Wavelet Technology in Quantum Mechanics and Nanophotonics

Earlier efforts in the group centered on making orthogonal wavelets a
practical means for solving the large-amplitude vibrational Schrödinger equation and Maxwell's
equations near plasmonic nanoparticles. The need for large-amplitude descriptions of vibrational
motion arise in many problems including specifically Dissociative Resonance Raman Spectroscopy (DRRS).
While a number of algorithms exist for solving the needed stationary and time-dependent
Schrödinger equations, many either waste effort in spatial regions where not much is happening
or else require considerable human coaxing. Similarly, the determination of the near EM fields
surrounding general nanostructures and nanoparticles is challenging and in need of accurate and
efficient methods that can bridge different scales. Compact support wavelets can be used to
approach a wide variety of problems with systematic accuracy, as has been a focus in work of this
group and is the basis of an object-oriented software program, MultiWavePack (not updated since
I began serving at NSF). Efficiency is a current focus, and is important if wavelets are to provide
a rational platform for the development of automated methods. Many of the algorithms we develop for
our specific problems, such as overcoming Gibbs-phenomenon interference in satisfying EM boundary
conditions, are really of interest in a wide variety of applications.

There are many papers and books introducing use of orthogonal wavelets,
but it is often difficult for the newcomer to figure out basic tasks such as creating a plot or
projecting a known function in a wavelet basis. Over the years of working with REU students at
RQI who came in with cursory knowledge of wavelets, we have developed our own introductory package
in Mathematica (OW.m) and an accompanying tutorial notebook
(TutorialOW.nb) to get people past this stage quickly. The current
version of the tutorial (Mathematica 8) was finished owing to an NSF IR/D grant which is
gratefully acknowledged.

Surface-Enhanced Raman Scattering from Metal Nanoparticles

This work aims at calculating molecular near-field response for molecules such as p-mercaptoaniline on silver nanoshells, including vibrational dynamics of multiple Raman-active modes, **directionality**, field polarization, and **polarizability tensors** derived from small molecule-metal cluster *ab initio* calculations. A density matrix formalism is used to include both radiative and non-radiative relaxation processes in spectral simulations of Surface-Enhanced Raman Scattering (SERS), and special attention is paid to the effects of near-resonance in the Raman processes.