4,759 research outputs found
Every planar graph with the Liouville property is amenable
We introduce a strengthening of the notion of transience for planar maps in
order to relax the standard condition of bounded degree appearing in various
results, in particular, the existence of Dirichlet harmonic functions proved by
Benjamini and Schramm. As a corollary we obtain that every planar non-amenable
graph admits Dirichlet harmonic functions
Exact Maps in Density Functional Theory for Lattice Models
In the present work, we employ exact diagonalization for model systems on a
real-space lattice to explicitly construct the exact density-to-potential and
for the first time the exact density-to-wavefunction map that underly the
Hohenberg-Kohn theorem in density functional theory. Having the explicit
wavefunction-to- density map at hand, we are able to construct arbitrary
observables as functionals of the ground-state density. We analyze the
density-to-potential map as the distance between the fragments of a system
increases and the correlation in the system grows. We observe a feature that
gradually develops in the density-to-potential map as well as in the
density-to-wavefunction map. This feature is inherited by arbitrary expectation
values as functional of the ground-state density. We explicitly show the
excited-state energies, the excited-state densities, and the correlation
entropy as functionals of the ground-state density. All of them show this exact
feature that sharpens as the coupling of the fragments decreases and the
correlation grows. We denominate this feature as intra-system steepening. We
show that for fully decoupled subsystems the intra-system steepening transforms
into the well-known inter-system derivative discontinuity. An important
conclusion is that for e.g. charge transfer processes between localized
fragments within the same system it is not the usual inter-system derivative
discontinuity that is missing in common ground-state functionals, but rather
the differentiable intra-system steepening that we illustrate in the present
work
Universal Dynamical Steps in the Exact Time-Dependent Exchange-Correlation Potential
We show that the exact exchange-correlation potential of time-dependent
density-functional theory displays dynamical step structures that have a
spatially non-local and time non-local dependence on the density. Using
one-dimensional two-electron model systems, we illustrate these steps for a
range of non-equilibrium dynamical situations relevant for modeling of
photo-chemical/physical processes: field-free evolution of a non-stationary
state, resonant local excitation, resonant complete charge-transfer, and
evolution under an arbitrary field. Lack of these steps in usual approximations
yield inaccurate dynamics, for example predicting faster dynamics and
incomplete charge transfer
Local reduced-density-matrix-functional theory: Incorporating static correlation effects in Kohn-Sham equations
We propose a novel scheme to bring reduced density matrix functional theory
(RDMFT) into the realm of density functional theory (DFT) that preserves the
accurate density functional description at equilibrium, while incorporating
accurately static and left-right correlation effects in molecules and keeping
the good computational performance of DFT-based schemes. The key ingredient is
to relax the requirement that the local potential is the functional derivative
of the energy with respect to the density. Instead, we propose to restrict the
search for the approximate natural orbitals within a domain where these
orbitals are eigenfunctions of a single-particle hamiltonian with a local
effective potential. In this way, fractional natural occupation numbers are
accommodated into Kohn-Sham equations allowing for the description of molecular
dissociation without breaking spin symmetry. Additionally, our scheme provides
a natural way to connect an energy eigenvalue spectrum to the approximate
natural orbitals and this spectrum is found to represent accurately the
ionization potentials of atoms and small molecules
Decay estimates for nonlinear nonlocal diffusion problems in the whole space
In this paper we obtain bounds for the decay rate in the L^r (\rr^d)-norm
for the solutions to a nonlocal and nolinear evolution equation, namely,
u_t(x,t) = \int_{\rr^d} K(x,y) |u(y,t)- u(x,t)|^{p-2} (u(y,t)- u(x,t)) \, dy,
with x \in \rr^d, . Here we consider a kernel of the form
, where is a bounded, nonnegative
function supported in the unit ball and is a linear function . To
obtain the decay rates we derive lower and upper bounds for the first
eigenvalue of a nonlocal diffusion operator of the form T(u) = - \int_{\rr^d}
K(x,y) |u(y)-u(x)|^{p-2} (u(y)-u(x)) \, dy, with . The
upper and lower bounds that we obtain are sharp and provide an explicit
expression for the first eigenvalue in the whole \rr^d: \lambda_{1,p}
(\rr^d) = 2(\int_{\rr^d} \psi (z) \, dz)|\frac{1}{|\det{A}|^{1/p}} -1|^p.
Moreover, we deal with the eigenvalue problem studying the limit as
of
Charge-transfer in time-dependent density-functional theory via spin-symmetry-breaking
Long-range charge-transfer excitations pose a major challenge for
time-dependent density functional approximations. We show that
spin-symmetry-breaking offers a simple solution for molecules composed of
open-shell fragments, yielding accurate excitations at large separations when
the acceptor effectively contains one active electron. Unrestricted
exact-exchange and self-interaction-corrected functionals are performed on
one-dimensional models and the real LiH molecule within the pseudopotential
approximation to demonstrate our results.Comment: 5 pages, 4 figure
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