Abstract Recent work has shown a deep connection between semilocal approximations in density functional theory and the asymptotics of the sum of the Wentzel–Kramers–Brillouin (WKB) semiclassical expansion for the eigenvalues. However, all examples studied to date have potentials with only real classical turning points. But systems with complex turning points generate subdominant (SD) terms beyond those in the WKB series. The simplest case is a pure quartic oscillator. We show how to generalize the asymptotics of eigenvalue sums to include SD contributions to the sums, if they are known for the eigenvalues. These corrections to WKB greatly improve accuracy for eigenvalue sums, especially for many levels. We obtain further improvements to the sums through hyperasymptotics. For the lowest level, our summation method has error below 2 × 10 −4 . For the sum of the lowest ten levels, our error is less than 10 −22 . We report all results to many digits and include copious details of the asymptotic expansions and their derivation.
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Calculation and interpretation of classical turning surfaces in solids
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.
more » « less- Award ID(s):
- 1939528
- PAR ID:
- 10212282
- Publisher / Repository:
- Nature Publishing Group
- Date Published:
- Journal Name:
- npj Computational Materials
- Volume:
- 7
- Issue:
- 1
- ISSN:
- 2057-3960
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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