Covalent molecules do not have forbidden regions between nuclei, but do in other directions. First, this shows that such systems differ fundamentally from their molecular cousins. The next most important feature is surprising: at equilibrium, covalent semiconductors also do not exhibit any CFRs. This is consistent with the original views of Kohn and Sham when developing DFT 1, and the subsequent use of gradient expansions starting from slowly varying densities. The single most important feature is that bulk metals at equilibrium have no classically forbidden regions (CFRs). We calculated many solids as a function of lattice parameter using the local spin density approximation (LSDA) and the PerdewвЂ“BurkeвЂ“Ernzerhof (PBE) generalized gradient approximation (GGA) 3, tracking the fraction of volume that is classically forbidden. Thus we might naively expect semiconductors with covalent bonds to have classical turning surfaces. Semiclassical reasoning suggests that solids that are entirely classically allowed should be metallic, while a significant volume of forbidden regions should accompany greater inhibition of conduction, i.e., a nonzero gap. 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. Physical intuition suggests that, in the limit of extreme compression, solids become metallic, while, in the less-physical limit of extreme expansion, they become insulators. But of greater interest is the nature of the turning surface itself in solids. We first find that equilibrium distances between nearest-neighbor atoms or atomic ions in solids can be estimated from the sum of the classical turning radii of the corresponding free atoms or atomic ions and the bond type (metallic, covalent, ionic, or van der Waals), much as found earlier for molecules. Here we augment existing codes to calculate classical turning surfaces for a variety of extended solids. Thus, unlike molecules, solids have the possibility of having no classical turning surfaces at all. But all finite systems have densities that decay to zero far from the nuclei, whereas the interiors of real solids extended over three dimensions do not. These turning surfaces are roughly ellipsoidal for covalent bonds, have seams for ionic bonds, and are bifurcated for van der Waals dimers. Recently, the classical turning surfaces at the energy of the highest occupied orbital of atoms and ions have provided considerable insight into the nature of chemical bonding 2. Modern KohnвЂ“Sham (KS) density functional theory (DFT) 1 calculations produce a KS potential, v s( r), which, while not a physical observable, has proven useful in providing physical and chemical insight. 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. Nonetheless, a strong correlation with insulating behavior can still be inferred. Surprisingly, many semiconductors have no CFR at equilibrium, a key finding for density functional construction. 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. Calculations with several approximate density functionals and codes confirm these behaviors. They can also appear at a monovacancy in a metal.

In all materials, CFRs appear or grow as the internuclear distances are uniformly expanded. 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. They are useful for understanding many chemical properties of molecules but need not exist in solids, where the density never decays to zero. Classical turning surfaces of KohnвЂ“Sham potentials separate classically allowed regions (CARs) from classically forbidden regions (CFRs).

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