Wavelet-Based Multiscale Methods for Electronic Structure Calculations

Summary In order to treat multiple energy- and length-scales in electronic structure calculations for extended systems, we have studied a wavelet based multiresolution analysis of electron correlations. Wavelets provide hierarchical basis sets that can be locally adapted to the length- and energy-scales of physical phenomena. The inherently high dimensional many-body problem can be kept tractable by using the sparse grid method for the construction of multivariate wavelets. These so called “hyperbolic” wavelets provide sparse representations for correlated wavefunctions and can be combined with diagrammatic techniques from quantum many-particle theory into a diagrammatic multiresolution analysis. Using sparsity features originating from the hierarchical structure and vanishing moments property of wavelet bases, this leads to many-particle methods with almost linear computational complexity for the treatment of electron correlations. We are aiming towards applications in semiconductor physics where quasi two-dimensional many-particle systems provide challenging computational problems. Such kind of systems are metallic slabs and interacting excitons confined in quantum wells of semiconductor heterostructures. As a first step we developed a multiresolution Hartree-Fock method suitable for quasi two-dimensional extended systems. Special emphasis has been laid on low rank tensor product decompositions of or-bitals, which take into account the strongly anisotropic character of these systems in one direction.