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Toward diagrammatic multiresolution analysis for electron correlations
Heinz-Jürgen Flad, Wolfgang Hackbusch, Hongjun Luo and Dietmar Kolb
We present a multiscale treatment of electron correlations based on hyperbolic wavelet expansions of Jastrow-type correlation functions. Wavelets provide hierarchical basis sets that can be locally adapted to the length- and energy-scales of physical phenomena. Combined with hyperbolic tensor products and local adaptive refinement near the inter-electron cusp, these wavelet bases enable sparse representations of Jastrow factors. The computational efficiency of wavelets in electronic structure calculations is demonstrated within the coupled electron-pair approximation (local ansatz). Based on a diagrammatic multiresolution analysis, we discuss various kinds of sparsity features for matrix elements required by the local ansatz. Sparsity originates from the hierarchical structure and vanishing moments property of wavelet bases. This led us to a recurrence scheme for the evaluation of matrix elements with almost linear computational complexity with respect to the size of the underlying isotropic 3d-wavelet basis. Numerical studies for selected diagrams are presented for a homogeneous electron gas.