exact KS potential may feature in the low-density interstices of stretched solids, well inside CFRs. Solids, as opposed to atoms and molecules, offer the possibility of metallic bonds, no classical turning surface This work presents calculations of turning surfaces for many simple solids at the LSDA and GGA levels of exchange–correlation approximations Both usually yield close approximations to more precise KS potentials in molecules (as both KS potential and εHO are typically too shallow by about the same amount). Even if a negative hydrostatic pressure could be achieved, extreme expansion of the lattice is unphysical: the work needed to stretch the lattice will eventually exceed the surface formation energy, signaling a transition to isolated clusters

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