This article reviews recent developments in multiresolution analysis whichmake it a powerful tool for the systematic treatment of the multiplelength-scales inherent in the electronic structure of matter. Although thearticle focuses on electronic structure, the advances described are useful fornon-linear problems in the physical sciences in general. The new language andnotations introduced are well- suited for both formal manipulations and thedevelopment of computer software using higher-level languages such as C++. Thediscussion is self-contained, and all needed algorithms are specifiedexplicitly in terms of simple operators and illustrated with straightforwarddiagrams which show the flow of data. Among the reviewed developments is theconstruction of_exact_ multiresolution representations from extremely limitedsamples of physical fields in real space. This new and profound result is thecritical advance in finally allowing systematic, all electron calculations tocompete in efficiency with state-of-the-art electronic structure calculationswhich depend for their celerity upon freezing the core electronic degrees offreedom. This review presents the theory of wavelets from a physicalperspective, provides a unified and self-contained treatment of non-linearcouplings and physical operators and introduces a modern framework foreffective single-particle theories of quantum mechanics.
展开▼
