Wavelet Technology

Wavelets provide a means to separate signals into independent contributions on different scales. They are localized, allowing fine scales to be used only where details are important. This can provide information compression advantages which can in principle be used to advantage in numerical quantum mechanical calculations. Bruce Johnson, Peter Nordlander, and James Kinsey are involved in a program to adapt the use of orthogonal wavelets (and multiwavelets, a newer variety) to the solution of the Schrödinger equation in a variety of problems. Applications of physical interest frequently possess multiple scales (e.g. electronic orbitals of widely varying diffuseness) that are challenging from a computational point of view, and it is in this area that wavelet methods are expected to be of most advantage. Areas of particular interest in the current work at Rice are large-amplitude vibrational motion, the dynamics of bond-breaking (e.g., laser photolysis of ozone), and surface-adsorbate interactions.

Edge member of a 2D curvilinear coordinate wavelet basis

While wavelet use has been restricted to simple Cartesian applications in the past, their use has now been extended to include coordinates that typically arise in more general molecular physics applications [B.R. Johnson, et al., J. Comp. Phys. 168, 356 (2001)]. The figure shows one of the 2D wavelet basis functions used for accurately solving the electronic equations for H₂+ in curvilinear coordinates. The x and y axes cover ranges of the two electron-proton distances. While appearing somewhat unusual, the function shown is purposely skewed and discontinuous in order to solve quantum boundary conditions in curvilinear coordinates. The same wavelet basis can be used either for Cartesian or non-Cartesian problems, providing a long-range target of highlyadaptive solution code for Schrödinger and other differential equations. The constuction of a C++ program named MultiWavePack is being undertaken for this purpose. The work within RQI particularly benefits from the presence at Rice of a number of wavelets researchers in other fields, reflecting the strongly multidisciplinary nature of wavelet applications.

Daubechies-8 wavelets at multiple scales