Implementation and benchmark of a long-range corrected functional in the density functional based tight-binding method

Bridging the gap between first principles methods and empirical schemes, the density functional based tight-binding method (DFTB) has become a versatile tool in predictive atomistic simulations over the past years. One of the major restrictions of this method is the limitation to local or gradient corrected exchange-correlation functionals. This excludes the important class of hybrid or long-range corrected functionals, which are advantageous in thermochemistry, as well as in the computation of vibrational, photoelectron and optical spectra. The present work provides a detailed account of the implementation of DFTB for a long-range corrected functional in generalized Kohn-Sham theory. We apply the method to a set of organic molecules and compare ionization potentials and electron affinities with the original DFTB method and higher level theory. The new scheme cures the significant overpolarization in electric fields found for local DFTB, which parallels the functional dependence in first principles density functional theory (DFT). At the same time the computational savings with respect to full DFT calculations are not compromised as evidenced by numerical benchmark data

The Potential of Restarts for ProbSAT

This work analyses the potential of restarts for probSAT, a quite successful algorithm for k-SAT, by estimating its runtime distributions on random 3-SAT instances that are close to the phase transition. We estimate an optimal restart time from empirical data, reaching a potential speedup factor of 1.39. Calculating restart times from fitted probability distributions reduces this factor to a maximum of 1.30. A spin-off result is that the Weibull distribution approximates the runtime distribution for over 93% of the used instances well. A machine learning pipeline is presented to compute a restart time for a fixed-cutoff strategy to exploit this potential. The main components of the pipeline are a random forest for determining the distribution type and a neural network for the distribution's parameters. ProbSAT performs statistically significantly better than Luby's restart strategy and the policy without restarts when using the presented approach. The structure is particularly advantageous on hard problems.Comment: Eurocast 201

Divide and colour models

Meester, R.W.J. [Promotor]Camia, F. [Copromotor

Superdiffusion in the periodic Lorentz gas

We prove a superdiffusive central limit theorem for the displacement of a test particle in the periodic Lorentz gas in the limit of large times $t$ and low scatterer densities (Boltzmann-Grad limit). The normalization factor is $\sqrt{t\log t}$, where $t$ is measured in units of the mean collision time. This result holds in any dimension and for a general class of finite-range scattering potentials. We also establish the corresponding invariance principle, i.e., the weak convergence of the particle dynamics to Brownian motion.Comment: 40 pages, 2 figures; this version also includes the invariance principl

The Polar Regions of Cassiopeia A: The Aftermath of a Gamma Ray Burst?

Probably not, but it is interesting nevertheless to investigate just how close Cas A might have come to generating such an event. Focusing on the northeast jet filaments, we analyze the polar regions of the recently acquired very deep 1 Ms Chandra X-ray observation. We infer that the so-called "jet" regions are indeed due to jets emanating from the explosion center, and not due to polar cavities in the circumstellar medium at the time of explosion. We place limits on the equivalent isotropic explosion energy in the polar regions (around 2.3 x 10^52 ergs), and the opening angle of the x-ray emitting ejecta (around 7 degrees), which give a total energy in the NE jet of order 10^50 ergs; an order of magnitude or more lower than inferred for "typical" GRBs. While the Cas A progenitor and explosion exhibit many of the features associated with GRB hosts, e.g. extensive presupernova mass loss and rotation, and jets associated with the explosion, we speculate that the recoil of the compact central object, with velocity 330 km/s, may have rendered the jet unstable. In such cases the jet rapidly becomes baryon loaded, if not truncated altogether. Although unlikely to have produced a gamma ray burst, the jets in Cas A suggest that such outflows may be common features of core-collapse SNe.Comment: 35 pages, 7 figures, accepted by Ap