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Information Ranking and Power Laws on Trees
We study the situations when the solution to a weighted stochastic recursion
has a power law tail. To this end, we develop two complementary approaches, the
first one extends Goldie's (1991) implicit renewal theorem to cover recursions
on trees; and the second one is based on a direct sample path large deviations
analysis of weighted recursive random sums. We believe that these methods may
be of independent interest in the analysis of more general weighted branching
processes as well as in the analysis of algorithms
Non-empirical hyper-generalized-gradient functionals constructed from the Lieb-Oxford bound
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.Comment: v2: Major revison. Added information on relation to the gradient
expansion and to local hybrids, improved discussion of size consistency and
of performance relative to other functional
How tight is the Lieb-Oxford bound?
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, , where
. 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, Hooke's atom and the Hubbard model).
All physically realistic density distributions investigated are consistent with
the tighter limit . For large classes of systems one can obtain
class-specific (but not fully universal) similar bounds. The Lieb-Oxford bound
with is a key ingredient in the construction of modern xc
functionals, and a substantial change in the prefactor will have
consequences for the performance of these functionals.Comment: 10 pages, 3 figure
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