Search

We combined a new parametrized density functional tight-binding (DFTB) theory (Fihey et al. 2015) with an unbiased modified basin hopping (MBH) optimization algorithm (Yen and Lai 2015) and applied it to calculate the lowest energy structures of Au clusters. From the calculated topologies and their conformational changes, we find that this DFTB/MBH method is a necessary procedure for a systematic study of the structural development of Au clusters but is somewhat insufficient for a quantitative study. As a result, we propose an extended hybridized algorithm. This improved algorithm proceeds in two steps. In the first step, the DFTB theory is employed to calculate the total energy of the cluster and this step (through running DFTB/MBH optimization for given Monte-Carlo steps) is meant to efficiently bring the Au cluster near to the region of the lowest energy minimum since the cluster as a whole has explicitly considered the interactions of valence electrons with ions, albeit semi-quantitatively. Then, in the second succeeding step, the energy-minimum searching process will continue with a skilledly replacement of the energy function calculated by the DFTB theory in the first step by one calculated in the full density functional theory (DFT). In these subsequent calculations, we couple the DFT energy also with the MBH strategy and proceed with the DFT/MBH optimization until the lowest energy value is found. We checked that this extended hybridized algorithm successfully predicts the twisted pyramidal structure for the Au 40 cluster and correctly confirms also the linear shape of C 8 which our previous DFTB/MBH method failed to do so. Perhaps more remarkable is the topological growth of Au n : it changes from a planar ( n =3–11) → an oblate-like cage ( n =12–15) → a hollow-shape cage ( n =16–18) and finally a pyramidal-like cage ( n =19, 20). These varied forms of the cluster’s shapes are consistent with those reported in the literature. [ABSTRACT FROM AUTHOR]

• We reveal a topological phase transition in the periodically forced Kuramoto model. • The transition separates states with oscillating and rotating collective dynamics. • The critical point is accompanied by an abrupt drop in the average cycling frequency. • Oscillating and rotating states have distinct winding numbers. • Winding numbers are defined relative to a singular point in the order-parameter space. A complete bifurcation analysis of explicit dynamical equations for the periodically forced Kuramoto model was performed in [L. M. Childs and S. H. Strogatz. Chaos 18 , 043128 (2008)], identifying all bifurcations within the model. We show that the phase diagram predicted by this analysis is incomplete. Our numerical analysis of the equations reveals that the model can also undergo an abrupt phase transition from oscillations to wobbly rotations of the order parameter under increasing field frequency or decreasing field strength. This transition was not revealed by bifurcation analysis because it is not caused by a bifurcation, and can neither be classified as first nor second order since it does not display critical phenomena characteristic of either transition. We discuss the topological origin of this transition and show that it is determined by a singular point in the order-parameter space. [ABSTRACT FROM AUTHOR]