8 research outputs found
Random Series and Discrete Path Integral methods: The Levy-Ciesielski implementation
We perform a thorough analysis of the relationship between discrete and
series representation path integral methods, which are the main numerical
techniques used in connection with the Feynman-Kac formula. First, a new
interpretation of the so-called standard discrete path integral methods is
derived by direct discretization of the Feynman-Kac formula. Second, we
consider a particular random series technique based upon the Levy-Ciesielski
representation of the Brownian bridge and analyze its main implementations,
namely the primitive, the partial averaging, and the reweighted versions. It is
shown that the n=2^k-1 subsequence of each of these methods can also be
interpreted as a discrete path integral method with appropriate short-time
approximations. We therefore establish a direct connection between the discrete
and the random series approaches. In the end, we give sharp estimates on the
rates of convergence of the partial averaging and the reweighted
Levy-Ciesielski random series approach for sufficiently smooth potentials. The
asymptotic rates of convergence are found to be O(1/n^2), in agreement with the
rates of convergence of the best standard discrete path integral techniques.Comment: 20 pages, 4 figures; the two equations before Eq. 14 are corrected;
other typos are remove
Ab initio calculations of optical properties of silver clusters: cross-over from molecular to nanoscale behavior
Electronic and optical properties of silver clusters were calculated using
two different \textit{ab initio} approaches: 1) based on all-electron
full-potential linearized-augmented plane-wave method and 2) local basis
function pseudopotential approach. Agreement is found between the two methods
for small and intermediate sized clusters for which the former method is
limited due to its all-electron formulation. The latter, due to non-periodic
boundary conditions, is the more natural approach to simulate small clusters.
The effect of cluster size is then explored using the local basis function
approach. We find that as the cluster size increases, the electronic structure
undergoes a transition from molecular behavior to nanoparticle behavior at a
cluster size of 140 atoms (diameter \,nm). Above this cluster size
the step-like electronic structure, evident as several features in the
imaginary part of the polarizability of all clusters smaller than
Ag, gives way to a dominant plasmon peak localized at
wavelengths 350\,nm 600\,nm. It is, thus, at this length-scale
that the conduction electrons' collective oscillations that are responsible for
plasmonic resonances begin to dominate the opto-electronic properties of silver
nanoclusters