Multiresolution analysis of electronic structure: semicardinal and wavelet bases

This article reviews recent developments in multiresolution analysis which make it a powerful tool for the systematic treatment of the multiple length-scales inherent in the electronic structure of matter. Although the article focuses on electronic structure, the advances described are useful for non-linear problems in the physical sciences in general. The new language and notations introduced are well- suited for both formal manipulations and the development of computer software using higher-level languages such as C++. The discussion is self-contained, and all needed algorithms are specified explicitly in terms of simple operators and illustrated with straightforward diagrams which show the flow of data. Among the reviewed developments is the construction of_exact_ multiresolution representations from extremely limited samples of physical fields in real space. This new and profound result is the critical advance in finally allowing systematic, all electron calculations to compete in efficiency with state-of-the-art electronic structure calculations which depend for their celerity upon freezing the core electronic degrees of freedom. This review presents the theory of wavelets from a physical perspective, provides a unified and self-contained treatment of non-linear couplings and physical operators and introduces a modern framework for effective single-particle theories of quantum mechanics.