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Density-functional theory requires ever better exchange-correlation (xc) functionals for the ever more precise description of many-body effects on electronic structure. Universal constraints on the xc energy are important ingredients in the construction of improved functionals. Here we investigate one such universal property of xc functionals: the Lieb-Oxford lower bound on the exchange-correlation energy, $E_{xc}[n] ge -C int d^3r n^{4/3}$, where $Cleq C_{LO}=1.68$. To this end, we perform a survey of available exact or near-exact data on xc energies of atoms, ions, molecules, solids, and some model Hamiltonians (the electron liquid, Hookes atom and the Hubbard model). All physically realistic density distributions investigated are consistent with the tighter limit $C leq 1$. For large classes of systems one can obtain class-specific (but not fully universal) similar bounds. The Lieb-Oxford bound with $C_{LO}=1.68$ is a key ingredient in the construction of modern xc functionals, and a substantial change in the prefactor $C$ will have consequences for the performance of these functionals.
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Universal properties of the Coulomb interaction energy apply to all many-electron systems. Bounds on the exchange-correlation energy, inparticular, are important for the construction of improved density functionals. Here we investigate one such universal property -- the Lieb-Oxford lower bound -- for ionic and molecular systems. In recent work [J. Chem. Phys. 127, 054106 (2007)], we observed that for atoms and electron liquids this bound may be substantially tightened. Calculations for a few ions and molecules suggested the same tendency, but were not conclusive due to the small number of systems considered. Here we extend that analysis to many different families of ions and molecules, and find that for these, too, the bound can be empirically tightened by a similar margin as for atoms and electron liquids. Tightening the Lieb-Oxford bound will have consequences for the performance of various approximate exchange-correlation functionals.
The Lieb-Oxford bound is a constraint upon approximate exchange-correlation functionals. We explore a non-empirical tightening of that bound in both universal and electron-number-dependent form. The test functional is PBE. Regarding both atomization energies (slightly worsened) and bond lengths (slightly bettered), we find the PBE functional to be remarkably insensitive to the value of the Lieb-Oxford bound. This both rationalizes the use of the original Lieb-Oxford constant in PBE and suggests that enhancement factors more sensitive to sharpened constraints await discovery.
A simple and completely general representation of the exact exchange-correlation functional of density-functional theory is derived from the universal Lieb-Oxford bound, which holds for any Coulomb-interacting system. This representation leads to an alternative point of view on popular hybrid functionals, providing a rationale for why they work and how they can be constructed. A similar representation of the exact correlation functional allows to construct fully non-empirical hyper-generalized-gradient approximations (HGGAs), radically departing from established paradigms of functional construction. Numerical tests of these HGGAs for atomic and molecular correlation energies and molecular atomization energies show that even simple HGGAs match or outperform state-of-the-art correlation functionals currently used in solid-state physics and quantum chemistry.
The Shannon lower bound is one of the few lower bounds on the rate-distortion function that holds for a large class of sources. In this paper, it is demonstrated that its gap to the rate-distortion function vanishes as the allowed distortion tends to zero for all sources having a finite differential entropy and whose integer part is finite. Conversely, it is demonstrated that if the integer part of the source has an infinite entropy, then its rate-distortion function is infinite for every finite distortion. Consequently, the Shannon lower bound provides an asymptotically tight bound on the rate-distortion function if, and only if, the integer part of the source has a finite entropy.
Transition metal dichalcogenides (TMDCs) have attracted significant attention for optoelectronic, photovoltaic and photoelectrochemical applications. The properties of TMDCs are highly dependent on the number of stacked atomic layers, which is usually counted post-fabrication, using a combination of optical methods and atomic force microscopy (AFM) height measurements. Here, we use photoluminescence spectroscopy and three different AFM methods to demonstrate significant discrepancies in height measurements of exfoliated MoSe$_2$ flakes on SiO$_2$ depending on the method used. We highlight that overlooking effects from electrostatic forces, contaminants and surface binding can be misleading when measuring the height of a MoSe$_2$ flake. These factors must be taken into account as a part of the protocol for counting TMDC layers.
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