On the piecewise convex or concave nature of ground state energy as a function of fractional number of electrons for approximate density functionals

Journal Article · Tue Feb 21 00:00:00 EST 2017 · Journal of Chemical Physics

DOI:https://doi.org/10.1063/1.4974988· OSTI ID:1465680

Li, Chen  ^[1]; Yang, Weitao ^[2]

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1. Duke Univ., Durham, NC (United States). Dept. of Chemistry
2. Duke Univ., Durham, NC (United States). Dept. of Chemistry; South China Normal Univ., Guangzhou (China). Key Lab. of Theoretical Chemistry of Environment. School of Chemistry and Environment

In this paper, we provide a rigorous proof that the Hartree Fock energy, as a function of the fractional electron number, E(N), is piecewise concave. Moreover, for semi-local density functionals, we show that the piecewise convexity of the E(N) curve, as stated in the literature, is not generally true for all fractions. By an analysis based on exchange-only local density approximation and careful examination of the E(N) curve, we find for some systems, there exists a very small concave region, corresponding to adding a small fraction of electrons to the integer system, while the remaining E(N) curve is convex. Several numerical examples are provided as verification. Although the E(N) curve is not convex everywhere in these systems, the previous conclusions on the consequence of the delocalization error in the commonly used density functional approximations, in particular, the underestimation of ionization potential, and the overestimation of electron affinity, and other related issues, remain unchanged. Finally, this suggests that instead of using the term convexity, a modified and more rigorous description for the delocalization error is that the E(N) curve lies below the straight line segment across the neighboring integer points for these approximate functionals.

• Research Organization: Energy Frontier Research Centers (EFRC) (United States). Center for Complex Materials from First Principles (CCM); Duke Univ., Durham, NC (United States)
• Sponsoring Organization: USDOE Office of Science (SC), Basic Energy Sciences (BES); National Science Foundation (NSF)
• Grant/Contract Number: SC0012575; CHE-13-62927
• OSTI ID: 1465680
• Alternate ID(s): OSTI ID: 1361757
• Journal Information: Journal of Chemical Physics, Vol. 146, Issue 7; ISSN 0021-9606
• Publisher: American Institute of Physics (AIP)
• Country of Publication: United States
• Language: English

Related Subjects

74 ATOMIC AND MOLECULAR PHYSICS
protons
Hilbert space
electron affinities
band gap
ground states
local density approximations
electron densities of states
many electron systems