Sums of the
This content will become publicly available on June 1, 2023
 Award ID(s):
 1919571
 Publication Date:
 NSFPAR ID:
 10340464
 Journal Name:
 AIP Advances
 Volume:
 12
 Issue:
 6
 Page Range or eLocationID:
 065012
 ISSN:
 21583226
 Sponsoring Org:
 National Science Foundation
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Abstract N lowest energy levels for quantum particles bound by potentials are calculated, emphasising the semiclassical regimeN ≫ 1. EulerMaclaurin summation, together with a regularisation, gives a formula for these energy sums, involving only the levelsN + 1,N + 2…. For the harmonic oscillator and the particle in a box, the formula is exact. For wells where the levels are known approximately (e.g. as a WKB series), with the higher levels being more accurate, the formula improves accuracy by avoiding the lower levels. For a linear potential, the formula gives the first Airy zero with an error of order 10^{−7}. For the Pöschl–Teller potential, regularisation is not immediately applicable but the energy sum can be calculated exactly; its semiclassical approximation depends on howN and the well depth are linked. In more dimensions, the Euler–Maclaurin technique is applied to give an analytical formula for the energy sum for a free particle on a torus, using levels determined by the smoothed spectral staircase plus some oscillatory corrections from short periodic orbits. 
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 widegap 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.

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We report vibrational spectra of the H 2 tagged, cryogenically cooled X − · HOCl (X = Cl, Br, and I) ion–molecule complexes and analyze the resulting band patterns with electronic structure calculations and an anharmonic theoretical treatment of nuclear motions on extended potential energy surfaces. The complexes are formed by “ligand exchange” reactions of X − · (H 2 O) n clusters with HOCl molecules at low pressure (∼10 −2 mbar) in a radio frequency ion guide. The spectra generally feature many bands in addition to the fundamentals expected at the double harmonic level. These “extra bands” appear in patterns that are similar to those displayed by the X − · HOD analogs, where they are assigned to excitations of nominally IR forbidden overtones and combination bands. The interactions driving these features include mechanical and electronic anharmonicities. Particularly intense bands are observed for the v = 0 → 2 transitions of the outofplane bending soft modes of the HOCl molecule relative to the ions. These involve displacements that act to break the strong Hbond to the ion, which give rise to large quadratic dependences of the electric dipoles (electronic anharmonicities) that drive the transition moments for the overtone bands. On the othermore »

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