Multiresolution analysis of electronic structure: semicardinal and wavelet bases

This article reviews recent developments in multiresolution analysis that make it a powerful tool for the systematic treatment of the multiple length scales inherent in the electroniic structure of matter. While the focus is on electronic structure, the advances described herein are useful for nonlinear problems in the physical sciences in general. Among the reviewed developments is the exact construction of 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 that 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 nonlinear couplings and physical operators, and introduces a modern framework for effective single-particle theories of quantum mechanics.